Plaintext
ACTIVE CALCULUS
2018 Edition - UPDATED
Matthew Boelkins
David Austin Steven Schlicker
Active Calculus
Active Calculus
Matthew Boelkins
Grand Valley State University
Contributing Authors
David Austin
Grand Valley State University
Steven Schlicker
Grand Valley State University
Production Editor
Mitchel T. Keller
Morningside College
July 2, 2019
Cover Photo: James Haefner Photography
Edition: 2018 Updated
Website: http://activecalculus.org
©2012–2019 Matthew Boelkins
Permission is granted to copy and (re)distribute this material in any format and/or adapt it
(even commercially) under the terms of the Creative Commons Attribution-ShareAlike 4.0
International License. The work may be used for free in any way by any party so long as at-
tribution is given to the author(s) and if the material is modified, the resulting contributions
are distributed under the same license as this original. All trademarks™ are the registered®
marks of their respective owners. The graphic
that may appear in other locations in the text shows that the work is licensed with the Cre-
ative Commons and that the work may be used for free by any party so long as attribution
is given to the author(s) and if the material is modified, the resulting contributions are dis-
tributed under the same license as this original. Full details may be found by visiting
https://creativecommons.org/licenses/by-sa/4.0/ or sending a letter to Creative Commons,
444 Castro Street, Suite 900, Mountain View, California, 94041, USA.
vi
Acknowledgements
This text began as my sabbatical project in the winter semester of 2012, during which I wrote
most of the material for the first four chapters. For the sabbatical leave, I am indebted to
Grand Valley State University for its support of the project, as well as to my colleagues in
the Department of Mathematics and the College of Liberal Arts and Sciences for their en-
dorsement of the project. I’m also grateful to the American Institute of Mathematics for their
support of free and open textbooks in general, and their support of this one in particular.
The beautiful full-color .eps graphics in the text are only possible because of David Austin
of GVSU and Bill Casselman of the University of British Columbia. Building on their long-
standing efforts to develop tools for high quality mathematical graphics, David wrote a li-
brary of Python routines that employ Bill’s PiScript program; David’s routines are so easy
to use that even I could generate graphics like the professionals that he and Bill are. I am
deeply grateful to them both.
The current .html version of the text is possible only because of the amazing work of Rob
Beezer and his development of the original Mathbook XML, now known as PreTeXt. My
ability to take advantage of Rob’s work is due in large part to the support of the American
Institute of Mathematics, which funded me to attend a weeklong workshop in Mathbook
XML in San Jose, CA, in April 2016, as well as the support of the ongoing user group. A
subsequent workshop in June 2019 has offered further support and more improvements to
the text.
David Farmer’s conversion script saved me hundreds of hours of work by taking my original
LATEX source and converting it to PreTeXt; David remains a major source of ongoing support
and advocacy. Alex Jordan of Portland Community College has also been a tremendous
help, and it is through Alex’s fantastic work that live WeBWorK exercises are not only possi-
ble, but also included from the 2017 version forward. Mitch Keller of Morningside College
agreed in early 2018 to serve as the book’s production editor; his technical expertise has
contributed to many aspects of the book, including the presence of answers to activities and
non- WeBWorK exercises and other supporting material for instructors.
For the 2018 edition, Kathy Yoshiwara of the AIM Editorial Board read the entire text and
contributed editorial suggestions for every section. In short, she made the prose cleaner,
more direct, and simply better. I’m deeply thankful for her time, effort, and insights.
Over my more than 20 years at GVSU, many of my colleagues have shared with me ideas
and resources for teaching calculus. I am particularly indebted to David Austin, Will Dickin-
son, Paul Fishback, Jon Hodge, and Steve Schlicker for their contributions that improved my
teaching of and thinking about calculus, including materials that I have modified and used
over many different semesters with students. Parts of these ideas can be found throughout
this text. In addition, Will Dickinson and Steve Schlicker provided me access to a large num-
ber of their electronic notes and activities from teaching of differential and integral calculus,
and those ideas and materials have similarly impacted my work and writing in positive ways,
with some of their problems and approaches finding parallel presentation here.
In the summer of 2012, David and Steve each agreed to write a chapter to support the com-
pletion of the material on integral calculus. David is the lead author of Chapter 7 and Steve
the lead author of Chapter 8. Along with our colleague Ted Sundstrom, Steve has also con-
tributed a large number of problem and activity solutions and answers. I’m especially grate-
ful for how the work of these friends and colleagues has made the text so much better.
Shelly Smith of GVSU and Matt Delong of Marian University both provided extensive com-
ments on the first few chapters of early drafts, feedback that was immensely helpful in im-
proving the text. As more and more people use the text, I am grateful to everyone who
reads, edits, and uses this book, and hence contributes to its improvement through ongoing
discussion.
Finally, I am grateful for all that my students have taught me over the years. Their responses
and feedback have helped to shape me as a teacher, and I appreciate their willingness to
wholeheartedly engage in the activities and approaches I’ve tried in class, to let me know
how those affect their learning, and to help me learn and grow as an instructor. Early on,
they also provided useful editorial feedback on this text.
Any and all remaining errors or inconsistencies are mine. I will gladly take reader and user
feedback to correct them, along with other suggestions to improve the text.
viii
Contributors
A large and growing number of people have generously contributed to the development or
improvement of the text. Contributing authors David Austin and Steven Schlicker have each
written drafts of at least one full chapter of the text. Production editor Mitchel Keller has
been an indispensable source of technological support and editorial counsel.
The following contributing editors have offered feedback that includes information about
typographical errors or suggestions to improve the exposition.
David Austin Patti Hunter
GVSU Westmont College
Rene Ardila
Mitchel Keller
GVSU
Morningside College
Allan Bickle
GVSU Sam Kolins
Lebanon Valley College
David Clark
GVSU
Dave Kung
Will Dickinson St. Mary’s College of Maryland
GVSU
Paul Latiolais
Nate Eldredge Portland State University
University of Northern Colorado
Charles Fortin Hugh McGuire
Champlain Regional College GVSU
St-Lambert, Quebec, Canada
Martin Mohlenkamp
Marcia Frobish
Ohio University
GVSU
Teresa Gonske Ray Rosentrater
University of Northwestern - St. Paul Westmont College
Luis Sanjuan Amy Stone
Conservatorio Profesional de Musica de Avila GVSU
Spain
Robert Talbert
Michael Santana
GVSU
GVSU
Greg Thull
Steven Schlicker
GVSU
GVSU
Michael Shulman Sue Van Hattum
University of San Diego Contra Costa College
Brian Stanley Kathy Yoshiwara
Foothill Community College AIM Editorial Board
x
Active Calculus: Our Goals
Several fundamental ideas in calculus are more than 2000 years old. As a formal subdisci-
pline of mathematics, calculus was first introduced and developed in the late 1600s, with
key independent contributions from Sir Isaac Newton and Gottfried Wilhelm Leibniz. The
subject has been understood rigorously since the work of Augustin Louis Cauchy and Karl
Weierstrass in the mid 1800s when the field of modern analysis was developed. As a body
of knowledge, calculus has been completely understood for at least 150 years. The disci-
pline is one of our great human intellectual achievements: among many spectacular ideas,
calculus models how objects fall under the forces of gravity and wind resistance, explains
how to compute areas and volumes of interesting shapes, enables us to work rigorously
with infinitely small and infinitely large quantities, and connects the varying rates at which
quantities change to the total change in the quantities themselves.
While each author of a calculus textbook certainly offers their own creative perspective on
the subject, it is hardly the case that many of the ideas they present are new. Indeed, the
mathematics community broadly agrees on what the main ideas of calculus are, as well as
their justification and their importance. In the 21st century and the age of the internet, no one
should be required to purchase a calculus text to read, to use for a class, or to find a coherent
collection of problems to solve. Calculus belongs to humankind, not any individual author
or publishing company. Thus, a primary purpose of this work is to present a calculus text
that is free. See https://activecalculus.org for links to both the .html and .pdf versions of the
text. In addition, instructors who are looking for a calculus text should have the opportunity
to download the source files and make modifications that they see fit; thus this text is open-
source. See GitHub for the source. Since August 2013, Active Calculus - Single Variable has
been endorsed by the American Institute of Mathematics and its Open Textbook Initiative.
In Active Calculus - Single Variable, we actively engage students in learning the subject through
an activity-driven approach in which the vast majority of the examples are generated by
students. Where many texts present a general theory followed by substantial collections of
worked examples, we instead pose problems or situations, consider possibilities, and then
ask students to investigate and explore. Following key activities or examples, the presen-
tation normally includes some overall perspective and a brief synopsis of general trends or
properties, followed by formal statements of rules or theorems. While we often offer plau-
sibility arguments for such results, rarely do we include formal proofs. It is not the intent of
this text for the instructor or author to demonstrate to students that the ideas of calculus are
coherent and true, but rather for students to encounter these ideas in a supportive, leading
manner that enables them to begin to understand calculus for themselves. This approach is
consistent with the scholarly consensus that calls for students to be interactively engaged in
class.
Moreover, this approach is consistent with the following goals:
• To have students engage in an active, inquiry-driven approach, where learners con-
struct solutions and approaches to ideas, with appropriate support through questions
posed, hints, and guidance from the instructor and text.
• To build in students intuition for why the main ideas in calculus are natural and true.
Often we do this through consideration of the instantaneous position and velocity of
a moving object.
• To challenge students to acquire deep, personal understanding of calculus through
reading the text and completing preview activities on their own, working on activities
in small groups in class, and doing substantial exercises outside of class time.
• To strengthen students’ written and oral communicating skills by having them write
about and explain aloud the key ideas of calculus.
xii
Features of the Text
Instructors and students alike will find several consistent features in the presentation, in-
cluding:
Motivating Questions At the start of each section, we list 2–3 motivating questions that pro-
vide motivation for why the following material is of interest to us. One overall goal of
each section is to answer each of the motivating questions.
Preview Activities Each section of the text begins with a short introduction, followed by a
preview activity. This brief reading and the preview activity are designed to foreshadow
the upcoming ideas in the remainder of the section; both the reading and preview
activity are intended to be accessible to students in advance of class, and to be completed
by students before the day on which a particular section is to be considered.
Activities A typical section in the text has at least three activities. These are designed to
engage students in an inquiry-based style that encourages them to construct solutions
to key examples on their own, working individually or in small groups.
Exercises There are dozens of calculus texts with (collectively) tens of thousands of exer-
cises. Rather than repeat standard and routine exercises in this text, we recommend
the use of WeBWorK with its access to the Open Problem Library and around 20,000
calculus problems. In this text, each section includes a small number of anonymous
WeBWorK exercises, as well as 3–4 challenging problems per section. The WeBWorK
exercises are best completed in the .html version of the text, as this provides students
with immediate feedback without penalty. Almost every non- WeBWorK exercise has
multiple parts, requires the student to connect several key ideas, and expects that the
student will do at least a modest amount of writing to answer the questions and explain
their findings. For instructors interested in a more conventional source of exercises,
consider the freely available APEX Calculus text by Greg Hartmann et al., available
from www.apexcalculus.com.
Graphics We strive to demonstrate key fundamental ideas visually, and to encourage stu-
dents to do the same. Throughout the text, we use full-color¹ graphics to exemplify
and magnify key ideas, and to use this graphical perspective alongside both numerical
and algebraic representations of calculus.
¹To keep cost low, the graphics in the print-on-demand version are in black and white. When the text itself refers
to color in images, one needs to view the .html or .pdf electronically.
Links to interactive graphics Many of the ideas of calculus are best understood dynami-
cally; java applets offer an often ideal format for investigations and demonstrations.
Relying primarily on the work of David Austin of Grand Valley State University and
Marc Renault of Shippensburg University, each of whom has developed a large li-
brary of applets for calculus, we frequently point the reader (through active links in
the electronic versions of the text) to applets that are relevant for key ideas under con-
sideration.
Summary of Key Ideas Each section concludes with a summary of the key ideas encoun-
tered in the preceding section; this summary normally reflects responses to the moti-
vating questions that began the section.
xiv
Students! Read this!
This book is different.
The text is available in three different formats: HTML, PDF, and print, each of which is
available via links on the landing page at https://activecalculus.org/single/. The first two
formats are free. If you are going to use the book electronically, the best mode is the HTML.
The HTML version looks great in any browser, including on a smartphone, and the links
are much easier to navigate in HTML than in PDF. Some particular direct suggestions about
using the HTML follow among the next few paragraphs; alternatively, you can watch this
short video from the author. It is also wise to download and save the PDF, since you can
use the PDF offline, while the HTML version requires an internet connection. A print copy
costs about $21 via Amazon.
This book is intended to be read sequentially and engaged with, much more than to be used
as a lookup reference. For example, each section begins with a short introduction and a
Preview Activity; you should read the short introduction and complete the Preview Activity
prior to class. Your instructor may require you to do this. Most Preview Activities can be
completed in 15-20 minutes and are intended to be accessible based on the understanding
you have from preceding sections. There are not answers provided to Preview Activities, as
these are designed simply to get you thinking about ideas that will be helpful in work on
upcoming new material.
As you use the book, think of it as a workbook, not a worked-book. There is a great deal
of scholarship that shows people learn better when they interactively engage and struggle
with ideas themselves, rather than passively watch others. Thus, instead of reading worked
examples or watching an instructor complete examples, you will engage with Activities that
prompt you to grapple with concepts and develop deep understanding. You should expect
to spend time in class working with peers on Activities and getting feedback from them
and from your instructor. You can purchase a separate Activities Workbook from Amazon
(Chapters 1-4, Chapters 5-8) in which to record your work on the activities, or you can ask
your instructor for a copy of the PDF file that has only the activities along with room to
record your work. Your goal should be to do all of the activities in the relevant sections of
the text and keep a careful record of your work. You can find answers to the activities in the
back matter.
Each section concludes with a Summary. Reading the Summary after you have read the
section and worked the Activities is a good way to find a short list of key ideas that are most
essential to take from the section. A good study habit is to write similar summaries in your
own words.
At the end of each section, you’ll find two types of Exercises. First, there are several anony-
mous WeBWorK exercises. These are online, interactive exercises that allow you to submit
answers for immediate feedback with unlimited attempts without penalty; to submit an-
swers, you have to be using the HTML version of the text (see this short video on the HTML
version that includes a WeBWorK demonstration). You should use these exercises as a way
to test your understanding of basic ideas in the preceding section. If your institution uses
WeBWorK, you may also need to log in to a server as directed by your instructor to complete
assigned WeBWorK sets as part of your course grade. The WeBWorK exercises included
in this text are ungraded and not connected to any individual account. Following the WeB-
WorK exercises there are 3-4 additional challenging exercises that are designed to encourage
you to connect ideas, investigate new situations, and write about your understanding. There
are answers to most of the non-WeBWorK exercises in the back matter.
You can find additional support for your work in learning calculus from the GVSU Math 201
YouTube Channel and GVSU Math 202 YouTube Channel where there are several short video
tutorials for each section of the text, numbered in alignment with the textbook sections.
Math 201 corresponds to Chapters 1-4 and Math 202 to Chapters 5-8; there are about 90
videos for each, totally more than 180.
The best way to be successful in mathematics generally and calculus specifically is to strive
to make sense of the main ideas. We make sense of ideas by asking questions, interacting
with others, attempting to solve problems, making mistakes, revising attempts, and writing
and speaking about our understanding. This text has been designed to help you make sense
of calculus; we wish you the very best as you undertake the large and challenging task of
doing so.
xvi
Instructors! Read this!
This book is different. Before you read further, first read “Students! Read this!”.
Chapters 1-4 are designed to correspond to what is often called differential calculus. Chap-
ters 5-8 correspond roughtly to what is often called integral calculus, including chapters on
differential equations and infinite series.
Among the three formats (HTML, PDF, print), the HTML is optimal for display in class if
you have a suitable projector. The HTML is also best for navigation, as links to internal and
external references are much more obvious. We recommend saving a downloaded version
of the PDF format as a backup in the event you don’t have internet access. It’s a good idea
for each student to have a printed version of the Activities Workbook, which is available on
Amazon (Chapters 1-4, Chapters 5-8) or as a PDF document by direct request to the author
(boelkinm at gvsu dot edu); many instructors use the PDF to have coursepacks printed for
students to purchase from their local bookstore.
The text is written so that, on average, one section corresponds to two hours of class meet-
ing time. A typical instructional sequence when starting a new section might look like the
following:
• Students complete a Preview Activity in advance of class. Class begins with a short
debrief among peers followed by all class discussion. (5-10 minutes)
• Brief lecture and discussion to build on the preview activity and set the stage for the
next activity. (5-10 minutes)�
• Students engage with peers to work on and discuss the first activity in the section.
(15-20 minutes)�
• Brief discussion and possibly lecture to reach closure on the preceding activity, fol-
lowed by transition to new ideas. (Varies, but 5-15 minutes)
• Possibly begin next activity.
The next hour of class would be similar, but without the Preview Activity to complete prior
to class: the principal focus of class will be completing 2 activities. Then rinse and repeat.
We recommend that instructors use appropriate incentives to encourage students to com-
plete Preview Activities prior to class. Having these be part of completion-based assign-
ments that count 5% of the semester grade usually results in the vast majority of students
completing the vast majority of the previews. If you’d like to see a sample syllabus for how
to organize a course and weight various assignments, you can request one via email to the
author.
Note that the WeBWorK exercises in the HTML version are anonymous and there’s not a
way to track students’ engagement with them. These are intended to be formative for stu-
dents and provide them with immediate feedback without penalty. If your institution is a
WeBWorK user, we have existing sets of .def files that correspond to the sections in the text;
these are available upon request to the author.
In the back matter of the text, you’ll find answers to the Activities and to non-WeBWorK
Exercises. Instructors interested in solutions to these should contact the author directly.
You and your students can find additional resources in the GVSU Math 201 YouTube Chan-
nel and GVSU Math 202 YouTube Channel where there are short video tutorials for every
section of the text. Math 201 (GVSU’s Calculus I) corresponds to Chapters 1-4 and Math 202
(GVSU’s Calculus II) to Chapters 5-8.
The PreTeXt source code for the text can be found on GitHub. If you find errors in the text or
have other suggestions, you can file an issue on GitHub, use the Feedback link in the HTML
version (found at the bottom left in the main menu), or email the author directly. To engage
with instructors who use the text, we maintain both an email list and the Open Calculus
blog; you can request that your address be added to the email list by contacting the author.
Finally, if you’re interested in a video presentation on using the text, you can see this online
video presentation to the MIT Electronic Seminar on Mathematics Education; at about the
17-minute mark, the portion begins where we demonstrate features of and how to use the
text.
Thank you for considering Active Calculus as a resource to help your students develop deep
understanding of the subject. I wish you the very best in your work and hope to hear from
you.
xviii
Contents
Acknowledgements vii
Contributors ix
Active Calculus: Our Goals xi
Features of the Text xiii
Students! Read this! xv
Instructors! Read this! xvii
1 Understanding the Derivative 1
1.1 How do we measure velocity? . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 The notion of limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.3 The derivative of a function at a point . . . . . . . . . . . . . . . . . . . . . . . 22
1.4 The derivative function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
1.5 Interpreting, estimating, and using the derivative . . . . . . . . . . . . . . . . 45
1.6 The second derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
1.7 Limits, Continuity, and Differentiability . . . . . . . . . . . . . . . . . . . . . . 68
1.8 The Tangent Line Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . 80
2 Computing Derivatives 89
2.1 Elementary derivative rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
2.2 The sine and cosine functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
2.3 The product and quotient rules . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
2.4 Derivatives of other trigonometric functions . . . . . . . . . . . . . . . . . . . . 115
Contents
2.5 The chain rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
2.6 Derivatives of Inverse Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
2.7 Derivatives of Functions Given Implicitly . . . . . . . . . . . . . . . . . . . . . 139
2.8 Using Derivatives to Evaluate Limits . . . . . . . . . . . . . . . . . . . . . . . . 146
3 Using Derivatives 157
3.1 Using derivatives to identify extreme values . . . . . . . . . . . . . . . . . . . 157
3.2 Using derivatives to describe families of functions . . . . . . . . . . . . . . . . 170
3.3 Global Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
3.4 Applied Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
3.5 Related Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
4 The Definite Integral 201
4.1 Determining distance traveled from velocity . . . . . . . . . . . . . . . . . . . 201
4.2 Riemann Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
4.3 The Definite Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228
4.4 The Fundamental Theorem of Calculus . . . . . . . . . . . . . . . . . . . . . . 244
5 Evaluating Integrals 259
5.1 Constructing Accurate Graphs of Antiderivatives . . . . . . . . . . . . . . . . . 259
5.2 The Second Fundamental Theorem of Calculus . . . . . . . . . . . . . . . . . . 270
5.3 Integration by Substitution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281
5.4 Integration by Parts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291
5.5 Other Options for Finding Algebraic Antiderivatives . . . . . . . . . . . . . . . 301
5.6 Numerical Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310
6 Using Definite Integrals 325
6.1 Using Definite Integrals to Find Area and Length . . . . . . . . . . . . . . . . . 325
6.2 Using Definite Integrals to Find Volume . . . . . . . . . . . . . . . . . . . . . . 334
6.3 Density, Mass, and Center of Mass . . . . . . . . . . . . . . . . . . . . . . . . . 344
6.4 Physics Applications: Work, Force, and Pressure . . . . . . . . . . . . . . . . . 354
6.5 Improper Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365
xx
7 Differential Equations 375
7.1 An Introduction to Differential Equations . . . . . . . . . . . . . . . . . . . . . 375
7.2 Qualitative behavior of solutions to DEs . . . . . . . . . . . . . . . . . . . . . . 386
7.3 Euler’s method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397
7.4 Separable differential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 407
7.5 Modeling with differential equations . . . . . . . . . . . . . . . . . . . . . . . . 415
7.6 Population Growth and the Logistic Equation . . . . . . . . . . . . . . . . . . . 423
8 Sequences and Series 435
8.1 Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435
8.2 Geometric Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443
8.3 Series of Real Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454
8.4 Alternating Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 470
8.5 Taylor Polynomials and Taylor Series . . . . . . . . . . . . . . . . . . . . . . . . 483
8.6 Power Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 498
A A Short Table of Integrals 509
B Answers to Activities 511
C Answers to Selected Exercises 565
Index 623
xxi
Contents
xxii
CHAPTER 1
Understanding the Derivative
1.1 How do we measure velocity?
Motivating Questions
• How is the average velocity of a moving object connected to the values of its position
function?
• How do we interpret the average velocity of an object geometrically on the graph of
its position function?
• How is the notion of instantaneous velocity connected to average velocity?
Calculus can be viewed broadly as the study of change. A natural and important question
to ask about any changing quantity is “how fast is the quantity changing?”
We begin with a simple problem: a ball is tossed straight up in the air. How is the ball
moving? Questions like this one are central to our study of differential calculus.
Preview Activity 1.1.1. Suppose that the height s of a ball at time t (in seconds) is
given in feet by the formula s(t) 64 − 16(t − 1)2 .
a. Construct a graph of y s(t) on the time interval 0 ≤ t ≤ 3. Label at least six
distinct points on the graph, including the three points showing when the ball
was released, when the ball reaches its highest point, and when the ball lands.
b. Describe the behavior of the ball on the time interval 0 < t < 1 and on time
interval 1 < t < 3. What occurs at the instant t 1?
c. Consider the expression
s(1) − s(0.5)
AV[0.5,1] .
1 − 0.5
Compute the value of AV[0.5,1] . What does this value measure on the graph?
What does this value tell us about the motion of the ball? In particular, what are
the units on AV[0.5,1] ?
Chapter 1 Understanding the Derivative
1.1.1 Position and average velocity
Any moving object has a position that can be considered a function of time. When the motion
is along a straight line, the position is given by a single variable, which we denote by s(t).
For example, s(t) might give the mile marker of a car traveling on a straight highway at time
t in hours. Similarly, the function s described in Preview Activity 1.1.1 is a position function,
where position is measured vertically relative to the ground.
On any time interval, a moving object also has an average velocity. For example, to compute
a car’s average velocity we divide the number of miles traveled by the time elapsed, which
gives the velocity in miles per hour. Similarly, the value of AV[0.5,1] in Preview Activity 1.1.1
gave the average velocity of the ball on the time interval [0.5, 1], measured in feet per second.
In general, we make the following definition:
Average Velocity.
For an object moving in a straight line with position function s(t), the average velocity
of the object on the interval from t a to t b, denoted AV[a,b] , is given by the formula
s(b) − s(a)
AV[a,b] .
b−a
Note well: the units on AV[a,b] are “units of s per unit of t,” such as “miles per hour” or “feet
per second.”
Activity 1.1.2. The following questions concern the position function given by s(t)
64 − 16(t − 1)2 , considered in Preview Activity 1.1.1.
a. Compute the average velocity of the ball on each of the following time inter-
vals: [0.4, 0.8], [0.7, 0.8], [0.79, 0.8], [0.799, 0.8], [0.8, 1.2], [0.8, 0.9], [0.8, 0.81],
[0.8, 0.801]. Include units for each value.
b. On the graph provided in Figure 1.1.1, sketch the line that passes through the
points A (0.4, s(0.4)) and B (0.8, s(0.8)). What is the meaning of the slope
of this line? In light of this meaning, what is a geometric way to interpret each
of the values computed in the preceding question?
c. Use a graphing utility to plot the graph of s(t) 64 − 16(t − 1)2 on an interval
containing the value t 0.8. Then, zoom in repeatedly on the point (0.8, s(0.8)).
What do you observe about how the graph appears as you view it more and
more closely?
d. What do you conjecture is the velocity of the ball at the instant t 0.8? Why?
2
1.1 How do we measure velocity?
feet
s
64
B
A
56
48
sec
0.4 0.8 1.2
Figure 1.1.1: A partial plot of s(t) 64 − 16(t − 1)2 .
1.1.2 Instantaneous Velocity
Whether we are driving a car, riding a bike, or throwing a ball, we have an intuitive sense that
a moving object has a velocity at any given moment -- a number that measures how fast the
object is moving right now. For instance, a car’s speedometer tells the driver the car’s velocity
at any given instant. In fact, the velocity on a speedometer is really an average velocity that
is computed over a very small time interval. If we let the time interval over which average
velocity is computed become shorter and shorter, we can progress from average velocity to
instantaneous velocity.
Informally, we define the instantaneous velocity of a moving object at time t a to be the
value that the average velocity approaches as we take smaller and smaller intervals of time
containing t a. We will develop a more formal definition of instantaneous velocity soon,
and this definition will be the foundation of much of our work in calculus. For now, it is fine
to think of instantaneous velocity as follows: take average velocities on smaller and smaller
time intervals around a specific point. If those average velocities approach a single number,
then that number will be the instantaneous velocity at that point.
Activity 1.1.3. Each of the following questions concern s(t) 64 − 16(t − 1)2 , the
position function from Preview Activity 1.1.1.
a. Compute the average velocity of the ball on the time interval [1.5, 2]. What is
different between this value and the average velocity on the interval [0, 0.5]?
b. Use appropriate computing technology to estimate the instantaneous velocity
of the ball at t 1.5. Likewise, estimate the instantaneous velocity of the ball at
t 2. Which value is greater?
3
Chapter 1 Understanding the Derivative
c. How is the sign of the instantaneous velocity of the ball related to its behavior
at a given point in time? That is, what does positive instantaneous velocity tell
you the ball is doing? Negative instantaneous velocity?
d. Without doing any computations, what do you expect to be the instantaneous
velocity of the ball at t 1? Why?
At this point we have started to see a close connection between average velocity and instan-
taneous velocity. Each is connected not only to the physical behavior of the moving object
but also to the geometric behavior of the graph of the position function. We are interested
in computing average velocities on the interval [a, b] for smaller and smaller intervals. In
order to make the link between average and instantaneous velocity more formal, think of
the value b as b a + h, where h is a small (non-zero) number that is allowed to vary. Then
the average velocity of the object on the interval [a, a + h] is
s(a + h) − s(a)
AV[a,a+h] ,
h
with the denominator being simply h because (a + h)− a h. Note that when h < 0, AV[a,a+h]
measures the average velocity on the interval [a + h, a].
To find the instantaneous velocity at t a, we investigate what happens as the value of h
approaches zero.
Example 1.1.2 Computing instantaneous velocity for a falling ball. The position function
for a falling ball is given by s(t) 16 − 16t 2 (where s is measured in feet and t in seconds).
a. Find an expression for the average velocity of the ball on a time interval of the form
[0.5, 0.5 + h] where −0.5 < h < 0.5 and h , 0.
b. Use this expression to compute the average velocity on [0.5, 0.75] and [0.4, 0.5].
c. Make a conjecture about the instantaneous velocity at t 0.5.
Solution.
a. We make the assumptions that −0.5 < h < 0.5 and h , 0 because h cannot be zero
(otherwise there is no interval on which to compute average velocity) and because the
function only makes sense on the time interval 0 ≤ t ≤ 1, as this is the duration of time
during which the ball is falling. We want to compute and simplify
s(0.5 + h) − s(0.5)
AV[0.5,0.5+h] .
(0.5 + h) − 0.5
We start by finding s(0.5 + h). To do so, we follow the rule that defines the function s.
s(0.5 + h) 16 − 16(0.5 + h)2
16 − 16(0.25 + h + h 2 )
16 − 4 − 16h − 16h 2
12 − 16h − 16h 2 .
4
1.1 How do we measure velocity?
Now, returning to our computation of the average velocity, we find that
s(0.5 + h) − s(0.5)
AV[0.5,0.5+h]
(0.5 + h) − 0.5
(12 − 16h − 16h 2 ) − (16 − 16(0.5)2 )
0.5 + h − 0.5
12 − 16h − 16h 2 − 12
h
−16h − 16h 2
.
h
At this point, we note two things: first, the expression for average velocity clearly de-
pends on h, which it must, since as h changes the average velocity will change. Further,
we note that since h can never equal zero, we may remove the common factor of h from
the numerator and denominator. It follows that
AV[0.5,0.5+h] −16 − 16h.
b. From this expression we can compute the average for any small positive or negative
value of h. For instance, to obtain the average velocity on [0.5, 0.75], we let h 0.25,
and the average velocity is −16 − 16(0.25) −20 ft/sec. To get the average velocity on
[0.4, 0.5], we let h −0.1, and compute the average velocity as
−16 − 16(−0.1) −14.4 ft/sec.
c. We can even explore what happens to AV[0.5,0.5+h] as h gets closer and closer to zero. As
h approaches zero, −16h will also approach zero, so it appears that the instantaneous
velocity of the ball at t 0.5 should be −16 ft/sec.
Activity 1.1.4. For the function given by s(t) 64 − 16(t − 1)2 from Preview Activ-
ity 1.1.1, find the most simplified expression you can for the average velocity of the
ball on the interval [2, 2 + h]. Use your result to compute the average velocity on
[1.5, 2] and to estimate the instantaneous velocity at t 2. Finally, compare your
earlier work in Activity 1.1.2.
1.1.3 Summary
• For an object moving in a straight line with position function s(t), the average velocity
of the object on the interval from t a to t b, denoted AV[a,b] , is given by the formula
s(b) − s(a)
AV[a,b] .
b−a
• The average velocity on [a, b] can be viewed geometrically as the slope of the line be-
tween the points (a, s(a)) and (b, s(b)) on the graph of y s(t), as shown in Figure 1.1.3.
5
Chapter 1 Understanding the Derivative
s
s(b)−s(a)
m= b−a
(b, s(b))
(a, s(a))
t
Figure 1.1.3: The graph of position function s together with the line through (a, s(a)) and
s(b)−s(a)
(b, s(b)) whose slope is m b−a . The line’s slope is the average rate of change of s on
the interval [a, b].
• Given a moving object whose position at time t is given by a function s, the average
s(b)−s(a)
velocity of the object on the time interval [a, b] is given by AV[a,b] b−a . View-
ing the interval [a, b] as having the form [a, a + h], we equivalently compute average
s(a+h)−s(a)
velocity by the formula AV[a,a+h] h .
• The instantaneous velocity of a moving object at a fixed time is estimated by consider-
ing average velocities on shorter and shorter time intervals that contain the instant of
interest.
1.1.4 Exercises
1. Average velocity from position. Consider a car whose position, s, is given by the table
t (s) 0 0.2 0.4 0.6 0.8 1
s (ft) 0 0.5 1.4 3.8 6.5 9.6
Find the average velocity over the interval 0 ≤ t ≤ 0.2. Estimate the velocity at t 0.2.
6
1.1 How do we measure velocity?
2. Rate of calorie consumption. The table below shows the number of calories used per
minute as a function of an individual’s body weight for three sports:
Activity 100 lb 120 lb 150 lb 170 lb 200 lb 220 lb
Walking 2.7 3.2 4 4.6 5.4 5.9
Bicycling 5.4 6.5 8.1 9.2 10.8 11.9
Swimming 5.8 6.9 8.7 9.8 11.6 12.7
a) Determine the number of calories that a 200 lb person uses in one half-hour of walk-
ing.
b) Who uses more calories, a 170 lb person swimming for one hour, or a 220 lb person
bicycling for a half-hour?
c) Does the number of calories of a person walking increase or decrease as weight in-
creases?
3. Average rate of change - quadratic function. Let f (x) 9 − x 2 .
a) Compute each of the following expressions and interpret each as an average rate of
change:
f (1)− f (0)
(i) 1−0
f (3)− f (1)
(ii) 3−1
f (3)− f (0)
(iii) 3−0
b) Based on the graph sketched below, match each of your answers in (i) - (iii) with one
of the lines labeled A - F. Type the corresponding letter of the line segment next to the
appropriate formula. Clearly not all letters will be used.
f (1) − f (0)
1−0
f (3) − f (1)
3−1
f (3) − f (0)
3−0
7
Chapter 1 Understanding the Derivative
4. Comparing average rate of change of two functions. Consider the graphs of f (x) and
1(x) below:
For each interval given below, decide whether the average rate of change of f (x) or 1(x)
is greater over that particular interval.
Interval Which function has GREATER average rate of change?
0≤x≤4 (□ f □ g □ both have an equal rate of change)
0≤x≤8 (□ f □ g □ both have an equal rate of change)
0 ≤ x ≤ 2.2 (□ f □ g □ both have an equal rate of change)
5.2 ≤ x ≤ 6.1 (□ f □ g □ both have an equal rate of change)
5.2 ≤ x ≤ 6.9 (□ f □ g □ both have an equal rate of change)
5. Matching a distance graph to velocity. A car is driven at a constant speed, starting at
noon. Which of the following could be a graph of the distance the car has traveled as a
function of time past noon?
1. 2. 3. 4.
5. 6. 7. 8.
8
1.1 How do we measure velocity?
6. A bungee jumper dives from a tower at time t 0. Her height h (measured in feet)
at time t (in seconds) is given by the graph in Figure 1.1.4. In this problem, you may
base your answers on estimates from the graph or use the fact that the jumper’s height
function is given by s(t) 100 cos(0.75t) · e −0.2t + 100.
200
s
150
100
50
t
5 10 15 20
Figure 1.1.4: A bungee jumper’s height function.
a. What is the change in vertical position of the bungee jumper between t 0 and
t 15?
b. Estimate the jumper’s average velocity on each of the following time intervals:
[0, 15], [0, 2], [1, 6], and [8, 10]. Include units on your answers.
c. On what time interval(s) do you think the bungee jumper achieves her greatest
average velocity? Why?
d. Estimate the jumper’s instantaneous velocity at t 5. Show your work and ex-
plain your reasoning, and include units on your answer.
e. Among the average and instantaneous velocities you computed in earlier ques-
tions, which are positive and which are negative? What does negative velocity
indicate?
7. A diver leaps from a 3 meter springboard. His feet leave the board at time t 0,
he reaches his maximum height of 4.5 m at t 1.1 seconds, and enters the water at
t 2.45. Once in the water, the diver coasts to the bottom of the pool (depth 3.5 m),
touches bottom at t 7, rests for one second, and then pushes off the bottom. From
there he coasts to the surface, and takes his first breath at t 13.
a. Let s(t) denote the function that gives the height of the diver’s feet (in meters)
above the water at time t. (Note that the “height” of the bottom of the pool is
−3.5 meters.) Sketch a carefully labeled graph of s(t) on the provided axes in
Figure 1.1.5. Include scale and units on the vertical axis. Be as detailed as possible.
b. Based on your graph in (a), what is the average velocity of the diver between
t 2.45 and t 7? Is his average velocity the same on every time interval within
[2.45, 7]?
9
Chapter 1 Understanding the Derivative
s v
t t
2 4 6 8 10 12 2 4 6 8 10 12
Figure 1.1.5: Axes for plotting s(t) Figure 1.1.6: Axes for plotting v(t)
in part (a). in part (c).
c. Let the function v(t) represent the instantaneous vertical velocity of the diver at time
t (i.e. the speed at which the height function s(t) is changing; note that velocity in
the upward direction is positive, while the velocity of a falling object is negative).
Based on your understanding of the diver’s behavior, as well as your graph of the
position function, sketch a carefully labeled graph of v(t) on the axes provided in
Figure 1.1.6. Include scale and units on the vertical axis. Write several sentences
that explain how you constructed your graph, discussing when you expect v(t)
to be zero, positive, negative, relatively large, and relatively small.
d. Is there a connection between the two graphs that you can describe? What can you
say about the velocity graph when the height function is increasing? decreasing?
Make as many observations as you can.
8. According to the U.S. census, the population of the city of Grand Rapids, MI, was
181,843 in 1980; 189,126 in 1990; and 197,800 in 2000.
a. Between 1980 and 2000, by how many people did the population of Grand Rapids
grow?
b. In an average year between 1980 and 2000, by how many people did the popula-
tion of Grand Rapids grow?
c. Just like we can find the average velocity of a moving body by computing change
in position over change in time, we can compute the average rate of change of any
function f . In particular, the average rate of change of a function f over an interval
[a, b] is the quotient
f (b) − f (a)
.
b−a
f (b)− f (a)
What does the quantity b−a measure on the graph of y f (x) over the inter-
val [a, b]?
d. Let P(t) represent the population of Grand Rapids at time t, where t is measured
in years from January 1, 1980. What is the average rate of change of P on the
interval t 0 to t 20? What are the units on this quantity?
10
1.1 How do we measure velocity?
e. If we assume the population of Grand Rapids is growing at a rate of approxi-
mately 4% per decade, we can model the population function with the formula
P(t) 181843(1.04)t/10 .
Use this formula to compute the average rate of change of the population on the
intervals [5, 10], [5, 9], [5, 8], [5, 7], and [5, 6].
f. How fast do you think the population of Grand Rapids was changing on January
1, 1985? Said differently, at what rate do you think people were being added to the
population of Grand Rapids as of January 1, 1985? How many additional people
should the city have expected in the following year? Why?
11
Chapter 1 Understanding the Derivative
1.2 The notion of limit
Motivating Questions
• What is the mathematical notion of limit and what role do limits play in the study of
functions?
• What is the meaning of the notation limx→a f (x) L?
• How do we go about determining the value of the limit of a function at a point?
• How do we manipulate average velocity to compute instantaneous velocity?
In Section 1.1 we used a function, s(t), to model the location of a moving object at a given
time. Functions can model other interesting phenomena, such as the rate at which an auto-
mobile consumes gasoline at a given velocity, or the reaction of a patient to a given dosage
of a drug. We can use calculus to study how a function value changes in response to changes
in the input variable.
Think about the falling ball whose position function is given by s(t) 64 − 16t 2 . Its average
velocity on the interval [1, x] is given by
s(x) − s(1) (64 − 16x 2 ) − (64 − 16) 16 − 16x 2
AV[1,x] .
x−1 x−1 x−1
2
Note that the average velocity is a function of x. That is, the function 1(x) 16−16x
x−1 tells us
the average velocity of the ball on the interval from t 1 to t x. To find the instantaneous
velocity of the ball when t 1, we need to know what happens to 1(x) as x gets closer and
closer to 1. But also notice that 1(1) is not defined, because it leads to the quotient 0/0.
This is where the notion of a limit comes in. By using a limit, we can investigate the behavior
of 1(x) as x gets arbitrarily close, but not equal, to 1. We first use the graph of a function to
explore points where interesting behavior occurs.
Preview Activity 1.2.1. Suppose that 1 is the function given by the graph below. Use
the graph in Figure 1.2.1 to answer each of the following questions.
a. Determine the values 1(−2), 1(−1), 1(0), 1(1), and 1(2), if defined. If the function
value is not defined, explain what feature of the graph tells you this.
b. For each of the values a −1, a 0, and a 2, complete the following sentence:
“As x gets closer and closer (but not equal) to a, 1(x) gets as close as we want to
em minus 4.5emem .”
c. What happens as x gets closer and closer (but not equal) to a 1? Does the
function 1(x) get as close as we would like to a single value?
12
1.2 The notion of limit
g
3
2
1
-2 -1 1 2 3
-1
Figure 1.2.1: Graph of y 1(x) for Preview Activity 1.2.1.
1.2.1 The Notion of Limit
Limits give us a way to identify a trend in the values of a function as its input variable
approaches a particular value of interest. We need a precise understanding of what it means
to say “a function f has limit L as x approaches a.” To begin, think about a recent example.
In Preview Activity 1.2.1, we saw that as x gets closer and closer (but not equal) to 0, 1(x)
gets as close as we want to the value 4. At first, this may feel counterintuitive, because the
value of 1(0) is 1, not 4. But limits describe the behavior of a function arbitrarily close to a
fixed input, and the value of the function at the fixed input does not matter. More formally,¹
we say the following.
Definition 1.2.2 Given a function f , a fixed input x a, and a real number L, we say that f
has limit L as x approaches a, and write
lim f (x) L
x→a
provided that we can make f (x) as close to L as we like by taking x sufficiently close (but
not equal) to a. If we cannot make f (x) as close to a single value as we would like as x
approaches a, then we say that f does not have a limit as x approaches a.
Example 1.2.3 For the function 1 pictured in Figure 1.2.1, we make the following observa-
tions:
lim 1(x) 3, lim 1(x) 4, and lim 1(x) 1.
x→−1 x→0 x→2
When working from a graph, it suffices to ask if the function approaches a single value from
each side of the fixed input. The function value at the fixed input is irrelevant. This reasoning
¹What follows here is not what mathematicians consider the formal definition of a limit. To be completely
precise, it is necessary to quantify both what it means to say “as close to L as we like” and “sufficiently close to a.”
That can be accomplished through what is traditionally called the epsilon-delta definition of limits. The definition
presented here is sufficient for the purposes of this text.
13
Chapter 1 Understanding the Derivative
explains the values of the three limits stated above.
However, 1 does not have a limit as x → 1. There is a jump in the graph at x 1. If we
approach x 1 from the left, the function values tend to get close to 3, but if we approach
x 1 from the right, the function values get close to 2. There is no single number that all of
these function values approach. This is why the limit of 1 does not exist at x 1.
For any function f , there are typically three ways to answer the question “does f have a
limit at x a, and if so, what is the limit?” The first is to reason graphically as we have just
done with the example from Preview Activity 1.2.1. If we have a formula for f (x), there are
two additional possibilities:
1 Evaluate the function at a sequence of inputs that approach a on either side (typically
using some sort of computing technology), and ask if the sequence of outputs seems
to approach a single value.
2 Use the algebraic form of the function to understand the trend in its output values as
the input values approach a.
The first approach produces only an approximation of the value of the limit, while the latter
can often be used to determine the limit exactly.
Example 1.2.4 Limits of Two Functions. For each of the following functions, we’d like to
know whether or not the function has a limit at the stated a-values. Use both numerical and
algebraic approaches to investigate and, if possible, estimate or determine the value of the
limit. Compare the results with a careful graph of the function on an interval containing the
points of interest. ( )
b. 1(x) sin πx ; a 3, a 0
2
a. f (x) 4−x
x+2 ; a −1, a −2
Solution. a. We first construct a graph of f along with tables of values near a −1 and
a −2.
From Table 1.2.5, it appears that we can make f as close as we want to 3 by taking x suf-
ficiently close to −1, which suggests that limx→−1 f (x) 3. This is also consistent with
the graph of f . To see this a bit more rigorously and from an algebraic point of view,
2
consider the formula for f : f (x) 4−xx+2 . As x → −1, (4 − x ) → (4 − (−1) ) 3, and
2 2
(x + 2) → (−1 + 2) 1, so as x → −1, the numerator of f tends to 3 and the denominator
tends to 1, hence limx→−1 f (x) 31 3.
The situation is more complicated when x → −2, because f (−2) is not defined. If we try
to use a similar algebraic argument regarding the numerator and denominator, we observe
that as x → −2, (4 − x 2 ) → (4 − (−2)2 ) 0, and (x + 2) → (−2 + 2) 0, so as x → −2, the
numerator and denominator of f both tend to 0. We call 0/0 an indeterminate form. This tells
us that there is somehow more work to do. From Table 1.2.6 and Figure 1.2.7, it appears that
f should have a limit of 4 at x −2.
14
1.2 The notion of limit
f
x f (x) x f (x)
5
−0.9 2.9 −1.9 3.9
−0.99 2.99 −1.99 3.99
−0.999 2.999 −1.999 3.999
−0.9999 2.9999 −1.9999 3.9999 3
−1.1 3.1 −2.1 4.1
−1.01 3.01 −2.01 4.01
−1.001 3.001 −2.001 4.001 1
−1.0001 3.0001 −2.0001 4.0001
-3 -1 1
Table 1.2.5: Table of Table 1.2.6: Table of
f values near x −1. f values near x −2.
Figure 1.2.7: Plot of f (x) on [−4, 2].
To see algebraically why this is the case, observe that
4 − x2
lim f (x) lim
x→−2 x→−2 x + 2
(2 − x)(2 + x)
lim .
x→−2 x+2
It is important to observe that, since we are taking the limit as x → −2, we are considering
x values that are close, but not equal, to −2. Because we never actually allow x to equal −2,
the quotient 2+x
x+2 has value 1 for every possible value of x. Thus, we can simplify the most
recent expression above, and find that
lim f (x) lim 2 − x.
x→−2 x→−2
This limit is now easy to determine, and its value clearly is 4. Thus, from several points of
view we’ve seen that limx→−2 f (x) 4.
b. Next we turn to the function 1, and construct two tables and a graph.
x 1(x) x 1(x) 2
2.9 0.84864 −0.1 0 g
2.99 0.86428 −0.01 0
2.999 0.86585 −0.001 0 -3 -1 1 3
2.9999 0.86601 −0.0001 0
3.1 0.88351 0.1 0 -2
3.01 0.86777 0.01 0
3.001 0.86620 0.001 0 Figure 1.2.10: Plot of 1(x) on [−4, 4].
3.0001 0.86604 0.0001 0
Table 1.2.8: Table of Table 1.2.9: Table of
1 values near x 3. 1 values near x 0.
First, as x → 3, it appears from the table values that the function is approaching a number
between 0.86601 and 0.86604. From the graph it appears that 1(x) → 1(3) as x → 3. The
15
Chapter 1 Understanding the Derivative
√
exact value of 1(3) sin( π3 ) is 3
2 , which is approximately 0.8660254038. This is convincing
evidence that √
3
lim 1(x) .
x→3 2
As x → 0, we observe that πx does not behave in an elementary way. When x is positive and
approaching zero, we are dividing by smaller and smaller positive values, and πx increases
without bound. When x is negative and approaching zero, πx decreases without bound. In
this sense, as we get close to x 0, the inputs to the sine function are growing rapidly, and
this leads to increasingly
( rapid
) oscillations in the graph of 1 betweem 1 and −1. If we plot
the function 1(x) sin πx with a graphing utility and then zoom in on x 0, we see that
the function never settles down to a single value near the origin, which suggests that 1 does
not have a limit at x 0.
How do we reconcile the graph with the righthand table above, which seems to suggest that
the limit of 1 as x approaches 0 may in fact be 0? The data misleads us because of the special
−k
nature of the sequence
( ) of input values {0.1, 0.01, 0.001, . . .}. When we evaluate 1(10 ), we
π
get 1(10−k ) sin 10−k
sin(10k π) 0 for each positive integer value of k. But if we take a
different sequence of values approaching zero, say {0.3, 0.03, 0.003, . . .}, then we find that
( ( k ) √
−k π ) 10 π 3
1(3 · 10 ) sin −k
sin ≈ 0.866025.
3 · 10 3 2
√
That sequence of function values suggests that the value of the limit is 23 . Clearly the func-
tion cannot have two different values for the limit, so 1 has no limit as x → 0.
An important lesson to take from Example 1.2.4 is that tables can be misleading when de-
termining the value of a limit. While a table of values is useful for investigating the possible
value of a limit, we should also use other tools to confirm the value.
Activity 1.2.2. Estimate the value of each of the following limits by constructing ap-
propriate tables of values. Then determine the exact value of the limit by using alge-
bra to simplify the function. Finally, plot each function on an appropriate interval to
check your result visually. √
2 −1 (2+x)3 −8
a. limx→1 xx−1 b. limx→0 x c. lim x→0
x+1−1
x
Recall that our primary motivation for considering limits of functions comes from our inter-
est in studying the rate of change of a function. To that end, we close this section by revisiting
our previous work with average and instantaneous velocity and highlighting the role that
limits play.
16
1.2 The notion of limit
1.2.2 Instantaneous Velocity
Suppose that we have a moving object whose position at time t is given by a function s. We
s(b)−s(a)
know that the average velocity of the object on the time interval [a, b] is AV[a,b] b−a .
We define the instantaneous velocity at a to be the limit of average velocity as b approaches
a. Note particularly that as b → a, the length of the time interval gets shorter and shorter
(while always including a). We will write IVta for the instantaneous velocity at t a, and
thus
s(b) − s(a)
IVta lim AV[a,b] lim .
b→a b→a b−a
Equivalently, if we think of the changing value b as being of the form b a + h, where h is
some small number, then we may instead write
s(a + h) − s(a)
IVta lim AV[a,a+h] lim .
h→0 h→0 h
Again, the most important idea here is that to compute instantaneous velocity, we take a
limit of average velocities as the time interval shrinks.
Activity 1.2.3. Consider a moving object whose position function is given by s(t) t 2 ,
where s is measured in meters and t is measured in minutes.
a. Determine the most simplified expression for the average velocity of the object
on the interval [3, 3 + h], where h > 0.
b. Determine the average velocity of the object on the interval [3, 3.2]. Include units
on your answer.
c. Determine the instantaneous velocity of the object when t 3. Include units on
your answer.
The closing activity of this section asks you to make some connections among average ve-
locity, instantaneous velocity, and slopes of certain lines.
Activity 1.2.4. For the moving object whose position s at time t is given by the graph
in Figure 1.2.11, answer each of the following questions. Assume that s is measured
in feet and t is measured in seconds.
17
Chapter 1 Understanding the Derivative
5 s
3
1
t
1 3 5
Figure 1.2.11: Plot of the position function y s(t) in Activity 1.2.4.
a. Use the graph to estimate the average velocity of the object on each of the fol-
lowing intervals: [0.5, 1], [1.5, 2.5], [0, 5]. Draw each line whose slope represents
the average velocity you seek.
b. How could you use average velocities or slopes of lines to estimate the instan-
taneous velocity of the object at a fixed time?
c. Use the graph to estimate the instantaneous velocity of the object when t 2.
Should this instantaneous velocity at t 2 be greater or less than the average
velocity on [1.5, 2.5] that you computed in (a)? Why?
1.2.3 Summary
• Limits enable us to examine trends in function behavior near a specific point. In partic-
ular, taking a limit at a given point asks if the function values nearby tend to approach
a particular fixed value.
• We read limx→a f (x) L, as “the limit of f as x approaches a is L,” which means that
we can make the value of f (x) as close to L as we want by taking x sufficiently close
(but not equal) to a.
• To find limx→a f (x) for a given value of a and a known function f , we can estimate this
value from the graph of f , or we can make a table of function values for x-values that
are closer and closer to a. If we want the exact value of the limit, we can work with the
function algebraically to understand how different parts of the formula for f change
as x → a.
• We find the instantaneous velocity of a moving object at a fixed time by taking the limit
of average velocities of the object over shorter and shorter time intervals containing the
time of interest.
18
1.2 The notion of limit
1.2.4 Exercises
1. Limits on a piecewise graph. Use the figure below, which gives a graph of the function
f (x), to give values for the indicated limits.
(a) lim f (x)
x→−1
(b) lim f (x)
x→0
(c) lim f (x)
x→1
(d) lim f (x)
x→4
2. Estimating a limit numerically. Use a graph to estimate the limit
sin(6θ)
lim .
θ→0 θ
Note: θ is measured in radians. All angles will be in radians in this class unless other-
wise specified.
3. Limits for a piecewise formula. For the function
x 2 − 4, 0≤x<4
f (x) 4, x4
3x + 0, 4 < x
use algebra to find each of the following limits:
lim f (x)
x→4+
lim f (x)
x→4−
lim f (x)
x→4
Sketch a graph of f (x) to confirm your answers.
4. Evaluating a limit algebraically. Evaluate the limit
x 2 − 49
lim
x→−7 x+7
19
Chapter 1 Understanding the Derivative
16−x 4
5. Consider the function whose formula is f (x) x 2 −4
.
a. What is the domain of f ?
b. Use a sequence of values of x near a 2 to estimate the value of limx→2 f (x), if
you think the limit exists. If you think the limit doesn’t exist, explain why.
16−x 4
c. Use algebra to simplify the expression x 2 −4
and hence work to evaluate lim f (x)
x→2
exactly, if it exists, or to explain how your work shows the limit fails to exist.
Discuss how your findings compare to your results in (b).
d. True or false: f (2) −8. Why?
4
e. True or false: 16−x
x 2 −4
−4 − x 2 . Why? How is this equality connected to your work
above with the function f ?
f. Based on all of your work above, construct an accurate, labeled graph of y f (x)
on the interval [1, 3], and write a sentence that explains what you now know about
4
limx→2 16−x
x 2 −4
.
|x+3|
6. Let 1(x) − x+3 .
a. What is the domain of 1?
b. Use a sequence of values near a −3 to estimate the value of limx→−3 1(x), if you
think the limit exists. If you think the limit doesn’t exist, explain why.
|x+3|
c. Use algebra to simplify the expression x+3 and hence work to evaluate lim 1(x)
x→−3
exactly, if it exists, or to explain how your work shows the limit fails to exist.
Discuss how your findings compare to your results in (b). (Hint: |a| a whenever
a ≥ 0, but |a| −a whenever a < 0.)
d. True or false: 1(−3) −1. Why?
|x+3|
e. True or false: − x+3 −1. Why? How is this equality connected to your work
above with the function 1?
f. Based on all of your work above, construct an accurate, labeled graph of y 1(x)
on the interval [−4, −2], and write a sentence that explains what you now know
about limx→−3 1(x).
7. For each of the following prompts, sketch a graph on the provided axes of a function
that has the stated properties.
a. y f (x) such that
• f (−2) 2 and limx→−2 f (x) 1
• f (−1) 3 and limx→−1 f (x) 3
• f (1) is not defined and limx→1 f (x) 0
• f (2) 1 and limx→2 f (x) does not exist.
20
1.2 The notion of limit
b. y 1(x) such that
• 1(−2) 3, 1(−1) −1, 1(1) −2, and 1(2) 3
• At x −2, −1, 1 and 2, 1 has a limit, and its limit equals the value of the
function at that point.
• 1(0) is not defined and limx→0 1(x) does not exist.
3 3
-3 3 -3 3
-3 -3
Figure 1.2.12: Axes for plotting y f (x) in (a) and y 1(x) in (b).
8. A bungee jumper dives from a tower at time t 0. Her height s in feet at time t in
seconds is given by s(t) 100 cos(0.75t) · e −0.2t + 100.
a. Write an expression for the average velocity of the bungee jumper on the interval
[1, 1 + h].
b. Use computing technology to estimate the value of the limit as h → 0 of the
quantity you found in (a).
c. What is the meaning of the value of the limit in (b)? What are its units?
21
Chapter 1 Understanding the Derivative
1.3 The derivative of a function at a point
Motivating Questions
• How is the average rate of change of a function on a given interval defined, and what
does this quantity measure?
• How is the instantaneous rate of change of a function at a particular point defined?
How is the instantaneous rate of change linked to average rate of change?
• What is the derivative of a function at a given point? What does this derivative value
measure? How do we interpret the derivative value graphically?
• How are limits used formally in the computation of derivatives?
The instantaneous rate of change of a function is an idea that sits at the foundation of calculus.
It is a generalization of the notion of instantaneous velocity and measures how fast a partic-
ular function is changing at a given point. If the original function represents the position of
a moving object, this instantaneous rate of change is precisely the velocity of the object. In
other contexts, instantaneous rate of change could measure the number of cells added to a
bacteria culture per day, the number of additional gallons of gasoline consumed by increas-
ing a car’s velocity one mile per hour, or the number of dollars added to a mortgage payment
for each percentage point increase in interest rate. The instantaneous rate of change can also
be interpreted geometrically on the function’s graph, and this connection is fundamental to
many of the main ideas in calculus.
Recall that for a moving object with position function s, its average velocity on the time
interval t a to t a + h is given by the quotient
s(a + h) − s(a)
AV[a,a+h] .
h
In a similar way, we make the following definition for an arbitrary function y f (x).
Definition 1.3.1 For a function f , the average rate of change of f on the interval [a, a + h] is
given by the value
f (a + h) − f (a)
AV[a,a+h] .
h
Equivalently, if we want to consider the average rate of change of f on [a, b], we compute
f (b) − f (a)
AV[a,b] .
b−a
It is essential that you understand how the average rate of change of f on an interval is
connected to its graph.
22
1.3 The derivative of a function at a point
Preview Activity 1.3.1. Suppose that f is the function given by the graph below and
that a and a + h are the input values as labeled on the x-axis. Use the graph in Fig-
ure 1.3.2 to answer the following questions.
y
f
x
a a+h
Figure 1.3.2: Plot of y f (x) for Preview Activity 1.3.1.
a. Locate and label the points (a, f (a)) and (a + h, f (a + h)) on the graph.
b. Construct a right triangle whose hypotenuse is the line segment from (a, f (a))
to (a + h, f (a + h)). What are the lengths of the respective legs of this triangle?
c. What is the slope of the line that connects the points (a, f (a)) and (a+h, f (a+h))?
d. Write a meaningful sentence that explains how the average rate of change of the
function on a given interval and the slope of a related line are connected.
1.3.1 The Derivative of a Function at a Point
Just as we defined instantaneous velocity in terms of average velocity, we now define the
instantaneous rate of change of a function at a point in terms of the average rate of change
of the function f over related intervals. This instantaneous rate of change of f at a is called
“the derivative of f at a,” and is denoted by f ′(a).
Definition 1.3.3 Let f be a function and x a a value in the function’s domain. We define
the derivative of f with respect to x evaluated at x a, denoted f ′(a), by the formula
f (a + h) − f (a)
f ′(a) lim ,
h→0 h
provided this limit exists.
Aloud, we read the symbol f ′(a) as either “ f -prime at a” or “the derivative of f evaluated
23
Chapter 1 Understanding the Derivative
at x a.” Much of the next several chapters will be devoted to understanding, computing,
applying, and interpreting derivatives. For now, we observe the following important things.
Note 1.3.4
• The derivative of f at the value x a is defined as the limit of the average rate of
change of f on the interval [a, a + h] as h → 0. This limit may not exist, so not every
function has a derivative at every point.
• We say that a function is differentiable at x a if it has a derivative at x a.
• The derivative is a generalization of the instantaneous velocity of a position function:
if y s(t) is a position function of a moving body, s ′(a) tells us the instantaneous
velocity of the body at time t a.
f (a+h)− f (a)
• Because the units on h are “units of f (x) per unit of x,” the derivative has
these very same units. For instance, if s measures position in feet and t measures time
in seconds, the units on s ′(a) are feet per second.
f (a+h)− f (a)
• Because the quantity h represents the slope of the line through (a, f (a)) and
(a + h, f (a + h)), when we compute the derivative we are taking the limit of a collec-
tion of slopes of lines. Thus, the derivative itself represents the slope of a particularly
important line.
We first consider the derivative at a given value as the slope of a certain line.
When we compute an instantaneous rate of change, we allow the interval [a, a + h] to shrink
as h → 0. We can think of one endpoint of the interval as “sliding towards” the other.
In particular, provided that f has a derivative at (a, f (a)), the point (a + h, f (a + h)) will
approach (a, f (a)) as h → 0. Because the process of taking a limit is a dynamic one, it can
be helpful to use computing technology to visualize it. One option is a java applet in which
the user is able to control the point that is moving. For a helpful collection of examples,
consider the work of David Austin of Grand Valley State University, and this particularly
relevant example. For applets that have been built in Geogebra¹, see Marc Renault’s library
via Shippensburg University, with this example being especially fitting for our work in this
section.
Figure 1.3.5 shows a sequence of figures with several different lines through the points
(a, f (a)) and (a + h, f (a + h)), generated by different values of h. These lines (shown in
the first three figures in magenta), are often called secant lines to the curve y f (x). A se-
cant line to a curve is simply a line that passes through two points on the curve. For each
f (a+h)− f (a)
such line, the slope of the secant line is m h , where the value of h depends on
the location of the point we choose. We can see in the diagram how, as h → 0, the secant
lines start to approach a single line that passes through the point (a, f (a)). If the limit of the
slopes of the secant lines exists, we say that the resulting value is the slope of the tangent
line to the curve. This tangent line (shown in the right-most figure in green) to the graph of
y f (x) at the point (a, f (a)) has slope m f ′(a).
¹You can even consider building your own examples; the fantastic program Geogebra is available for free down-
load and is easy to learn and use.
24
1.3 The derivative of a function at a point
y y y y
f f f f
x x x x
a a a a
Figure 1.3.5: A sequence of secant lines approaching the tangent line to f at (a, f (a)).
If the tangent line at x a exists, the graph of f looks like a straight line when viewed up
close at (a, f (a)). In Figure 1.3.6 we combine the four graphs in Figure 1.3.5 into the single
one on the left, and zoom in on the box centered at (a, f (a)) on the right. Note how the
tangent line sits relative to the curve y f (x) at (a, f (a)) and how closely it resembles the
curve near x a.
y
f
x
a
Figure 1.3.6: A sequence of secant lines approaching the tangent line to f at (a, f (a)). At
right, we zoom in on the point (a, f (a)). The slope of the tangent line (in green) to f at
(a, f (a)) is given by f ′(a).
Note 1.3.7 The instantaneous rate of change of f with respect to x at x a, f ′(a), also mea-
sures the slope of the tangent line to the curve y f (x) at (a, f (a)).
The following example demonstrates several key ideas involving the derivative of a function.
Example 1.3.8 Using the limit definition of the derivative. For the function f (x) x − x 2 ,
use the limit definition of the derivative to compute f ′(2). In addition, discuss the meaning
of this value and draw a labeled graph that supports your explanation.
Solution. From the limit definition, we know that
f (2 + h) − f (2)
f ′(2) lim .
h→0 h
Now we use the rule for f , and observe that f (2) 2−22 −2 and f (2+ h) (2+ h)−(2+ h)2 .
25
Chapter 1 Understanding the Derivative
Substituting these values into the limit definition, we have that
(2 + h) − (2 + h)2 − (−2)
f ′(2) lim .
h→0 h
In order to let h → 0, we must simplify the quotient. Expanding and distributing in the
numerator,
2 + h − 4 − 4h − h 2 + 2
f ′(2) lim .
h→0 h
Combining like terms, we have
−3h − h 2
f ′(2) lim .
h→0 h
Next, we remove a common factor of h in both the numerator and denominator and find
that
f ′(2) lim (−3 − h).
h→0
Finally, we are able to take the limit as h → 0, and thus conclude that f ′(2) −3. We note
that f ′(2) is the instantaneous rate of change of f at the point (2, −2). It is also the slope of
the tangent line to the graph of y x − x 2 at the point (2, −2). Figure 1.3.9 shows both the
function and the line through (2, −2) with slope m f ′(2) −3.
m = f ′ (2)
1 2
-2
-4
y = x − x2
Figure 1.3.9: The tangent line to y x − x 2 at the point (2, −2).
The following activities will help you explore a variety of key ideas related to derivatives.
26
1.3 The derivative of a function at a point
Activity 1.3.2. Consider the function f whose formula is f (x) 3 − 2x.
a. What familiar type of function is f ? What can you say about the slope of f at
every value of x?
b. Compute the average rate of change of f on the intervals [1, 4], [3, 7], and [5, 5 +
h]; simplify each result as much as possible. What do you notice about these
quantities?
c. Use the limit definition of the derivative to compute the exact instantaneous rate
of change of f with respect to x at the value a 1. That is, compute f ′(1) using
the limit definition. Show your work. Is your result surprising?
d. Without doing any additional computations, what are the values of f ′(2), f ′(π),
√
and f ′(− 2)? Why?
Activity 1.3.3. A water balloon is tossed vertically in the air from a window. The
balloon’s height in feet at time t in seconds after being launched is given by s(t)
−16t 2 + 16t + 32. Use this function to respond to each of the following questions.
a. Sketch an accurate, labeled graph of s on the axes provided in Figure 1.3.10. You
should be able to do this without using computing technology.
y
32
16
t
1 2
Figure 1.3.10: Axes for plotting y s(t) in Activity 1.3.3.
b. Compute the average rate of change of s on the time interval [1, 2]. Include units
on your answer and write one sentence to explain the meaning of the value you
found.
c. Use the limit definition to compute the instantaneous rate of change of s with
respect to time, t, at the instant a 1. Show your work using proper notation,
include units on your answer, and write one sentence to explain the meaning of
the value you found.
27
Chapter 1 Understanding the Derivative
d. On your graph in (a), sketch two lines: one whose slope represents the average
rate of change of s on [1, 2], the other whose slope represents the instantaneous
rate of change of s at the instant a 1. Label each line clearly.
e. For what values of a do you expect s ′(a) to be positive? Why? Answer the same
questions when “positive” is replaced by “negative” and “zero.”
Activity 1.3.4. A rapidly growing city in Arizona has its population P at time t,
where t is the number of decades after the year 2010, modeled by the formula P(t)
25000e t/5 . Use this function to respond to the following questions.
a. Sketch an accurate graph of P for t 0 to t 5 on the axes provided in Fig-
ure 1.3.11. Label the scale on the axes carefully.
y
t
Figure 1.3.11: Axes for plotting y P(t) in Activity 1.3.4.
b. Compute the average rate of change of P between 2030 and 2050. Include units
on your answer and write one sentence to explain the meaning (in everyday
language) of the value you found.
c. Use the limit definition to write an expression for the instantaneous rate of
change of P with respect to time, t, at the instant a 2. Explain why this limit
is difficult to evaluate exactly.
d. Estimate the limit in (c) for the instantaneous rate of change of P at the instant
a 2 by using several small h values. Once you have determined an accurate
estimate of P ′(2), include units on your answer, and write one sentence (using
everyday language) to explain the meaning of the value you found.
e. On your graph above, sketch two lines: one whose slope represents the average
rate of change of P on [2, 4], the other whose slope represents the instantaneous
rate of change of P at the instant a 2.
28
1.3 The derivative of a function at a point
f. In a carefully-worded sentence, describe the behavior of P ′(a) as a increases in
value. What does this reflect about the behavior of the given function P?
1.3.2 Summary
f (b)− f (a)
• The average rate of change of a function f on the interval [a, b] is b−a . The units
on the average rate of change are units of f (x) per unit of x, and the numerical value
of the average rate of change represents the slope of the secant line between the points
(a, f (a)) and (b, f (b)) on the graph of y f (x). If we view the interval as being [a, a+h]
instead of [a, b], the meaning is still the same, but the average rate of change is now
f (a+h)− f (a)
computed by h .
• The instantaneous rate of change with respect to x of a function f at a value x a
is denoted f ′(a) (read “the derivative of f evaluated at a” or “ f -prime at a”) and is
defined by the formula
f (a + h) − f (a)
f ′(a) lim ,
h→0 h
provided the limit exists. Note particularly that the instantaneous rate of change at
x a is the limit of the average rate of change on [a, a + h] as h → 0.
• Provided the derivative f ′(a) exists, its value tells us the instantaneous rate of change
of f with respect to x at x a, which geometrically is the slope of the tangent line to
the curve y f (x) at the point (a, f (a)). We even say that f ′(a) is the “slope of the
curve” at the point (a, f (a)).
• Limits allow us to move from the rate of change over an interval to the rate of change
at a single point.
29
Chapter 1 Understanding the Derivative
1.3.3 Exercises
1. Estimating derivative values graphically. Consider the function y f (x) graphed
below.
Give the x-coordinate of a point where:
A. the derivative of the function is negative
B. the value of the function is negative
C. the derivative of the function is smallest
(most negative)
D. the derivative of the function is zero
E. the derivative of the function is approxi-
mately the same as the derivative at x 2.25
(be sure that you give a point that is distinct
from x 2.25!)
2. Tangent line to a curve. The figure below shows a function 1(x) and its tangent line at
the point B (6.8, 2). If the point A on the tangent line is (6.74, 2.05), fill in the blanks
below to complete the statements about the function 1 at the point B.
1( )
1′( )
30
1.3 The derivative of a function at a point
3. Interpreting values and slopes from a graph. Consider the graph of the function f (x)
shown below. Using this graph, for each of the following pairs of numbers decide
which is larger. Be sure that you can explain your answer.
A. f (6) (□ < □ = □ >) f (8)
B. f (6) − f (4) (□ < □ = □ >) f (4) − f (2)
f (4)− f (2) f (6)− f (2)
C. 4−2 (□ < □ = □ >) 6−2
D. f ′(2) (□ < □ = □ >) f ′(8)
4. Finding an exact derivative value algebraically. Find the derivative of 1(t) 2t 2 + 2t
at t 7 algebraically.
5. Estimating a derivative from the limit definition. Estimate f ′(3) for f (x) 6x . Be sure
your answer is accurate to within 0.1 of the actual value.
6. Consider the graph of y f (x) provided in Figure 1.3.12.
a. On the graph of y f (x), sketch y
and label the following quantities: 4
f
• the secant line to y f (x) on
the interval [−3, −1] and the
secant line to y f (x) on the
interval [0, 2]. x
• the tangent line to y f (x) at -4 4
x −3 and the tangent line to
y f (x) at x 0.
b. What is the approximate value of
the average rate of change of f on
-4
[−3, −1]? On [0, 2]? How are these
values related to your work in (a)?
c. What is the approximate value of Figure 1.3.12: Plot of y f (x).
the instantaneous rate of change of
f at x −3? At x 0? How are
these values related to your work in
(a)?
7. For each of the following prompts, sketch a graph on the provided axes in Figure 1.3.13
of a function that has the stated properties.
31
Chapter 1 Understanding the Derivative
3 3
-3 3 -3 3
-3 -3
Figure 1.3.13: Axes for plotting y f (x) in (a) and y 1(x) in (b).
a. y f (x) such that
• the average rate of change of f on [−3, 0] is −2 and the average rate of change
of f on [1, 3] is 0.5, and
• the instantaneous rate of change of f at x −1 is −1 and the instantaneous
rate of change of f at x 2 is 1.
b. y 1(x) such that
1(3)−1(−2) 1(1)−1(−1)
• 5 0 and 2 −1, and
• 1 ′(2) 1 and 1 ′(−1) 0
8. Suppose that the population, P, of China (in billions) can be approximated by the func-
tion P(t) 1.15(1.014)t where t is the number of years since the start of 1993.
a. According to the model, what was the total change in the population of China
between January 1, 1993 and January 1, 2000? What will be the average rate of
change of the population over this time period? Is this average rate of change
greater or less than the instantaneous rate of change of the population on Janu-
ary 1, 2000? Explain and justify, being sure to include proper units on all your
answers.
b. According to the model, what is the average rate of change of the population of
China in the ten-year period starting on January 1, 2012?
c. Write an expression involving limits that, if evaluated, would give the exact in-
stantaneous rate of change of the population on today’s date. Then estimate the
value of this limit (discuss how you chose to do so) and explain the meaning (in-
cluding units) of the value you have found.
d. Find an equation for the tangent line to the function y P(t) at the point where
the t-value is given by today’s date.
32
1.3 The derivative of a function at a point
9. The goal of this problem is to compute the value of the derivative at a point for several
different functions, where for each one we do so in three different ways, and then to
compare the results to see that each produces the same value.
For each of the following functions, use the limit definition of the derivative to compute
the value of f ′(a) using three different approaches: strive to use the algebraic approach
first (to compute the limit exactly), then test your result using numerical evidence (with
small values of h), and finally plot the graph of y f (x) near (a, f (a)) along with the
appropriate tangent line to estimate the value of f ′(a) visually. Compare your findings
among all three approaches; if you are unable to complete the algebraic approach, still
work numerically and graphically.
a. f (x) x 2 − 3x, a 2 d. f (x) 2 − |x − 1|, a 1
b. f (x) 1
x, a1
√ π
c. f (x) x, a 1 e. f (x) sin(x), a 2
33
Chapter 1 Understanding the Derivative
1.4 The derivative function
Motivating Questions
• How does the limit definition of the derivative of a function f lead to an entirely new
(but related) function f ′?
• What is the difference between writing f ′(a) and f ′(x)?
• How is the graph of the derivative function f ′(x) related to the graph of f (x)?
• What are some examples of functions f for which f ′ is not defined at one or more
points?
We now know that the instantaneous rate of change of a function f (x) at x a, or equiva-
lently the slope of the tangent line to the graph of y f (x) at x a, is given by the value
f ′(a). In all of our examples so far, we have identified a particular value of a as our point
of interest: a 1, a 3, etc. But it is not hard to imagine that we will often be interested
in the derivative value for more than just one a-value, and possibly for many of them. In
this section, we explore how we can move from computing the derivative at a single point to
computing a formula for f ′(a) at any point a. Indeed, the process of “taking the derivative”
generates a new function, denoted by f ′(x), derived from the original function f (x).
Preview Activity 1.4.1. Consider the function f (x) 4x − x 2 .
a. Use the limit definition to compute the derivative values: f ′(0), f ′(1), f ′(2), and
f ′(3).
b. Observe that the work to find f ′(a) is the same, regardless of the value of a.
Based on your work in (a), what do you conjecture is the value of f ′(4)? How
about f ′(5)? (Note: you should not use the limit definition of the derivative to
find either value.)
c. Conjecture a formula for f ′(a) that depends only on the value a. That is, in the
same way that we have a formula for f (x) (recall f (x) 4x − x 2 ), see if you can
use your work above to guess a formula for f ′(a) in terms of a.
1.4.1 How the derivative is itself a function
In your work in Preview Activity 1.4.1 with f (x) 4x − x 2 , you may have found several
patterns. One comes from observing that f ′(0) 4, f ′(1) 2, f ′(2) 0, and f ′(3) −2.
That sequence of values leads us naturally to conjecture that f ′(4) −4 and f ′(5) −6. We
also observe that the particular value of a has very little effect on the process of computing
the value of the derivative through the limit definition. To see this more clearly, we compute
f ′(a), where a represents a number to be named later. Following the now standard process
34
1.4 The derivative function
of using the limit definition of the derivative,
f (a + h) − f (a) 4(a + h) − (a + h)2 − (4a − a 2 )
f ′(a) lim lim
h→0 h h→0 h
4a + 4h − a − 2ha − h − 4a + a
2 2 2 4h − 2ha − h 2
lim lim
h→0 h h→0 h
h(4 − 2a − h)
lim lim (4 − 2a − h).
h→0 h h→0
Here we observe that neither 4 nor 2a depend on the value of h, so as h → 0, (4 − 2a − h) →
(4 − 2a). Thus, f ′(a) 4 − 2a.
This result is consistent with the specific values we found above: e.g., f ′(3) 4 − 2(3) −2.
And indeed, our work confirms that the value of a has almost no bearing on the process of
computing the derivative. We note further that the letter being used is immaterial: whether
we call it a, x, or anything else, the derivative at a given value is simply given by “4 minus
2 times the value.” We choose to use x for consistency with the original function given by
y f (x), as well as for the purpose of graphing the derivative function. For the function
f (x) 4x − x 2 , it follows that f ′(x) 4 − 2x.
Because the value of the derivative function is linked to the graph of the original function,
it makes sense to look at both of these functions plotted on the same domain.
m=0 (0, 4)
4 4
m=2 m = −2
3 3
(1, 2)
2 2
1 1
m=4 m = −4 (2, 0)
1 2 3 4 1 2 3 4
-1 -1
(3, −2)
-2 -2
y = f (x) y = f ′ (x)
-3 -3
-4 -4
(4, −4)
Figure 1.4.1: The graphs of f (x) 4x − x 2 (at left) and f ′(x) 4 − 2x (at right). Slopes on
the graph of f correspond to heights on the graph of f ′.
In Figure 1.4.1, on the left we show a plot of f (x) 4x−x 2 together with a selection of tangent
lines at the points we’ve considered above. On the right, we show a plot of f ′(x) 4 − 2x
with emphasis on the heights of the derivative graph at the same selection of points. Notice
35
Chapter 1 Understanding the Derivative
the connection between colors in the left and right graphs: the green tangent line on the
original graph is tied to the green point on the right graph in the following way: the slope of
the tangent line at a point on the lefthand graph is the same as the height at the corresponding
point on the righthand graph. That is, at each respective value of x, the slope of the tangent
line to the original function is the same as the height of the derivative function. Do note,
however, that the units on the vertical axes are different: in the left graph, the vertical units
are simply the output units of f . On the righthand graph of y f ′(x), the units on the
vertical axis are units of f per unit of x.
An excellent way to explore how the graph of f (x) generates the graph of f ′(x) is through a
java applet. See, for instance, the applets at http://gvsu.edu/s/5C or http://gvsu.edu/s/
5D, via the sites of Austin and Renault¹.
In Section 1.3 when we first defined the derivative, we wrote the definition in terms of a
value a to find f ′(a). As we have seen above, the letter a is merely a placeholder, and it
often makes more sense to use x instead. For the record, here we restate the definition of the
derivative.
Definition 1.4.2 Let f be a function and x a value in the function’s domain. We define the
f (x+h)− f (x)
derivative of f , a new function called f ′, by the formula f ′(x) limh→0 h , provided
this limit exists.
We now have two different ways of thinking about the derivative function:
1 given a graph of y f (x), how does this graph lead to the graph of the derivative
function y f ′(x)? and
2 given a formula for y f (x), how does the limit definition of derivative generate a
formula for y f ′(x)?
Both of these issues are explored in the following activities.
Activity 1.4.2. For each given graph of y f (x), sketch an approximate graph of its
derivative function, y f ′(x), on the axes immediately below. The scale of the grid
for the graph of f is 1 × 1; assume the horizontal scale of the grid for the graph of f ′
is identical to that for f . If necessary, adjust and label the vertical scale on the axes for
f ′.
When you are finished with all 8 graphs, write several sentences that describe your
overall process for sketching the graph of the derivative function, given the graph
the original function. What are the values of the derivative function that you tend
to identify first? What do you do thereafter? How do key traits of the graph of the
derivative function exemplify properties of the graph of the original function?
¹David Austin, http://gvsu.edu/s/5r; Marc Renault, http://gvsu.edu/s/5p.
36
1.4 The derivative function
f g
x
x
f′ g′
x x
p q
x
x
p′ q′
x x
37
Chapter 1 Understanding the Derivative
r s
x x
r′ s′
x x
w z
x x
w′ z′
x x
For a dynamic investigation that allows you to experiment with graphing f ′ when given the
graph of f , see http://gvsu.edu/s/8y.²
Now, recall the opening example of this section: we began with the function y f (x) 4x −
²Marc Renault, Calculus Applets Using Geogebra.
38
1.4 The derivative function
x 2 and used the limit definition of the derivative to show that f ′(a) 4 − 2a, or equivalently
that f ′(x) 4−2x. We subsequently graphed the functions f and f ′ as shown in Figure 1.4.1.
Following Activity 1.4.2, we now understand that we could have constructed a fairly accurate
graph of f ′(x) without knowing a formula for either f or f ′. At the same time, it is useful to
know a formula for the derivative function whenever it is possible to find one.
In the next activity, we further explore the more algebraic approach to finding f ′(x): given a
formula for y f (x), the limit definition of the derivative will be used to develop a formula
for f ′(x).
Activity 1.4.3. For each of the listed functions, determine a formula for the derivative
function. For the first two, determine the formula for the derivative by thinking about
the nature of the given function and its slope at various points; do not use the limit
definition. For the latter four, use the limit definition. Pay careful attention to the
function names and independent variables. It is important to be comfortable with us-
ing letters other than f and x. For example, given a function p(z), we call its derivative
p ′(z).
a. f (x) 1 c. p(z) z 2 e. F(t) 1t
√
b. 1(t) t d. q(s) s 3 f. G(y) y
1.4.2 Summary
f (x+h)− f (x)
• The limit definition of the derivative, f ′(x) limh→0 h , produces a value for
each x at which the derivative is defined, and this leads to a new function y f ′(x).
It is especially important to note that taking the derivative is a process that starts with
a given function ( f ) and produces a new, related function ( f ′).
• There is essentially no difference between writing f ′(a) (as we did regularly in Sec-
tion 1.3) and writing f ′(x). In either case, the variable is just a placeholder that is used
to define the rule for the derivative function.
• Given the graph of a function y f (x), we can sketch an approximate graph of its
derivative y f ′(x) by observing that heights on the derivative’s graph correspond to
slopes on the original function’s graph.
• In Activity 1.4.2, we encountered some functions that had sharp corners on their graphs,
such as the shifted absolute value function. At such points, the derivative fails to exist,
and we say that f is not differentiable there. For now, it suffices to understand this as
a consequence of the jump that must occur in the derivative function at a sharp corner
on the graph of the original function.
39
Chapter 1 Understanding the Derivative
1.4.3 Exercises
1. The derivative function graphically. Consider the function f (x) shown in the graph
below.
Carefully sketch the derivative function of the
given function (you will want to estimate val-
ues on the derivative function at different x val-
ues as you do this). Use your derivative func-
tion graph to estimate the following values on
the derivative function.
at x -3 -1 1 3
the derivative is
2. Applying the limit definition of the derivative. Find a formula for the derivative of
the function 1(x) 4x 2 − 8 using difference quotients.
3. Sketching the derivative. For the function f (x) shown in the graph below, sketch a
graph of the derivative. You will then be picking which of the following is the correct
derivative graph, but should be sure to first sketch the derivative yourself.
Which of the following graphs is the derivative of f (x)?
40
1.4 The derivative function
1. 2. 3. 4.
5. 6. 7. 8.
4. Comparing function and derivative values. The graph of a function f is shown below.
At which of the labeled x-values is f (x) least? f (x) greatest? f ′(x) least? f ′(x) greatest?
5. Limit definition of the derivative for a rational function. Let
1
f (x)
x−4
Find (i) f ′(3), (ii) f ′(5), (iii) f ′(6), and (iv) f ′(8).
6. Let f be a function with the following properties: f is differentiable at every value of
x (that is, f has a derivative at every point), f (−2) 1, and f ′(−2) −2, f ′(−1) −1,
f ′(0) 0, f ′(1) 1, and f ′(2) 2.
a. On the axes provided at left in Figure 1.4.3, sketch a possible graph of y f (x).
Explain why your graph meets the stated criteria.
b. Conjecture a formula for the function y f (x). Use the limit definition of the
derivative to determine the corresponding formula for y f ′(x). Discuss both
graphical and algebraic evidence for whether or not your conjecture is correct.
41
Chapter 1 Understanding the Derivative
3 3
-3 3 -3 3
-3 -3
Figure 1.4.3: Axes for plotting y f (x) in (a) and y f ′(x) in (b).
7. Consider the function 1(x) x 2 − x + 3.
a. Use the limit definition of the derivative to determine a formula for 1 ′(x).
b. Use a graphing utility to plot both y 1(x) and your result for y 1 ′(x); does
your formula for 1 ′(x) generate the graph you expected?
c. Use the limit definition of the derivative to find a formula for p ′(x) where p(x)
5x 2 − 4x + 12.
d. Compare and contrast the formulas for 1 ′(x) and p ′(x) you have found. How do
the constants 5, 4, 12, and 3 affect the results?
8. Let 1 be a continuous function (that is, one with no jumps or holes in the graph) and
suppose that a graph of y 1 ′(x) is given by the graph on the right in Figure 1.4.4.
2 2
-2 2 -2 2
-2 -2
Figure 1.4.4: Axes for plotting y 1(x) and, at right, the graph of y 1 ′(x).
42
1.4 The derivative function
a. Observe that for every value of x that satisfies 0 < x < 2, the value of 1 ′(x) is
constant. What does this tell you about the behavior of the graph of y 1(x) on
this interval?
b. On what intervals other than 0 < x < 2 do you expect y 1(x) to be a linear
function? Why?
c. At which values of x is 1 ′(x) not defined? What behavior does this lead you to
expect to see in the graph of y 1(x)?
d. Suppose that 1(0) 1. On the axes provided at left in Figure 1.4.4, sketch an
accurate graph of y 1(x).
9. For each graph that provides an original function y f (x) in Figure 1.4.5, your task
is to sketch an approximate graph of its derivative function, y f ′(x), on the axes
immediately below. View the scale of the grid for the graph of f as being 1 × 1, and
assume the horizontal scale of the grid for the graph of f ′ is identical to that for f . If
you need to adjust the vertical scale on the axes for the graph of f ′, you should label
that accordingly.
43
Chapter 1 Understanding the Derivative
f
f
x x
f′ f′
x x
f f
x x
f′ f′
x x
Figure 1.4.5: Graphs of y f (x) and grids for plotting the corresponding graph of
y f ′(x).
44
1.5 Interpreting, estimating, and using the derivative
1.5 Interpreting, estimating, and using the derivative
Motivating Questions
• In contexts other than the position of a moving object, what does the derivative of a
function measure?
• What are the units on the derivative function f ′, and how are they related to the units
of the original function f ?
• What is a central difference, and how can one be used to estimate the value of the
derivative at a point from given function data?
• Given the value of the derivative of a function at a point, what can we infer about
how the value of the function changes nearby?
It is a powerful feature of mathematics that it can be studied both as abstract discipline and
as an applied one. For instance, calculus can be developed almost entirely as an abstract col-
lection of ideas that focus on properties of functions. At the same time, if we consider func-
tions that represent meaningful processes, calculus can describe our experience of physical
reality. We have already seen that for the position function y s(t) of a ball being tossed
straight up in the air, the derivative of the position function, v(t) s ′(t), gives the ball’s
velocity at time t.
In this section, we investigate several functions with specific physical meaning, and consider
how the units on the independent variable, dependent variable, and the derivative function
add to our understanding. To start, we consider the familiar problem of a position function
of a moving object.
Preview Activity 1.5.1. One of the longest stretches of straight (and flat) road in North
America can be found on the Great Plains in the state of North Dakota on state high-
way 46, which lies just south of the interstate highway I-94 and runs through the town
of Gackle. A car leaves town (at time t 0) and heads east on highway 46; its posi-
tion in miles from Gackle at time t in minutes is given by the graph of the function in
Figure 1.5.1. Three important points are labeled on the graph; where the curve looks
linear, assume that it is indeed a straight line.
a. In everyday language, describe the behavior of the car over the provided time
interval. In particular, discuss what is happening on the time intervals [57, 68]
and [68, 104].
b. Find the slope of the line between the points (57, 63.8) and (104, 106.8). What
are the units on this slope? What does the slope represent?
c. Find the average rate of change of the car’s position on the interval [68, 104].
Include units on your answer.
d. Estimate the instantaneous rate of change of the car’s position at the moment
t 80. Write a sentence to explain your reasoning and the meaning of this
45
Chapter 1 Understanding the Derivative
value.
(104, 106.8)
s
100
80
(57, 63.8)
60
(68, 63.8)
40
20
t
20 40 60 80 100
Figure 1.5.1: The graph of y s(t), the position of the car along highway 46, which
tells its distance in miles from Gackle, ND, at time t in minutes.
1.5.1 Units of the derivative function
As we now know, the derivative of the function f at a fixed value x is given by
f (x + h) − f (x)
f ′(x) lim ,
h→0 h
and this value has several different interpretations. If we set x a, one meaning of f ′(a) is
the slope of the tangent line at the point (a, f (a)).
df dy
We also sometimes write dx or dx instead of f ′(x), and these alternate notations help us see
the units (and thus the meaning) of the derivative as the instantaneous rate of change of f with
f (x+h)− f (x)
respect to x. The units on the slope of the secant line, h , are “units of y per unit of
x,” and when we take the limit as h goes to zero, the derivative f ′(x) has the same units:
units of y per unit of x. It is helpful to remember that the units on the derivative function
are “units of output per unit of input,” for the variables of the original function.
For example, suppose that the function y P(t) measures the population of a city (in
thousands) at the start of year t (where t 0 corresponds to 2010 AD). We are told that
P ′(2) 21.37. What is the meaning of this value? Well, since P is measured in thousands
and t is measured in years, we can say that the instantaneous rate of change of the city’s
population with respect to time at the start of 2012 is 21.37 thousand people per year. We
therefore expect that in the coming year, about 21,370 people will be added to the city’s
population.
46
1.5 Interpreting, estimating, and using the derivative
1.5.2 Toward more accurate derivative estimates
f (x+h)− f (x)
Recall that to estimate the value of f ′(x) at a given x, we calculate a difference quotient h
with a relatively small value of h. We should use both positive and negative values of h in
order to account for the behavior of the function on both sides of the point of interest. To
that end, we introduce the notion of a central difference and its role in estimating derivatives.
Example 1.5.2 Suppose that y f (x) is a function for which three values are known: f (1)
2.5, f (2) 3.25, and f (3) 3.625. Estimate f ′(2).
f (2+h)− f (2)
Solution. We know that f ′(2) limh→0 h . But since we don’t have a graph or a
formula for the function, we can neither sketch a tangent line nor evaluate the limit alge-
braically. We can’t even use smaller and smaller values of h to estimate the limit. Instead,
we have just two choices: using h −1 or h 1, depending on which point we pair with
(2, 3.25).
So, one estimate is
f (1) − f (2) 2.5 − 3.25
f ′(2) ≈ 0.75.
1−2 −1
The other is
f (3) − f (2) 3.625 − 3.25
f ′(2) ≈ 0.375.
3−2 1
Because the first approximation looks backward from the point (2, 3.25) and the second ap-
proximation looks forward, it makes sense to average these two estimates in order to account
for behavior on both sides of x 2. Doing so, we find that
0.75 + 0.375
f ′(2) ≈ 0.5625.
2
The intuitive approach to average the two estimates found in Example 1.5.2 is in fact the
best possible way estimate to a derivative when we have just two function values for f on
opposite sides of the point of interest.
To see why, we think about the diagram in Figure 1.5.3. On the left, we see the two secant
f (1)− f (2)
lines with slopes that come from computing the backward difference 1−2 0.75 and from
f (3)− f (2)
the forward difference 3−2 0.375. Note how the first slope over-estimates the slope of
the tangent line at (2, f (2)), while the second slope underestimates f ′(2). On the right, we
see the secant line whose slope is given by the central difference
f (3) − f (1) 3.625 − 2.5 1.125
0.5625.
3−1 2 2
Note that this central difference has the same value as the average of the forward and back-
ward differences (and it is straightforward to explain why this always holds). The central
difference yields a very good approximation to the derivative’s value, because it yields a line
closer to being parallel to the tangent line.
47
Chapter 1 Understanding the Derivative
3 3
2 2
1 1
1 2 3 1 2 3
Figure 1.5.3: At left, the graph of y f (x) along with the secant line through (1, 2.5) and
(2, 3.25), the secant line through (2, 3.25) and (3, 3.625), as well as the tangent line. At right,
the same graph along with the secant line through (1, 2.5) and (3, 3.625), plus the tangent
line.
The central difference approximation to the value of the first derivative is given by
f (a + h) − f (a − h)
f ′(a) ≈ .
2h
This quantity measures the slope of the secant line to y f (x) through the points (a−h, f (a−
h)) and (a + h, f (a + h)).
Activity 1.5.2. A potato is placed in an oven, and the potato’s temperature F (in de-
grees Fahrenheit) at various points in time is taken and recorded in the following
table. Time t is measured in minutes.
t 0 15 30 45 60 75 90
F(t) 70 180.5 251 296 324.5 342.8 354.5
Table 1.5.4: Temperature data in degrees Fahrenheit.
a. Use a central difference to estimate the instantaneous rate of change of the tem-
perature of the potato at t 30. Include units on your answer.
b. Use a central difference to estimate the instantaneous rate of change of the tem-
perature of the potato at t 60. Include units on your answer.
c. Without doing any calculation, which do you expect to be greater: F′(75) or
F′(90)? Why?
d. Suppose it is given that F(64) 330.28 and F′(64) 1.341. What are the units
on these two quantities? What do you expect the temperature of the potato to
48
1.5 Interpreting, estimating, and using the derivative
be when t 65? when t 66? Why?
e. Write a couple of careful sentences that describe the behavior of the temper-
ature of the potato on the time interval [0, 90], as well as the behavior of the
instantaneous rate of change of the temperature of the potato on the same time
interval.
Activity 1.5.3. A company manufactures rope, and the total cost of producing r feet
of rope is C(r) dollars.
a. What does it mean to say that C(2000) 800?
b. What are the units of C′(r)?
c. Suppose that C(2000) 800 and C′(2000) 0.35. Estimate C(2100), and justify
your estimate by writing at least one sentence that explains your thinking.
d. Do you think C′(2000) is less than, equal to, or greater than C′(3000)? Why?
e. Suppose someone claims that C′(5000) −0.1. What would the practical mean-
ing of this derivative value tell you about the approximate cost of the next foot
of rope? Is this possible? Why or why not?
Activity 1.5.4. Researchers at a major car company have found a function that relates
gasoline consumption to speed for a particular model of car. In particular, they have
determined that the consumption C, in liters per kilometer, at a given speed s, is given
by a function C f (s), where s is the car’s speed in kilometers per hour.
a. Data provided by the car company tells us that f (80) 0.015, f (90) 0.02,
and f (100) 0.027. Use this information to estimate the instantaneous rate of
change of fuel consumption with respect to speed at s 90. Be as accurate as
possible, use proper notation, and include units on your answer.
b. By writing a complete sentence, interpret the meaning (in the context of fuel
consumption) of “ f (80) 0.015.”
c. Write at least one complete sentence that interprets the meaning of the value of
f ′(90) that you estimated in (a).
In Section 1.4, we learned how use to the graph of a given function f to plot the graph
of its derivative, f ′. It is important to remember that when we do so, the scale and the
units on the vertical axis often have to change to represent f ′. For example, suppose that
P(t) 400 − 330e −0.03t tells us the temperature in degrees Fahrenheit of a potato in an oven
at time t in minutes. In Figure 1.5.5, we sketch the graph of P on the left and the graph of P ′
on the right.
Notice that the vertical scales are different in size and different in units, as the units of P are
◦ F, while those of P ′ are ◦ F/min.
49
Chapter 1 Understanding the Derivative
◦F ◦ F/min
400 16
300 y = P(t) 12
200 8 y = P′ (t)
100 4
min min
20 40 60 80 20 40 60 80
Figure 1.5.5: Plot of P(t) 400 − 330e −0.03t at left, and its derivative P ′(t) at right.
1.5.3 Summary
• The derivative of a given function y f (x) measures the instantaneous rate of change
of the output variable with respect to the input variable.
• The units on the derivative function y f ′(x) are units of y per unit of x. Again, this
measures how fast the output of the function f changes when the input of the function
changes.
• The central difference approximation to the value of the first derivative is given by
f (a + h) − f (a − h)
f ′(a) ≈ .
2h
This quantity measures the slope of the secant line to y f (x) through the points (a −
h, f (a − h)) and (a + h, f (a + h)). The central difference generates a good approximation
of the derivative’s value.
1.5.4 Exercises
1. A cooling cup of coffee. The temperature, H, in degrees Celsius, of a cup of coffee
placed on the kitchen counter is given by H f (t), where t is in minutes since the
coffee was put on the counter.
(a) Is f ′(t) positive or negative? (Be sure that you are able to give a reason for your answer.)
(b) What are the units of f ′(35)?
50
1.5 Interpreting, estimating, and using the derivative
Suppose that | f ′(35)| 1.5 and f (35) 68. Fill in the blanks (including units where
needed) and select the appropriate terms to complete the following statement about
the temperature of the coffee in this case.
At minutes after the coffee was put on the counter, its (□ derivative □ temperature
□ change in temperature) is and will (□ increase □ decrease)
by about in the next 30 seconds.
2. A cost function. The cost, C (in dollars) to produce 1 gallons of ice cream can be ex-
pressed as C f (1).
(a) In the expression f (100) 250, what are the units of 100? What are the units of 250?
(b) In the expression f ′(100) 1.2, what are the units of 100? What are the units of 1.2?
(Be sure that you can carefully put into words the meanings of each of these statement in terms
of ice cream and money.)
3. Weight as a function of calories. A laboratory study investigating the relationship
between diet and weight in adult humans found that the weight of a subject, W, in
pounds, was a function, W f (c), of the average number of Calories, c, consumed by
the subject in a day.
(a) In the statement f (1600) 165 what are the units of 1600? What are the units of
165?
(Think about what this statement means in terms of the weight of the subject and the number of
calories that the subject consumes.)
(b) In the statement f ′(2000) 0, what are the units of 2000? What are the units of 0?
(Think about what this statement means in terms of the weight of the subject and the number of
calories that the subject consumes.)
(c) In the statement f −1 (173) 2400, what are the units of 173? What are the units of
2400?
(Think about what this statement means in terms of the weight of the subject and the number of
calories that the subject consumes.)
(d) What are the units of f ′(c) dW/dc?
(e) Suppose that Sam reads about f ′ in this study and draws the following conclusion:
If Sam increases her average calorie intake from 2800 to 2840 calories per day, then her
weight will increase by approximately 0.8 pounds.
Fill in the blanks below so that the equation supports her conclusion.
( )
f′
4. Displacement and velocity. The displacement (in meters) of a particle moving in a
straight line is given by s t 2 − 5t + 16, where t is measured in seconds.
(A)
(i) Find the average velocity over the time interval [3,4].
51
Chapter 1 Understanding the Derivative
(ii) Find the average velocity over the time interval [3.5,4].
(iii) Find the average velocity over the time interval [4,5].
(iv) Find the average velocity over the time interval [4,4.5].
(B) Find the instantaneous velocity when t 4.
5. A cup of coffee has its temperature F (in degrees Fahrenheit) at time t given by the
function F(t) 75 + 110e −0.05t , where time is measured in minutes.
a. Use a central difference with h 0.01 to estimate the value of F′(10).
b. What are the units on the value of F′(10) that you computed in (a)? What is the
practical meaning of the value of F′(10)?
c. Which do you expect to be greater: F′(10) or F′(20)? Why?
d. Write a sentence that describes the behavior of the function y F′(t) on the time
interval 0 ≤ t ≤ 30. How do you think its graph will look? Why?
6. The temperature change T (in Fahrenheit degrees), in a patient, that is generated by a
dose q (in milliliters), of a drug, is given by the function T f (q).
a. What does it mean to say f (50) 0.75? Write a complete sentence to explain,
using correct units.
b. A person’s sensitivity, s, to the drug is defined by the function s(q) f ′(q). What
are the units of sensitivity?
c. Suppose that f ′(50) −0.02. Write a complete sentence to explain the meaning
of this value. Include in your response the information given in (a).
7. The velocity of a ball that has been tossed vertically in the air is given by v(t) 16 − 32t,
where v is measured in feet per second, and t is measured in seconds. The ball is in the
air from t 0 until t 2.
a. When is the ball’s velocity greatest?
b. Determine the value of v ′(1). Justify your thinking.
c. What are the units on the value of v ′(1)? What does this value and the corre-
sponding units tell you about the behavior of the ball at time t 1?
d. What is the physical meaning of the function v ′(t)?
8. The value, V, of a particular automobile (in dollars) depends on the number of miles,
m, the car has been driven, according to the function V h(m).
a. Suppose that h(40000) 15500 and h(55000) 13200. What is the average rate of
change of h on the interval [40000, 55000], and what are the units on this value?
b. In addition to the information given in (a), say that h(70000) 11100. Deter-
mine the best possible estimate of h ′(55000) and write one sentence to explain the
meaning of your result, including units on your answer.
c. Which value do you expect to be greater: h ′(30000) or h ′(80000)? Why?
d. Write a sentence to describe the long-term behavior of the function V h(m),
52
1.5 Interpreting, estimating, and using the derivative
plus another sentence to describe the long-term behavior of h ′(m). Provide your
discussion in practical terms regarding the value of the car and the rate at which
that value is changing.
53
Chapter 1 Understanding the Derivative
1.6 The second derivative
Motivating Questions
• How does the derivative of a function tell us whether the function is increasing or
decreasing on an interval?
• What can we learn by taking the derivative of the derivative (the second derivative)
of a function f ?
• What does it mean to say that a function is concave up or concave down? How are
these characteristics connected to certain properties of the derivative of the function?
• What are the units of the second derivative? How do they help us understand the
rate of change of the rate of change?
Given a differentiable function y f (x), we know that its derivative, y f ′(x), is a related
function whose output at x a tells us the slope of the tangent line to y f (x) at the point
(a, f (a)). That is, heights on the derivative graph tell us the values of slopes on the original
function’s graph.
At a point where f ′(x) is positive, the slope of the tangent line to f is positive. Therefore, on
an interval where f ′(x) is positive, the function f is increasing (or rising). Similarly, if f ′(x)
is negative on an interval, the graph of f is decreasing (or falling).
The derivative of f tells us not only whether the
function f is increasing or decreasing on an inter-
val, but also how the function f is increasing or
decreasing. Look at the two tangent lines shown
in Figure 1.6.1. We see that near point A the value
of f ′(x) is positive and relatively close to zero, and
near that point the graph is rising slowly. By con-
trast, near point B, the derivative is negative and
relatively large in absolute value, and f is decreas- A
ing rapidly near B. B
Besides asking whether the value of the deriva-
tive function is positive or negative and whether
it is large or small, we can also ask “how is the
derivative changing?”
Because the derivative, y f ′(x), is itself a func- Figure 1.6.1: Two tangent lines on a
tion, we can consider taking its derivative — the graph.
derivative of the derivative — and ask “what does
the derivative of the derivative tell us about how
the original function behaves?” We start with an
investigation of a moving object.
54
1.6 The second derivative
Preview Activity 1.6.1. The position of a car driving along a straight road at time t
in minutes is given by the function y s(t) that is pictured in Figure 1.6.2. The car’s
position function has units measured in thousands of feet. For instance, the point
(2, 4) on the graph indicates that after 2 minutes, the car has traveled 4000 feet.
y s
14
10
6
2
t
2 6 10
Figure 1.6.2: The graph of y s(t), the position of the car (measured in thousands of
feet from its starting location) at time t in minutes.
a. In everyday language, describe the behavior of the car over the provided time
interval. In particular, you should carefully discuss what is happening on each
of the time intervals [0, 1], [1, 2], [2, 3], [3, 4], and [4, 5], plus provide commen-
tary overall on what the car is doing on the interval [0, 12].
b. On the lefthand axes provided in Figure 1.6.3, sketch a careful, accurate graph
of y s ′(t).
c. What is the meaning of the function y s ′(t) in the context of the given prob-
lem? What can we say about the car’s behavior when s ′(t) is positive? when
s ′(t) is zero? when s ′(t) is negative?
d. Rename the function you graphed in (b) to be called y v(t). Describe the
behavior of v in words, using phrases like “v is increasing on the interval . . .”
and “v is constant on the interval . . ..”
e. Sketch a graph of the function y v ′(t) on the righthand axes provide in Fig-
ure 1.6.3. Write at least one sentence to explain how the behavior of v ′(t) is
connected to the graph of y v(t).
55
Chapter 1 Understanding the Derivative
y y
t t
2 6 10 2 6 10
Figure 1.6.3: Axes for plotting y v(t) s ′(t) and y v ′(t).
1.6.1 Increasing or decreasing
So far, we have used the words increasing and decreasing intuitively to describe a function’s
graph. Here we define these terms more formally.
Definition 1.6.4 Given a function f (x) defined on the interval (a, b), we say that f is in-
creasing on (a, b) provided that for all x, y in the interval (a, b), if x < y, then f (x) < f (y).
Similarly, we say that f is decreasing on (a, b) provided that for all x, y in the interval (a, b),
if x < y, then f (x) > f (y).
Simply put, an increasing function is one that is rising as we move from left to right along
the graph, and a decreasing function is one that falls as the value of the input increases.
If the function has a derivative, the sign of the derivative tells us whether the function is
increasing or decreasing.
Let f be a function that is differentiable on an interval (a, b). It is possible to show that that if
f ′(x) > 0 for every x such that a < x < b, then f is increasing on (a, b); similarly, if f ′(x) < 0
on (a, b), then f is decreasing on (a, b).
For example, the function pictured in Figure 1.6.5 is increasing on the entire interval −2 <
x < 0, and decreasing on the interval 0 < x < 2. Note that the value x 0 is not included in
either interval since at this location, the function is changing from increasing to decreasing.
56
1.6 The second derivative
A
2
-2 2
y = f (x) -2 B
Figure 1.6.5: A function that is decreasing on the intervals −3 < x < −2 and 0 < x < 2 and
increasing on −2 < x < 0 and 2 < x < 3.
1.6.2 The Second Derivative
We are now accustomed to investigating the behavior of a function by examining its deriv-
ative. The derivative of a function f is a new function given by the rule
f (x + h) − f (x)
f ′(x) lim .
h→0 h
Because f ′ is itself a function, it is perfectly feasible for us to consider the derivative of the
derivative, which is the new function y [ f ′(x)]′. We call this resulting function the second
derivative of y f (x), and denote the second derivative by y f ′′(x). Consequently, we will
sometimes call f ′ “the first derivative” of f , rather than simply “the derivative” of f .
Definition 1.6.6 The second derivative is defined by the limit definition of the derivative of
the first derivative. That is,
f ′(x + h) − f ′(x)
f ′′(x) lim .
h→0 h
The meaning of the derivative function still holds, so when we compute y f ′′(x), this new
function measures slopes of tangent lines to the curve y f ′(x), as well as the instantaneous
rate of change of y f ′(x). In other words, just as the first derivative measures the rate at
which the original function changes, the second derivative measures the rate at which the
first derivative changes. The second derivative will help us understand how the rate of
change of the original function is itself changing.
57
Chapter 1 Understanding the Derivative
1.6.3 Concavity
In addition to asking whether a function is increasing or decreasing, it is also natural to in-
quire how a function is increasing or decreasing. There are three basic behaviors that an
increasing function can demonstrate on an interval, as pictured in Figure 1.6.7: the function
can increase more and more rapidly, it can increase at the same rate, or it can increase in
a way that is slowing down. Fundamentally, we are beginning to think about how a par-
ticular curve bends, with the natural comparison being made to lines, which don’t bend at
all. More than this, we want to understand how the bend in a function’s graph is tied to
behavior characterized by the first derivative of the function.
Figure 1.6.7: Three functions that are all increasing, but doing so at an increasing rate, at a
constant rate, and at a decreasing rate, respectively.
On the leftmost curve in Figure 1.6.7, draw a sequence of tangent lines to the curve. As we
move from left to right, the slopes of those tangent lines will increase. Therefore, the rate
of change of the pictured function is increasing, and this explains why we say this function
is increasing at an increasing rate. For the rightmost graph in Figure 1.6.7, observe that as x
increases, the function increases, but the slopes of the tangent lines decrease. This function
is increasing at a decreasing rate.
Similar options hold for how a function can decrease. Here we must be extra careful with our
language, because decreasing functions involve negative slopes. Negative numbers present
an interesting tension between common language and mathematical language. For example,
it can be tempting to say that “−100 is bigger than −2.” But we must remember that “greater
than” describes how numbers lie on a number line: x > y provided that x lies to the right
of y. So of course, −100 is less than −2. Informally, it might be helpful to say that “−100 is
more negative than −2.” When a function’s values are negative, and those values get more
negative as the input increases, the function must be decreasing. Now consider the three
graphs shown in Figure 1.6.8. Clearly the middle graph depicts a function decreasing at a
constant rate. Now, on the first curve, draw a sequence of tangent lines. We see that the
slopes of these lines get less and less negative as we move from left to right. That means that
the values of the first derivative, while all negative, are increasing, and thus we say that the
leftmost curve is decreasing at an increasing rate.
This leaves only the rightmost curve in Figure 1.6.8 to consider. For that function, the slopes
58
1.6 The second derivative
Figure 1.6.8: From left to right, three functions that are all decreasing, but doing so in
different ways.
of the tangent lines are negative throughout the pictured interval, but as we move from left
to right, the slopes get more and more negative. Hence the slope of the curve is decreasing,
and we say that the function is decreasing at a decreasing rate.
We now introduce the notion of concavity which provides simpler language to describe these
behaviors. When a curve opens upward on a given interval, like the parabola y x 2 or the
exponential growth function y e x , we say that the curve is concave up on that interval.
Likewise, when a curve opens down, like the parabola y −x 2 or the opposite of the ex-
ponential function y −e x , we say that the function is concave down. Concavity is linked to
both the first and second derivatives of the function.
Figure 1.6.9: At left, a function that is concave up; at right, one that is concave down.
In Figure 1.6.9, we see two functions and a sequence of tangent lines to each. On the lefthand
plot, where the function is concave up, observe that the tangent lines always lie below the
curve itself, and the slopes of the tangent lines are increasing as we move from left to right.
In other words, the function f is concave up on the interval shown because its derivative,
f ′, is increasing on that interval. Similarly, on the righthand plot in Figure 1.6.9, where the
59
Chapter 1 Understanding the Derivative
function shown is concave down, we see that the tangent lines alway lie above the curve,
and the slopes of the tangent lines are decreasing as we move from left to right. The fact that
its derivative, f ′, is decreasing makes f concave down on the interval.
We state these most recent observations formally as the definitions of the terms concave up
and concave down.
Definition 1.6.10 Let f be a differentiable function on an interval (a, b). Then f is concave
up on (a, b) if and only if f ′ is increasing on (a, b); f is concave down on (a, b) if and only if
f ′ is decreasing on (a, b).
Activity 1.6.2. The position of a car driving along a straight road at time t in minutes
is given by the function y s(t) that is pictured in Figure 1.6.11. The car’s position
function has units measured in thousands of feet. Remember that you worked with
this function and sketched graphs of y v(t) s ′(t) and y v ′(t) in Preview Activ-
ity 1.6.1.
y s
14
10
6
2
t
2 6 10
Figure 1.6.11: The graph of y s(t), the position of the car (measured in thousands
of feet from its starting location) at time t in minutes.
a. On what intervals is the position function y s(t) increasing? decreasing?
Why?
b. On which intervals is the velocity function y v(t) s ′(t) increasing? decreas-
ing? neither? Why?
c. Acceleration is defined to be the instantaneous rate of change of velocity, as the
acceleration of an object measures the rate at which the velocity of the object is
changing. Say that the car’s acceleration function is named a(t). How is a(t)
computed from v(t)? How is a(t) computed from s(t)? Explain.
d. What can you say about s ′′ whenever s ′ is increasing? Why?
60
1.6 The second derivative
e. Using only the words increasing, decreasing, constant, concave up, concave down,
and linear, complete the following sentences. For the position function s with
velocity v and acceleration a,
• on an interval where v is positive, s is .
• on an interval where v is negative, s is .
• on an interval where v is zero, s is .
• on an interval where a is positive, v is .
• on an interval where a is negative, v is .
• on an interval where a is zero, v is .
• on an interval where a is positive, s is .
• on an interval where a is negative, s is .
• on an interval where a is zero, s is .
Exploring the context of position, velocity, and acceleration is an excellent way to understand
how a function, its first derivative, and its second derivative are related to one another. In
Activity 1.6.2, we can replace s, v, and a with an arbitrary function f and its derivatives f ′
and f ′′, and essentially all the same observations hold. In particular, note that the following
are equivalent: on an interval where the graph of f is concave up, f ′ is increasing and f ′′
is positive. Likewise, on an interval where the graph of f is concave down, f ′ is decreasing
and f ′′ is negative.
Activity 1.6.3. A potato is placed in an oven, and the potato’s temperature F (in de-
grees Fahrenheit) at various points in time is taken and recorded in the following
table. Time t is measured in minutes. In Activity 1.5.2, we computed approximations
to F′(30) and F′(60) using central differences. Those values and more are provided in
the second table below, along with several others computed in the same way.
t F(t) t F′(t)
0 70 0 NA
15 180.5 15 6.03
30 251 30 3.85
45 296 45 2.45
60 324.5 60 1.56
75 342.8 75 1.00
90 354.5 90 NA
Table 1.6.12: Select values of F(t). Table 1.6.13: Select values of F′(t).
a. What are the units on the values of F′(t)?
b. Use a central difference to estimate the value of F′′(30).
c. What is the meaning of the value of F′′(30) that you have computed in (b) in
terms of the potato’s temperature? Write several careful sentences that discuss,
61
Chapter 1 Understanding the Derivative
with appropriate units, the values of F(30), F′(30), and F′′(30), and explain the
overall behavior of the potato’s temperature at this point in time.
d. Overall, is the potato’s temperature increasing at an increasing rate, increasing
at a constant rate, or increasing at a decreasing rate? Why?
Activity 1.6.4. This activity builds on our experience and understanding of how to
sketch the graph of f ′ given the graph of f .
In Figure 1.6.14, given the respective graphs of two different functions f , sketch the
corresponding graph of f ′ on the first axes below, and then sketch f ′′ on the second
set of axes. In addition, for each, write several careful sentences in the spirit of those
in Activity 1.6.2 that connect the behaviors of f , f ′, and f ′′. For instance, write some-
thing such as
f ′ is on the interval , which is connected to the fact that f
is on the same interval , and f ′′ is on the interval.
but of course with the blanks filled in. Throughout, view the scale of the grid for the
graph of f as being 1 × 1, and assume the horizontal scale of the grid for the graph of
f ′ is identical to that for f . If you need to adjust the vertical scale on the axes for the
graph of f ′ or f ′′, you should label that accordingly.
1.6.4 Summary
• A differentiable function f is increasing on an interval whenever its first derivative is
positive, and decreasing whenever its first derivative is negative.
• By taking the derivative of the derivative of a function f , we arrive at the second deriv-
ative, f ′′. The second derivative measures the instantaneous rate of change of the first
derivative. The sign of the second derivative tells us whether the slope of the tangent
line to f is increasing or decreasing.
• A differentiable function is concave up whenever its first derivative is increasing (or
equivalently whenever its second derivative is positive), and concave down whenever
its first derivative is decreasing (or equivalently whenever its second derivative is neg-
ative). Examples of functions that are everywhere concave up are y x 2 and y e x ;
examples of functions that are everywhere concave down are y −x 2 and y −e x .
• The units on the second derivative are “units of output per unit of input per unit of
input.” They tell us how the value of the derivative function is changing in response to
changes in the input. In other words, the second derivative tells us the rate of change
of the rate of change of the original function.
62
1.6 The second derivative
f f
x x
f′ f′
x x
f ′′ f ′′
x x
Figure 1.6.14: Two given functions f , with axes provided for plotting f ′ and f ′′ below.
63
Chapter 1 Understanding the Derivative
1.6.5 Exercises
1. Comparing f , f ′ , f ′′ values. Consider the function f (x) graphed below.
For this function, are the following
nonzero quantities positive or negative?
f (0.5), f ′(0.5), f ′′(0.5)
2. Signs of f , f ′ , f ′′ values. At exactly two of the labeled points in the figure below, which
shows a function f , the derivative f ′ is zero; the second derivative f ′′ is not zero at any
of the labeled points. Give the sign for each of f , f ′ and f ′′ at each marked point.
3. Acceleration from velocity. Suppose that an accelerating car goes from 0 mph to 64.1
mph in five seconds. Its velocity is given in the following table, converted from miles
per hour to feet per second, so that all time measurements are in seconds. (Note: 1
mph is 22/15 ft/sec.) Find the average acceleration of the car over each of the first two
seconds.
t (s) 0 1 2 3 4 5
v(t) (ft/s) 0.00 32.05 55.55 72.64 85.45 94.00
average acceleration over the first second =
average acceleration over the second second =
64
1.6 The second derivative
4. Rates of change of stock values. Let P(t) represent the price of a share of stock of a
corporation at time t. What does each of the following statements tell us about the
signs of the first and second derivatives of P(t)?
(a) The price of the stock is falling slower and slower.
The first derivative of P(t) is (□ positive □ zero □ negative)
The second derivative of P(t) is (□ positive □ zero □ negative)
(b) The price of the stock is close to bottoming out.
The first derivative of P(t) is (□ positive □ zero □ negative)
The second derivative of P(t) is (□ positive □ zero □ negative)
5. Interpreting a graph of f ′. The graph of f ′ (not f ) is given below.
(Note that this is a graph of f ′, not a graph
of f .)
At which of the marked values of x is
A. f (x) greatest?
B. f (x) least?
C. f ′(x) greatest?
D. f ′(x) least?
E. f ′′(x) greatest?
F. f ′′(x) least?
6. Suppose that y f (x) is a twice-differentiable function such that f ′′ is continuous for
which the following information is known: f (2) −3, f ′(2) 1.5, f ′′(2) −0.25.
a. Is f increasing or decreasing near x 2? Is f concave up or concave down near
x 2?
b. Do you expect f (2.1) to be greater than −3, equal to −3, or less than −3? Why?
c. Do you expect f ′(2.1) to be greater than 1.5, equal to 1.5, or less than 1.5? Why?
d. Sketch a graph of y f (x) near (2, f (2)) and include a graph of the tangent line.
65
Chapter 1 Understanding the Derivative
7. For a certain function y 1(x), its derivative is given by the function pictured in Fig-
ure 1.6.15.
y = g′ (x)
4
2
-3 -1 1 3
Figure 1.6.15: The graph of y 1 ′(x).
a. What is the approximate slope of the tangent line to y 1(x) at the point (2, 1(2))?
b. How many real number solutions can there be to the equation 1(x) 0? Justify
your conclusion fully and carefully by explaining what you know about how the
graph of 1 must behave based on the given graph of 1 ′.
c. On the interval −3 < x < 3, how many times does the concavity of 1 change?
Why?
d. Use the provided graph to estimate the value of 1 ′′(2).
8. A bungee jumper’s height h (in feet ) at time t (in seconds) is given in part by the table:
t 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
h(t) 200 184.2 159.9 131.9 104.7 81.8 65.5 56.8 55.5 60.4 69.8
t 5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.5 10.0
h(t) 81.6 93.7 104.4 112.6 117.7 119.4 118.2 114.8 110.0 104.7
a. Use the given data to estimate h ′(4.5), h ′(5), and h ′(5.5). At which of these times
is the bungee jumper rising most rapidly?
b. Use the given data and your work in (a) to estimate h ′′(5).
c. What physical property of the bungee jumper does the value of h ′′(5) measure?
What are its units?
d. Based on the data, on what approximate time intervals is the function y h(t)
concave down? What is happening to the velocity of the bungee jumper on these
time intervals?
66
1.6 The second derivative
9. For each prompt that follows, sketch a possible graph of a function on the interval
−3 < x < 3 that satisfies the stated properties.
a. y f (x) such that f is increasing on −3 < x < 3, concave up on −3 < x < 0, and
concave down on 0 < x < 3.
b. y 1(x) such that 1 is increasing on −3 < x < 3, concave down on −3 < x < 0,
and concave up on 0 < x < 3.
c. y h(x) such that h is decreasing on −3 < x < 3, concave up on −3 < x < −1,
neither concave up nor concave down on −1 < x < 1, and concave down on
1 < x < 3.
d. y p(x) such that p is decreasing and concave down on −3 < x < 0 and is
increasing and concave down on 0 < x < 3.
67
Chapter 1 Understanding the Derivative
1.7 Limits, Continuity, and Differentiability
Motivating Questions
• What does it mean graphically to say that f has limit L as x → a? How is this con-
nected to having a left-hand limit at x a and having a right-hand limit at x a?
• What does it mean to say that a function f is continuous at x a? What role do limits
play in determining whether or not a function is continuous at a point?
• What does it mean graphically to say that a function f is differentiable at x a? How
is this connected to the function being locally linear?
• How are the characteristics of a function having a limit, being continuous, and being
differentiable at a given point related to one another?
In Section 1.2, we learned how limits can be used to study the trend of a function near a
fixed input value. In this section, we aim to quantify how the function acts and how its
values change near a particular point. If the function has a limit L at x a, we will consider
how the value of the function f (a) is related to limx→a f (x), and whether or not the function
has a derivative f ′(a) at x a.
Preview Activity 1.7.1. A function f defined on −4 < x < 4 is given by the graph in
Figure 1.7.1. Use the graph to answer each of the following questions. Note: to the
right of x 2, the graph of f is exhibiting infinite oscillatory behavior similar to the
function sin( πx ) that we encountered in the key example early in Section 1.2.
f
3
2
1
-3 -2 -1 1 2 3
-1
-2
-3
Figure 1.7.1: The graph of y f (x).
a. For each of the values a −3, −2, −1, 0, 1, 2, 3, determine whether or not lim f (x)
x→a
exists. If the function has a limit L at a given point, state the value of the limit
68
1.7 Limits, Continuity, and Differentiability
using the notation limx→a f (x) L. If the function does not have a limit at a
given point, write a sentence to explain why.
b. For each of the values of a from part (a) where f has a limit, determine the value
of f (a) at each such point. In addition, for each such a value, does f (a) have the
same value as limx→a f (x)?
c. For each of the values a −3, −2, −1, 0, 1, 2, 3, determine whether or not f ′(a)
exists. In particular, based on the given graph, ask yourself if it is reasonable
to say that f has a tangent line at (a, f (a)) for each of the given a-values. If so,
visually estimate the slope of the tangent line to find the value of f ′(a).
1.7.1 Having a limit at a point
In Section 1.2, we learned that f has limit L as x approaches a provided that we can make
the value of f (x) as close to L as we like by taking x sufficiently close (but not equal to) a. If
so, we write limx→a f (x) L.
Essentially there are two behaviors that a function can exhibit near a point where it fails to
have a limit. In Figure 1.7.3, at left we see a function f whose graph shows a jump at a 1.
If we let x approach 1 from the left side, the value of f approaches 2, but if we let x approach
1 from the right, the value of f tends to 3. Because the value of f does not approach a single
number as x gets arbitrarily close to 1 from both sides, we know that f does not have a limit
at a 1.
For such cases, we introduce the notion of left and right (or one-sided) limits.
Definition 1.7.2 We say that f has limit L1 as x approaches a from the left and write
lim f (x) L1
x→a −
provided that we can make the value of f (x) as close to L1 as we like by taking x sufficiently
close to a while always having x < a. We call L1 the left-hand limit of f as x approaches a.
Similarly, we say L2 is the right-hand limit of f as x approaches a and write
lim f (x) L2
x→a +
provided that we can make the value of f (x) as close to L2 as we like by taking x sufficiently
close to a while always having x > a.
In the graph of the function f in Figure 1.7.3, we see that
lim f (x) 2 and lim+ f (x) 3.
x→1− x→1
Precisely because the left and right limits are not equal, the overall limit of f as x → 1 fails
to exist.
69
Chapter 1 Understanding the Derivative
f g
3 3
2 2
1
1 1
Figure 1.7.3: Functions f and 1 that each fail to have a limit at a 1.
For the function 1 pictured at right in Figure 1.7.3, the function fails to have a limit at a 1
for a different reason. While the function does not have a jump in its graph at a 1, it is
still not the case that 1 approaches a single value as x approaches 1. In particular, due to the
infinitely oscillating behavior of 1 to the right of a 1, we say that the right-hand limit of 1
as x → 1+ does not exist, and thus limx→1 1(x) does not exist.
To summarize, if either a left- or right-hand limit fails to exist or if the left- and right-hand
limits are not equal to each other, the overall limit does not exist.
A function f has limit L as x → a if and only if
lim f (x) L lim+ f (x).
x→a − x→a
That is, a function has a limit at x a if and only if both the left- and right-hand limits at
x a exist and have the same value.
In Preview Activity 1.7.1, the function f given in Figure 1.7.1 fails to have a limit at only two
values: at a −2 (where the left- and right-hand limits are 2 and −1, respectively) and at
x 2, where limx→2+ f (x) does not exist). Note well that even at values such as a −1 and
a 0 where there are holes in the graph, the limit still exists.
70
1.7 Limits, Continuity, and Differentiability
Activity 1.7.2. Consider a function that is piecewise-defined according to the formula
3(x + 2) + 2 for −3 < x < −2
2
(x + 2) + 1 for −2 ≤ x < −1
3
f (x) 2
(x + 2) + 1 for −1 < x < 1
3
for x 1
2
4 − x
for x > 1
Use the given formula to answer the following questions.
3
2
1
-2 -1 1 2
-1
Figure 1.7.4: Axes for plotting the function y f (x) in Activity 1.7.2.
a. For each of the values a −2, −1, 0, 1, 2, compute f (a).
b. For each of the values a −2, −1, 0, 1, 2, determine lim− f (x) and lim+ f (x).
x→a x→a
c. For each of the values a −2, −1, 0, 1, 2, determine limx→a f (x). If the limit fails
to exist, explain why by discussing the left- and right-hand limits at the relevant
a-value.
d. For which values of a is the following statement true?
lim f (x) , f (a)
x→a
e. On the axes provided in Figure 1.7.4, sketch an accurate, labeled graph of y
f (x). Be sure to carefully use open circles (◦) and filled circles (•) to represent
key points on the graph, as dictated by the piecewise formula.
71
Chapter 1 Understanding the Derivative
1.7.2 Being continuous at a point
Intuitively, a function is continuous if we can draw its graph without ever lifting our pencil
from the page. Alternatively, we might say that the graph of a continuous function has no
jumps or holes in it. In Figure 1.7.5 we consider three functions that have a limit at a 1,
and use them to make the idea of continuity more precise.
f g h
3 3 3
2 2 2
1 1 1
Figure 1.7.5: Functions f , 1, and h that demonstrate subtly different behaviors at a 1.
First consider the function in the left-most graph. Note that f (1) is not defined, which leads
to the resulting hole in the graph of f at a 1. We will naturally say that f is not continuous
at a 1. For the function 1, we observe that while limx→1 1(x) 3, the value of 1(1) 2,
and thus the limit does not equal the function value. Here, too, we will say that 1 is not
continuous, even though the function is defined at a 1. Finally, the function h appears to
be the most well-behaved of all three, since at a 1 its limit and its function value agree.
That is,
lim h(x) 3 h(1).
x→1
With no hole or jump in the graph of h at a 1, we say that h is continuous there. More
formally, we make the following definition.
Definition 1.7.6 A function f is continuous at x a provided that
a. f has a limit as x → a,
b. f is defined at x a, and
c. limx→a f (x) f (a).
Conditions (a) and (b) are technically contained implicitly in (c), but we state them explicitly
to emphasize their individual importance. The definition says that a function is continuous
at x a provided that its limit as x → a exists and equals its function value at x a. If a
function is continuous at every point in an interval [a, b], we say the function is “continuous
on [a, b].” If a function is continuous at every point in its domain, we simply say the function
is “continuous.” Thus, continuous functions are particularly nice: to evaluate the limit of a
continuous function at a point, all we need to do is evaluate the function.
For example, consider p(x) x 2 − 2x + 3. It can be proved that every polynomial is a
72
1.7 Limits, Continuity, and Differentiability
continuous function at every real number, and thus if we would like to know limx→2 p(x),
we simply compute
lim (x 2 − 2x + 3) 22 − 2 · 2 + 3 3.
x→2
This route of substituting an input value to evaluate a limit works whenever we know that
the function being considered is continuous. Besides polynomial functions, all exponential
functions and the sine and cosine functions are continuous at every point, as are many other
familiar functions and combinations thereof.
Activity 1.7.3. This activity builds on your work in Preview Activity 1.7.1, using the
same function f as given by the graph that is repeated in Figure 1.7.7.
f
3
2
1
-3 -2 -1 1 2 3
-1
-2
-3
Figure 1.7.7: The graph of y f (x) for Activity 1.7.3.
a. At which values of a does limx→a f (x) not exist?
b. At which values of a is f (a) not defined?
c. At which values of a does f have a limit, but limx→a f (x) , f (a)?
d. State all values of a for which f is not continuous at x a.
e. Which condition is stronger, and hence implies the other: f has a limit at x a
or f is continuous at x a? Explain, and hence complete the following sentence:
“If f at x a, then f at x a,” where you complete the blanks
with has a limit and is continuous, using each phrase once.
1.7.3 Being differentiable at a point
We recall that a function f is said to be differentiable at x a if f ′(a) exists. Moreover,
for f ′(a) to exist, we know that the function y f (x) must have a tangent line at the point
73
Chapter 1 Understanding the Derivative
(a, f (a)), since f ′(a) is precisely the slope of this line. In order to even ask if f has a tangent
line at (a, f (a)), it is necessary that f be continuous at x a: if f fails to have a limit at x a,
if f (a) is not defined, or if f (a) does not equal the value of limx→a f (x), then it doesn’t make
sense to talk about a tangent line to the curve at this point.
Indeed, it can be proved formally that if a function f is differentiable at x a, then it must
be continuous at x a. So, if f is not continuous at x a, then it is automatically the
case that f is not differentiable there. For example, in Figure 1.7.5, both f and 1 fail to be
differentiable at x 1 because neither function is continuous at x 1. But can a function
fail to be differentiable at a point where the function is continuous?
In Figure 1.7.8, the function has a sharp corner at a point. For the pictured function f , we
observe that f is clearly continuous at a 1, since limx→1 f (x) 1 f (1).
f
1 (1, 1)
1
Figure 1.7.8: A function f that is continuous at a 1 but not differentiable at a 1; at right,
we zoom in on the point (1, 1) in a magnified version of the box in the left-hand plot.
But the function f in Figure 1.7.8 is not differentiable at a 1 because f ′(1) fails to exist.
One way to see this is to observe that f ′(x) −1 for every value of x that is less than 1, while
f ′(x) +1 for every value of x that is greater than 1. That makes it seem that either +1 or −1
would be equally good candidates for the value of the derivative at x 1. Alternately, we
could use the limit definition of the derivative to attempt to compute f ′(1), and discover that
the derivative does not exist. Finally, we can see visually that the function f in Figure 1.7.8
does not have a tangent line. When we zoom in on (1, 1) on the graph of f , no matter how
closely we examine the function, it will always look like a “V”, and never like a single line,
which tells us there is no possibility for a tangent line there.
If a function does have a tangent line at a given point, when we zoom in on the point of
tangency, the function and the tangent line should appear essentially indistinguishable¹.
Conversely, if we zoom in on a point and the function looks like a single straight line, then
the function should have a tangent line there, and thus be differentiable. Hence, a function
that is differentiable at x a will, up close, look more and more like its tangent line at
(a, f (a)). Therefore, we say that a function that is differentiable at x a is locally linear.
To summarize the preceding discussion of differentiability and continuity, we make several
¹See, for instance, http://gvsu.edu/s/6J for an applet (due to David Austin, GVSU) where zooming in shows
the increasing similarity between the tangent line and the curve.
74
1.7 Limits, Continuity, and Differentiability
important observations.
• If f is differentiable at x a, then f is continuous at x a. Equivalently, if f
fails to be continuous at x a, then f will not be differentiable at x a.
• A function can be continuous at a point, but not be differentiable there. In
particular, a function f is not differentiable at x a if the graph has a sharp
corner (or cusp) at the point (a, f (a)).
• If f is differentiable at x a, then f is locally linear at x a. That is, when
a function is differentiable, it looks linear when viewed up close because it re-
sembles its tangent line there.
Activity 1.7.4. In this activity, we explore two different functions and classify the
points at which each is not differentiable. Let 1 be the function given by the rule
1(x) |x|, and let f be the function that we have previously explored in Preview
Activity 1.7.1, whose graph is given again in Figure 1.7.9.
f
3
2
1
-3 -2 -1 1 2 3
-1
-2
-3
Figure 1.7.9: The graph of y f (x) for Activity 1.7.4.
a. Reasoning visually, explain why 1 is differentiable at every point x such that
x , 0.
|h|
b. Use the limit definition of the derivative to show that 1 ′(0) limh→0 h .
c. Explain why 1 ′(0) fails to exist by using small positive and negative values of h.
d. State all values of a for which f is not differentiable at x a. For each, provide
a reason for your conclusion.
e. True or false: if a function p is differentiable at x b, then limx→b p(x) must
exist. Why?
75
Chapter 1 Understanding the Derivative
1.7.4 Summary
• A function f has limit L as x → a if and only if f has a left-hand limit at x a, f has
a right-hand limit at x a, and the left- and right-hand limits are equal. Visually, this
means that there can be a hole in the graph at x a, but the function must approach
the same single value from either side of x a.
• A function f is continuous at x a whenever f (a) is defined, f has a limit as x → a,
and the value of the limit and the value of the function agree. This guarantees that
there is not a hole or jump in the graph of f at x a.
• A function f is differentiable at x a whenever f ′(a) exists, which means that f has
a tangent line at (a, f (a)) and thus f is locally linear at x a. Informally, this means
that the function looks like a line when viewed up close at (a, f (a)) and that there is
not a corner point or cusp at (a, f (a)).
• Of the three conditions discussed in this section (having a limit at x a, being con-
tinuous at x a, and being differentiable at x a), the strongest condition is being
differentiable, and the next strongest is being continuous. In particular, if f is differ-
entiable at x a, then f is also continuous at x a, and if f is continuous at x a,
then f has a limit at x a.
1.7.5 Exercises
1. Limit values of a piecewise graph. Use the figure below, which gives a graph of the
function f (x), to give values for the indicated limits. If a limit does not exist, enter none.
(a) lim f (x)
x→−1
(b) lim f (x)
x→0
(c) lim f (x)
x→1
(d) lim f (x)
x→4
2. Limit values of a piecewise formula. For the function
3x − 2, 0≤x<1
f (x) 5, x1
x 2 − 2x + 2, 1 < x
use algebra to find each of the following limits:
lim f (x)
x→1+
lim f (x)
x→1−
76
1.7 Limits, Continuity, and Differentiability
lim f (x)
x→1
Sketch a graph of f (x) to confirm your answers.
3. Continuity and differentiability of a graph. Consider the function graphed below.
At what x-values does the function appear to not be continuous?
At what x-values does the function appear to not be differentiable?
4. Continuity of a piecewise formula. Find k so that the following function is continuous:
{
kx if 0 ≤ x < 2
f (x)
5x 2 if 2 ≤ x.
5. Consider the graph of the function y p(x) that is provided in Figure 1.7.10. Assume
that each portion of the graph of p is a straight line, as pictured.
a. State all values of a for which limx→a p(x) does not exist.
b. State all values of a for which p is not continuous at a.
c. State all values of a for which p is not differentiable at x a.
d. On the axes provided in Figure 1.7.10, sketch an accurate graph of y p ′(x).
77
Chapter 1 Understanding the Derivative
p
3 3
-3 3 -3 3
-3 -3
Figure 1.7.10: At left, the piecewise linear function y p(x). At right, axes for plotting
y p ′(x).
6. For each of the following prompts, give an example of a function that satisfies the stated
criteria. A formula or a graph, with reasoning, is sufficient for each. If no such example
is possible, explain why.
a. A function f that is continuous at a 2 but not differentiable at a 2.
b. A function 1 that is differentiable at a 3 but does not have a limit at a 3.
c. A function h that has a limit at a −2, is defined at a −2, but is not continuous
at a −2.
d. A function p that satisfies all of the following:
• p(−1) 3 and limx→−1 p(x) 2
• p(0) 1 and p ′(0) 0
• limx→1 p(x) p(1) and p ′(1) does not exist
7. Let h(x) be a function whose derivative y h ′(x) is given by the graph on the right in
Figure 1.7.11.
a. Based on the graph of y h ′(x), what can you say about the behavior of the
function y h(x)?
b. At which values of x is y h ′(x) not defined? What behavior does this lead you
to expect to see in the graph of y h(x)?
c. Is it possible for y h(x) to have points where h is not continuous? Explain your
answer.
d. On the axes provided at left, sketch at least two distinct graphs that are possible
functions y h(x) that each have a derivative y h ′(x) that matches the provided
graph at right. Explain why there are multiple possibilities for y h(x).
78
1.7 Limits, Continuity, and Differentiability
3 3
2
y = h′ (x)
1
-3 3 -3 -2 -1 1 2 3
-1
-2
-3 -3
Figure 1.7.11: Axes for plotting y h(x) and, at right, the graph of y h ′(x).
√
8. Consider the function 1(x) |x|.
a. Use a graph to explain visually why 1 is not differentiable at x 0.
b. Use the limit definition of the derivative to show that
√
′ |h|
1 (0) lim .
h→0 h
c. Investigate the value of 1 ′(0) by estimating the limit in (b) using small positive
√
|−0.01|
and negative values of h. For instance, you might compute 0.01 . Be sure to use
several different values of h (both positive and negative), including ones closer to
0 than 0.01. What do your results tell you about 1 ′(0)?
d. Use your graph in (a) to sketch an approximate graph of y 1 ′(x).
79
Chapter 1 Understanding the Derivative
1.8 The Tangent Line Approximation
Motivating Questions
• What is the formula for the general tangent line approximation to a differentiable
function y f (x) at the point (a, f (a))?
• What is the principle of local linearity and what is the local linearization of a differ-
entiable function f at a point (a, f (a))?
• How does knowing just the tangent line approximation tell us information about the
behavior of the original function itself near the point of approximation? How does
knowing the second derivative’s value at this point provide us additional knowledge
of the original function’s behavior?
Among all functions, linear functions are simplest. One of the powerful consequences of
a function y f (x) being differentiable at a point (a, f (a)) is that, up close, the function
y f (x) is locally linear and looks like its tangent line at that point. In certain circum-
stances, this allows us to approximate the original function f with a simpler function L that
is linear: this can be advantageous when we have limited information about f or when f
is computationally or algebraically complicated. We will explore all of these situations in
what follows.
It is essential to recall that when f is differentiable at x a, the value of f ′(a) provides the
slope of the tangent line to y f (x) at the point (a, f (a)). If we know both a point on the line
and the slope of the line we can find the equation of the tangent line and write the equation
in point-slope form¹.
Preview Activity 1.8.1. Consider the function y 1(x) −x 2 + 3x + 2.
a. Use the limit definition of the derivative to compute a formula for y 1 ′(x).
b. Determine the slope of the tangent line to y 1(x) at the value x 2.
c. Compute 1(2).
d. Find an equation for the tangent line to y 1(x) at the point (2, 1(2)). Write
your result in point-slope form.
e. On the axes provided in Figure 1.8.1, sketch an accurate, labeled graph of y
1(x) along with its tangent line at the point (2, 1(2)).
¹Recall that a line with slope m that passes through (x 0 , y0 ) has equation y − y0 m(x − x0 ), and this is the
point-slope form of the equation.
80
1.8 The Tangent Line Approximation
Figure 1.8.1: Axes for plotting y 1(x) and its tangent line to the point (2, 1(2)).
1.8.1 The tangent line
Given a function f that is differentiable at x a, we know that we can determine the slope
of the tangent line to y f (x) at (a, f (a)) by computing f ′(a). The equation of the resulting
tangent line is given in point-slope form by
y − f (a) f ′(a)(x − a) or y f ′(a)(x − a) + f (a).
Note well: there is a major difference between f (a) and f (x) in this context. The former is
a constant that results from using the given fixed value of a, while the latter is the general
expression for the rule that defines the function. The same is true for f ′(a) and f ′(x): we
must carefully distinguish between these expressions. Each time we find the tangent line,
we need to evaluate the function and its derivative at a fixed a-value.
In Figure 1.8.2, we see the graph of a function f and its tangent line at the point (a, f (a)).
Notice how when we zoom in we see the local linearity of f more clearly highlighted. The
function and its tangent line are nearly indistinguishable up close. Local linearity can also
be seen dynamically in the java applet at http://gvsu.edu/s/6J.
1.8.2 The local linearization
A slight change in perspective and notation will enable us to be more precise in discussing
how the tangent line approximates f near x a. By solving for y, we can write the equation
for the tangent line as
y f ′(a)(x − a) + f (a)
This line is itself a function of x. Replacing the variable y with the expression L(x), we call
L(x) f ′(a)(x − a) + f (a)
81
Chapter 1 Understanding the Derivative
y
y = f (x)
y = f (x)
(a, f (a))
(a, f (a))
y = f ′ (a)(x − a) + f (a) y = L(x)
a x
Figure 1.8.2: A function y f (x) and its tangent line at the point (a, f (a)): at left, from a
distance, and at right, up close. At right, we label the tangent line function by y L(x) and
observe that for x near a, f (x) ≈ L(x).
the local linearization of f at the point (a, f (a)). In this notation, L(x) is nothing more than a
new name for the tangent line. As we saw above, for x close to a, f (x) ≈ L(x).
Example 1.8.3 Suppose that a function y f (x) has its tangent line approximation given by
L(x) 3 − 2(x − 1) at the point (1, 3), but we do not know anything else about the function f .
To estimate a value of f (x) for x near 1, such as f (1.2), we can use the fact that f (1.2) ≈ L(1.2)
and hence
f (1.2) ≈ L(1.2) 3 − 2(1.2 − 1) 3 − 2(0.2) 2.6.
We emphasize that y L(x) is simply a new name for the tangent line function. Using this
new notation and our observation that L(x) ≈ f (x) for x near a, it follows that we can write
f (x) ≈ f (a) + f ′(a)(x − a) for x near a.
Activity 1.8.2. Suppose it is known that for a given differentiable function y 1(x),
its local linearization at the point where a −1 is given by L(x) −2 + 3(x + 1).
a. Compute the values of L(−1) and L′(−1).
b. What must be the values of 1(−1) and 1 ′(−1)? Why?
c. Do you expect the value of 1(−1.03) to be greater than or less than the value of
1(−1)? Why?
d. Use the local linearization to estimate the value of 1(−1.03).
e. Suppose that you also know that 1 ′′(−1) 2. What does this tell you about the
graph of y 1(x) at a −1?
82
1.8 The Tangent Line Approximation
f. For x near −1, sketch the graph of the local linearization y L(x) as well as a
possible graph of y 1(x) on the axes provided in Figure 1.8.4.
Figure 1.8.4: Axes for plotting y L(x) and y 1(x).
From Activity 1.8.2, we see that the local linearization y L(x) is a linear function that shares
two important values with the function y f (x) that it is derived from. In particular,
• because L(x) f (a) + f ′(a)(x − a), it follows that L(a) f (a); and
• because L is a linear function, its derivative is its slope.
Hence, L′(x) f ′(a) for every value of x, and specifically L′(a) f ′(a). Therefore, we see
that L is a linear function that has both the same value and the same slope as the function f
at the point (a, f (a)).
Thus, if we know the linear approximation y L(x) for a function, we know the original
function’s value and its slope at the point of tangency. What remains unknown, however, is
the shape of the function f at the point of tangency. There are essentially four possibilities,
as shown in Figure 1.8.5.
Figure 1.8.5: Four possible graphs for a nonlinear differentiable function and how it can be
situated relative to its tangent line at a point.
83
Chapter 1 Understanding the Derivative
These possible shapes result from the fact that there are three options for the value of the
second derivative: either f ′′(a) < 0, f ′′(a) 0, or f ′′(a) > 0.
• If f ′′(a) > 0, then we know the graph of f is concave up, and we see the first possibility
on the left, where the tangent line lies entirely below the curve.
• If f ′′(a) < 0, then f is concave down and the tangent line lies above the curve, as shown
in the second figure.
• If f ′′(a) 0 and f ′′ changes sign at x a, the concavity of the graph will change, and
we will see either the third or fourth figure.².
• A fifth option (which is not very interesting) can occur if the function f itself is linear,
so that f (x) L(x) for all values of x.
The plots in Figure 1.8.5 highlight yet another important thing that we can learn from the
concavity of the graph near the point of tangency: whether the tangent line lies above or
below the curve itself. This is key because it tells us whether or not the tangent line ap-
proximation’s values will be too large or too small in comparison to the true value of f . For
instance, in the first situation in the leftmost plot in Figure 1.8.5 where f ′′(a) > 0, because
the tangent line falls below the curve, we know that L(x) ≤ f (x) for all values of x near a.
Activity 1.8.3. This activity concerns a function f (x) about which the following infor-
mation is known:
• f is a differentiable function defined at every real number x
• f (2) −1
• y f ′(x) has its graph given in Figure 1.8.6
y = f ′ (x)
2 2 2
x x x
2 2 2
Figure 1.8.6: At center, a graph of y f ′(x); at left, axes for plotting y f (x); at
right, axes for plotting y f ′′(x).
Your task is to determine as much information as possible about f (especially near
the value a 2) by responding to the questions below.
²It is possible that f ′′ (a) 0 but f ′′ does not change sign at x a, in which case the graph will look like one of
the first two options.
84
1.8 The Tangent Line Approximation
a. Find a formula for the tangent line approximation, L(x), to f at the point (2, −1).
b. Use the tangent line approximation to estimate the value of f (2.07). Show your
work carefully and clearly.
c. Sketch a graph of y f ′′(x) on the righthand grid in Figure 1.8.6; label it ap-
propriately.
d. Is the slope of the tangent line to y f (x) increasing, decreasing, or neither
when x 2? Explain.
e. Sketch a possible graph of y f (x) near x 2 on the lefthand grid in Fig-
ure 1.8.6. Include a sketch of y L(x) (found in part (a)). Explain how you
know the graph of y f (x) looks like you have drawn it.
f. Does your estimate in (b) over- or under-estimate the true value of f (2.07)?
Why?
The idea that a differentiable function looks linear and can be well-approximated by a lin-
ear function is an important one that finds wide application in calculus. For example, by
approximating a function with its local linearization, it is possible to develop an effective
algorithm to estimate the zeroes of a function. Local linearity also helps us to make further
sense of certain challenging limits. For instance, we have seen that the limit
sin(x)
lim
x→0 x
is indeterminate, because both its numerator and denominator tend to 0. While there is no
sin(x)
algebra that we can do to simplify x , it is straightforward to show that the linearization of
f (x) sin(x) at the point (0, 0) is given by L(x) x. Hence, for values of x near 0, sin(x) ≈ x,
and therefore
sin(x) x
≈ 1,
x x
which makes plausible the fact that
sin(x)
lim 1.
x→0 x
1.8.3 Summary
• The tangent line to a differentiable function y f (x) at the point (a, f (a)) is given in
point-slope form by the equation
y − f (a) f ′(a)(x − a).
• The principle of local linearity tells us that if we zoom in on a point where a function
y f (x) is differentiable, the function will be indistinguishable from its tangent line.
85
Chapter 1 Understanding the Derivative
That is, a differentiable function looks linear when viewed up close. We rename the
tangent line to be the function y L(x), where L(x) f (a) + f ′(a)(x − a). Thus,
f (x) ≈ L(x) for all x near x a.
• If we know the tangent line approximation L(x) f (a) + f ′(a)(x − a) to a function
y f (x), then because L(a) f (a) and L′(a) f ′(a), we also know the values of both
the function and its derivative at the point where x a. In other words, the linear
approximation tells us the height and slope of the original function. If, in addition, we
know the value of f ′′(a), we then know whether the tangent line lies above or below
the graph of y f (x), depending on the concavity of f .
1.8.4 Exercises
√ √
1. Approximating x. Use linear approximation to approximate 36.1 as follows.
√
Let f (x) x. The equation of the tangent line to f (x) at x 36 can be written in the
√
form y mx + b. Compute m and b. Using this find the approximation for 36.1.
2. Local linearization of a graph. The figure below shows f (x) and its local linearization
at x a, y 4x − 4. (The local linearization is shown in blue.)
What is the value of a? What is the value of f (a)? Use the linearization to approximate
the value of f (3.2). Is the approximation an under- or overestimate?
3. Estimating with the local linearization. Suppose that f (x) is a function with f (130)
46 and f ′(130) 1. Estimate f (125.5).
4. Predicting behavior from the local linearization. The temperature, H, in degrees Cel-
sius, of a cup of coffee placed on the kitchen counter is given by H f (t), where t is in
minutes since the coffee was put on the counter.
(a) Is f ′(t) positive or negative?
(b) What are the units of f ′(30)?
Suppose that | f ′(30)| 0.9 and f (30) 51. Fill in the blanks (including units where
needed) and select the appropriate terms to complete the following statement about
the temperature of the coffee in this case.
At minutes after the coffee was put on the counter, its (□ derivative □ temperature
□ change in temperature) is and will (□ increase □ decrease)
86
1.8 The Tangent Line Approximation
by about in the next 75 seconds.
5. A certain function y p(x) has its local linearization at a 3 given by L(x) −2x + 5.
a. What are the values of p(3) and p ′(3)? Why?
b. Estimate the value of p(2.79).
c. Suppose that p ′′(3) 0 and you know that p ′′(x) < 0 for x < 3. Is your estimate
in (b) too large or too small?
d. Suppose that p ′′(x) > 0 for x > 3. Use this fact and the additional information
above to sketch an accurate graph of y p(x) near x 3. Include a sketch of
y L(x) in your work.
6. A potato is placed in an oven, and the potato’s temperature F (in degrees Fahrenheit) at
various points in time is taken and recorded in the following table. Time t is measured
in minutes.
t F(t)
0 70
15 180.5
30 251
45 296
60 324.5
75 342.8
90 354.5
Table 1.8.7: Temperature data for the potato.
a. Use a central difference to estimate F′(60). Use this estimate as needed in subse-
quent questions.
b. Find the local linearization y L(t) to the function y F(t) at the point where
a 60.
c. Determine an estimate for F(63) by employing the local linearization.
d. Do you think your estimate in (c) is too large or too small? Why?
7. An object moving along a straight line path has a differentiable position function y
s(t); s(t) measures the object’s position relative to the origin at time t. It is known that
at time t 9 seconds, the object’s position is s(9) 4 feet (i.e., 4 feet to the right of
the origin). Furthermore, the object’s instantaneous velocity at t 9 is −1.2 feet per
second, and its acceleration at the same instant is 0.08 feet per second per second.
a. Use local linearity to estimate the position of the object at t 9.34.
b. Is your estimate likely too large or too small? Why?
c. In everyday language, describe the behavior of the moving object at t 9. Is it
moving toward the origin or away from it? Is its velocity increasing or decreasing?
87
Chapter 1 Understanding the Derivative
For a certain function f , its derivative is known to be f ′(x) (x − 1)e −x . Note that you
2
8.
do not know a formula for y f (x).
a. At what x-value(s) is f ′(x) 0? Justify your answer algebraically, but include a
graph of f ′ to support your conclusion.
b. Reasoning graphically, for what intervals of x-values is f ′′(x) > 0? What does
this tell you about the behavior of the original function f ? Explain.
c. Assuming that f (2) −3, estimate the value of f (1.88) by finding and using the
tangent line approximation to f at x 2. Is your estimate larger or smaller than
the true value of f (1.88)? Justify your answer.
88
CHAPTER 2
Computing Derivatives
2.1 Elementary derivative rules
Motivating Questions
• What are alternate notations for the derivative?
• How can we use the algebraic structure of a function f (x) to compute a formula for
f ′(x)?
• What is the derivative of a power function of the form f (x) x n ? What is the deriv-
ative of an exponential function of form f (x) a x ?
• If we know the derivative of y f (x), what is the derivative of y k f (x), where k is
a constant?
• If we know the derivatives of y f (x) and y 1(x), how do we compute the deriv-
ative of y f (x) + 1(x)?
In Chapter 1, we developed the concept of the derivative of a function. We now know that
the derivative f ′ of a function f measures the instantaneous rate of change of f with respect
to x. The derivative also tells us the slope of the tangent line to y f (x) at any given value
of x. So far, we have focused on interpreting the derivative graphically or, in the context of
a physical setting, as a meaningful rate of change. To calculate the value of the derivative at
a specific point, we have relied on the limit definition of the derivative,
f (x + h) − f (x)
f ′(x) lim .
h→0 h
In this chapter, we investigate how the limit definition of the derivative leads to interesting
patterns and rules that enable us to find a formula for f ′(x) quickly, without using the limit
definition directly. For example, we would like to apply shortcuts to differentiate a function
such as 1(x) 4x 7 − sin(x) + 3e x
Chapter 2 Computing Derivatives
Preview Activity 2.1.1. Functions of the form f (x) x n , where n 1, 2, 3, . . ., are of-
ten called power functions. The first two questions below revisit work we did earlier
in Chapter 1, and the following questions extend those ideas to higher powers of x.
a. Use the limit definition of the derivative to find f ′(x) for f (x) x 2 .
b. Use the limit definition of the derivative to find f ′(x) for f (x) x 3 .
c. Use the limit definition of the derivative to find f ′(x) for f (x) x 4 . (Hint:
(a + b)4 a 4 + 4a 3 b + 6a 2 b 2 + 4ab 3 + b 4 . Apply this rule to (x + h)4 within
the limit definition.)
d. Based on your work in (a), (b), and (c), what do you conjecture is the derivative
of f (x) x 5 ? Of f (x) x 13 ?
e. Conjecture a formula for the derivative of f (x) x n that holds for any positive
integer n. That is, given f (x) x n where n is a positive integer, what do you
think is the formula for f ′(x)?
2.1.1 Some Key Notation
In addition to our usual f ′ notation, there are other ways to denote the derivative of a func-
tion, as well as the instruction to take the derivative. If we are thinking about the relationship
between y and x, we sometimes denote the derivative of y with respect to x by the symbol
dy
dx
which we read “dee-y dee-x.” For example, if y x 2 , we’ll write that the derivative is
dy
dx 2x. This notation comes from the fact that the derivative is related to the slope of a
∆y ∆y
line, and slope is measured by ∆x . Note that while we read ∆x as “change in y over change
dy
in x,” we view dx as a single symbol, not a quotient of two quantities.
We use a variant of this notation as the instruction to take the derivative. In particular,
d
[□]
dx
means “take the derivative of the quantity in □ with respect to x.” For example, we may
d
write dx [x 2 ] 2x.
It is important to note that the independent variable can be different from x. If we have
dy
f (z) z 2 , we then write f ′(z) 2z. Similarly, if y t 2 , we say dt 2t. And it is also true that
[ ]
df d2 f
dq [q ]
d 2 2q. This notation may also be used for second derivatives: f ′′(z) d
dz dz dz 2
.
In what follows, we’ll build a repertoire of functions for which we can quickly compute the
derivative.
90
2.1 Elementary derivative rules
2.1.2 Constant, Power, and Exponential Functions
So far, we know the derivative formula for two important classes of functions: constant
functions and power functions. If f (x) c is a constant function, its graph is a horizontal
d
line with slope zero at every point. Thus, dx [c] 0. We summarize this with the following
rule.
Constant Functions.
For any real number c, if f (x) c, then f ′(x) 0.
√
Example 2.1.1 If f (x) 7, then f ′(x) 0. Similarly, dx [
d
3] 0.
In your work in Preview Activity 2.1.1, you conjectured that for any positive integer n, if
f (x) x n , then f ′(x) nx n−1 . This rule can be formally proved for any positive integer n,
and also for any nonzero real number (positive or negative).
Power Functions.
For any nonzero real number n, if f (x) x n , then f ′(x) nx n−1 .
Example 2.1.2 Using the rule for power functions, we can compute the following derivatives.
If 1(z) z −3 , then 1 ′(z) −3z −4 . Similarly, if h(t) t 7/5 , then dh
dt 5 t
7 2/5 d
, and dq [q π ]
πq π−1 .
It will be instructive to have a derivative formula for one more type of basic function. For
now, we simply state this rule without explanation or justification; we will explore why this
rule is true in one of the exercises. And we will encounter graphical reasoning for why the
rule is plausible in Preview Activity 2.2.1.
Exponential Functions.
For any positive real number a, if f (x) a x , then f ′(x) a x ln(a).
Example 2.1.3 If f (x) 2x , then f ′(x) 2x ln(2). Similarly, for p(t) 10t , p ′(t) 10t ln(10).
It is especially important to note that when a e, where e is the base of the natural logarithm
function, we have that
d x
[e ] e x ln(e) e x
dx
since ln(e) 1. This is an extremely important property of the function e x : its derivative
function is itself!
Note carefully the distinction between power functions and exponential functions: in power
functions, the variable is in the base, as in x 2 , while in exponential functions, the variable is
in the power, as in 2x . As we can see from the rules, this makes a big difference in the form
of the derivative.
91
Chapter 2 Computing Derivatives
Activity 2.1.2. Use the three rules above to determine the derivative of each of the
following functions. For each, state your answer using full and proper notation, la-
beling the derivative with its name. For example, if you are given a function h(z), you
should write “h ′(z) ” or “ dh
dz ” as part of your response.
a. f (t) π d. p(x) 31/2 f. s(q) q −1
b. 1(z) 7z
√
c. h(w) w 3/4 e. r(t) ( 2)t g. m(t) 1
t3
2.1.3 Constant Multiples and Sums of Functions
Next we will learn how to compute the derivative of a function constructed as an algebraic
combination of basic functions. For instance, we’d like to be able to take the derivative of a
polynomial function such as
p(t) 3t 5 − 7t 4 + t 2 − 9,
which is a sum of constant multiples of powers of t. To that end, we develop two new rules:
the Constant Multiple Rule and the Sum Rule.
How is the derivative of y k f (x) related to the derivative of y f (x)? Recall that when
we multiply a function by a constant k, we vertically stretch the graph by a factor of |k| (and
reflect the graph across y 0 if k < 0). This vertical stretch affects the slope of the graph, so
the slope of the function y k f (x) is k times as steep as the slope of y f (x). Thus, when
we multiply a function by a factor of k, we change the value of its derivative by a factor of k
as well.¹,
The Constant Multiple Rule.
For any real number k, if f (x) is a differentiable function with derivative f ′(x), then
′
dx [k f (x)] k f (x).
d
In words, this rule says that “the derivative of a constant times a function is the constant
times the derivative of the function.”
Example 2.1.4 If 1(t) 3 · 5t , we have 1 ′(t) 3 · 5t ln(5). Similarly, −2
dz [5z ]
d
5(−2z −3 ).
Next we examine a sum of two functions. If we have y f (x) and y 1(x), we can compute
a new function y ( f + 1)(x) by adding the outputs of the two functions: ( f + 1)(x)
f (x) + 1(x). Not only is the value of the new function the sum of the values of the two
known functions, but the slope of the new function is the sum of the slopes of the known
functions. Therefore², we arrive at the following Sum Rule for derivatives:
¹The Constant Multiple Rule can be formally proved as a consequence of properties of limits, using the limit
definition of the derivative.
²Like the Constant Multiple Rule, the Sum Rule can be formally proved as a consequence of properties of limits,
using the limit definition of the derivative.
92
2.1 Elementary derivative rules
The Sum Rule.
If f (x) and 1(x) are differentiable functions with derivatives f ′(x) and 1 ′(x) respec-
d
tively, then dx [ f (x) + 1(x)] f ′(x) + 1 ′(x).
In words, the Sum Rule tells us that “the derivative of a sum is the sum of the derivatives.”
It also tells us that a sum of two differentiable functions is also differentiable. Furthermore,
because we can view the difference function y ( f − 1)(x) f (x) − 1(x) as y f (x) + (−1 ·
1(x)), the Sum Rule and Constant Multiple Rules together tell us that dx d
[ f (x) + (−1 · 1(x))]
′ ′
f (x) − 1 (x), or that “the derivative of a difference is the difference of the derivatives.” We
can now compute derivatives of sums and differences of elementary functions.
Example 2.1.5 Using the sum rule, dw d
(2w + w 2 ) 2w ln(2) + 2w. Using both the sum and
constant multiple rules, if h(q) 3q − 4q −3 , then h ′(q) 3(6q 5 ) − 4(−3q −4 ) 18q 5 + 12q −4 .
6
Activity 2.1.3. Use only the rules for constant, power, and exponential functions, to-
gether with the Constant Multiple and Sum Rules, to compute the derivative of each
function below with respect to the given independent variable. Note well that we do
not yet know any rules for how to differentiate the product or quotient of functions.
This means that you may have to do some algebra first on the functions below before
you can actually use existing rules to compute the desired derivative formula. In each
case, label the derivative you calculate with its name using proper notation such as
f ′(x), h ′(z), dr/dt, etc.
a. f (x) x 5/3 − x 4 + 2x e. s(y) (y 2 + 1)(y 2 − 1)
b. 1(x) 14e x + 3x 5 − x
√ x 3 −x+2
f. q(x)
c. h(z) z + z14 + 5z x
√
d. r(t) 53 t 7 − πe t + e 4 g. p(a) 3a 4 − 2a 3 + 7a 2 − a + 12
In the same way that we have shortcut rules to help us find derivatives, we introduce some
language that is simpler and shorter. Often, rather than say “take the derivative of f ,” we’ll
instead say simply “differentiate f .” Similarly, if the derivative exists at a point, we say “ f
is differentiable at that point,” or that f can be differentiated.
As we work with the algebraic structure of functions, it is important to develop a big picture
view of what we are doing. Here, we make several general observations based on the rules
we have so far.
• The derivative of any polynomial function will be another polynomial function, and
that the degree of the derivative is one less than the degree of the original function.
For instance, if p(t) 7t 5 − 4t 3 + 8t, p is a degree 5 polynomial, and its derivative,
p ′(t) 35t 4 − 12t 2 + 8, is a degree 4 polynomial.
• The derivative of any exponential function is another exponential function: for exam-
ple, if 1(z) 7 · 2z , then 1 ′(z) 7 · 2z ln(2), which is also exponential.
• We should not lose sight of the fact that all of the meaning of the derivative that we
developed in Chapter 1 still holds. The derivative measures the instantaneous rate of
93
Chapter 2 Computing Derivatives
change of the original function, as well as the slope of the tangent line at any selected
point on the curve.
Activity 2.1.4. Each of the following questions asks you to use derivatives to answer
key questions about functions. Be sure to think carefully about each question and to
use proper notation in your responses.
√
a. Find the slope of the tangent line to h(z) z + 1z at the point where z 4.
b. A population of cells is growing in such a way that its total number in millions
is given by the function P(t) 2(1.37)t + 32, where t is measured in days.
i. Determine the instantaneous rate at which the population is growing on
day 4, and include units on your answer.
ii. Is the population growing at an increasing rate or growing at a decreasing
rate on day 4? Explain.
c. Find an equation for the tangent line to the curve p(a) 3a 4 − 2a 3 + 7a 2 − a + 12
at the point where a −1.
d. What is the difference between being asked to find the slope of the tangent line
(asked in (a)) and the equation of the tangent line (asked in (c))?
2.1.4 Summary
• Given a differentiable function y f (x), we can express the derivative of f in several
d f dy
different notations: f ′(x), dx , dx , and dx
d
[ f (x)].
• The limit definition of the derivative leads to patterns among certain families of func-
tions that enable us to compute derivative formulas without resorting directly to the
limit definition. For example, if f is a power function of the form f (x) x n , then
f ′(x) nx n−1 for any real number n other than 0. This is called the Rule for Power
Functions.
• We have stated a rule for derivatives of exponential functions in the same spirit as the
rule for power functions: for any positive real number a, if f (x) a x , then f ′(x)
a x ln(a).
• If we are given a constant multiple of a function whose derivative we know, or a sum
of functions whose derivatives we know, the Constant Multiple and Sum Rules make
it straightforward to compute the derivative of the overall function. More formally,
if f (x) and 1(x) are differentiable with derivatives f ′(x) and 1 ′(x) and a and b are
constants, then
d [ ]
a f (x) + b1(x) a f ′(x) + b1 ′(x).
dx
94
2.1 Elementary derivative rules
2.1.5 Exercises
1. Derivative of a power function. Find the derivative of y x 15/16 .
1
2. Derivative of a rational function. Find the derivative of f (x) .
x 19
√
3. Derivative of a root function. Find the derivative of y x.
4. Derivative of a quadratic. Find the derivative of f (t) 3t 2 − 7t + 2.
√
5. Derivative of a sum of power functions. Find the derivative of y 6t 6 − 9 t + 7t .
√
6. Simplifying a product before differentiating. Find the derivative of y x(x 3 + 9).
x6 + 9
7. Simplifying a quotient before differentiating. Find the derivative of y .
x
8. Finding a tangent line equation. Find an equation for the line tangent to the graph of
f at (3, 76), where f is given by f (x) 4x 3 − 4x 2 + 4.
9. Determining where f ′(x) 0. If f (x) x 3 + 6x 2 − 288x + 5, find analytically all values
of x for which f ′(x) 0.
10. Let f and 1 be differentiable functions for which the following information is known:
f (2) 5, 1(2) −3, f ′(2) −1/2, 1 ′(2) 2.
a. Let h be the new function defined by the rule h(x) 3 f (x) − 41(x). Determine
h(2) and h ′(2).
b. Find an equation for the tangent line to y h(x) at the point (2, h(2)).
c. Let p be the function defined by the rule p(x) −2 f (x) + 12 1(x). Is p increasing,
decreasing, or neither at a 2? Why?
d. Estimate the value of p(2.03) by using the local linearization of p at the point
(2, p(2)).
11. Let functions p and q be the piecewise linear functions given by their respective graphs
in Figure 2.1.6. Use the graphs to answer the following questions.
a. At what values of x is p not differentiable? At what values of x is q not differen-
tiable? Why?
b. Let r(x) p(x) + 2q(x). At what values of x is r not differentiable? Why?
c. Determine r ′(−2) and r ′(0).
d. Find an equation for the tangent line to y r(x) at the point (2, r(2)).
95
Chapter 2 Computing Derivatives
p
3
2
1
-3 -2 -1 1 2 3
-1
q
-2
-3
Figure 2.1.6: The graphs of p (in blue) and q (in green).
12. Consider the functions r(t) t t and s(t) arccos(t), for which you are given the
facts that r ′(t) t t (ln(t) + 1) and s ′(t) − √ 1 2 . Do not be concerned with where these
1−t
derivative formulas come from. We restrict our interest in both functions to the domain
0 < t < 1.
a. Let w(t) 3t t − 2 arccos(t). Determine w ′(t).
b. Find an equation for the tangent line to y w(t) at the point ( 12 , w( 21 )).
c. Let v(t) t t + arccos(t). Is v increasing or decreasing at the instant t 12 ? Why?
13. Let f (x) a x . The goal of this problem is to explore how the value of a affects the
derivative of f (x), without assuming we know the rule for dx
d
[a x ] that we have stated
and used in earlier work in this section.
a. Use the limit definition of the derivative to show that
ax · ah − ax
f ′(x) lim .
h→0 h
b. Explain why it is also true that
ah − 1
f ′(x) a x · lim .
h→0 h
c. Use computing technology and small values of h to estimate the value of
ah − 1
L lim
h→0 h
when a 2. Do likewise when a 3.
d. Note that it would be ideal if the value of the limit L was 1, for then f would be
a particularly special function: its derivative would be simply a x , which would
96
2.1 Elementary derivative rules
mean that its derivative is itself. By experimenting with different values of a be-
tween 2 and 3, try to find a value for a for which
ah − 1
L lim 1.
h→0 h
e. Compute ln(2) and ln(3). What does your work in (b) and (c) suggest is true about
dx [2 ] and dx [3 ]?
d x d x
f. How do your investigations in (d) lead to a particularly important fact about the
function f (x) e x ?
97
Chapter 2 Computing Derivatives
2.2 The sine and cosine functions
Motivating Questions
dx [a ] a x ln(a)?
d x
• What is a graphical justification for why
• What do the graphs of y sin(x) and y cos(x) suggest as formulas for their re-
spective derivatives?
• Once we know the derivatives of sin(x) and cos(x), how do previous derivative rules
work when these functions are involved?
Throughout Chapter 2, we will develop shortcut derivative rules to help us bypass the limit
definition and quickly compute f ′(x) from a formula for f (x). In Section 2.1, we stated the
rule for power functions,
if f (x) x n , then f ′(x) nx n−1 ,
and the rule for exponential functions,
if a is a positive real number and f (x) a x , then f ′(x) a x ln(a).
Later in this section, we will use a graphical argument to conjecture derivative formulas for
the sine and cosine functions.
Preview Activity 2.2.1. Consider the function 1(x) 2x , which is graphed in Fig-
ure 2.2.1.
a. At each of x −2, −1, 0, 1, 2, use a straightedge to sketch an accurate tangent
line to y 1(x).
b. Use the provided grid to estimate the slope of the tangent line you drew at each
point in (a).
c. Use the limit definition of the derivative to estimate 1 ′(0) by using small values
of h, and compare the result to your visual estimate for the slope of the tangent
line to y 1(x) at x 0 in (b).
d. Based on your work in (a), (b), and (c), sketch an accurate graph of y 1 ′(x) on
the axes adjacent to the graph of y 1(x).
e. Write at least one sentence that explains why it is reasonable to think that 1 ′(x)
c1(x), where c is a constant. In addition, calculate ln(2), and then discuss how
this value, combined with your work above, reasonably suggests that 1 ′(x)
2x ln(2).
98
2.2 The sine and cosine functions
7 7
6 6
5 5
4 4
3 3
2 2
1 1
-2 -1 1 2 -2 -1 1 2
Figure 2.2.1: At left, the graph of y 1(x) 2x . At right, axes for plotting y 1 ′(x).
2.2.1 The sine and cosine functions
The sine and cosine functions are among the most important functions in all of mathemat-
ics. Sometimes called the circular functions due to their definition on the unit circle, these
periodic functions play a key role in modeling repeating phenomena such as tidal eleva-
tions, the behavior of an oscillating mass attached to a spring, or the location of a point
on a bicycle tire. Like polynomial and exponential functions, the sine and cosine functions
are considered basic functions, ones that are often used in building more complicated func-
tions. As such, we would like to know formulas for dx d
[sin(x)] and dx
d
[cos(x)], and the next
two activities lead us to that end.
Activity 2.2.2. Consider the function f (x) sin(x), which is graphed in Figure 2.2.2
below. Note carefully that the grid in the diagram does not have boxes that are 1 × 1,
but rather approximately 1.57 × 1, as the horizontal scale of the grid is π/2 units per
box.
1 1
2π
−2π −π
-1
π −2π −π
-1
π 2π
Figure 2.2.2: At left, the graph of y f (x) sin(x).
π π
a. At each of x −2π, − 3π
2 , −π, − 2 , 0, 2 , π, 2 , 2π,
3π
use a straightedge to sketch an
accurate tangent line to y f (x).
b. Use the provided grid to estimate the slope of the tangent line you drew at each
point. Pay careful attention to the scale of the grid.
99
Chapter 2 Computing Derivatives
c. Use the limit definition of the derivative to estimate f ′(0) by using small values
of h, and compare the result to your visual estimate for the slope of the tangent
line to y f (x) at x 0 in (b). Using periodicity, what does this result suggest
about f ′(2π)? about f ′(−2π)?
d. Based on your work in (a), (b), and (c), sketch an accurate graph of y f ′(x) on
the axes adjacent to the graph of y f (x).
e. What familiar function do you think is the derivative of f (x) sin(x)?
Activity 2.2.3. Consider the function 1(x) cos(x), which is graphed in Figure 2.2.3
below. Note carefully that the grid in the diagram does not have boxes that are 1 × 1,
but rather approximately 1.57 × 1, as the horizontal scale of the grid is π/2 units per
box.
1 1
−2π −π
-1
π 2π −2π −π
-1
π 2π
Figure 2.2.3: At left, the graph of y 1(x) cos(x).
π π
a. At each of x −2π, − 3π
2 , −π, − 2 , 0, 2 , π, 2 , 2π,
3π
use a straightedge to sketch an
accurate tangent line to y 1(x).
b. Use the provided grid to estimate the slope of the tangent line you drew at each
point. Again, note the scale of the axes and grid.
c. Use the limit definition of the derivative to estimate 1 ′( π2 ) by using small values
of h, and compare the result to your visual estimate for the slope of the tangent
line to y 1(x) at x π2 in (b). Using periodicity, what does this result sug-
gest about 1 ′(− 3π
2 )? can symmetry on the graph help you estimate other slopes
easily?
d. Based on your work in (a), (b), and (c), sketch an accurate graph of y 1 ′(x) on
the axes adjacent to the graph of y 1(x).
e. What familiar function do you think is the derivative of 1(x) cos(x)?
The results of the two preceding activities suggest that the sine and cosine functions not only
have beautiful connections such as the identities sin2 (x)+cos2 (x) 1 and cos(x − π2 ) sin(x),
but that they are even further linked through calculus, as the derivative of each involves the
other. The following rules summarize the results of the activities¹.
¹These two rules may be formally proved using the limit definition of the derivative and the expansion identities
for sin(x + h) and cos(x + h).
100
2.2 The sine and cosine functions
Sine and Cosine Functions.
For all real numbers x,
d d
[sin(x)] cos(x) and [cos(x)] − sin(x).
dx dx
We have now added the sine and cosine functions to our library of basic functions whose
derivatives we know. The constant multiple and sum rules still hold, of course, as well as all
of the inherent meaning of the derivative.
Activity 2.2.4. Answer each of the following questions. Where a derivative is re-
quested, be sure to label the derivative function with its name using proper notation.
a. Determine the derivative of h(t) 3 cos(t) − 4 sin(t).
sin(x)
b. Find the exact slope of the tangent line to y f (x) 2x + 2 at the point
where x π6 .
c. Find the equation of the tangent line to y 1(x) x 2 + 2 cos(x) at the point
where x π2 .
d. Determine the derivative of p(z) z 4 + 4z + 4 cos(z) − sin( π2 ).
e. The function P(t) 24 + 8 sin(t) represents a population of a particular kind of
animal that lives on a small island, where P is measured in hundreds and t is
measured in decades since January 1, 2010. What is the instantaneous rate of
change of P on January 1, 2030? What are the units of this quantity? Write a
sentence in everyday language that explains how the population is behaving at
this point in time.
2.2.2 Summary
• For an exponential function f (x) a x (a > 1), the graph of f ′(x) appears to be a scaled
version of the original function. In particular, careful analysis of the graph of f (x)
d
2x , suggests that dx [2x ] 2x ln(2), which is a special case of the rule we stated in
Section 2.1.
• By carefully analyzing the graphs of y sin(x) and y cos(x), and by using the
d
limit definition of the derivative at select points, we found that dx [sin(x)] cos(x) and
d
dx [cos(x)] − sin(x).
• We note that all previously encountered derivative rules still hold, but now may also
be applied to functions involving the sine and cosine. All of the established meaning
of the derivative applies to these trigonometric functions as well.
101
Chapter 2 Computing Derivatives
2.2.3 Exercises
1. Suppose that V(t) 24 · 1.07t + 6 sin(t) represents the value of a person’s investment
portfolio in thousands of dollars in year t, where t 0 corresponds to January 1, 2010.
a. At what instantaneous rate is the portfolio’s value changing on January 1, 2012?
Include units on your answer.
b. Determine the value of V ′′(2). What are the units on this quantity and what does
it tell you about how the portfolio’s value is changing?
c. On the interval 0 ≤ t ≤ 20, graph the function V(t) 24 · 1.07t + 6 sin(t) and
describe its behavior in the context of the problem. Then, compare the graphs of
the functions A(t) 24 · 1.07t and V(t) 24 · 1.07t + 6 sin(t), as well as the graphs
of their derivatives A′(t) and V ′(t). What is the impact of the term 6 sin(t) on the
behavior of the function V(t)?
2. Let f (x) 3 cos(x) − 2 sin(x) + 6.
π
a. Determine the exact slope of the tangent line to y f (x) at the point where a 4.
b. Determine the tangent line approximation to y f (x) at the point where a π.
π
c. At the point where a 2, is f increasing, decreasing, or neither?
d. At the point where a 3π
2 , does the tangent line to y f (x) lie above the curve,
below the curve, or neither? How can you answer this question without even
graphing the function or the tangent line?
3. In this exercise, we explore how the limit definition of the derivative more formally
shows that dxd
[sin(x)] cos(x). Letting f (x) sin(x), note that the limit definition of
the derivative tells us that
sin(x + h) − sin(x)
f ′(x) lim .
h→0 h
a. Recall the trigonometric identity for the sine of a sum of angles α and β: sin(α +
β) sin(α) cos(β) + cos(α) sin(β). Use this identity and some algebra to show that
sin(x)(cos(h) − 1) + cos(x) sin(h)
f ′(x) lim .
h→0 h
b. Next, note that as h changes, x remains constant. Explain why it therefore makes
sense to say that
cos(h) − 1 sin(h)
f ′(x) sin(x) · lim + cos(x) · lim .
h→0 h h→0 h
c. Finally, use small values of h to estimate the values of the two limits in (c):
cos(h) − 1 sin(h)
lim and lim .
h→0 h h→0 h
d. What do your results in (b) and (c) thus tell you about f ′(x)?
e. By emulating the steps taken above, use the limit definition of the derivative to
d
argue convincingly that dx [cos(x)] − sin(x).
102
2.2 The sine and cosine functions
103
Chapter 2 Computing Derivatives
2.3 The product and quotient rules
Motivating Questions
• How does the algebraic structure of a function guide us in computing its derivative
using shortcut rules?
• How do we compute the derivative of a product of two basic functions in terms of
the derivatives of the basic functions?
• How do we compute the derivative of a quotient of two basic functions in terms of
the derivatives of the basic functions?
• How do the product and quotient rules combine with the sum and constant multiple
rules to expand the library of functions we can differentiate quickly?
So far, we can differentiate power functions (x n ), exponential functions (a x ), and the two
fundamental trigonometric functions (sin(x) and cos(x)). With the sum rule and constant
multiple rules, we can also compute the derivative of combined functions.
Example 2.3.1 Differentiate
√
f (x) 7x 11 − 4 · 9x + π sin(x) − 3 cos(x)
Because f is a sum of basic functions, we can now quickly say that f ′(x) 77x 10 −4·9x ln(9)+
√
π cos(x) + 3 sin(x).
What about a product or quotient of two basic functions, such as
p(z) z 3 cos(z),
or
sin(t)
q(t) ?
2t
While the derivative of a sum is the sum of the derivatives, it turns out that the rules for
computing derivatives of products and quotients are more complicated.
Preview Activity 2.3.1. Let f and 1 be the functions defined by f (t) 2t 2 and 1(t)
t 3 + 4t.
a. Determine f ′(t) and 1 ′(t).
b. Let p(t) 2t 2 (t 3 + 4t) and observe that p(t) f (t) · 1(t). Rewrite the formula for
p by distributing the 2t 2 term. Then, compute p ′(t) using the sum and constant
multiple rules.
c. True or false: p ′(t) f ′(t) · 1 ′(t).
d. Let q(t) t 2t+4t
3 1(t)
2 and observe that q(t) f (t) . Rewrite the formula for q by
dividing each term in the numerator by the denominator and simplify to write
104
2.3 The product and quotient rules
q as a sum of constant multiples of powers of t. Then, compute q ′(t) using the
sum and constant multiple rules.
1 ′ (t)
e. True or false: q ′(t) f ′ (t)
.
2.3.1 The product rule
As part (b) of Preview Activity 2.3.1 shows, it is not true in general that the derivative of a
product of two functions is the product of the derivatives of those functions. To see why this
is the case, we consider an example involving meaningful functions.
Say that an investor is regularly purchasing stock in a particular company. Let N(t) represent
the number of shares owned on day t, where t 0 represents the first day on which shares
were purchased. Let S(t) give the value of one share of the stock on day t; note that the units
on S(t) are dollars per share. To compute the total value of the stock on day t, we take the
product
V(t) N(t) shares · S(t) dollars per share.
Observe that over time, both the number of shares and the value of a given share will vary.
The derivative N ′(t) measures the rate at which the number of shares is changing, while
S′(t) measures the rate at which the value per share is changing. How do these respective
rates of change affect the rate of change of the total value function?
To help us understand the relationship among changes in N, S, and V, let’s consider some
specific data.
• Suppose that on day 100, the investor owns 520 shares of stock and the stock’s current
value is $27.50 per share. This tells us that N(100) 520 and S(100) 27.50.
• On day 100, the investor purchases an additional 12 shares (so the number of shares
held is rising at a rate of 12 shares per day).
• On that same day the price of the stock is rising at a rate of 0.75 dollars per share per
day.
In calculus notation, the latter two facts tell us that N ′(100) 12 (shares per day) and
S′(100) 0.75 (dollars per share per day). At what rate is the value of the investor’s to-
tal holdings changing on day 100?
Observe that the increase in total value comes from two sources: the growing number of
shares, and the rising value of each share. If only the number of shares is increasing (and
the value of each share is constant), the rate at which which total value would rise is the
product of the current value of the shares and the rate at which the number of shares is
changing. That is, the rate at which total value would change is given by
dollars shares dollars
S(100) · N ′(100) 27.50 · 12 330 .
share day day
105
Chapter 2 Computing Derivatives
Note particularly how the units make sense and show the rate at which the total value V is
changing, measured in dollars per day.
If instead the number of shares is constant, but the value of each share is rising, the rate
at which the total value would rise is the product of the number of shares and the rate of
change of share value. The total value is rising at a rate of
dollars per share dollars
N(100) · S′(100) 520 shares · 0.75 390 .
day day
Of course, when both the number of shares and the value of each share are changing, we
have to include both of these sources. In that case the rate at which the total value is rising
is
dollars
V ′(100) S(100) · N ′(100) + N(100) · S′(100) 330 + 390 720 .
day
We expect the total value of the investor’s holdings to rise by about $720 on the 100th day.¹
Next, we expand our perspective from the specific example above to the more general and
abstract setting of a product p of two differentiable functions, f and 1. If P(x) f (x) · 1(x),
our work above suggests that P ′(x) f (x)1 ′(x) + 1(x) f ′(x). Indeed, a formal proof using
the limit definition of the derivative can be given to show that the following rule, called the
product rule, holds in general.
Product Rule.
If f and 1 are differentiable functions, then their product P(x) f (x) · 1(x) is also a
differentiable function, and
P ′(x) f (x)1 ′(x) + 1(x) f ′(x).
In light of the earlier example involving shares of stock, the product rule also makes sense
intuitively: the rate of change of P should take into account both how fast f and 1 are chang-
ing, as well as how large f and 1 are at the point of interest. In words the product rule says:
if P is the product of two functions f (the first function) and 1 (the second), then “the deriv-
ative of P is the first times the derivative of the second, plus the second times the derivative
of the first.” It is often a helpful mental exercise to say this phrasing aloud when executing
the product rule.
Example 2.3.2 If P(z) z 3 · cos(z), we can use the product rule to differentiate P. The first
function is z 3 and the second function is cos(z). By the product rule, P ′ will be given by
the first, z 3 , times the derivative of the second, − sin(z), plus the second, cos(z), times the
¹While this example highlights why the product rule is true, there are some subtle issues to recognize. For one,
if the stock’s value really does rise exactly $0.75 on day 100, and the number of shares really rises by 12 on day
100, then we’d expect that V(101) N(101) · S(101) 532 · 28.25 15029. If, as noted above, we expect the total
value to rise by $720, then with V(100) N(100) · S(100) 520 · 27.50 14300, then it seems we should find that
V(101) V(100) + 720 15020. Why do the two results differ by 9? One way to understand why this difference
occurs is to recognize that N ′ (100) 12 represents an instantaneous rate of change, while our (informal) discussion
has also thought of this number as the total change in the number of shares over the course of a single day. The
formal proof of the product rule reconciles this issue by taking the limit as the change in the input tends to zero.
106
2.3 The product and quotient rules
derivative of the first, 3z 2 . That is,
P ′(z) z 3 (− sin(z)) + cos(z)3z 2 −z 3 sin(z) + 3z 2 cos(z).
Activity 2.3.2. Use the product rule to answer each of the questions below. Through-
out, be sure to carefully label any derivative you find by name. It is not necessary to
algebraically simplify any of the derivatives you compute.
a. Let m(w) 3w 17 4w . Find m ′(w).
b. Let h(t) (sin(t) + cos(t))t 4 . Find h ′(t).
c. Determine the slope of the tangent line to the curve y f (x) at the point where
a 1 if f is given by the rule f (x) e x sin(x).
d. Find the tangent line approximation L(x) to the function y 1(x) at the point
where a −1 if 1 is given by the rule 1(x) (x 2 + x)2x .
2.3.2 The quotient rule
Because quotients and products are closely linked, we can use the product rule to under-
stand how to take the derivative of a quotient. Let Q(x) be defined by Q(x) f (x)/1(x),
where f and 1 are both differentiable functions. It turns out that Q is differentiable every-
where that 1(x) , 0. We would like a formula for Q ′ in terms of f , 1, f ′, and 1 ′. multiplying
both sides of the formula Q f /1 by 1, we observe that
f (x) Q(x) · 1(x).
Now we can use the product rule to differentiate f .
f ′(x) Q(x)1 ′(x) + 1(x)Q ′(x).
We want to know a formula for Q ′, so we solve this equation for Q ′(x).
Q ′(x)1(x) f ′(x) − Q(x)1 ′(x)
and dividing both sides by 1(x), we have
f ′(x) − Q(x)1 ′(x)
Q ′(x) .
1(x)
f (x)
Finally, we recall that Q(x) 1(x)
. Substituting this expression in the preceding equation,
we have
f (x) ′
f ′(x) − 1(x)
1 (x)
′
Q (x)
1(x)
f (x) ′
f ′(x) − 1(x)
1 (x) 1(x)
·
1(x) 1(x)
107
Chapter 2 Computing Derivatives
1(x) f ′(x) − f (x)1 ′(x)
.
1(x)2
This calculation gives us the quotient rule.
Quotient Rule.
f (x)
If f and 1 are differentiable functions, then their quotient Q(x) 1(x)
is also a dif-
ferentiable function for all x where 1(x) , 0 and
1(x) f ′(x) − f (x)1 ′(x)
Q ′(x) .
1(x)2
As with the product rule, it can be helpful to think of the quotient rule verbally. If a function
Q is the quotient of a top function f and a bottom function 1, then Q ′ is given by “the bottom
times the derivative of the top, minus the top times the derivative of the bottom, all over the
bottom squared.”
Example 2.3.3 If Q(t) sin(t)/2t , we call sin(t) the top function and 2t the bottom function.
By the quotient rule, Q ′ is given by the bottom, 2t , times the derivative of the top, cos(t),
minus the top, sin(t), times the derivative of the bottom, 2t ln(2), all over the bottom squared,
(2t )2 . That is,
2t cos(t) − sin(t)2t ln(2)
Q ′(t) .
(2t )2
In this particular example, it is possible to simplify Q ′(t) by removing a factor of 2t from
both the numerator and denominator, so that
cos(t) − sin(t) ln(2)
Q ′(t) .
2t
In general, we must be careful in doing any such simplification, as we don’t want to execute
the quotient rule correctly but then make an algebra error.
Activity 2.3.3. Use the quotient rule to answer each of the questions below. Through-
out, be sure to carefully label any derivative you find by name. That is, if you’re given
a formula for f (x), clearly label the formula you find for f ′(x). It is not necessary to
algebraically simplify any of the derivatives you compute.
3z
a. Let r(z) z 4 +1
. Find r ′(z).
Find v ′(t).
sin(t)
b. Let v(t) cos(t)+t 2
.
x 2 − 2x − 8
c. Determine the slope of the tangent line to the curve R(x) at the
x2 − 9
point where x 0.
d. When a camera flashes, the intensity I of light seen by the eye is given by the
108
2.3 The product and quotient rules
function
100t
I(t)
,
et
where I is measured in candles and t is measured in milliseconds. Compute
I ′(0.5), I ′(2), and I ′(5); include appropriate units on each value; and discuss the
meaning of each.
2.3.3 Combining rules
In order to apply the derivative shortcut rules correctly we must recognize the fundamental
structure of a function.
Example 2.3.4 Determine the derivative of the function
x2
f (x) x sin(x) + .
cos(x) + 2
How do we decide which rules to apply? Our first task is to recognize the structure of the
function. This function f is a sum of two slightly less complicated functions, so we can apply
the sum rule² to get
[ ]
d x2
f ′(x) x sin(x) +
dx cos(x) + 2
[ ]
d d x2
[x sin(x)] +
dx dx cos(x) + 2
Now, the left-hand term above is a product, so the product rule is needed there, while the
right-hand term is a quotient, so the quotient rule is required. Applying these rules respec-
tively, we find that
(cos(x) + 2)2x − x 2 (− sin(x))
f ′(x) (x cos(x) + sin(x)) +
(cos(x) + 2)2
2x cos(x) + 4x 2 + x 2 sin(x)
x cos(x) + sin(x) + .
(cos(x) + 2)2
Example 2.3.5 Differentiate
y · 7y
.s(y)
y2 + 1
The function s is a quotient of two simpler functions, so the quotient rule will be needed. To
d
begin, we set up the quotient rule and use the notation dy to indicate the derivatives of the
numerator and denominator. Thus,
[ ] [ ]
(y 2 + 1) · d
dy y · 7y − y · 7y · d
dy y2 + 1
s ′(y) .
(y 2 + 1)2
²When taking a derivative that involves the use of multiple derivative rules, it is often helpful to use the notation
d
dx [ ] to wait to apply subsequent rules. This is demonstrated in each of the two examples presented here.
109
Chapter 2 Computing Derivatives
[ ]
Now, there remain two derivatives to calculate. The first one, d
y · 7 y calls for use of the
[ ] dy
product rule, while the second, dy d
y 2 + 1 needs only the sum rule. Applying these rules,
we now have
(y 2 + 1)[y · 7 y ln(7) + 7 y · 1] − y · 7 y [2y]
s ′(y) .
(y 2 + 1)2
While some simplification is possible, we are content to leave s ′(y) in its current form.
Success in applying derivative rules begins with recognizing the structure of the function,
followed by the careful and diligent application of the relevant derivative rules. The best
way to become proficient at this process is to do a large number of examples.
Activity 2.3.4. Use relevant derivative rules to answer each of the questions below.
Throughout, be sure to use proper notation and carefully label any derivative you
find by name.
a. Let f (r) (5r 3 + sin(r))(4r − 2 cos(r)). Find f ′(r).
cos(t)
b. Let p(t) . Find p ′(t).
t 6 · 6t
c. Let 1(z) 3z 7 e z − 2z 2 sin(z) + z
z 2 +1
. Find 1 ′(z).
d. A moving particle has its position in feet at time t in seconds given by the func-
3 cos(t)−sin(t)
tion s(t) et
. Find the particle’s instantaneous velocity at the moment
t 1.
e. Suppose that f (x) and 1(x) are differentiable functions and it is known that
f (3) −2, f ′(3) 7, 1(3) 4, and 1 ′(3) −1. If p(x) f (x) · 1(x) and
f (x)
q(x) , calculate p ′(3) and q ′(3).
1(x)
As the algebraic complexity of the functions we are able to differentiate continues to increase,
it is important to remember that all of the derivative’s meaning continues to hold. Regardless
of the structure of the function f , the value of f ′(a) tells us the instantaneous rate of change
of f with respect to x at the moment x a, as well as the slope of the tangent line to y f (x)
at the point (a, f (a)).
2.3.4 Summary
• If a function is a sum, product, or quotient of simpler functions, then we can use the
sum, product, or quotient rules to differentiate it in terms of the simpler functions and
their derivatives.
• The product rule tells us that if P is a product of differentiable functions f and 1 ac-
cording to the rule P(x) f (x)1(x), then
P ′(x) f (x)1 ′(x) + 1(x) f ′(x).
110
2.3 The product and quotient rules
• The quotient rule tells us that if Q is a quotient of differentiable functions f and 1
f (x)
according to the rule Q(x) 1(x) , then
1(x) f ′(x) − f (x)1 ′(x)
Q ′(x) .
1(x)2
• Along with the constant multiple and sum rules, the product and quotient rules enable
us to compute the derivative of any function that consists of sums, constant multiples,
products, and quotients of basic functions. For instance, if F has the form
2a(x) − 5b(x)
F(x) ,
c(x) · d(x)
then F is a quotient, in which the numerator is a sum of constant multiples and the de-
nominator is a product. This, the derivative of F can be found by applying the quotient
rule and then using the sum and constant multiple rules to differentiate the numerator
and the product rule to differentiate the denominator.
2.3.5 Exercises
1. Derivative of a basic product. Find the derivative of the function f (x), below. It may
be to your advantage to simplify first.
f (x) x · 13x
2. Derivative of a product. Find the derivative of the function f (x), below. It may be to
your advantage to simplify first.
√
f (x) (x 9 − 3 x)2x
3. Derivative of a quotient of linear functions. Find the derivative of the function z,
below. It may be to your advantage to simplify first.
2t + 7
z
8t + 7
4. Derivative of a rational function. Find the derivative of the function h(r), below. It
may be to your advantage to simplify first.
r3
h(r)
9r + 13
5. Derivative of a product of trigonometric functions. Find the derivative of s(q)
6 cos q sin q.
6. Derivative of a product of power and trigonmetric functions. Find the derivative of
f (x) x 5 cos x
7. Derivative of a sum that involves a product. Find the derivative of h(t) t sin t + tan t
8. Product and quotient rules with graphs. Let h(x) f (x) · 1(x), and k(x) f (x)/1(x).
Use the figures below to find the exact values of the indicated derivatives.
111
Chapter 2 Computing Derivatives
f (x) 1(x)
A. h ′(1)
B. k ′(−2)
9. Product and quotient rules with given function values. Let F(4) 4, F′(4) 5, H(4)
4, H ′(4) 5.
A. If G(z) F(z) · H(z), then G′(4)
B. If G(w) F(w)/H(w), then G′(4)
10. Let f and 1 be differentiable functions for which the following information is known:
f (2) 5, 1(2) −3, f ′(2) −1/2, 1 ′(2) 2.
a. Let h be the new function defined by the rule h(x) 1(x) · f (x). Determine h(2)
and h ′(2).
b. Find an equation for the tangent line to y h(x) at the point (2, h(2)) (where h is
the function defined in (a)).
1(x)
c. Let r be the function defined by the rule r(x) f (x)
. Is r increasing, decreasing,
or neither at a 2? Why?
d. Estimate the value of r(2.06) (where r is the function defined in (c)) by using the
local linearization of r at the point (2, r(2)).
11. Consider the functions r(t) t t and s(t) arccos(t), for which you are given the
facts that r ′(t) t t (ln(t) + 1) and s ′(t) − √ 1 2 . Do not be concerned with where these
1−t
derivative formulas come from. We restrict our interest in both functions to the domain
0 < t < 1.
a. Let w(t) t t arccos(t). Determine w ′(t).
b. Find an equation for the tangent line to y w(t) at the point ( 12 , w( 21 )).
tt
c. Let v(t) arccos(t)
. Is v increasing or decreasing at the instant t 12 ? Why?
112
2.3 The product and quotient rules
12. Let functions p and q be the piecewise linear functions given by their respective graphs
in Figure 2.3.6. Use the graphs to answer the following questions.
a. Let r(x) p(x) · q(x). Determine p
r ′(−2) and r ′(0). 3
b. Are there values of x for which 2
r ′(x) does not exist? If so, which
values, and why? 1
c. Find an equation for the tangent
line to y r(x) at the point (2, r(2)). -3 -2 -1 1 2 3
q(x) -1
d. Let z(x) p(x)
. Determine z ′(0)
and z ′(2).
q
-2
e. Are there values of x for which
-3
z ′(x) does not exist? If so, which
values, and why?
Figure 2.3.6: The graphs of p (in blue)
and q (in green).
13. A farmer with large land holdings has historically grown a wide variety of crops. With
the price of ethanol fuel rising, he decides that it would be prudent to devote more and
more of his acreage to producing corn. As he grows more and more corn, he learns
efficiencies that increase his yield per acre. In the present year, he used 7000 acres of
his land to grow corn, and that land had an average yield of 170 bushels per acre. At
the current time, he plans to increase his number of acres devoted to growing corn at
a rate of 600 acres/year, and he expects that right now his average yield is increasing
at a rate of 8 bushels per acre per year. Use this information to answer the following
questions.
a. Say that the present year is t 0, that A(t) denotes the number of acres the farmer
devotes to growing corn in year t, Y(t) represents the average yield in year t (mea-
sured in bushels per acre), and C(t) is the total number of bushels of corn the
farmer produces. What is the formula for C(t) in terms of A(t) and Y(t)? Why?
b. What is the value of C(0)? What does it measure?
c. Write an expression for C′(t) in terms of A(t), A′(t), Y(t), and Y ′(t). Explain your
thinking.
d. What is the value of C′(0)? What does it measure?
e. Based on the given information and your work above, estimate the value of C(1).
14. Let f (v) be the gas consumption (in liters/km) of a car going at velocity v (in km/hour).
In other words, f (v) tells you how many liters of gas the car uses to go one kilometer
if it is traveling at v kilometers per hour. In addition, suppose that f (80) 0.05 and
f ′(80) 0.0004.
a. Let 1(v) be the distance the same car goes on one liter of gas at velocity v. What
is the relationship between f (v) and 1(v)? Hence find 1(80) and 1 ′(80).
113
Chapter 2 Computing Derivatives
b. Let h(v) be the gas consumption in liters per hour of a car going at velocity v. In
other words, h(v) tells you how many liters of gas the car uses in one hour if it
is going at velocity v. What is the algebraic relationship between h(v) and f (v)?
Hence find h(80) and h ′(80).
c. How would you explain the practical meaning of these function and derivative
values to a driver who knows no calculus? Include units on each of the function
and derivative values you discuss in your response.
114
2.4 Derivatives of other trigonometric functions
2.4 Derivatives of other trigonometric functions
Motivating Questions
• What are the derivatives of the tangent, cotangent, secant, and cosecant functions?
• How do the derivatives of tan(x), cot(x), sec(x), and csc(x) combine with other de-
rivative rules we have developed to expand the library of functions we can quickly
differentiate?
One of the powerful themes in trigonometry comes from a very simple idea: locating a point
on the unit circle.
(x, y)
1
sin(θ )
θ
cos(θ )
Figure 2.4.1: The unit circle and the definition of the sine and cosine functions.
Because each angle θ in standard position corresponds to one and only one point (x, y) on
the unit circle, the x- and y-coordinates of this point are each functions of θ. In fact, this is the
very definition of cos(θ) and sin(θ): cos(θ) is the x-coordinate of the point on the unit circle
corresponding to the angle θ, and sin(θ) is the y-coordinate. From this simple definition,
all of trigonometry is founded. For instance, the Fundamental Trigonometric Identity,
sin2 (θ) + cos2 (θ) 1,
is a restatement of the Pythagorean Theorem, applied to the right triangle shown in Fig-
ure 2.4.1.
There are four other trigonometric functions, each defined in terms of the sine and/or cosine
functions.
sin(θ)
• The tangent function is defined by tan(θ) cos(θ)
;
cos(θ)
• the cotangent function is its reciprocal: cot(θ) sin(θ)
.
115
Chapter 2 Computing Derivatives
• The secant function is the reciprocal of the cosine function, sec(θ) 1
cos(θ)
;
• and the cosecant function is the reciprocal of the sine function, csc(θ) 1
sin(θ)
.
These six trigonometric functions together offer us a wide range of flexibility in problems
involving right triangles.
Because we know the derivatives of the sine and cosine function, we can now develop short-
cut differentiation rules for the tangent, cotangent, secant, and cosecant functions. In this
section’s preview activity, we work through the steps to find the derivative of y tan(x).
Preview Activity 2.4.1. Consider the function f (x) tan(x), and remember that
sin(x)
tan(x) cos(x) .
a. What is the domain of f ?
b. Use the quotient rule to show that one expression for f ′(x) is
cos(x) cos(x) + sin(x) sin(x)
f ′(x) .
cos2 (x)
c. What is the Fundamental Trigonometric Identity? How can this identity be used
to find a simpler form for f ′(x)?
d. Recall that sec(x) 1
cos(x)
. How can we express f ′(x) in terms of the secant
function?
e. For what values of x is f ′(x) defined? How does this set compare to the domain
of f ?
2.4.1 Derivatives of the cotangent, secant, and cosecant functions
In Preview Activity 2.4.1, we found that the derivative of the tangent function can be ex-
pressed in several ways, but most simply in terms of the secant function. Next, we develop
the derivative of the cotangent function.
Let 1(x) cot(x). To find 1 ′(x), we observe that 1(x)
cos(x)
sin(x)
and apply the quotient rule.
Hence
sin(x)(− sin(x)) − cos(x) cos(x)
1 ′(x)
sin2 (x)
sin2 (x) + cos2 (x)
−
sin2 (x)
By the Fundamental Trigonometric Identity, we see that 1 ′(x) − sin12 (x) , and recalling that
csc(x) 1
sin(x)
, it follows that we can express 1 ′ by the rule
1 ′(x) − csc2 (x).
116
2.4 Derivatives of other trigonometric functions
Note that neither 1 nor 1 ′ is defined when sin(x) 0, which occurs at every integer multiple
of π. Hence we have the following rule.
Cotangent Function.
For all real numbers x such that x , kπ, where k 0, ±1, ±2, . . .,
d
[cot(x)] − csc2 (x).
dx
Notice that the derivative of the cotangent function is very similar to the derivative of the
tangent function we discovered in Preview Activity 2.4.1.
Tangent Function.
(2k+1)π
For all real numbers x such that x , 2 , where k ±1, ±2, . . .,
d
[tan(x)] sec2 (x).
dx
In the next two activities, we develop the rules for differentiating the secant and cosecant
functions.
Activity 2.4.2. Let h(x) sec(x) and recall that sec(x) 1
cos(x)
.
a. What is the domain of h?
b. Use the quotient rule to develop a formula for h ′(x) that is expressed completely
in terms of sin(x) and cos(x).
c. How can you use other relationships among trigonometric functions to write
h ′(x) only in terms of tan(x) and sec(x)?
d. What is the domain of h ′? How does this compare to the domain of h?
Activity 2.4.3. Let p(x) csc(x) and recall that csc(x) 1
sin(x)
.
a. What is the domain of p?
b. Use the quotient rule to develop a formula for p ′(x) that is expressed completely
in terms of sin(x) and cos(x).
c. How can you use other relationships among trigonometric functions to write
p ′(x) only in terms of cot(x) and csc(x)?
d. What is the domain of p ′? How does this compare to the domain of p?
Using the quotient rule we have determined the derivatives of the tangent, cotangent, secant,
and cosecant functions, expanding our overall library of functions we can differentiate. Ob-
serve that just as the derivative of any polynomial function is a polynomial, and the deriv-
117
Chapter 2 Computing Derivatives
ative of any exponential function is another exponential function, so it is that the derivative
of any basic trigonometric function is another function that consists of basic trigonometric
functions. This makes sense because all trigonometric functions are periodic, and hence
their derivatives will be periodic, too.
The derivative retains all of its fundamental meaning as an instantaneous rate of change and
as the slope of the tangent line to the function under consideration.
Activity 2.4.4. Answer each of the following questions. Where a derivative is re-
quested, be sure to label the derivative function with its name using proper notation.
a. Let f (x) 5 sec(x) − 2 csc(x). Find the slope of the tangent line to f at the point
where x π3 .
b. Let p(z) z 2 sec(z) − z cot(z). Find the instantaneous rate of change of p at the
point where z π4 .
tan(t)
c. Let h(t) − 2e t cos(t). Find h ′(t).
t2 + 1
r sec(r)
d. Let 1(r) . Find 1 ′(r).
5r
e. When a mass hangs from a spring and is set in motion, the object’s position
oscillates in a way that the size of the oscillations decrease. This is usually called
a damped oscillation. Suppose that for a particular object, its displacement from
equilibrium (where the object sits at rest) is modeled by the function
15 sin(t)
s(t) .
et
Assume that s is measured in inches and t in seconds. Sketch a graph of this
function for t ≥ 0 to see how it represents the situation described. Then compute
ds/dt, state the units on this function, and explain what it tells you about the
object’s motion. Finally, compute and interpret s ′(2).
2.4.2 Summary
• The derivatives of the other four trigonometric functions are
d d
[tan(x)] sec2 (x), [cot(x)] − csc2 (x),
dx dx
d d
[sec(x)] sec(x) tan(x), and [csc(x)] − csc(x) cot(x).
dx dx
Each derivative exists and is defined on the same domain as the original function. For
example, both the tangent function and its derivative are defined for all real numbers
x such that x , kπ
2 , where k ±1, ±2, . . ..
118
2.4 Derivatives of other trigonometric functions
• The four rules for the derivatives of the tangent, cotangent, secant, and cosecant can be
used along with the rules for power functions, exponential functions, and the sine and
cosine, as well as the sum, constant multiple, product, and quotient rules, to quickly
differentiate a wide range of different functions.
2.4.3 Exercises
1. A sum and product involving tan(x). Find the derivative of h(t) t tan t + cos t
5 tan(x)
2. A quotient involving tan(t). Let f (x) . Find f ′(x) and f ′(4).
x
tan(x) − 2
3. A quotient of trigonometric functions. Let f (x) . Find f ′(x) and f ′(1).
sec(x)
2x 2 tan(x)
4. A quotient that involves a product. Let f (x) . Find f ′(x) and f ′(4).
sec(x)
5. Finding a tangent line equation. Find the equation of the tangent line to the curve
y 3 tan x at the point (π/4, 3). The equation of this tangent line can be written in the
form y mx + b. Find m and b.
6. An object moving vertically has its height at time t (measured in feet, with time in
2 cos(t)
seconds) given by the function h(t) 3 + 1.2t .
a. What is the object’s instantaneous velocity when t 2?
b. What is the object’s acceleration at the instant t 2?
c. Describe in everyday language the behavior of the object at the instant t 2.
7. Let f (x) sin(x) cot(x).
a. Use the product rule to find f ′(x).
b. True or false: for all real numbers x, f (x) cos(x).
c. Explain why the function that you found in (a) is almost the opposite of the sine
function, but not quite. (Hint: convert all of the trigonometric functions in (a) to
sines and cosines, and work to simplify. Think carefully about the domain of f
and the domain of f ′.)
8. Let p(z) be given by the rule
z tan(z)
p(z) + 3e z + 1.
+1
z 2 sec(z)
a. Determine p ′(z).
b. Find an equation for the tangent line to p at the point where z 0.
c. At z 0, is p increasing, decreasing, or neither? Why?
119
Chapter 2 Computing Derivatives
2.5 The chain rule
Motivating Questions
• What is a composite function and how do we recognize its structure algebraically?
• Given a composite function C(x) f (1(x)) that is built from differentiable functions
f and 1, how do we compute C′(x) in terms of f , 1, f ′, and 1 ′? What is the statement
of the Chain Rule?
In addition to learning how to differentiate a variety of basic functions, we have also been
developing our ability to use rules to differentiate certain algebraic combinations of them.
Example 2.5.1 State the rule(s) to find the derivative of each of the following combinations
of f (x) sin(x) and 1(x) x 2 :
s(x) 3x 2 − 5 sin(x),
p(x) x 2 sin(x), and
sin(x)
q(x) .
x2
Solution. Finding s ′ uses the sum and constant multiple rules, because s(x) 31(x) −
5 f (x). Determining p ′ requires the product rule, because p(x) 1(x) · f (x). To calculate q ′
f (x)
we use the quotient rule, because q(x) 1(x) .
There is one more natural way to combine basic functions algebraically, and that is by com-
posing them. For instance, let’s consider the function
C(x) sin(x 2 ),
and observe that any input x passes through a chain of functions. In the process that defines
the function C(x), x is first squared, and then the sine of the result is taken. Using an arrow
diagram,
x −→ x 2 −→ sin(x 2 ).
In terms of the elementary functions f and 1, we observe that x is the input for the function
1, and the result is used as the input for f . We write
C(x) f (1(x)) sin(x 2 )
and say that C is the composition of f and 1. We will refer to 1, the function that is first
applied to x, as the inner function, while f , the function that is applied to the result, is the
outer function.
Given a composite function C(x) f (1(x)) that is built from differentiable functions f and 1,
how do we compute C′(x) in terms of f , 1, f ′, and 1 ′? In the same way that the rate of change
of a product of two functions, p(x) f (x) · 1(x), depends on the behavior of both f and 1, it
makes sense intuitively that the rate of change of a composite function C(x) f (1(x)) will
120
2.5 The chain rule
also depend on some combination of f and 1 and their derivatives. The rule that describes
how to compute C′ in terms of f and 1 and their derivatives is called the chain rule.
But before we can learn what the chain rule says and why it works, we first need to be
comfortable decomposing composite functions so that we can correctly identify the inner
and outer functions, as we did in the example above with C(x) sin(x 2 ).
Preview Activity 2.5.1. For each function given below, identify its fundamental alge-
braic structure. In particular, is the given function a sum, product, quotient, or com-
position of basic functions? If the function is a composition of basic functions, state a
formula for the inner function 1 and the outer function f so that the overall compos-
ite function can be written in the form f (1(x)). If the function is a sum, product, or
quotient of basic functions, use the appropriate rule to determine its derivative.
a. h(x) tan(2x ) d. m(x) e tan(x)
√
b. p(x) 2x tan(x) e. w(x) x + tan(x)
√
c. r(x) (tan(x))2 f. z(x) tan(x)
2.5.1 The chain rule
Often a composite function cannot be written in an alternate algebraic form. For instance,
the function C(x) sin(x 2 ) cannot be expanded or otherwise rewritten, so it presents no al-
ternate approaches to taking the derivative. But some composite functions can be expanded
or simplified, and these provide a way to explore how the chain rule works.
Example 2.5.2 Let f (x) −4x + 7 and 1(x) 3x − 5. Determine a formula for C(x) f (1(x))
and compute C′(x). How is C′ related to f and 1 and their derivatives?
Solution. By the rules given for f and 1,
C(x) f (1(x))
f (3x − 5)
− 4(3x − 5) + 7
− 12x + 20 + 7
− 12x + 27.
Thus, C′(x) −12. Noting that f ′(x) −4 and 1 ′(x) 3, we observe that C′ appears to be
the product of f ′ and 1 ′.
It may seem that Example 2.5.2 is too elementary to illustrate how to differentiate a composite
fuction. Linear functions are the simplest of all functions, and composing linear functions
yields another linear function. While this example does not illustrate the full complexity of
a composition of nonlinear functions, at the same time we remember that any differentiable
function is locally linear, and thus any function with a derivative behaves like a line when
viewed up close. The fact that the derivatives of the linear functions f and 1 are multiplied
121
Chapter 2 Computing Derivatives
to find the derivative of their composition turns out to be a key insight.
We now consider a composition involving a nonlinear function.
Example 2.5.3 Let C(x) sin(2x). Use the double angle identity to rewrite C as a product of
basic functions, and use the product rule to find C′. Rewrite C′ in the simplest form possible.
Solution. Using the double angle identity for the sine function, we write
C(x) sin(2x) 2 sin(x) cos(x).
Applying the product rule and simplifying, we find
C′(x) 2 sin(x)(− sin(x)) + cos(x)(2 cos(x)) 2(cos2 (x) − sin2 (x)).
Next, we recall that the double angle identity for the cosine,
cos(2x) cos2 (x) − sin2 (x).
Substituting this result into our expression for C′(x), we now have that
C′(x) 2 cos(2x).
In Example 2.5.3, if we let 1(x) 2x and f (x) sin(x), we observe that C(x) f (1(x)). Now,
1 ′(x) 2 and f ′(x) cos(x), so we can view the structure of C′(x) as
C′(x) 2 cos(2x) 1 ′(x) f ′(1(x)).
In this example, as in the example involving linear functions, we see that the derivative of
the composite function C(x) f (1(x)) is found by multiplying the derivatives of f and 1,
but with f ′ evaluated at 1(x).
It makes sense intuitively that these two quantities are involved in the rate of change of a
composite function: if we ask how fast C is changing at a given x value, it clearly matters
how fast 1 is changing at x, as well as how fast f is changing at the value of 1(x). It turns
out that this structure holds for all differentiable functions¹ as is stated in the Chain Rule.
Chain Rule.
If 1 is differentiable at x and f is differentiable at 1(x), then the composite function
C defined by C(x) f (1(x)) is differentiable at x and
C′(x) f ′(1(x))1 ′(x).
As with the product and quotient rules, it is often helpful to think verbally about what the
chain rule says: “If C is a composite function defined by an outer function f and an inner
function 1, then C′ is given by the derivative of the outer function evaluated at the inner
function, times the derivative of the inner function.”
¹Like other differentiation rules, the Chain Rule can be proved formally using the limit definition of the deriv-
ative.
122
2.5 The chain rule
It is helpful to identify clearly the inner function 1 and outer function f , compute their
derivatives individually, and then put all of the pieces together by the chain rule.
Example 2.5.4 Determine the derivative of the function
r(x) (tan(x))2 .
Solution. The function r is composite, with inner function 1(x) tan(x) and outer function
f (x) x 2 . Organizing the key information involving f , 1, and their derivatives, we have
f (x) x 2 1(x) tan(x)
f ′(x) 2x 1 ′(x) sec2 (x)
f ′(1(x)) 2 tan(x)
Applying the chain rule, we find that
r ′(x) f ′(1(x))1 ′(x) 2 tan(x) sec2 (x).
As a side note, we remark that r(x) is usually written as tan2 (x). This is common notation for
powers of trigonometric functions: cos4 (x), sin5 (x), and sec2 (x) are all composite functions,
with the outer function a power function and the inner function a trigonometric one.
Activity 2.5.2. For each function given below, identify an inner function 1 and outer
function f to write the function in the form f (1(x)). Determine f ′(x), 1 ′(x), and
f ′(1(x)), and then apply the chain rule to determine the derivative of the given func-
tion.
a. h(x) cos(x 4 ) d. z(x) cot5 (x)
√
b. p(x) tan(x)
c. s(x) 2sin(x) e. m(x) (sec(x) + e x )9
2.5.2 Using multiple rules simultaneously
The chain rule now joins the sum, constant multiple, product, and quotient rules in our
collection of techniques for finding the derivative of a function through understanding its
algebraic structure and the basic functions that constitute it. It takes practice to get comfort-
able applying multiple rules to differentiate a single function, but using proper notation and
taking a few extra steps will help.
2 +2t
Example 2.5.5 Find a formula for the derivative of h(t) 3t sec4 (t).
Solution. We first observe that h is the product of two functions: h(t) a(t) · b(t), where
a(t) 3t +2t and b(t) sec4 (t). We will need to use the product rule to differentiate h. And
2
because a and b are composite functions, we will need the chain rule. We therefore begin by
computing a ′(t) and b ′(t).
2 +2t
Writing a(t) f (1(t)) 3t , and finding the derivatives of f and 1, we have
123
Chapter 2 Computing Derivatives
f (t) 3t 1(t) t 2 + 2t
f ′(t) 3t ln(3) 1 ′(t) 2t + 2
f ′(1(t)) 3t +2t ln(3)
2
Thus, by the chain rule, it follows that a ′(t) f ′(1(t))1 ′(t) 3t
2 +2t
ln(3)(2t + 2).
Turning next to b, we write b(t) r(s(t)) sec4 (t) and find the derivatives of r and s.
r(t) t 4 s(t) sec(t)
r ′(t) 4t 3 s ′(t) sec(t) tan(t)
r ′(s(t)) 4 sec3 (t)
By the chain rule,
b ′(t) r ′(s(t))s ′(t) 4 sec3 (t) sec(t) tan(t) 4 sec4 (t) tan(t).
Now we are finally ready to compute the derivative of the function h. Recalling that h(t)
3t +2t sec4 (t), by the product rule we have
2
d d
h ′(t) 3t
2 +2t
[sec4 (t)] + sec4 (t) [3t +2t ].
2
dt dt
From our work above with a and b, we know the derivatives of 3t +2t and sec4 (t), and there-
2
fore
h ′(t) 3t +2t 4 sec4 (t) tan(t) + sec4 (t)3t +2t ln(3)(2t + 2).
2 2
Activity 2.5.3. For each of the following functions, find the function’s derivative. State
the rule(s) you use, label relevant derivatives appropriately, and be sure to clearly
identify your √overall answer.
2
a. p(r) 4 r 6 + 2e r d. s(z) 2z sec(z)
b. m(v) sin(v 2 ) cos(v 3 )
cos(10y) 2
c. h(y) e 4y +1
e. c(x) sin(e x )
The chain rule now adds substantially to our ability to compute derivatives. Whether we
are finding the equation of the tangent line to a curve, the instantaneous velocity of a mov-
ing particle, or the instantaneous rate of change of a certain quantity, if the function under
consideration is a composition, the chain rule is indispensable.
Activity 2.5.4. Use known derivative rules, including the chain rule, as needed to
answer each of the following questions.
√
a. Find an equation for the tangent line to the curve y e x + 3 at the point where
x 0.
1
b. If s(t) represents the position function of a particle moving horizon-
(t 2
+ 1)3
tally along an axis at time t (where s is measured in inches and t in seconds),
find the particle’s instantaneous velocity at t 1. Is the particle moving to the
124
2.5 The chain rule
left or right at that instant?
c. At sea level, air pressure is 30 inches of mercury. At an altitude of h feet above
sea level, the air pressure, P, in inches of mercury, is given by the function
P 30e −0.0000323h . Compute dP/dh and explain what this derivative function
tells you about air pressure, including a discussion of the units on dP/dh. In
addition, determine how fast the air pressure is changing for a pilot of a small
plane passing through an altitude of 1000 feet.
d. Suppose that f (x) and 1(x) are differentiable functions and that the following
information about them is known:
x f (x) f ′(x) 1(x) 1 ′(x)
−1 2 −5 −3 4
2 −3 4 −1 2
Table 2.5.6: Data for functions f and 1.
If C(x) is a function given by the formula f (1(x)), determine C′(2). In addition,
if D(x) is the function f ( f (x)), find D ′(−1).
2.5.3 The composite version of basic function rules
As we gain more experience with differention, we will become more comfortable in simply
writing down the derivative without taking multiple steps. This is particularly simple when
the inner function is linear, since the derivative of a linear function is a constant.
Example 2.5.7 Use the chain rule to differentiate each of the following composite functions
whose inside function is linear:
d [ ]
(5x + 7)10 10(5x + 7)9 · 5,
dx
d
[tan(17x)] 17 sec2 (17x), and
dx
d [ −3x ]
e −3e −3x .
dx
More generally, following is an excellent exercise for getting comfortable with the derivative
rules. Write down a list of all the basic functions whose derivatives we know, and list the
derivatives. Then write a composite function with the inner function being an unknown
function u(x) and the outer function being a basic function. Finally, write the chain rule for
the composite function. The following example illustrates this for two different functions.
Example 2.5.8 To determine
d
[sin(u(x))],
dx
where u is a differentiable function of x, we use the chain rule with the sine function as the
125
Chapter 2 Computing Derivatives
outer function. Applying the chain rule, we find that
d
[sin(u(x))] cos(u(x)) · u ′(x).
dx
dx [sin(x)] cos(x).
d
This rule is analogous to the basic derivative rule that
dx [a ] a x ln(a), it follows by the chain rule that
d x
Similarly, since
d u(x)
[a ] a u(x) ln(a) · u ′(x).
dx
dx [a ] a x ln(a).
d x
This rule is analogous to the basic derivative rule that
2.5.4 Summary
• A composite function is one where the input variable x first passes through one func-
tion, and then the resulting output passes through another. For example, the function
h(x) 2sin(x) is composite since x −→ sin(x) −→ 2sin(x) .
• Given a composite function C(x) f (1(x)) where f and 1 are differentiable functions,
the chain rule tells us that
C′(x) f ′(1(x))1 ′(x).
2.5.5 Exercises
1. Mixing rules: chain, product, sum. Find the derivative of f (x) e 5x (x 2 + 7x ).
2. Mixing rules: chain and product. Find the derivative of v(t) t 6 e −ct . Assume that c
is a constant.
√
3. Using the chain rule repeatedly. Find the derivative of y e −5t 2 + 9.
4. Derivative involving arbitrary constants a and b. Find the derivative of the function
f (x) axe −bx+12 . Assume that a and b are constants.
5. Chain rule with graphs. Use the graph below to find exact values of the indicated
derivatives, or state that they do not exist. If a derivative does not exist, enter dne in the
answer blank. The graph of f (x) is black and has a sharp corner at x 2. The graph of
1(x) is blue.
126
2.5 The chain rule
Let h(x) f (1(x)). Find h ′(1), h ′(2), and h ′(3) or explain why they do not exist.
6. Chain rule with function values. Given F(4) 1, F′(4) 5, F(5) 4, F′(5) 6 and
G(1) 3, G′(1) 4, G(4) 5, G′(4) 6, find each of the following.
A. H(4) if H(x) F(G(x))
B. H ′(4) if H(x) F(G(x))
C. H(4) if H(x) G(F(x))
D. H ′(4) if H(x) G(F(x))
E. H ′(4) if H(x) F(x)/G(x)
7. A product involving a composite function. Find the derivative of f (x) 2x sin(6x).
8. Consider the basic functions f (x) x 3 and 1(x) sin(x).
a. Let h(x) f (1(x)). Find the exact instantaneous rate of change of h at the point
where x π4 .
b. Which function is changing most rapidly at x 0.25: h(x) f (1(x)) or r(x)
1( f (x))? Why?
c. Let h(x) f (1(x)) and r(x) 1( f (x)). Which of these functions has a derivative
that is periodic? Why?
9. Let u(x) be a differentiable function. For each of the following functions, determine the
derivative. Each response will involve u and/or u ′.
a. p(x) e u(x) d. s(x) u(cot(x))
b. q(x) u(e x ) e. a(x) u(x 4 )
c. r(x) cot(u(x)) f. b(x) u 4 (x)
127
Chapter 2 Computing Derivatives
10. Let functions p and q be the piecewise linear functions given by their respective graphs
in Figure 2.5.9. Use the graphs to answer the following questions.
p
3
2
1
-3 -2 -1 1 2 3
-1
q
-2
-3
Figure 2.5.9: The graphs of p (in blue) and q (in green).
a. Let C(x) p(q(x)). Determine C′(0) and C′(3).
b. Find a value of x for which C′(x) does not exist. Explain your thinking.
c. Let Y(x) q(q(x)) and Z(x) q(p(x)). Determine Y ′(−2) and Z′(0).
11. If a spherical tank of radius 4 feet has h feet of water present in the tank, then the volume
of water in the tank is given by the formula
π 2
V h (12 − h).
3
a. At what instantaneous rate is the volume of water in the tank changing with re-
spect to the height of the water at the instant h 1? What are the units on this
quantity?
b. Now suppose that the height of water in the tank is being regulated by an inflow
and outflow (e.g., a faucet and a drain) so that the height of the water at time t is
given by the rule h(t) sin(πt) + 1, where t is measured in hours (and h is still
measured in feet). At what rate is the height of the water changing with respect
to time at the instant t 2?
c. Continuing under the assumptions in (b), at what instantaneous rate is the vol-
ume of water in the tank changing with respect to time at the instant t 2?
d. What are the main differences between the rates found in (a) and (c)? Include a
discussion of the relevant units.
128
2.6 Derivatives of Inverse Functions
2.6 Derivatives of Inverse Functions
Motivating Questions
• What is the derivative of the natural logarithm function?
• What are the derivatives of the inverse trigonometric functions arcsin(x) and arctan(x)?
• If 1 is the inverse of a differentiable function f , how is 1 ′ computed in terms of f , f ′,
and 1?
Much of mathematics centers on the notion of function. Indeed, throughout our study of cal-
culus, we are investigating the behavior of functions, with particular emphasis on how fast
the output of the function changes in response to changes in the input. Because each func-
tion represents a process, a natural question to ask is whether or not the particular process
can be reversed. That is, if we know the output that results from the function, can we deter-
mine the input that led to it? And if we know how fast a particular process is changing, can
we determine how fast the inverse process is changing?
One of the most important functions in all of mathematics is the natural exponential function
f (x) e x . Its inverse, the natural logarithm, 1(x) ln(x), is similarly important. One of our
goals in this section is to learn how to differentiate the logarithm function. First, we review
some of the basic concepts surrounding functions and their inverses.
Preview Activity 2.6.1. The equation y 95 (x − 32) relates a temperature given in x
degrees Fahrenheit to the corresponding temperature y measured in degrees Celcius.
a. Solve the equation y 59 (x − 32) for x to write x (Fahrenheit temperature) in
terms of y (Celcius temperature).
b. Let C(x) 59 (x − 32) be the function that takes a Fahrenheit temperature as
input and produces the Celcius temperature as output. In addition, let F(y)
be the function that converts a temperature given in y degrees Celcius to the
temperature F(y) measured in degrees Fahrenheit. Use your work in (a) to write
a formula for F(y).
c. Next consider the new function defined by p(x) F(C(x)). Use the formulas
for F and C to determine an expression for p(x) and simplify this expression as
much as possible. What do you observe?
d. Now, let r(y) C(F(y)). Use the formulas for F and C to determine an expres-
sion for r(y) and simplify this expression as much as possible. What do you
observe?
e. What is the value of C′(x)? of F′(y)? How do these values appear to be related?
129
Chapter 2 Computing Derivatives
2.6.1 Basic facts about inverse functions
A function f : A → B is a rule that associates each element in the set A to one and only
one element in the set B. We call A the domain of f and B the codomain of f . If there exists
a function 1 : B → A such that 1( f (a)) a for every possible choice of a in the set A and
f (1(b)) b for every b in the set B, then we say that 1 is the inverse of f .
We often use the notation f −1 (read “ f -inverse”) to denote the inverse of f . The inverse
function undoes the work of f . Indeed, if y f (x), then
f −1 (y) f −1 ( f (x)) x.
Thus, the equations y f (x) and x f −1 (y) say the same thing. The only difference be-
tween the two equations is one of perspective — one is solved for x, while the other is solved
for y.
Here we briefly remind ourselves of some key facts about inverse functions.
Note 2.6.1 For a function f : A → B,
• f has an inverse if and only if f is one-to-one ¹ and onto ²;
• provided f −1 exists, the domain of f −1 is the codomain of f , and the codomain of f −1
is the domain of f ;
• f −1 ( f (x)) x for every x in the domain of f and f ( f −1 (y)) y for every y in the
codomain of f ;
• y f (x) if and only if x f −1 (y).
The last fact reveals a special relationship between the graphs of f and f −1 . If a point (x, y)
that lies on the graph of y f (x), then it is also true that x f −1 (y), which means that
the point (y, x) lies on the graph of f −1 . This shows us that the graphs of f and f −1 are
the reflections of each other across the line y x, because this reflection is precisely the
geometric action that swaps the coordinates in an ordered pair. In Figure 2.6.2, we see this
illustrated by the function y f (x) 2x and its inverse, with the points (−1, 21 ) and ( 12 , −1)
highlighting the reflection of the curves across y x. To close our review of important
facts about inverses, we recall that the natural exponential function y f (x) e x has an
inverse function, namely the natural logarithm, x f −1 (y) ln(y). Thus, writing y e x is
interchangeable with x ln(y), plus ln(e x ) x for every real number x and e ln(y) y for
every positive real number y.
2.6.2 The derivative of the natural logarithm function
In what follows, we find a formula for the derivative of 1(x) ln(x). To do so, we take
advantage of the fact that we know the derivative of the natural exponential function, the
inverse of 1. In particular, we know that writing 1(x) ln(x) is equivalent to writing e 1(x)
¹A function f is one-to-one provided that no two distinct inputs lead to the same output.
²A function f is onto provided that every possible element of the codomain can be realized as an output of the
function for some choice of input from the domain.
130
2.6 Derivatives of Inverse Functions
y = f (x)
2
(−1, 21 )
-2 2
( 12 , −1)
-2
y = f −1 (x)
y=x
Figure 2.6.2: A graph of a function y f (x) along with its inverse, y f −1 (x).
x. Now we differentiate both sides of this equation and observe that
d [ 1(x) ] d
e [x].
dx dx
The righthand side is simply 1; by applying the chain rule to the left side, we find that
e 1(x) 1 ′(x) 1.
Next we solve for 1 ′(x), to get
1
1 ′(x) .
e 1(x)
Finally, we recall that 1(x) ln(x), so e 1(x) e ln(x) x, and thus
1
1 ′(x) .
x
Natural Logarithm.
dx [ln(x)] x1 .
d
For all positive real numbers x,
This rule for the natural logarithm function now joins our list of basic derivative rules. Note
that this rule applies only to positive values of x, as these are the only values for which ln(x)
is defined.
Also notice that for the first time in our work, differentiating a basic function of a particular
type has led to a function of a very different nature: the derivative of the natural logarithm
is not another logarithm, nor even an exponential function, but rather a rational one.
131
Chapter 2 Computing Derivatives
Derivatives of logarithms may now be computed in concert with all of the rules known to
date. For instance, if f (t) ln(t 2 + 1), then by the chain rule, f ′(t) t 21+1 · 2t.
There are interesting connections between the graphs of f (x) e x and f −1 (x) ln(x).
In Figure 2.6.3, we are reminded that
since the natural exponential function y = ex
has the property that its derivative is it- 8
self, the slope of the tangent to y e x is
equal to the height of the curve at that
point. For instance, at the point A B
(ln(0.5), 0.5), the slope of the tangent line 4
is m A 0.5, and at B (ln(5), 5), the tan-
gent line’s slope is m B 5. B′
At the corresponding points A′ and B′ on
A
the graph of the natural logarithm func-
tion (which come from reflecting A and -4 4 8
B across the line y x), we know that A′
the slope of the tangent line is the rec-
iprocal of the x-coordinate of the point y = ln(x)
(since dxd
[ln(x)] x1 ). Thus, at A′
-4
(0.5, ln(0.5)), we have m A′ 0.51
2, and
′
at B (5, ln(5)), m B′ 5 .
1
Figure 2.6.3: A graph of the function y e x
along with its inverse, y ln(x), where both
functions are viewed using the input variable
x.
In particular, we observe that m A′ m1A and m B′ m1B . This is not a coincidence, but in fact
holds for any curve y f (x) and its inverse, provided the inverse exists. This is due to the
reflection across y x. It changes the roles of x and y, thus reversing the rise and run, so
the slope of the inverse function at the reflected point is the reciprocal of the slope of the
original function.
Activity 2.6.2. For each function given below, find its derivative.
a. h(x) x 2 ln(x) d. z(x) tan(ln(x))
ln(t)
b. p(t) e t +1
c. s(y) ln(cos(y) + 2) e. m(z) ln(ln(z))
2.6.3 Inverse trigonometric functions and their derivatives
Trigonometric functions are periodic, so they fail to be one-to-one, and thus do not have
inverse functions. However, we can restrict the domain of each trigonometric function so
that it is one-to-one on that domain.
For instance, consider the sine function on the domain [− π2 , π2 ]. Because no output of the
sine function is repeated on this interval, the function is one-to-one and thus has an inverse.
132
2.6 Derivatives of Inverse Functions
Thus, the function f (x) sin(x) with [− π2 , π2 ] and codomain [−1, 1] has an inverse function
f −1 such that
π π
f −1 : [−1, 1] → [− , ].
2 2
We call f −1 the arcsine (or inverse sine)
function and write f −1 (y) arcsin(y). It
is especially important to remember that (1, π2 )
π
2
y sin(x) and x arcsin(y) f −1
say the same thing. “The arcsine of y” f ( π2 , 1)
means “the angle whose sine is y.” For ex-
ample, arcsin( 21 ) π6 means that π6 is the
angle whose sine is 12 , which is equivalent − π2 π
2
to writing sin( π6 ) 12 .
Next, we determine the derivative of the
arcsine function. Letting h(x) arcsin(x), − π2
our goal is to find h ′(x). Since h(x) is the an-
gle whose sine is x, it is equivalent to write
sin(h(x)) x.
Figure 2.6.4: A graph of f (x) sin(x) (in
blue), restricted to the domain [− π2 , π2 ],
along with its inverse, f −1 (x) arcsin(x)
(in magenta).
Differentiating both sides of the previous equation, we have
d d
[sin(h(x))] [x].
dx dx
The righthand side is simply 1, and by applying the chain rule applied to the left side,
cos(h(x))h ′(x) 1.
Solving for h ′(x), it follows that
1
h ′(x) .
cos(h(x))
Finally, we recall that h(x) arcsin(x), so the denominator of h ′(x) is the function cos(arcsin(x)),
or in other words, “the cosine of the angle whose sine is x.” A bit of right triangle trigonom-
etry allows us to simplify this expression considerably.
133
Chapter 2 Computing Derivatives
Let’s say that θ arcsin(x), so that θ is the angle
whose sine is x. We can picture θ as an angle in a right
triangle with hypotenuse 1 and a vertical leg of length
x, as shown in Figure 2.6.5. The horizontal leg must be
√
1 − x 2 , by the Pythagorean Theorem.
Now, because θ arcsin(x), the expression for 1 x
cos(arcsin(x)) is equivalent to cos(θ). From the figure,
√
cos(arcsin(x)) cos(θ) 1 − x 2 .
Substituting this expression into our formula, h ′(x) θ
1 √
cos(arcsin(x))
, we have now shown that 1 − x2
1 Figure 2.6.5: The right triangle
h ′(x) √ .
1 − x2 that corresponds to the angle
θ arcsin(x).
Inverse sine.
For all real numbers x such that −1 < x < 1,
d 1
[arcsin(x)] √ .
dx 1 − x2
Activity 2.6.3. The following prompts in this activity will lead you to develop the
derivative of the inverse tangent function.
a. Let r(x) arctan(x). Use the relationship between the arctangent and tangent
functions to rewrite this equation using only the tangent function.
b. Differentiate both sides of the equation you found in (a). Solve the resulting
equation for r ′(x), writing r ′(x) as simply as possible in terms of a trigonometric
function evaluated at r(x).
c. Recall that r(x) arctan(x). Update your expression for r ′(x) so that it only
involves trigonometric functions and the independent variable x.
d. Introduce a right triangle with angle θ so that θ arctan(x). What are the three
sides of the triangle?
e. In terms of only x and 1, what is the value of cos(arctan(x))?
f. Use the results of your work above to find an expression involving only 1 and x
for r ′(x).
While derivatives for other inverse trigonometric functions can be established similarly, for
now we limit ourselves to the arcsine and arctangent functions.
134
2.6 Derivatives of Inverse Functions
Activity 2.6.4. Determine the derivative of each of the following functions.
a. f (x) x 3 arctan(x) + e x ln(x)
b. p(t) 2t arcsin(t)
c. h(z) (arcsin(5z) + arctan(4 − z))27
d. s(y) cot(arctan(y))
e. m(v) ln(sin2 (v) + 1)
( )
ln(w)
f. 1(w) arctan
1 + w2
2.6.4 The link between the derivative of a function and the derivative of
its inverse
In Figure 2.6.3, we saw an interesting relationship between the slopes of tangent lines to
the natural exponential and natural logarithm functions at points reflected across the line
y x. In particular, we observed that at the point (ln(2), 2) on the graph of f (x) e x , the
slope of the tangent line is f ′(ln(2)) 2, while at the corresponding point (2, ln(2)) on the
graph of f −1 (x) ln(x), the slope of the tangent line is ( f −1 )′(2) 12 , which is the reciprocal
of f ′(ln(2)).
That the two corresponding tangent lines have reciprocal slopes is not a coincidence. If f and
1 are differentiable inverse functions, then y f (x) if and only if x 1(y), then f (1(x)) x
for every x in the domain of f −1 . Differentiating both sides of this equation, we have
d d
[ f (1(x))] [x],
dx dx
and by the chain rule,
f ′(1(x))1 ′(x) 1.
Solving for 1 ′(x), we have 1 ′(x) f ′ (1(x))
1
. Here we see that the slope of the tangent line to the
inverse function 1 at the point (x, 1(x)) is precisely the reciprocal of the slope of the tangent
line to the original function f at the point (1(x), f (1(x))) (1(x), x). To see this more clearly,
consider the graph of the function y f (x) shown in Figure 2.6.6, along with its inverse
y 1(x). Given a point (a, b) that lies on the graph of f , we know that (b, a) lies on the
graph of 1; because f (a) b and 1(b) a. Now, applying the rule that 1 ′(x) 1/ f ′(1(x)) to
the value x b, we have
1 1
1 ′(b) ′ ′ ,
f (1(b)) f (a)
which is precisely what we see in the figure: the slope of the tangent line to 1 at (b, a) is the
reciprocal of the slope of the tangent line to f at (a, b), since these two lines are reflections
of one another across the line y x.
135
Chapter 2 Computing Derivatives
m = f ′ (a)
y = f (x) (a, b)
m = g′ (b)
(b, a)
y = g(x)
Figure 2.6.6: A graph of function y f (x) along with its inverse, y 1(x) f −1 (x).
Observe that the slopes of the two tangent lines are reciprocals of one another.
Derivative of an inverse function.
Suppose that f is a differentiable function with inverse 1 and that (a, b) is a point
that lies on the graph of f at which f ′(a) , 0. Then
1
1 ′(b) .
f ′(a)
More generally, for any x in the domain of 1 ′, we have 1 ′(x) 1/ f ′(1(x)).
The rules we derived for ln(x), arcsin(x), and arctan(x) are all just specific examples of this
general property of the derivative of an inverse function. For example, with 1(x) ln(x)
and f (x) e x , it follows that
1 1 1
1 ′(x) .
f ′(1(x)) e ln(x) x
2.6.5 Summary
dx [ln(x)] x1 .
d
• For all positive real numbers x,
• For all real numbers x such that −1 < x < 1, dx [arcsin(x)]
d
√ 1 . In addition, for all
1−x 2
dx [arctan(x)]
d 1
real numbers x, 1+x 2
.
• If 1 is the inverse of a differentiable function f , then for any point x in the domain of
1 ′, 1 ′(x) f ′ (1(x))
1
.
136
2.6 Derivatives of Inverse Functions
2.6.6 Exercises
1. Composite function involving logarithms and polynomials. Find the derivative of
the function f (t) ln(t 3 + 3).
2. Composite function involving trigonometric functions and logarithms. Find the de-
rivative of the function 1(t) cos(ln(t)).
3. Product involving arcsin(w). Find the derivative of the function h(w) 5w arcsin w
4. Derivative involving arctan(x). For x > 0, find and simplify the derivative of f (x)
arctan x + arctan(1/x). (What does your result tell you about f )?
5. Composite function from a graph. Let (x0 , y0 ) (2, 6) and (x1 , y1 ) (2.1, 6.2). Use the
following graph of the function f to find the indicated derivatives.
If h(x) ( f (x))5 , find h ′(2).
If 1(x) f −1 (x), find 1 ′(6).
6. Composite function involving an inverse trigonometric function. Let
( )
f (x) 7 sin−1 x 3 .
Find f ′(x).
7. Mixing rules: product, chain, and inverse trig. If f (x) 8x 4 arctan(3x 3 ), find f ′(x).
8. Mixing rules: product and inverse trig. Let f (x) 8 cos(x) sin−1 (x). Find f ′(x).
9. Determine the derivative of each of the following functions. Use proper notation and
clearly identify the derivative rules you use.
a. f (x) ln(2 arctan(x) + 3 arcsin(x) + 5)
b. r(z) arctan(ln(arcsin(z)))
c. q(t) arctan2 (3t) arcsin4 (7t)
( )
arctan(v)
d. 1(v) ln arcsin(v)+v 2
137
Chapter 2 Computing Derivatives
10. Consider the graph of y f (x) provided in Figure 2.6.7 and use it to answer the fol-
lowing questions.
a. Use the provided graph to estimate the value
of f ′(1).
b. Sketch an approximate graph of y f −1 (x).
Label at least three distinct points on the
graph that correspond to three points on the
graph of f .
y = f (x)
c. Based on your work in (a), what is the value
of ( f −1 )′(−1)? Why?
Figure 2.6.7: A function
y f (x)
11. Let f (x) 14 x 3 + 4.
a. Sketch a graph of y f (x) and explain why f is an invertible function.
b. Let 1 be the inverse of f and determine a formula for 1.
c. Compute f ′(x), 1 ′(x), f ′(2), and 1 ′(6). What is the special relationship between
f ′(2) and 1 ′(6)? Why?
12. Let h(x) x + sin(x).
a. Sketch a graph of y h(x) and explain why h must be invertible.
b. Explain why it does not appear to be algebraically possible to determine a formula
for h −1 .
c. Observe that the point ( π2 , π
2 + 1) lies on the graph of y h(x). Determine the
value of (h −1 )′( π2 + 1).
138
2.7 Derivatives of Functions Given Implicitly
2.7 Derivatives of Functions Given Implicitly
Motivating Questions
• What does it mean to say that a curve is an implicit function of x, rather than an
explicit function of x?
dy
• How does implicit differentiation enable us to find a formula for dx when y is an
implicit function of x?
dy
• In the context of an implicit curve, how can we use dx to answer important questions
about the tangent line to the curve?
In all of our studies with derivatives so far, we have worked with functions whose formula
is given explicitly in terms of x. But there are many interesting curves whose equations
involving x and y are impossible to solve for y in terms of x.
x2 + y2 = 16
4
x3 − y3 = 6xy
A
x
-4 4
B
-4
Figure 2.7.1: At left, the circle given by x 2 + y 2 16. In the middle, the portion of the circle
x 2 + y 2 16 that has been highlighted in the box at left. And at right, the lemniscate given
by x 3 − y 3 6x y.
Perhaps the simplest and most natural of all such curves are circles. Because of the circle’s
symmetry, for each x value strictly between the endpoints of the horizontal diameter, there
√
are two corresponding y-values. For instance, in Figure 2.7.1, we have labeled A (−3, 7)
√
and B (−3, − 7), and these points demonstrate that the circle fails the vertical line test.
Hence, it is impossible to represent the circle through a single function of the form y
f (x). But portions of the circle can be represented explicitly as a function of x, such as the
highlighted arc that is magnified in the center of Figure 2.7.1. Moreover, it is evident that
the circle is locally linear, so we ought to be able to find a tangent line to the curve at every
dy
point. Thus, it makes sense to wonder if we can compute dx at any point on the circle, even
though we cannot write y explicitly as a function of x.
We say that the equation x 2 + y 2 16 defines y implicitly as a function of x. The graph
of the equation can be broken into pieces where each piece can be defined by an explicit
139
Chapter 2 Computing Derivatives
function of x. For the circle, we could choose to take the top half as one function of x, namely
√ √
y 16 − x 2 and the bottom half as y − 16 − x 2 . The equation for the circle defines two
implicit functions of x.
The righthand curve in Figure 2.7.1 is called a lemniscate and is just one of many fascinating
possibilities for implicitly given curves.
dy
How can we find an equation for dx without an explicit formula for y in terms of x? The
following preview activity reminds us of some ways we can compute derivatives of functions
in settings where the function’s formula is not known.
Preview Activity 2.7.1. Let f be a differentiable function of x (whose formula is not
known) and recall that dx d
[ f (x)] and f ′(x) are interchangeable notations. Determine
each of the following derivatives of combinations of explicit functions of x, the un-
known function
[ 2 f , ]and an arbitrary constant c. [ ]
a. dx x + f (x)
d
d. dxd
f (x 2 )
[ ]
b. d
dx x 2 f (x)
[ ] [ ]
c. d
dx c + x + f (x)2 e. d
dx x f (x) + f (cx) + c f (x)
2.7.1 Implicit Differentiation
We begin our exploration of implicit differentiation with the example of the circle given by
dy
x 2 + y 2 16. How can we find a formula for dx ?
By viewing y as an implicit function of x, we think of y as some function whose formula f (x)
is unknown, but which we can differentiate. Just as y represents an unknown formula, so
dy
too its derivative with respect to x, dx , will be (at least temporarily) unknown.
So we view y as an unknown differentiable function of x and differentiate both sides of the
equation with respect to x.
d [ 2 ] d
x + y2 [16] .
dx dx
On the right, the derivative of the constant 16 is 0, and on the left we can apply the sum rule,
so it follows that
d [ 2] d [ 2]
x + y 0.
dx dx
Note carefully ] different roles being played by x and y. Because x is the independent
[ 2the
d
variable, dx x 2x. But y is the dependent variable and y is an implicit function of x.
d
Recall Preview Activity 2.7.1, where we computed dx [ f (x)2 ]. Computing dx
d
[y 2 ] is the same,
dy
dx [y ] 2y 1 dx . We now have that
d 2
and requires the chain rule, by which we find that
dy
2x + 2y 0.
dx
140
2.7 Derivatives of Functions Given Implicitly
dy
We solve this equation for dx by subtracting 2x from both sides and dividing by 2y.
dy 2x x
− − .
dx 2y y
mt = − ba
There are several important things to observe (a, b)
dy
about the result that dx − xy . First, this ex-
pression for the derivative involves both x and
y. This makes sense because there are two b
corresponding points on the circle for each mr = a
value of x between −4 and 4, and the slope
of the tangent line is different at each of these
points.Second, this formula is entirely consis-
tent with our understanding of circles. The
slope of the radius from the origin to the point
(a, b) is m r ba . The tangent line to the circle at
(a, b) is perpendicular to the radius, and thus
has slope m t − ba , as shown in Figure 2.7.2.
In particular, the slope of the tangent line is
zero at (0, 4) and (0, −4), and is undefined at Figure 2.7.2: The circle given by
(−4, 0) and (4, 0). All of these values are con- x 2 + y 2 16 with point (a, b) on the circle
dy
sistent with the formula dx − xy . and the tangent line at that point, with
labeled slopes of the radial line, m r , and
tangent line, m t .
Example 2.7.3 For the curve given implicitly by x 3 + y 2 − 2x y 2, shown in Figure 2.7.4, find
the slope of the tangent line at (−1, 1).
y
3
x
-3 3
-3
Figure 2.7.4: The curve x 3 + y 2 − 2x y 2.
141
Chapter 2 Computing Derivatives
Solution. We begin by differentiating the curve’s equation implicitly. Taking the derivative
of each side with respect to x,
d [ 3 ] d
x + y 2 − 2x y [2] ,
dx dx
by the sum rule and the fact that the derivative of a constant is zero, we have
d 3 d 2 d
[x ] + [y ] − [2x y] 0.
dx dx dx
For the three derivatives we now must execute, the first uses the simple power rule, the
second requires the chain rule (since y is an implicit function of x), and the third necessitates
the product rule (again since y is a function of x). Applying these rules, we now find that
dy dy
3x 2 + 2y − [2x + 2y] 0.
dx dx
dy dy
We want to solve this equation for dx . To do so, we first collect all of the terms involving dx
on one side of the equation.
dy dy
2y − 2x 2y − 3x 2 .
dx dx
dy
Then we factor the left side to isolate dx .
dy
(2y − 2x) 2y − 3x 2 .
dx
Finally, we divide both sides by (2y − 2x) and conclude that
dy 2y − 3x 2
.
dx 2y − 2x
dy
Note that the expression for dx depends on both x and y. To find the slope of the tangent
dy
line at (−1, 1), we substitute the coordinates into the formula for dx , using the notation
dy 2(1) − 3(−1)2 1
− .
dx (−1,1) 2(1) − 2(−1) 4
This value matches our visual estimate of the slope of the tangent line shown in Figure 2.7.4.
Example 2.7.3 shows that it is possible when differentiating implicitly to have multiple terms
dy dy
involving dx . We use addition and subtraction to collect all terms involving dx on one side
dy dy
of the equation, then factor to get a single term of dx . Finally, we divide to solve for dx .
We use the notation
dy
dx (a,b)
142
2.7 Derivatives of Functions Given Implicitly
dy
to denote the evaluation of dx at the point (a, b). This is analogous to writing f ′(a) when f ′
depends on a single variable.
d dy
There is a big difference between writing dx and dx . For example,
d 2
[x + y 2 ]
dx
gives an instruction to take the derivative with respect to x of the quantity x 2 + y 2 , presum-
ably where y is a function of x. On the other hand,
dy 2
(x + y 2 )
dx
means the product of the derivative of y with respect to x with the quantity x 2 + y 2 . Under-
standing this notational subtlety is essential.
Activity 2.7.2. Consider the curve defined by the equation x y 5 − 5y 3 + 4y, whose
graph is pictured in Figure 2.7.5.
y
a. Explain why it is not possible to 3
express y as an explicit function of
x.
b. Use implicit differentiation to find
a formula for dy/dx. x
-3 3
c. Use your result from part (b) to
find an equation of the line tangent
to the graph of x y 5 − 5y 3 + 4y
at the point (0, 1).
-3
d. Use your result from part (b) to de-
termine all of the points at which
the graph of x y 5 − 5y 3 + 4y has
Figure 2.7.5: The curve
a vertical tangent line.
x y 5 − 5y 3 + 4y.
It is natural to ask where the tangent line to a curve is vertical or horizontal. The slope of a
horizontal tangent line must be zero, while the slope of a vertical tangent line is undefined.
dy
Often the formula for dx is expressed as a quotient of functions of x and y, say
dy p(x, y)
.
dx q(x, y)
The tangent line is horizontal precisely when the numerator is zero and the denominator is
nonzero, making the slope of the tangent line zero. If we can solve the equation p(x, y) 0
for either x and y in terms of the other, we can substitute that expression into the original
equation for the curve. This gives an equation in a single variable, and if we can solve that
143
Chapter 2 Computing Derivatives
equation we can find the point(s) on the curve where p(x, y) 0. At those points, the tangent
line is horizontal.
Similarly, the tangent line is vertical whenever q(x, y) 0 and p(x, y) , 0, making the slope
undefined.
Activity 2.7.3. Consider the curve defined by the equation y(y 2 − 1)(y − 2) x(x −
1)(x − 2), whose graph is pictured in Figure 2.7.6. Through implicit differentiation, it
can be shown that
dy (x − 1)(x − 2) + x(x − 2) + x(x − 1)
2 .
dx (y − 1)(y − 2) + 2y 2 (y − 2) + y(y 2 − 1)
Use this fact to answer each of the following questions.
y
a. Determine all points (x, y) at
which the tangent line to the 2
curve is horizontal. (Use tech-
nology appropriately to find the
needed zeros of the relevant 1
polynomial function.)
x
b. Determine all points (x, y) at
which the tangent line is vertical. 1 2 3
(Use technology appropriately
to find the needed zeros of the -1
relevant polynomial function.)
c. Find the equation of the tangent
line to the curve at one of the Figure 2.7.6:
points where x 1. y(y 2 − 1)(y − 2) x(x − 1)(x − 2).
Activity 2.7.4. For each of the following curves, use implicit differentiation to find
dy/dx and determine the equation of the tangent line at the given point.
a. x 3 − y 3 6x y, (−3, 3) c. 3xe −x y y 2 , (0.619061, 1)
b. sin(y) + y x 3 + x, (0, 0)
2.7.2 Summary
• In an equation involving x and y where portions of the graph can be defined by explicit
functions of x, we say that y is an implicit function of x. A good example of such a
curve is the unit circle.
• We use implicit differentiation to differentiate an implicitly defined function. We dif-
ferentiate both sides of the equation with respect to x, treating y as a function of x by
144
2.7 Derivatives of Functions Given Implicitly
dy
applying the chain rule. If possible, we subsequently solve for dx using algebra.
dy dy
• While dx may now involve both the variables x and y, dx still gives the slope of the
tangent line to the curve. It may be used to decide where the tangent line is horizontal
dy dy
( dx 0) or vertical ( dx is undefined), or to find the equation of the tangent line at a
particular point on the curve.
2.7.3 Exercises
1. Implicit differentiaion in a polynomial equation. Find dy/dx in terms of x and y if
x 2 y − x − 5y − 11 0.
dy
2. Implicit differentiation in an equation with logarithms. Find in terms of x and y
dx
if x ln y + y 3 3 ln x.
3. Implicit differentiation in an equation with inverse trigonometric functions. Find
dy/dx in terms of x and y if arctan(x 3 y) x y 3 .
4. Slope of the tangent line to an implicit curve. Find the slope of the tangent to the
curve x 3 + x y + y 2 31 at (1, 5).
5. Equation of the tangent line to an implicit curve. Use implicit differentiation to find
an equation of the tangent line to the curve 3x y 3 + x y 16 at the point (4, 1).
6. Consider the curve given by the equation 2y 3 + y 2 − y 5 x 4 − 2x 3 + x 2 . Find all points at
which the tangent line to the curve is horizontal or vertical. Be sure to use a graphing
utility to plot this implicit curve and to visually check the results of algebraic reasoning
that you use to determine where the tangent lines are horizontal and vertical.
7. For the curve given by the equation sin(x + y) + cos(x − y) 1, find the equation of the
tangent line to the curve at the point ( π2 , π2 ).
8. Implicit differentiation enables us a different perspective from which to see why the
d
rule dx [a x ] a x ln(a) holds, if we assume that dx
d
[ln(x)] x1 . This exercise leads you
through the key steps to do so.
a. Let y a x . Rewrite this equation using the natural logarithm function to write x
in terms of y (and the constant a).
b. Differentiate both sides of the equation you found in (a) with respect to x, keeping
in mind that y is implicitly a function of x.
dy
c. Solve the equation you found in (b) for dx , and then use the definition of y to
dy
write dx solely in terms of x. What have you found?
145
Chapter 2 Computing Derivatives
2.8 Using Derivatives to Evaluate Limits
Motivating Questions
• How can derivatives be used to help us evaluate indeterminate limits of the form 00 ?
• What does it mean to say that limx→∞ f (x) L and limx→a f (x) ∞?
∞
• How can derivatives assist us in evaluating indeterminate limits of the form ∞?
Because differential calculus is based on the definition of the derivative, and the definition
of the derivative involves a limit, there is a sense in which all of calculus rests on limits. In
addition, the limit involved in the definition of the derivative always generates the indeter-
minate form 00 . If f is a differentiable function, then in the definition
f (x + h) − f (x)
f ′(x) lim ,
h→0 h
not only does h → 0 in the denominator, but also ( f (x + h) − f (x)) → 0 in the numerator,
since f is continuous. Remember, saying that a limit has an indeterminate form only means
that we don’t yet know its value and have more work to do: indeed, limits of the form 00 can
take on any value, as is evidenced by evaluating f ′(x) for varying values of x for a function
such as f ′(x) x 2 .
We have learned many techniques for evaluating the limits that result from the derivative
definition, including a large number of shortcut rules. In this section, we turn the situa-
tion upside-down: instead of using limits to evaluate derivatives, we explore how to use
derivatives to evaluate certain limits.
x 5 +x−2
Preview Activity 2.8.1. Let h be the function given by h(x) x 2 −1
.
a. What is the domain of h?
x5 + x − 2
b. Explain why lim results in an indeterminate form.
x→1 x2 − 1
c. Next we will investigate the behavior of both the numerator and denominator
of h near the point where x 1. Let f (x) x 5 + x − 2 and 1(x) x 2 − 1. Find
the local linearizations of f and 1 at a 1, and call these functions L f (x) and
L 1 (x), respectively.
L f (x)
d. Explain why h(x) ≈ L 1 (x)
for x near a 1.
e. Using your work from (c) and (d), evaluate
L f (x)
lim .
x→1 L 1 (x)
What do you think your result tells us about limx→1 h(x)?
146
2.8 Using Derivatives to Evaluate Limits
f. Investigate the function h(x) graphically and numerically near x 1. What do
you think is the value of limx→1 h(x)?
2.8.1 Using derivatives to evaluate indeterminate limits of the form 00 .
f
Lf
Lf ≈ f
a a
Lg Lg ≈ g
g
Figure 2.8.1: At left, the graphs of f and 1 near the value a, along with their tangent line
approximations L f and L 1 at x a. At right, zooming in on the point a and the four
graphs.
The idea demonstrated in Preview Activity 2.8.1 — that we can evaluate an indeterminate
limit of the form 00 by replacing each of the numerator and denominator with their local lin-
earizations at the point of interest — can be generalized in a way that enables us to evaluate
f (x)
a wide range of limits. We have a function h(x) that can be written as a quotient h(x) 1(x) ,
where f and 1 are both differentiable at x a and for which f (a) 1(a) 0. We would
like to evaluate the indeterminate limit given by limx→a h(x). Figure 2.8.1 illustrates the sit-
uation. We see that both f and 1 have an x-intercept at x a. Their respective tangent line
approximations L f and L 1 at x a are also shown in the figure. We can take advantage
of the fact that a function and its tangent line approximation become indistinguishable as
x → a.
First, let’s recall that L f (x) f ′(a)(x − a) + f (a) and L 1 (x) 1 ′(a)(x − a) + 1(a). Because x is
getting arbitrarily close to a when we take the limit, we can replace f with L f and replace 1
with L 1 , and thus we observe that
f (x) L f (x)
lim lim
x→a 1(x) x→a L 1 (x)
147
Chapter 2 Computing Derivatives
f ′(a)(x − a) + f (a)
lim .
x→a 1 ′(a)(x − a) + 1(a)
Next, we remember that both f (a) 0 and 1(a) 0, which is precisely what makes the
original limit indeterminate. Substituting these values for f (a) and 1(a) in the limit above,
we now have
f (x) f ′(a)(x − a)
lim lim ′
x→a 1(x) x→a 1 (a)(x − a)
f ′(a)
lim ′ ,
x→a 1 (a)
where the latter equality holds because x−a
x−a 1 when x is approaching (but not equal to) a.
f ′ (a)
Finally, we note that 1 ′ (a)
is constant with respect to x, and thus
f (x) f ′(a)
lim ′ .
x→a 1(x) 1 (a)
This result holds as long as 1 ′(a) is not equal to zero. The formal name of the result is
L’Hôpital’s Rule.
L’Hôpital’s Rule.
Let f and 1 be differentiable at x a, and suppose that f (a) 1(a) 0 and that
f (x) f ′ (a)
1 ′(a) , 0. Then limx→a 1(x) 1′ (a) .
In practice, we typically work with a slightly more general version of L’Hôpital’s Rule, which
states that (under the identical assumptions as the boxed rule above and the extra assump-
tion that 1 ′ is continuous at x a)
f (x) f ′(x)
lim lim ′ ,
x→a 1(x) x→a 1 (x)
provided the righthand limit exists. This form reflects the basic idea of L’Hôpital’s Rule: if
f (x)
1(x)
produces an indeterminate limit of form 00 as x → a, that limit is equivalent to the limit
f ′ (x)
of the quotient of the two functions’ derivatives, 1 ′ (x)
.
For example, if we consider the limit from Preview Activity 2.8.1,
x5 + x − 2
lim ,
x→1 x2 − 1
by L’Hôpital’s Rule we have that
x5 + x − 2 5x 4 + 1 6
lim lim 3.
x→1 x2 − 1 x→1 2x 2
By replacing the numerator and denominator with their respective derivatives, we often
replace an indeterminate limit with one whose value we can easily determine.
148
2.8 Using Derivatives to Evaluate Limits
Activity 2.8.2. Evaluate each of the following limits. If you use L’Hôpital’s Rule, in-
dicate where it was used, and be certain its hypotheses are met before you apply it.
ln(1+x) 2 ln(x)
a. limx→0 x c. limx→1 1−e x−1
cos(x) sin(x)−x
b. limx→π x
d. limx→0 cos(2x)−1
While L’Hôpital’s Rule can be applied in an entirely algebraic way, it is important to re-
member that the justification of the rule is graphical: the main idea is that the slopes of the
f (x)
tangent lines to f and 1 at x a determine the value of the limit of 1(x) as x → a.
f
m = f ′ (a)
m = f ′ (a)
a m = g′ (a) a m = g′ (a)
g
Figure 2.8.2: Two functions f and 1 that satisfy L’Hôpital’s Rule.
We see this in Figure 2.8.2, where we can see from the grid that f ′(a) 2 and 1 ′(a) −1,
hence by L’Hôpital’s Rule,
f (x) f ′(a) 2
lim ′ −2.
x→a 1(x) 1 (a) −1
It’s not the fact that f and 1 both approach zero that matters most, but rather the rate at
which each approaches zero that determines the value of the limit. This is a good way to
f (x)
remember what L’Hôpital’s Rule says: if f (a) 1(a) 0, the the limit of 1(x) as x → a is
given by the ratio of the slopes of f and 1 at x a.
Activity 2.8.3. In this activity, we reason graphically from the following figure to
evaluate limits of ratios of functions about which some information is known.
149
Chapter 2 Computing Derivatives
g p
2 2 2
f
1 1 1
q r
1 2 3 4 1 2 3 4 1 2 3 4
-1 -1 -1
s
-2 -2 -2
Figure 2.8.3: Three graphs referenced in the questions of Activity 2.8.3.
a. Use the left-hand graph to determine the values of f (2), f ′(2), 1(2), and 1 ′(2).
f (x)
Then, evaluate lim 1(x) .
x→2
b. Use the middle graph to find p(2), p ′(2), q(2), and q ′(2). Then, determine the
p(x)
value of lim q(x) .
x→2
c. Assume that r and s are functions whose for which r ′′(2) , 0 and s ′′(2) , 0
Use the right-hand graph to compute r(2), r ′(2), s(2), s ′(2). Explain why you
r(x)
cannot determine the exact value of lim s(x) without further information being
x→2
r(x)
provided, but that you can determine the sign of lim . In addition, state what
x→2 s(x)
the sign of the limit will be, with justification.
2.8.2 Limits involving ∞
The concept of infinity, denoted ∞, arises naturally in calculus, as it does in much of math-
ematics. It is important to note from the outset that ∞ is a concept, but not a number itself.
Indeed, the notion of ∞ naturally invokes the idea of limits. Consider, for example, the
function f (x) x1 , whose graph is pictured in Figure 2.8.4.
We note that x 0 is not in the domain of f , so we may naturally wonder what happens
as x → 0. As x → 0+ , we observe that f (x) increases without bound. That is, we can make
the value of f (x) as large as we like by taking x closer and closer (but not equal) to 0, while
keeping x > 0. This is a good way to think about what infinity represents: a quantity is
tending to infinity if there is no single number that the quantity is always less than. Recall
that the statement limx→a f (x) L, means that can make f (x) as close to L as we’d like by
taking x sufficiently close (but not equal) to a. We now expand this notation and language to
include the possibility that either L or a can be ∞. For instance, for f (x) x1 , we now write
1
lim+ ∞,
x→0 x
1
by which we mean that we can make x as large as we like by taking x sufficiently close (but
150
2.8 Using Derivatives to Evaluate Limits
1
f (x) = x
1
1
Figure 2.8.4: The graph of f (x) x1 .
not equal) to 0. In a similar way, we write
1
lim 0,
x→∞ x
since we can make x1 as close to 0 as we’d like by taking x sufficiently large (i.e., by letting x
increase without bound).
In general, the notation limx→a f (x) ∞ means that we can make f (x) as large as we like by
taking x sufficiently close (but not equal) to a, and the notation limx→∞ f (x) L means that
we can make f (x) as close to L as we like by taking x sufficiently large. This notation also
applies to left- and right-hand limits, and to limits involving −∞. For example, returning to
Figure 2.8.4 and f (x) x1 , we can say that
1 1
lim −∞ and lim 0.
x→0− x x→−∞ x
Finally, we write
lim f (x) ∞
x→∞
if we can make the value of f (x) as large as we’d like by taking x sufficiently large. For
example,
lim x 2 ∞.
x→∞
Limits involving infinity identify vertical and horizontal asymptotes of a function. If lim f (x)
x→a
∞, then x a is a vertical asymptote of f , while if limx→∞ f (x) L, then y L is a horizontal
asymptote of f . Similar statements can be made using −∞, and with left- and right-hand
limits as x → a − or x → a + .
In precalculus classes, it is common to study the end behavior of certain families of functions,
by which we mean the behavior of a function as x → ∞ and as x → −∞. Here we briefly
examine some familiar functions and note the values of several limits involving ∞.
151
Chapter 2 Computing Derivatives
y = ex
8 64 y = sin(x)
y = f (x) 1
4
-2 2 10
-4 4 8
-64
y = ln(x)
-4 y = g(x)
Figure 2.8.5: Graphs of some familiar functions whose end behavior as x → ±∞ is known.
In the middle graph, f (x) x 3 − 16x and 1(x) x 4 − 16x 2 − 8.
For the natural exponential function e x , we note that limx→∞ e x ∞ and limx→−∞ e x
0. For the exponential decay function e −x , these limits are reversed, with limx→∞ e −x 0
and limx→−∞ e −x ∞. Turning to the natural logarithm function, we have limx→0+ ln(x)
−∞ and limx→∞ ln(x) ∞. While both e x and ln(x) grow without bound as x → ∞, the
exponential function does so much more quickly than the logarithm function does. We’ll
soon use limits to quantify what we mean by “quickly.”
For polynomial functions of the form
p(x) a n x n + a n−1 x n−1 + · · · a1 x + a0 ,
the end behavior depends on the sign of a n and whether the highest power n is even or odd.
If n is even and a n is positive, then limx→∞ p(x) ∞ and limx→−∞ p(x) ∞, as in the plot of
1 in Figure 2.8.5. If instead a n is negative, then limx→∞ p(x) −∞ and limx→−∞ p(x) −∞.
In the situation where n is odd, then either limx→∞ p(x) ∞ and limx→−∞ p(x) −∞ (which
occurs when a n is positive, as in the graph of f in Figure 2.8.5), or limx→∞ p(x) −∞ and
limx→−∞ p(x) ∞ (when a n is negative).
A function can fail to have a limit as x → ∞. For example, consider the plot of the sine
function at right in Figure 2.8.5. Because the function continues oscillating between −1 and
1 as x → ∞, we say that limx→∞ sin(x) does not exist.
Finally, it is straightforward to analyze the behavior of any rational function as x → ∞.
Example 2.8.6 Determine the limit of the function
3x 2 − 4x + 5
q(x)
7x 2 + 9x − 10
as x → ∞.
Note that both (3x 2 − 4x + 5) → ∞ as x → ∞ and (7x 2 + 9x − 10) → ∞ as x → ∞. Here
∞
we say that limx→∞ q(x) has indeterminate form ∞ . We can determine the value of this limit
through a standard algebraic approach. Multiplying the numerator and denominator each
152
2.8 Using Derivatives to Evaluate Limits
1
by x2
, we find that
1
3x 2 − 4x + 5 x2
lim q(x) lim ·
x→∞ x→∞ 7x 2 + 9x − 10 1
x2
3 − 4 x1 + 5 x12 3
lim
x→∞ 7+ 9 x1 − 10 x12 7
since x12 → 0 and x1 → 0 as x → ∞. This shows that the rational function q has a horizontal
asymptote at y 73 . A similar approach can be used to determine the limit of any rational
function as x → ∞.
But how should we handle a limit such as
x2
lim ?
x→∞ ex
Here, both x 2 → ∞ and e x → ∞, but there is not an obvious algebraic approach that enables
us to find the limit’s value. Fortunately, it turns out that L’Hôpital’s Rule extends to cases
involving infinity.
L’Hôpital’s Rule (∞).
If f and 1 are differentiable and both approach zero or both approach ±∞ as x → a
(where a is allowed to be ∞) , then
f (x) f ′(x)
lim lim ′ ,
x→a 1(x) x→a 1 (x)
provided the righthand limit exists.
(To be technically correct, we need to add the additional hypothesis that 1 ′(x) , 0 on an
open interval that contains a or in every neighborhood of infinity if a is ∞; this is almost
always met in practice.)
2
To evaluate limx→∞ xe x , we can apply L’Hôpital’s Rule, since both x 2 → ∞ and e x → ∞.
Doing so, it follows that
x2 2x
lim x lim x .
x→∞ e x→∞ e
∞
This updated limit is still indeterminate and of the form ∞ , but it is simpler since 2x has
replaced x 2 . Hence, we can apply L’Hôpital’s Rule again, and find that
x2 2x 2
lim lim lim x .
x→∞ e x x→∞ e x x→∞ e
Now, since 2 is constant and e x → ∞ as x → ∞, it follows that 2
ex → 0 as x → ∞, which
shows that
x2
lim x 0.
x→∞ e
153
Chapter 2 Computing Derivatives
Activity 2.8.4. Evaluate each of the following limits. If you use L’Hôpital’s Rule, in-
dicate where it was used, and be certain its hypotheses are met before you apply it.
x tan(x)
a. limx→∞ ln(x) d. limx→ π2 − x− π
2
e x +x
b. limx→∞ 2e x +x 2
ln(x)
c. limx→0+ 1
x
e. limx→∞ xe −x
f (x)
To evaluate the limit of a quotient of two functions 1(x) that results in an indeterminate form
∞
of ∞ , in essence we are asking which function is growing faster without bound. We say that
the function 1 dominates the function f as x → ∞ provided that
f (x)
lim 0,
x→∞ 1(x)
f (x) f (x)
whereas f dominates 1 provided that limx→∞ 1(x) ∞. Finally, if the value of limx→∞ 1(x)
is finite and nonzero, we say that f and 1 grow at the same rate. For example, we saw that
2 3x 2 −4x+5
limx→∞ xe x 0, so e x dominates x 2 , while limx→∞ 7x 2 +9x−10 7 , so f (x) 3x − 4x + 5 and
3 2
1(x) 7x 2 + 9x − 10 grow at the same rate.
2.8.3 Summary
• Derivatives can be used to help us evaluate indeterminate limits of the form 00 through
L’Hôpital’s Rule, by replacing the functions in the numerator and denominator with
their tangent line approximations. In particular, if f (a) 1(a) 0 and f and 1 are
differentiable at a, L’Hôpital’s Rule tells us that
f (x) f ′(x)
lim lim ′ .
x→a 1(x) x→a 1 (x)
• When we write x → ∞, this means that x is increasing without bound. Thus, lim f (x)
x→∞
L means that we can make f (x) as close to L as we like by choosing x to be sufficiently
large. Similarly, limx→a f (x) ∞, means that we can make f (x) as large as we like by
choosing x sufficiently close to a.
• A version of L’Hôpital’s Rule also helps us evaluate indeterminate limits of the form
∞
∞ . If f and 1 are differentiable and both approach zero or both approach ±∞ as x → a
(where a is allowed to be ∞), then
f (x) f ′(x)
lim lim ′ .
x→a 1(x) x→a 1 (x)
154
2.8 Using Derivatives to Evaluate Limits
2.8.4 Exercises
f (x)
1. L’Hôpital’s Rule with graphs. For the figures below, determine if lim is positive,
x→a 1(x)
negative, zero, or undefined when f (x) is shown as the blue curve and 1(x) as the black
curve.
ln(x/4)
2. L’Hôpital’s Rule to evaluate a limit. Find the limit: lim .
x→4 x 2 − 16
3. Determining if L’Hôpital’s Rule applies. Compute the following limits using
l’Hôpital’s rule if appropriate.
1 − cos(7x) 4x − 3x − 1
lim lim
x→0 1 − cos(3x) x→1 x2 − 1
4. Using L’Hôpital’s Rule multiple times. Evaluate the limit using L’Hopital’s rule.
15x 3
lim
x→∞ e 2x
5. Let f and 1 be differentiable functions about which the following information is known:
f (3) 1(3) 0, f ′(3) 1 ′(3) 0, f ′′(3) −2, and 1 ′′(3) 1. Let a new function h be
f (x)
given by the rule h(x) 1(x) . On the same set of axes, sketch possible graphs of f and
1 near x 3, and use the provided information to determine the value of
lim h(x).
x→3
Provide explanation to support your conclusion.
6. Find all vertical and horizontal asymptotes of the function
3(x − a)(x − b)
R(x) ,
5(x − a)(x − c)
where a, b, and c are distinct, arbitrary constants. In addition, state all values of x for
which R is not continuous. Sketch a possible graph of R, clearly labeling the values of
155
Chapter 2 Computing Derivatives
a, b, and c.
7. Consider the function 1(x) x 2x , which is defined for all x > 0. Observe that lim+ 1(x)
x→0
is indeterminate due to its form of 00 . (Think about how we know that 0k 0 for all
k > 0, while b 0 1 for all b , 0, but that neither rule can apply to 00 .)
a. Let h(x) ln(1(x)). Explain why h(x) 2x ln(x).
2 ln(x)
b. Next, explain why it is equivalent to write h(x) 1 .
x
c. Use L’Hôpital’s Rule and your work in (b) to compute limx→0+ h(x).
d. Based on the value of limx→0+ h(x), determine limx→0+ 1(x).
8. Recall we say that function 1 dominates function f provided that limx→∞ f (x) ∞,
f (x)
limx→∞ 1(x) ∞, and limx→∞ 1(x) 0.
√
a. Which function dominates the other: ln(x) or x?
√
b. Which function dominates the other: ln(x) or n x? (n can be any positive integer)
c. Explain why e x will dominate any polynomial function.
d. Explain why x n will dominate ln(x) for any positive integer n.
e. Give any example of two nonlinear functions such that neither dominates the
other.
156
CHAPTER 3
Using Derivatives
3.1 Using derivatives to identify extreme values
Motivating Questions
• What are the critical numbers of a function f and how are they connected to identi-
fying the most extreme values the function achieves?
• How does the first derivative of a function reveal important information about the
behavior of the function, including the function’s extreme values?
• How can the second derivative of a function be used to help identify extreme values
of the function?
In many different settings, we are interested in knowing where a function achieves its least
and greatest values. These can be important in applications — say to identify a point at
which maximum profit or minimum cost occurs — or in theory to characterize the behavior
of a function or a family of related functions.
Consider the simple and familiar example of a parabolic function such as s(t) −16t 2 +32t +
48 (shown at left in Figure 3.1.2) that represents the height of an object tossed vertically: its
maximum value occurs at the vertex of the parabola and represents the greatest height the
object reaches. This maximum value is an especially important point on the graph, the point
at which the curve changes from increasing to decreasing.
Definition 3.1.1 Given a function f , we say that f (c) is a global or absolute maximum of f
provided that f (c) ≥ f (x) for all x in the domain of f , and similarly we call f (c) a global or
absolute minimum of f whenever f (c) ≤ f (x) for all x in the domain of f .
For instance, in Figure 3.1.2, 1 has a global maximum of 1(c), but 1 does not appear to have
a global minimum, as the graph of 1 seems to decrease without bound. Note that the point
(c, 1(c)) marks a fundamental change in the behavior of 1, where 1 changes from increasing
to decreasing; similar things happen at both (a, 1(a)) and (b, 1(b)), although these points are
not global minima or maxima.
Definition 3.1.3 We say that f (c) is a local maximum or relative maximum of f provided
that f (c) ≥ f (x) for all x near c, and f (c) is called a local or relative minimum of f whenever
Chapter 3 Using Derivatives
V y = s(t) (c, g(c))
40
30 (a, g(a))
20
(b, g(b))
10
y = g(x)
1 2
Figure 3.1.2: At left, s(t) −16t 2 + 24t + 32 whose vertex is ( 34 , 41); at right, a function 1
that demonstrates several high and low points.
f (c) ≤ f (x) for all x near c.
For example, in Figure 3.1.2, 1 has a relative minimum of 1(b) at the point (b, 1(b)) and a
relative maximum of 1(a) at (a, 1(a)). We have already identified the global maximum of 1
as 1(c); it can also be considered a relative maximum. Any maximum or minimum may also
be called an extreme value of f .
We would like to use calculus ideas to identify and classify key function behavior, including
the location of relative extremes. Of course, if we are given a graph of a function, it is often
straightforward to locate these important behaviors visually.
Preview Activity 3.1.1. Consider the function h given by the graph in Figure 3.1.4.
Use the graph to answer each of the following questions.
a. Identify all of the values of c such that −3 < c < 3 for which h(c) is a local
maximum of h.
b. Identify all of the values of c such that −3 < c < 3 for which h(c) is a local
minimum of h.
c. Does h have a global maximum on the interval [−3, 3]? If so, what is the value
of this global maximum?
d. Does h have a global minimum on the interval [−3, 3]? If so, what is its value?
e. Identify all values of c for which h ′(c) 0.
f. Identify all values of c for which h ′(c) does not exist.
g. True or false: every relative maximum and minimum of h occurs at a point
where h ′(c) is either zero or does not exist.
158
3.1 Using derivatives to identify extreme values
h. True or false: at every point where h ′(c) is zero or does not exist, h has a relative
maximum or minimum.
y = h(x)
2
1
-2 -1 1 2
-1
-2
Figure 3.1.4: The graph of a function h on the interval [−3, 3].
3.1.1 Critical numbers and the first derivative test
If a continuous function has a relative maximum at c, then it is both necessary and sufficient
that the function change from being increasing just before c to decreasing just after c. A
continuous function has a relative minimum at c if and only if the function changes from
decreasing to increasing at c. (See Figure 3.1.6.) There are only two possible ways for these
changes in behavior to occur: either f ′(c) 0 or f ′(c) is undefined. Because these values of
c are so important, we call them critical numbers.
Definition 3.1.5 We say that a function f has a critical number at x c provided that c is in
the domain of f , and f ′(c) 0 or f ′(c) is undefined.
Critical numbers are the only possible locations where the function f may have relative
extremes. Note that not every critical number produces a maximum or minimum; in the
middle graph of Figure 3.1.6, the function pictured there has a horizontal tangent line at the
noted point, but the function is increasing before and increasing after, so the critical number
does not yield a maximum or minimum.
When c is a critical number, we say that (c, f (c)) is a critical point of the function, or that f (c)
is a critical value . The first derivative test summarizes how sign changes in the first derivative
(which can only occur at critical numbers) indicate the presence of a local maximum or
minimum for a given function.
159
Chapter 3 Using Derivatives
Figure 3.1.6: From left to right, a function with a relative maximum where its derivative is
zero; a function with a relative maximum where its derivative is undefined; a function
with neither a maximum nor a minimum at a point where its derivative is zero; a function
with a relative minimum where its derivative is zero; and a function with a relative
minimum where its derivative is undefined.
First Derivative Test.
If p is a critical number of a continuous function f that is differentiable near p (except
possibly at x p), then f has a relative maximum at p if and only¹ if f ′ changes sign
from positive to negative at p, and f has a relative minimum at p if and only if f ′
changes sign from negative to positive at p.
Example 3.1.7 Let f be a function whose derivative is given by the formula f ′(x) e −2x (3 −
x)(x + 1)2 . Determine all critical numbers of f and decide whether a relative maximum,
relative minimum, or neither occurs at each.
Solution. Since we already have f ′(x) written in factored form, it is straightforward to find
the critical numbers of f . Because f ′(x) is defined for all values of x, we need only determine
where f ′(x) 0. From the equation
e −2x (3 − x)(x + 1)2 0
and the zero product property, it follows that x 3 and x −1 are critical numbers of f .
(There is no value of x that makes e −2x 0.)
Next, to apply the first derivative test, we’d like to know the sign of f ′(x) at inputs near the
critical numbers. Because the critical numbers are the only locations at which f ′ can change
sign, it follows that the sign of the derivative is the same on each of the intervals created
by the critical numbers: for instance, the sign of f ′ must be the same for every x < −1. We
create a first derivative sign chart to summarize the sign of f ′ on the relevant intervals, along
with the corresponding behavior of f .
To produce the first derivative sign chart in Figure 3.1.8 we identify the sign of each factor of
f ′(x) at one selected point in each interval. For instance, for x < −1, we could determine the
sign of e −2x , (3 − x), and (x + 1)2 at the value x −2. We note that both e −2x and (x + 1)2 are
positive regardless of the value of x, while (3 − x) is also positive at x −2. Hence, each of
the three terms in f ′ is positive, which we indicate by writing “+ + +.” Taking the product
of three positive terms results in a positive value for f ′, which we denote by the “+” in the
¹Technically, we also have to assume that f is not piecewise constant on any intervals. This is because every
point on a horizontal line is a relative maximum (and relative minimum) despite the fact that the derivative doesn’t
change sign at any point along the horizontal line.
160
3.1 Using derivatives to identify extreme values
interval to the left of x −1. And, since f ′ is positive on that interval, we know that f is
increasing, so we write “INC” to represent the behavior of f . In a similar way, we find that
f ′ is positive and f is increasing on −1 < x < 3, and f ′ is negative and f is decreasing for
x > 3.
f ′ (x) = e−2x (3 − x)(x + 1)2
+++ +++ +−+
sign( f ′ ) + + −
behav( f ) INC −1 INC 3 DEC
Figure 3.1.8: The first derivative sign chart for a function f whose derivative is given by the
formula f ′(x) e −2x (3 − x)(x + 1)2 .
Now we look for critical numbers at which f ′ changes sign. In this example, f ′ changes sign
only at x 3, from positive to negative, so f has a relative maximum at x 3. Although
f has a critical number at x −1, since f is increasing both before and after x −1, f has
neither a minimum nor a maximum at x −1.
Activity 3.1.2. Suppose that 1(x) is a function continuous for every value of x , 2
(x+4)(x−1)2
whose first derivative is 1 ′(x) x−2 . Further, assume that it is known that 1 has
a vertical asymptote at x 2.
a. Determine all critical numbers of 1.
b. By developing a carefully labeled first derivative sign chart, decide whether 1
has as a local maximum, local minimum, or neither at each critical number.
c. Does 1 have a global maximum? global minimum? Justify your claims.
d. What is the value of limx→∞ 1 ′(x)? What does the value of this limit tell you
about the long-term behavior of 1?
e. Sketch a possible graph of y 1(x).
3.1.2 The second derivative test
Recall that the second derivative of a function tells us several important things about the
behavior of the function itself. For instance, if f ′′ is positive on an interval, then we know
that f ′ is increasing on that interval and, consequently, that f is concave up, so throughout
that interval the tangent line to y f (x) lies below the curve at every point. At a point where
f ′(p) 0, the sign of the second derivative determines whether f has a local minimum or
161
Chapter 3 Using Derivatives
local maximum at the critical number p.
In Figure 3.1.9, we see the four possibilities for a function f that has a critical number p at
which f ′(p) 0, provided f ′′(p) is not zero on an interval including p (except possibly at p).
On either side of the critical number, f ′′ can be either positive or negative, and hence f can
be either concave up or concave down. In the first two graphs, f does not change concavity
at p, and in those situations, f has either a local minimum or local maximum. In particular,
if f ′(p) 0 and f ′′(p) < 0, then f is concave down at p with a horizontal tangent line, so
f has a local maximum there. This fact, along with the corresponding statement for when
f ′′(p) is positive, is the substance of the second derivative test.
Figure 3.1.9: Four possible graphs of a function f with a horizontal tangent line at a critical
point.
Second Derivative Test.
If p is a critical number of a continuous function f such that f ′(p) 0 and f ′′(p) , 0,
then f has a relative maximum at p if and only if f ′′(p) < 0, and f has a relative
minimum at p if and only if f ′′(p) > 0.
In the event that f ′′(p) 0, the second derivative test is inconclusive. That is, the test doesn’t
provide us any information. This is because if f ′′(p) 0, it is possible that f has a local
minimum, local maximum, or neither.²
Just as a first derivative sign chart reveals all of the increasing and decreasing behavior of
a function, we can construct a second derivative sign chart that demonstrates all of the im-
portant information involving concavity.
Example 3.1.10 Let f (x) be a function whose first derivative is f ′(x) 3x 4 − 9x 2 . Construct
both first and second derivative sign charts for f , fully discuss where f is increasing and de-
creasing and concave up and concave down, identify all relative extreme values, and sketch
a possible graph of f .
Solution. Since we know f ′(x) 3x 4 −9x 2 , we can find the critical numbers of f by solving
3x 4 − 9x 2 0. Factoring, we observe that
√ √
0 3x 2 (x 2 − 3) 3x 2 (x + 3)(x − 3),
√
so that x 0, ± 3 are the three critical numbers of f . The first derivative sign chart for f is
²Consider the functions f (x) x 4 , 1(x) −x 4 , and h(x) x 3 at the critical point p 0.
162
3.1 Using derivatives to identify extreme values
given in Figure 3.1.11.
√ √
f ′ (x) = 3x2 (x + 3)(x − 3)
+−− ++− ++− +++
sign( f ′ ) + − − +
behav( f ) INC √
− 3 DEC 0 DEC √
3 INC
Figure 3.1.11: The first derivative sign chart for f when f ′(x) 3x 4 − 9x 2 3x 2 (x 2 − 3).
√ √
We see that f is increasing on the intervals (−∞, − 3) and ( 3, ∞), and f is decreasing on
√ √
(− 3, 0) and (0, 3). By the first derivative test, this information tells us that f has a local
√ √
maximum at x − 3 and a local minimum at x 3. Although f also has a critical
number at x 0, neither a maximum nor minimum occurs there since f ′ does not change
sign at x 0.
Next, we move on to investigate concavity. Differentiating f ′(x) 3x 4 − 9x 2 , we see that
f ′′(x) 12x 3 − 18x. Since we are interested in knowing the intervals on which f ′′ is positive
and negative, we first find where f ′′(x) 0. Observe that
( ) ( √ )( √ )
3 3 3
0 12x 3 − 18x 12x x 2 − 12x x + x− .
2 2 2
√
This equation has solutions x 0, ± 32 . Building a sign chart for f ′′ in the exact same way
we do for f ′, we see the result shown in Figure 3.1.12.
q q
f ′′ (x) = 12x x + 32 x − 32
−−− −+− ++− +++
sign( f ′′ ) − + − +
behav( f ) CCD q CCU 0 CCD q 3 CCU
− 32 2
Figure 3.1.12: The second derivative
( ) sign chart for f when
f ′′(x) 12x 3 − 18x 12x 2 x 2 − 32 .
163
Chapter 3 Using Derivatives
√ √
Therefore, f is concave down on the intervals (−∞, − 2)
3
and (0, 2 ),
3
and concave up on
√ √
(− 2 , 0)
3
and ( 2 , ∞).
3
Putting all of this information together, we now see a complete and accurate possible graph
of f in Figure 3.1.13.
A
B
f
C
D
E
√ √ √ √
− 3 − 1.5 1.5 3
Figure 3.1.13: A possible graph of the function f in Example 3.1.10.
√ √
The point A (− 3, f (− 3)) is a local maximum, because f is increasing prior to A and
√ √
decreasing after; similarly, the point E ( 3, f ( 3) is a local minimum. Note, too, that f is
concave down at A and concave up at B, which is consistent both with our second derivative
sign chart and the second derivative test. At points B and D, concavity changes, as we saw
in the results of the second derivative sign chart in Figure 3.1.12. Finally, at point C, f has a
critical point with a horizontal tangent line, but neither a maximum nor a minimum occurs
there, since f is decreasing both before and after C. It is also the case that concavity changes
at C.
While we completely understand where f is increasing and decreasing, where f is concave
up and concave down, and where f has relative extremes, we do not know any specific
information about the y-coordinates√of points on the curve. For instance, while we know
that f has a local maximum
√ at x − 3, we don’t know the value of that maximum because
we do not know f (− 3). Any vertical translation of our sketch of f in Figure 3.1.13 would
satisfy the given criteria for f .
Points B, C, and D in Figure 3.1.13 are locations at which the concavity of f changes. We
give a special name to any such point.
Definition 3.1.14 If p is a value in the domain of a continuous function f at which f changes
concavity, then we say that (p, f (p)) is an inflection point (or point of inflection) of f .
Just as we look for locations where f changes from increasing to decreasing at points where
164
3.1 Using derivatives to identify extreme values
f ′(p) 0 or f ′(p) is undefined, so too we find where f ′′(p) 0 or f ′′(p) is undefined to see
if there are points of inflection at these locations.
At this point in our study, it is important to remind ourselves of the big picture that de-
rivatives help to paint: the sign of the first derivative f ′ tells us whether the function f is
increasing or decreasing, while the sign of the second derivative f ′′ tells us how the function
f is increasing or decreasing.
Activity 3.1.3. Suppose that 1 is a function whose second derivative, 1 ′′, is given by
the graph in Figure 3.1.15.
g′′ 2
1
1 2
Figure 3.1.15: The graph of y 1 ′′(x).
a. Find the x-coordinates of all points of inflection of 1.
b. Fully describe the concavity of 1 by making an appropriate sign chart.
c. Suppose you are given that 1 ′(−1.67857351) 0. Is there is a local maximum,
local minimum, or neither (for the function 1) at this critical number of 1, or is
it impossible to say? Why?
d. Assuming that 1 ′′(x) is a polynomial (and that all important behavior of 1 ′′ is
seen in the graph above), what degree polynomial do you think 1(x) is? Why?
As we will see in more detail in the following section, derivatives also help us to understand
families of functions that differ only by changing one or more parameters. For instance,
we might be interested in understanding the behavior of all functions of the form f (x)
a(x − h)2 + k where a, h, and k are parameters. Each parameter has considerable impact on
how the graph appears.
165
Chapter 3 Using Derivatives
Activity 3.1.4. Consider the family of functions given by h(x) x 2 + cos(kx), where k
is an arbitrary positive real number.
a. Use a graphing utility to sketch the graph of h for several different k-values,
including k 1, 3, 5, 10. Plot h(x) x 2 + cos(3x) on the axes provided. What
is the smallest value of k at which you think you can see (just by looking at the
graph) at least one inflection point on the graph of h?
12
8
4
-2 2
Figure 3.1.16: Axes for plotting y h(x).
√
b. Explain why the graph of h has no inflection points if k ≤ 2, but infinitely
√
many inflection points if k > 2.
c. Explain why, no matter the value of k, h can only have finitely many critical
numbers.
3.1.3 Summary
• The critical numbers of a continuous function f are the values of p for which f ′(p) 0
or f ′(p) does not exist. These values are important because they identify horizontal
tangent lines or corner points on the graph, which are the only possible locations at
which a local maximum or local minimum can occur.
• Given a differentiable function f , whenever f ′ is positive, f is increasing; whenever
f ′ is negative, f is decreasing. The first derivative test tells us that at any point where
f changes from increasing to decreasing, f has a local maximum, while conversely at
any point where f changes from decreasing to increasing f has a local minimum.
• Given a twice differentiable function f , if we have a horizontal tangent line at x p
and f ′′(p) is nonzero, the sign of f ′′ tells us the concavity of f and hence whether f has
a maximum or minimum at x p. In particular, if f ′(p) 0 and f ′′(p) < 0, then f is
concave down at p and f has a local maximum there, while if f ′(p) 0 and f ′′(p) > 0,
166
3.1 Using derivatives to identify extreme values
then f has a local minimum at p. If f ′(p) 0 and f ′′(p) 0, then the second derivative
does not tell us whether f has a local extreme at p or not.
3.1.4 Exercises
1. Finding critical points and inflection points. Use a graph below of f (x) ln(2x 2 + 1)
to estimate the x-values of any critical points and inflection points of f (x). Next, use
derivatives to find the x-values of any critical points and inflection points exactly.
2. Finding inflection points. Find the inflection points of f (x) 4x 4 + 55x 3 − 21x 2 + 3.
3. Matching graphs of f , f ′ , f ′′. The following shows graphs of three functions, A (in
black), B (in blue), and C (in green). If these are the graphs of three functions f , f ′, and
f ′′, identify which is which.
4. This problem concerns a function about which the following information is known:
• f is a differentiable function defined at every real number x
• f (0) −1/2
• y f ′(x) has its graph given at center in Figure 3.1.17
167
Chapter 3 Using Derivatives
f′
2
x x x
1 1 1
Figure 3.1.17: At center, a graph of y f ′(x); at left, axes for plotting y f (x); at
right, axes for plotting y f ′′(x).
a. Construct a first derivative sign chart for f . Clearly identify all critical numbers
of f , where f is increasing and decreasing, and where f has local extrema.
b. On the right-hand axes, sketch an approximate graph of y f ′′(x).
c. Construct a second derivative sign chart for f . Clearly identify where f is concave
up and concave down, as well as all inflection points.
d. On the left-hand axes, sketch a possible graph of y f (x).
5. Suppose that 1 is a differentiable function and 1 ′(2) 0. In addition, suppose that on
1 < x < 2 and 2 < x < 3 it is known that 1 ′(x) is positive.
a. Does 1 have a local maximum, local minimum, or neither at x 2? Why?
b. Suppose that 1 ′′(x) exists for every x such that 1 < x < 3. Reasoning graphically,
describe the behavior of 1 ′′(x) for x-values near 2.
c. Besides being a critical number of 1, what is special about the value x 2 in terms
of the behavior of the graph of 1?
168
3.1 Using derivatives to identify extreme values
6. Suppose that h is a differentiable function whose first derivative is given by the graph
in Figure 3.1.18.
a. How many real number solutions
can the equation h(x) 0 have? h′
Why?
b. If h(x) 0 has two distinct real so-
lutions, what can you say about the
signs of the two solutions? Why?
c. Assume that limx→∞ h ′(x) 3,
as appears to be indicated in Fig-
ure 3.1.18. How will the graph of
y h(x) appear as x → ∞? Why?
d. Describe the concavity of y h(x)
as fully as you can from the pro-
vided information.
Figure 3.1.18: The graph of y h ′(x).
7. Let p be a function whose second derivative is p ′′(x) (x + 1)(x − 2)e −x .
a. Construct a second derivative sign chart for p and determine all inflection points
of p.
√
5−1
b. Suppose you also know that x 2 is a critical number of p. Does p have a
√
5−1
local minimum, local maximum, or neither at x 2 ? Why?
c. If the point (2, 12
e2
) lies on the graph of y p(x) and p ′(2) − e52 , find the equation
of the tangent line to y p(x) at the point where x 2. Does the tangent line lie
above the curve, below the curve, or neither at this value? Why?
169
Chapter 3 Using Derivatives
3.2 Using derivatives to describe families of functions
Motivating Questions
• Given a family of functions that depends on one or more parameters, how does the
shape of the graph of a typical function in the family depend on the value of the
parameters?
• How can we construct first and second derivative sign charts of functions that depend
on one or more parameters while allowing those parameters to remain arbitrary con-
stants?
Mathematicians are often interested in making general observations, say by describing pat-
terns that hold in a large number of cases. Think about the Pythagorean Theorem: it doesn’t
tell us something about a single right triangle, but rather a fact about every right triangle. In
the next part of our studies, we use calculus to make general observations about families of
functions that depend on one or more parameters. People who use applied mathematics,
such as engineers and economists, often encounter the same types of functions where only
small changes to certain constants occur. These constants are called parameters.
d +a
d
f (t) = a sin(b(t − c)) + d
c c + 2π
b
Figure 3.2.1: The graph of f (t) a sin(b(t − c)) + d based on parameters a, b, c, and d.
You are already familiar with certain families of functions. For example, f (t) a sin(b(t −
c)) + d is a stretched and shifted version of the sine function with amplitude a, period 2π b ,
phase shift c, and vertical shift d. We know that a affects the size of the oscillation, b the
rapidity of oscillation, and c where the oscillation starts, as shown in Figure 3.2.1, while d
affects the vertical positioning of the graph.
As another example, every function of the form y mx + b is a line with slope m and y-
intercept (0, b). The value of m affects the line’s steepness, and the value of b situates the line
170
3.2 Using derivatives to describe families of functions
vertically on the coordinate axes. These two parameters describe all possible non-vertical
lines.
For other less familiar families of functions, we can use calculus to discover where key behav-
ior occurs: where members of the family are increasing or decreasing, concave up or concave
down, where relative extremes occur, and more, all in terms of the parameters involved. To
get started, we revisit a common collection of functions to see how calculus confirms things
we already know.
Preview Activity 3.2.1. Let a, h, and k be arbitrary real numbers with a , 0, and let f
be the function given by the rule f (x) a(x − h)2 + k.
a. What familiar type of function is f ? What information do you know about f
just by looking at its form? (Think about the roles of a, h, and k.)
b. Next we use some calculus to develop familiar ideas from a different perspec-
tive. To start, treat a, h, and k as constants and compute f ′(x).
c. Find all critical numbers of f . (These will depend on at least one of a, h, and k.)
d. Assume that a < 0. Construct a first derivative sign chart for f .
e. Based on the information you’ve found above, classify the critical values of f as
maxima or minima.
3.2.1 Describing families of functions in terms of parameters
Our goal is to describe the key characteristics of the overall behavior of each member of a
family of functions in terms of its parameters. By finding the first and second derivatives
and constructing sign charts (each of which may depend on one or more of the parameters),
we can often make broad conclusions about how each member of the family will appear.
Example 3.2.2 Consider the two-parameter family of functions given by 1(x) axe −bx ,
where a and b are positive real numbers. Fully describe the behavior of a typical mem-
ber of the family in terms of a and b, including the location of all critical numbers, where 1
is increasing, decreasing, concave up, and concave down, and the long term behavior of 1.
Solution. We begin by computing 1 ′(x). By the product rule,
d [ −bx ] d
1 ′(x) ax e + e −bx [ax].
dx dx
By applying the chain rule and constant multiple rule, we find that
1 ′(x) axe −bx (−b) + e −bx (a).
To find the critical numbers of 1, we solve the equation 1 ′(x) 0. By factoring 1 ′(x), we find
0 ae −bx (−bx + 1).
171
Chapter 3 Using Derivatives
Since we are given that a , 0 and we know that e −bx , 0 for all values of x, the only way this
equation can hold is when −bx + 1 0. Solving for x, we find x 1b , and this is therefore
the only critical number of 1.
We construct the first derivative sign chart for 1 that is shown in Figure 3.2.3.
g′ (x) = ae−bx (1 − bx)
++ +−
sign(g′ ) + −
behav(g) INC DEC
1
b
Figure 3.2.3: The first derivative sign chart for 1(x) axe −bx .
Because the factor ae −bx is always positive, the sign of 1 ′ depends on the linear factor (1−bx),
which is positive for x < 1b and negative for x > 1b . Hence we can not only conclude that 1 is
always increasing for x < 1b and decreasing for x > 1b , but also that 1 has a global maximum
at ( 1b , 1( 1b )) and no local minimum.
We turn next to analyzing the concavity of 1. With 1 ′(x) −abxe −bx + ae −bx , we differentiate
to find that
1 ′′(x) −abxe −bx (−b) + e −bx (−ab) + ae −bx (−b).
Combining like terms and factoring, we now have
1 ′′(x) ab 2 xe −bx − 2abe −bx abe −bx (bx − 2).
We observe that abe −bx is always positive, and thus the sign of 1 ′′ depends on the sign of
(bx − 2), which is zero when x 2b . Since b is positive, the value of (bx − 2) is negative for
x < 2b and positive for x > 2b . The sign chart for 1 ′′ is shown in Figure 3.2.4. Thus, 1 is
concave down for all x < 2b and concave up for all x > 2b .
Finally, we analyze the long term behavior of 1 by considering two limits. First, we note that
ax
lim 1(x) lim axe −bx lim .
x→∞ x→∞ x→∞ e bx
This limit has indeterminate form ∞
∞ , so we apply L’Hôpital’s Rule and find that lim 1(x) 0.
x→∞
In the other direction,
lim 1(x) lim axe −bx −∞,
x→−∞ x→−∞
because ax → −∞ and e −bx → ∞ as x → −∞. Hence, as we move left on its graph, 1
decreases without bound, while as we move to the right, 1(x) → 0.
172
3.2 Using derivatives to describe families of functions
g′′ (x) = abe−bx (bx − 2)
+− ++
sign(g′′ ) − +
behav(g) CCD CCU
2
b
Figure 3.2.4: The second derivative sign chart for 1(x) axe −bx .
All of this information now helps us produce the graph of a typical member of this family
of functions without using a graphing utility (and without choosing particular values for a
and b), as shown in Figure 3.2.5.
global max
a −1
be inflection pt
g(x) = axe−bx
1 2
b b
Figure 3.2.5: The graph of 1(x) axe −bx .
Note that the value of b controls the horizontal location of the global maximum and the
inflection point, as neither depends on a. The value of a affects the vertical stretch of the
graph. For example, the global maximum occurs at the point ( 1b , 1( 1b )) ( 1b , ba e −1 ), so the
larger the value of a, the greater the value of the global maximum.
The work we’ve completed in Example 3.2.2 can often be replicated for other families of
functions that depend on parameters. Normally we are most interested in determining all
critical numbers, a first derivative sign chart, a second derivative sign chart, and the limit
of the function as x → ∞. Throughout, we prefer to work with the parameters as arbitrary
constants. In addition, we can experiment with some particular values of the parameters
present to reduce the algebraic complexity of our work. The following activities offer several
key examples where we see that the values of the parameters substantially affect the behavior
of individual functions within a given family.
173
Chapter 3 Using Derivatives
Activity 3.2.2. Consider the family of functions defined by p(x) x 3 − ax, where a , 0
is an arbitrary constant.
a. Find p ′(x) and determine the critical numbers of p. How many critical numbers
does p have?
b. Construct a first derivative sign chart for p. What can you say about the over-
all behavior of p if the constant a is positive? Why? What if the constant a is
negative? In each case, describe the relative extremes of p.
c. Find p ′′(x) and construct a second derivative sign chart for p. What does this
tell you about the concavity of p? What role does a play in determining the
concavity of p?
d. Without using a graphing utility, sketch and label typical graphs of p(x) for the
cases where a > 0 and a < 0. Label all inflection points and local extrema.
e. Finally, use a graphing utility to test your observations above by entering and
plotting the function p(x) x 3 − ax for at least four different values of a. Write
several sentences to describe your overall conclusions about how the behavior
of p depends on a.
Activity 3.2.3. Consider the two-parameter family of functions of the form h(x)
a(1 − e −bx ), where a and b are positive real numbers.
a. Find the first derivative and the critical numbers of h. Use these to construct a
first derivative sign chart and determine for which values of x the function h is
increasing and decreasing.
b. Find the second derivative and build a second derivative sign chart. For which
values of x is a function in this family concave up? concave down?
c. What is the value of limx→∞ a(1 − e −bx )? limx→−∞ a(1 − e −bx )?
d. How does changing the value of b affect the shape of the curve?
e. Without using a graphing utility, sketch the graph of a typical member of this
family. Write several sentences to describe the overall behavior of a typical func-
tion h and how this behavior depends on a and b.
Activity 3.2.4. Let L(t) A
1+ce −kt
, where A, c, and k are all positive real numbers.
a. Observe that we can equivalently write L(t) A(1 + ce −kt )−1 . Find L′(t) and
explain why L has no critical numbers. Is L always increasing or always de-
creasing? Why?
b. Given the fact that
ce −kt − 1
L′′(t) Ack 2 e −kt ,
(1 + ce −kt )3
174
3.2 Using derivatives to describe families of functions
find all values of t such that L′′(t) 0 and hence construct a second derivative
sign chart. For which values of t is a function in this family concave up? concave
down?
A A
c. What is the value of limt→∞ 1+ce −kt
? limt→−∞ 1+ce −kt
?
d. Find the value of L(x) at the inflection point found in (b).
e. Without using a graphing utility, sketch the graph of a typical member of this
family. Write several sentences to describe the overall behavior of a typical func-
tion L and how this behavior depends on A, c, and k number.
f. Explain why it is reasonable to think that the function L(t) models the growth
of a population over time in a setting where the largest possible population the
surrounding environment can support is A.
3.2.2 Summary
• Given a family of functions that depends on one or more parameters, by investigating
how critical numbers and locations where the second derivative is zero depend on the
values of these parameters, we can often accurately describe the shape of the function
in terms of the parameters.
• In particular, just as we can created first and second derivative sign charts for a single
function, we often can do so for entire families of functions where critical numbers
and possible inflection points depend on arbitrary constants. These sign charts then
reveal where members of the family are increasing or decreasing, concave up or con-
cave down, and help us to identify relative extremes and inflection points.
3.2.3 Exercises
1. Drug dosage with a parameter. For some positive constant C, a patient’s tempera-
( )
ture change, T, due to a dose, D, of a drug is given by T C2 − D3 D 2 . What dosage
maximizes the temperature change?
The sensitivity of the body to the drug is defined as dT/dD. What dosage maximizes
sensitivity?
2. Using the graph of 1 ′. The figure below gives the behavior of the derivative of 1(x)
on −2 ≤ x ≤ 2. Sketch a graph of 1(x) and use your sketch to answer the following
questions.
A. Where does the graph of 1(x) have inflection points?
B. Where are the global maxima and minima of 1 on [−2, 2]?
C. If 1(−2) −8, what are possible values for 1(0)? How is the value of 1(2) related to
the value of 1(0)?
175
Chapter 3 Using Derivatives
Graph of 1 ′(x) (not 1(x))
3. Consider the one-parameter family of functions given by p(x) x 3 − ax 2 , where a > 0.
a. Sketch a plot of a typical member of the family, using the fact that each is a cubic
polynomial with a repeated zero at x 0 and another zero at x a.
b. Find all critical numbers of p.
c. Compute p ′′ and find all values for which p ′′(x) 0. Hence construct a second
derivative sign chart for p.
d. Describe how the location of the critical numbers and the inflection point of p
change as a changes. That is, if the value of a is increased, what happens to the
critical numbers and inflection point?
e −x
4. Let q(x) x−c be a one-parameter family of functions where c > 0.
a. Explain why q has a vertical asymptote at x c.
b. Determine limx→∞ q(x) and limx→−∞ q(x).
c. Compute q ′(x) and find all critical numbers of q.
d. Construct a first derivative sign chart for q and determine whether each critical
number leads to a local minimum, local maximum, or neither for the function q.
e. Sketch a typical member of this family of functions with important behaviors
clearly labeled.
(x−m)2
−
5. Let E(x) e 2s 2 , where m is any real number and s is a positive real number.
a. Compute E′(x) and hence find all critical numbers of E.
b. Construct a first derivative sign chart for E and classify each critical number of
the function as a local minimum, local maximum, or neither.
c. It can be shown that E′′(x) is given by the formula
( )
′′ −
(x−m)2 (x − m)2 − s 2
E (x) e 2s 2 .
s4
Find all values of x for which E′′(x) 0.
176
3.2 Using derivatives to describe families of functions
d. Determine limx→∞ E(x) and limx→−∞ E(x).
e. Construct a labeled graph of a typical function E that clearly shows how impor-
tant points on the graph of y E(x) depend on m and s.
177
Chapter 3 Using Derivatives
3.3 Global Optimization
Motivating Questions
• What are the differences between finding relative extreme values and global extreme
values of a function?
• How is the process of finding the global maximum or minimum of a function over
the function’s entire domain different from determining the global maximum or min-
imum on a restricted domain?
• For a function that is guaranteed to have both a global maximum and global min-
imum on a closed, bounded interval, what are the possible points at which these
extreme values occur?
We have seen that we can use the first derivative of a function to determine where the func-
tion is increasing or decreasing, and the second derivative to know where the function is
concave up or concave down. This information helps us determine the overall shape and
behavior of the graph, as well as whether the function has relative extrema.
Remember the difference between a relative maximum and a global maximum: there is a
relative maximum of f at x p if f (p) ≥ f (x) for all x near p, while there is a global maximum
at p if f (p) ≥ f (x) for all x in the domain of f .
For instance, in Figure 3.3.1, we see a func-
tion f that has a global maximum at x c
and a relative maximum at x a, since f (c) global max
is greater than f (x) for every value of x, while
f (a) is only greater than the value of f (x) for
x near a. Since the function appears to de- relative max
crease without bound, f has no global min-
imum, though clearly f has a relative mini- relative min
mum at x b. f
Our emphasis in this section is on finding the
global extreme values of a function (if they ex-
a b c
ist), either over its entire domain or on some
restricted portion.
Figure 3.3.1: A function f with a global
maximum, but no global minimum.
178
3.3 Global Optimization
Preview Activity 3.3.1. Let f (x) 2 + 3
1+(x+1)2
.
a. Determine all of the critical numbers of f .
b. Construct a first derivative sign chart for f and thus determine all intervals on
which f is increasing or decreasing.
c. Does f have a global maximum? If so, why, and what is its value and where is
the maximum attained? If not, explain why.
d. Determine limx→∞ f (x) and limx→−∞ f (x).
e. Explain why f (x) > 2 for every value of x.
f. Does f have a global minimum? If so, why, and what is its value and where is
the minimum attained? If not, explain why.
3.3.1 Global Optimization
In Figure 3.3.1 and Preview Activity 3.3.1, we were interested in finding the global minimum
and global maximum for f on its entire domain. At other times, we might focus on some
restriction of the domain.
For example, rather than considering f (x) 2+ 1+(x+1)
3
2 for every value of x, perhaps instead
we are only interested in those x for which 0 ≤ x ≤ 4, and we would like to know which
values of x in the interval [0, 4] produce the largest possible and smallest possible values of
f . We are accustomed to critical numbers playing a key role in determining the location of
extreme values of a function; now, by restricting the domain to an interval, it makes sense
that the endpoints of the interval will also be important to consider, as we see in the following
activity. When limiting ourselves to a particular interval, we will often refer to the absolute
maximum or minimum value, rather than the global maximum or minimum.
Activity 3.3.2. Let 1(x) 13 x 3 − 2x + 2.
a. Find all critical numbers of 1 that lie in the interval −2 ≤ x ≤ 3.
b. Use a graphing utility to construct the graph of 1 on the interval −2 ≤ x ≤ 3.
c. From the graph, determine the x-values at which the absolute minimum and
absolute maximum of 1 occur on the interval [−2, 3].
d. How do your answers change if we instead consider the interval −2 ≤ x ≤ 2?
e. What if we instead consider the interval −2 ≤ x ≤ 1?
In Activity 3.3.2, we saw how the absolute maximum and absolute minimum of a function on
a closed, bounded interval [a, b], depend not only on the critical numbers of the function, but
also on the values of a and b. These observations demonstrate several important facts that
hold more generally. First, we state an important result called the Extreme Value Theorem.
179
Chapter 3 Using Derivatives
The Extreme Value Theorem.
If f is a continuous function on a closed interval [a, b], then f attains both an absolute
minimum and absolute maximum on [a, b]. That is, for some value x m such that
a ≤ x m ≤ b, it follows that f (x m ) ≤ f (x) for all x in [a, b]. Similarly, there is a
value x M in [a, b] such that f (x M ) ≥ f (x) for all x in [a, b]. Letting m f (x m ) and
M f (x M ), it follows that m ≤ f (x) ≤ M for all x in [a, b].
The Extreme Value Theorem tells us that on any closed interval [a, b], a continuous function
has to achieve both an absolute minimum and an absolute maximum. The theorem does not
tell us where these extreme values occur, but rather only that they must exist. As we saw
in Activity 3.3.2, the only possible locations for relative extremes are at the endpoints of the
interval or at a critical number.
Note 3.3.2 Thus, we have the following approach to finding the absolute maximum and
minimum of a continuous function f on the interval [a, b]:
• find all critical numbers of f that lie in the interval;
• evaluate the function f at each critical number in the interval and at each endpoint of
the interval;
• from among those function values, the smallest is the absolute minimum of f on the
interval, while the largest is the absolute maximum.
Activity 3.3.3. Find the exact absolute maximum and minimum of each function on
the stated interval.
a. h(x) xe −x , [0, 3]
b. p(t) sin(t) + cos(t), [− π2 , π2 ]
x2
c. q(x) x−2 , [3, 7]
d. f (x) 4 − e −(x−2) , (−∞, ∞)
2
e. h(x) xe −ax , [0, 2a ] (a > 0)
f. f (x) b − e −(x−a) , (−∞, ∞), a, b > 0
2
The interval we choose has nearly the same influence on extreme values as the function un-
der consideration. Consider, for instance, the function pictured in Figure 3.3.3. In sequence,
from left to right, the interval under consideration is changed from [−2, 3] to [−2, 2] to [−2, 1].
• On the interval [−2, 3], there are two critical numbers, with the absolute minimum at
one critical number and the absolute maximum at the right endpoint.
• On the interval [−2, 2], both critical numbers are in the interval, with the absolute
minimum and maximum at the two critical numbers.
• On the interval [−2, 1], just one critical number lies in the interval, with the absolute
maximum at one critical number and the absolute minimum at one endpoint.
180
3.3 Global Optimization
g g g
2 2 2
-2 3 -2 2 -2 1
Figure 3.3.3: A function 1 considered on three different intervals.
Remember to consider only the critical numbers that lie within the interval.
3.3.2 Moving toward applications
We conclude this section with an example of an applied optimization problem. It highlights
the role that a closed, bounded domain can play in finding absolute extrema.
Example 3.3.4 A 20 cm piece of wire is cut into two pieces. One piece is used to form a square
and the other to form an equilateral triangle. How should the wire be cut to maximize the
total area enclosed by the square and triangle? to minimize the area?
Solution. We begin by sketching a picture that illustrates the situation. The variable in the
problem is where we decide to cut the wire. We thus label the cut point at a distance x from
one end of the wire, and note that the remaining portion of the wire then has length 20 − x
As shown in Figure 3.3.5, we see that the x cm of wire that is used to form the equilateral
triangle with three sides of length x3 . For the remaining 20 − x cm of wire, the square that
results will have each side of length 20−x
4 .
x 20 − x
x
3 20−x
4
Figure 3.3.5: A 20 cm piece of wire cut into two pieces, one of which forms an equilateral
triangle, the other which yields a square.
At this point, we note that there are obvious restrictions on x: in particular, 0 ≤ x ≤ 20. In
181
Chapter 3 Using Derivatives
the extreme cases, all of the wire is being used to make just one figure. For instance, if x 0,
then all 20 cm of wire are used to make a square that is 5 × 5.
Now, our overall goal is to find the minimum and maximum areas that can be enclosed.
√
Because the height of an equilateral triangle is 3 times half the length of the base, the area
of the triangle is
√
1 1 x x 3
A∆ bh · · .
2 2 3 6
( 20−x ) 2
The area of the square is A□ s 2 4 . Therefore, the total area function is
√ ( )2
3x 2 20 − x
A(x) + .
36 4
Remember that we are considering this function only on the restricted domain [0, 20].
Differentiating A(x), we have
√ ( )( ) √
′ 3x 20 − x 1 3 1 5
A (x) +2 − x+ x− .
18 4 4 18 8 2
When we set A′(x) 0, we find that x 180
√ ≈ 11.3007 is the only critical number of A in
4 3+9
the interval [0, 20].
Evaluating A at the critical number and endpoints, we see that
( ) √ ( )2
3( 180 2
√ ) 20− 180
√
• A 180
√ 4 3+9
4 + 4 3+9
4 ≈ 10.8741
4 3+9
• A(0) 25
√
3
√
• A(20) 36 (400) 100
9 3 ≈ 19.2450
Thus, the absolute minimum occurs when x ≈ 11.3007 and results in the minimum area of
approximately 10.8741 square centimeters. The absolute maximum occurs when we invest
all of the wire in the square (and none in the triangle), resulting in 25 square centimeters of
area. These results are confirmed by a plot of y A(x) on the interval [0, 20], as shown in
Figure 3.3.6.
182
3.3 Global Optimization
25
20
15 y = A(x)
10
5
5 10 15 20
Figure 3.3.6: A plot of the area function from Example 3.3.4.
Activity 3.3.4. A piece of cardboard that is 10 × 15 (each measured in inches) is being
made into a box without a top. To do so, squares are cut from each corner of the box
and the remaining sides are folded up. If the box needs to be at least 1 inch deep and
no more than 3 inches deep, what is the maximum possible volume of the box? what
is the minimum volume? Justify your answers using calculus.
a. Draw a labeled diagram that shows the given information. What variable should
we introduce to represent the choice we make in creating the box? Label the di-
agram appropriately with the variable, and write a sentence to state what the
variable represents.
b. Determine a formula for the function V (that depends on the variable in (a)) that
tells us the volume of the box.
c. What is the domain of the function V? That is, what values of x make sense for
input? Are there additional restrictions provided in the problem?
d. Determine all critical numbers of the function V.
e. Evaluate V at each of the endpoints of the domain and at any critical numbers
that lie in the domain.
f. What is the maximum possible volume of the box? the minimum?
Example 3.3.4 and Activity 3.3.4 illustrate standard steps that we undertake in almost every
applied optimization problem: we draw a picture to demonstrate the situation, introduce
one or more variables to represent quantities that are changing, find a function that models
the quantity to be optimized, and then decide on an appropriate domain for that function.
Once that is done, we are in the familiar situation of finding the absolute minimum and
maximum of a function over a particular domain, so we apply the calculus ideas that we
have been studying to this point in Chapter 3.
183
Chapter 3 Using Derivatives
3.3.3 Summary
• To find relative extreme values of a function, we use a first derivative sign chart and
classify all of the function’s critical numbers. If instead we are interested in absolute
extreme values, we first decide whether we are considering the entire domain of the
function or a particular interval.
• In the case of finding global extremes over the function’s entire domain, we again use
a first or second derivative sign chart. If we are working to find absolute extremes on
a restricted interval, then we first identify all critical numbers of the function that lie
in the interval.
• For a continuous function on a closed, bounded interval, the only possible points at
which absolute extreme values occur are the critical numbers and the endpoints. Thus,
we simply evaluate the function at each endpoint and each critical number in the in-
terval, and compare the results to decide which is largest (the absolute maximum) and
which is smallest (the absolute minimum).
3.3.4 Exercises
1. Based on the given information about each function, decide whether the function has
global maximum, a global minimum, neither, both, or that it is not possible to say with-
out more information. Assume that each function is twice differentiable and defined
for all real numbers, unless noted otherwise. In each case, write one sentence to explain
your conclusion.
a. f is a function such that f ′′(x) < 0 for every x.
b. 1 is a function with two critical numbers a and b (where a < b), and 1 ′(x) < 0 for
x < a, 1 ′(x) < 0 for a < x < b, and 1 ′(x) > 0 for x > b.
c. h is a function with two critical numbers a and b (where a < b), and h ′(x) < 0
for x < a, h ′(x) > 0 for a < x < b, and h ′(x) < 0 for x > b. In addition,
limx→∞ h(x) 0 and limx→−∞ h(x) 0.
d. p is a function differentiable everywhere except at x a and p ′′(x) > 0 for x < a
and p ′′(x) < 0 for x > a.
2. For each family of functions that depends on one or more parameters, determine the
function’s absolute maximum and absolute minimum on the given interval.
a. p(x) x 3 − a 2 x, [0, a] (a > 0)
b. r(x) axe −bx , [ 2b , b ] (a > 0, b > 1)
1 2
c. w(x) a(1 − e −bx ), [b, 3b] (a, b > 0)
[π ]
d. s(x) sin(kx), 3k , 6k
5π
(k > 0)
184
3.3 Global Optimization
3. For each of the functions described below (each continuous on [a, b]), state the loca-
tion of the function’s absolute maximum and absolute minimum on the interval [a, b],
or say there is not enough information provided to make a conclusion. Assume that
any critical numbers mentioned in the problem statement represent all of the critical
numbers the function has in [a, b]. In each case, write one sentence to explain your
answer.
a. f ′(x) ≤ 0 for all x in [a, b]
b. 1 has a critical number at c such that a < c < b and 1 ′(x) > 0 for x < c and
1 ′(x) < 0 for x > c
c. h(a) h(b) and h ′′(x) < 0 for all x in [a, b]
d. p(a) > 0, p(b) < 0, and for the critical number c such that a < c < b, p ′(x) < 0 for
x < c and p ′(x) > 0 for x > c
4. Let s(t) 3 sin(2(t − π6 )) + 5. Find the exact absolute maximum and minimum of s on
the provided intervals by testing the endpoints and finding and evaluating all relevant
critical numbers of s.
a. [ π6 , 7π
6 ] c. [0, 2π]
b. [0, π2 ] d. [ π3 , 6 ]
5π
185
Chapter 3 Using Derivatives
3.4 Applied Optimization
Motivating Questions
• In a setting where a situation is described for which optimal parameters are sought,
how do we develop a function that models the situation and use calculus to find the
desired maximum or minimum?
Near the conclusion of Section 3.3, we considered two optimization problems where deter-
mining the function to be optimized was part of the problem. In Example 3.3.4, we sought
to use a single piece of wire to build an equilateral triangle and square in order to maximize
the total combined area enclosed. In the subsequent Activity 3.3.4, we investigated how
the volume of a box constructed from a piece of cardboard by removing squares from each
corner and folding up the sides depends on the size of the squares removed.
In neither of these problems was a function to optimize explicitly provided. Rather, we
first tried to understand the problem by drawing a figure and introducing variables, and
then sought to develop a formula for a function that modeled the quantity to be optimized.
Once the function was established, we then considered what domain was appropriate. At
that point, we were finally ready to apply the ideas of calculus to determine the absolute
minimum or maximum.
Throughout what follows in the current section, the primary emphasis is on the reader solv-
ing problems. Initially, some substantial guidance is provided, with the problems progress-
ing to require greater independence as we move along.
Preview Activity 3.4.1. According to U.S. postal regulations, the girth plus the length
of a parcel sent by mail may not exceed 108 inches, where by “girth” we mean the
perimeter of the smallest end. What is the largest possible volume of a rectangular
parcel with a square end that can be sent by mail? What are the dimensions of the
package of largest volume?
Figure 3.4.1: A rectangular parcel with a square end.
a. Let x represent the length of one side of the square end and y the length of
the longer side. Label these quantities appropriately on the image shown in
186
3.4 Applied Optimization
Figure 3.4.1.
b. What is the quantity to be optimized in this problem? Find a formula for this
quantity in terms of x and y.
c. The problem statement tells us that the parcel’s girth plus length may not ex-
ceed 108 inches. In order to maximize volume, we assume that we will actually
need the girth plus length to equal 108 inches. What equation does this produce
involving x and y?
d. Solve the equation you found in (c) for one of x or y (whichever is easier).
e. Now use your work in (b) and (d) to determine a formula for the volume of the
parcel so that this formula is a function of a single variable.
f. Over what domain should we consider this function? Note that both x and y
must be positive; how does the constraint that girth plus length is 108 inches
produce intervals of possible values for x and y?
g. Find the absolute maximum of the volume of the parcel on the domain you
established in (f) and hence also determine the dimensions of the box of greatest
volume. Justify that you’ve found the maximum using calculus.
3.4.1 More applied optimization problems
Many of the steps in Preview Activity 3.4.1 are ones that we will execute in any applied opti-
mization problem. We briefly summarize those here to provide an overview of our approach
in subsequent questions.
Note 3.4.2
• Draw a picture and introduce variables. It is essential to first understand what quan-
tities are allowed to vary in the problem and then to represent those values with vari-
ables. Constructing a figure with the variables labeled is almost always an essential
first step. Sometimes drawing several diagrams can be especially helpful to get a sense
of the situation. A nice example of this can be seen at http://gvsu.edu/s/99, where
the choice of where to bend a piece of wire into the shape of a rectangle determines
both the rectangle’s shape and area.
• Identify the quantity to be optimized as well as any key relationships among the vari-
able quantities. Essentially this step involves writing equations that involve the vari-
ables that have been introduced: one to represent the quantity whose minimum or
maximum is sought, and possibly others that show how multiple variables in the prob-
lem may be interrelated.
• Determine a function of a single variable that models the quantity to be optimized;
this may involve using other relationships among variables to eliminate one or more
variables in the function formula. For example, in Preview Activity 3.4.1, we initially
found that V x 2 y, but then the additional relationship that 4x + y 108 (girth plus
187
Chapter 3 Using Derivatives
length equals 108 inches) allows us to relate x and y and thus observe equivalently that
y 108 − 4x. Substituting for y in the volume equation yields V(x) x 2 (108 − 4x),
and thus we have written the volume as a function of the single variable x.
• Decide the domain on which to consider the function being optimized. Often the phys-
ical constraints of the problem will limit the possible values that the independent vari-
able can take on. Thinking back to the diagram describing the overall situation and
any relationships among variables in the problem often helps identify the smallest
and largest values of the input variable.
• Use calculus to identify the absolute maximum and/or minimum of the quantity be-
ing optimized. This always involves finding the critical numbers of the function first.
Then, depending on the domain, we either construct a first derivative sign chart (for
an open or unbounded interval) or evaluate the function at the endpoints and critical
numbers (for a closed, bounded interval), using ideas we’ve studied so far in Chapter 3.
• Finally, we make certain we have answered the question: does the question seek the
absolute maximum of a quantity, or the values of the variables that produce the max-
imum? That is, finding the absolute maximum volume of a parcel is different from
finding the dimensions of the parcel that produce the maximum.
Activity 3.4.2. A soup can in the shape of a right circular cylinder is to be made from
two materials. The material for the side of the can costs $0.015 per square inch and the
material for the lids costs $0.027 per square inch. Suppose that we desire to construct
a can that has a volume of 16 cubic inches. What dimensions minimize the cost of the
can?
a. Draw a picture of the can and label its dimensions with appropriate variables.
b. Use your variables to determine expressions for the volume, surface area, and
cost of the can.
c. Determine the total cost function as a function of a single variable. What is the
domain on which you should consider this function?
d. Find the absolute minimum cost and the dimensions that produce this value.
Familiarity with common geometric formulas is particularly helpful in problems such as the
one in Activity 3.4.2. Sometimes those involve perimeter, area, volume, or surface area. At
other times, the constraints of a problem introduce right triangles (where the Pythagorean
Theorem applies) or other functions whose formulas provide relationships among the vari-
ables.
Activity 3.4.3. A hiker starting at a point P on a straight road walks east towards point
Q, which is on the road and 3 kilometers from point P.
Two kilometers due north of point Q is a cabin. The hiker will walk down the road
for a while, at a pace of 8 kilometers per hour. At some point Z between P and Q, the
hiker leaves the road and makes a straight line towards the cabin through the woods,
hiking at a pace of 3 kph, as pictured in Figure 3.4.3. In order to minimize the time to
188
3.4 Applied Optimization
go from P to Z to the cabin, where should the hiker turn into the forest?
cabin
2
P 3 Z Q
Figure 3.4.3: A hiker walks from P to Z to the cabin, as pictured.
In more geometric problems, we often use curves or functions to provide natural constraints.
For instance, we could investigate which isosceles triangle that circumscribes a unit circle has
the smallest area, which you can explore for yourself at http://gvsu.edu/s/9b. Or similarly,
for a region bounded by a parabola, we might seek the rectangle of largest area that fits
beneath the curve, as shown at http://gvsu.edu/s/9c. The next activity is similar to the
latter problem.
Activity 3.4.4. Consider the region in the x-y plane that is bounded by the x-axis
and the function f (x) 25 − x 2 . Construct a rectangle whose base lies on the x-axis
and is centered at the origin, and whose sides extend vertically until they intersect the
curve y 25− x 2 . Which such rectangle has the maximum possible area? Which such
rectangle has the greatest perimeter? Which has the greatest combined perimeter and
area? (Challenge: answer the same questions in terms of positive parameters a and b
for the function f (x) b − ax 2 .)
Activity 3.4.5. A trough is being constructed by bending a 4 × 24 (measured in feet)
rectangular piece of sheet metal.
Two symmetric folds 2 feet apart will be made parallel to the longest side of the rec-
tangle so that the trough has cross-sections in the shape of a trapezoid, as pictured
in Figure 3.4.4. At what angle should the folds be made to produce the trough of
maximum volume?
3.4.2 Summary
• While there is no single algorithm that works in every situation where optimization
is used, in most of the problems we consider, the following steps are helpful: draw a
picture and introduce variables; identify the quantity to be optimized and find rela-
tionships among the variables; determine a function of a single variable that models
the quantity to be optimized; decide the domain on which to consider the function
189
Chapter 3 Using Derivatives
1 1
θ
2
Figure 3.4.4: A cross-section of the trough formed by folding to an angle of θ.
being optimized; use calculus to identify the absolute maximum and/or minimum of
the quantity being optimized.
3.4.3 Exercises
1. Maximizing the volume of a box. An open box is to be made out of a 10-inch by 18-
inch piece of cardboard by cutting out squares of equal size from the four corners and
bending up the sides. Find the dimensions of the resulting box that has the largest
volume.
2. Minimizing the cost of a container. A rectangular storage container with an open top
is to have a volume of 26 cubic meters. The length of its base is twice the width. Material
for the base costs 11 dollars per square meter. Material for the sides costs 9 dollars per
square meter. Find the cost of materials for the cheapest such container.
3. Maximizing area contained by a fence. An ostrich farmer wants to enclose a rectangu-
lar area and then divide it into six pens with fencing parallel to one side of the rectangle
(see the figure below). There are 620 feet of fencing available to complete the job. What
is the largest possible total area of the six pens?
4. Minimizing the area of a poster. The top and bottom margins of a poster are 8 cm and
the side margins are each 6 cm. If the area of printed material on the poster is fixed at
388 square centimeters, find the dimensions of the poster with the smallest area.
printe d
mate rial
5. Maximizing the area of a rectangle. A rectangle is inscribed with its base on the x-axis
and its upper corners on the parabola y 1 − x 2 . What are the dimensions of such a
rectangle with the greatest possible area?
6. A rectangular box with a square bottom and closed top is to be made from two mate-
rials. The material for the side costs $1.50 per square foot and the material for the top
190
3.4 Applied Optimization
and bottom costs $3.00 per square foot. If you are willing to spend $15 on the box, what
is the largest volume it can contain? Justify your answer completely using calculus.
7. A farmer wants to start raising cows, horses, goats, and sheep, and desires to have
a rectangular pasture for the animals to graze in. However, no two different kinds
of animals can graze together. In order to minimize the amount of fencing she will
need, she has decided to enclose a large rectangular area and then divide it into four
equally sized pens by adding three segments of fence inside the large rectangle that are
parallel to two existing sides. She has decided to purchase 7500 ft of fencing. What is
the maximum possible area that each of the four pens will enclose?
8. Two vertical poles of heights 60 ft and 80 ft stand on level ground, with their bases 100
ft apart. A cable that is stretched from the top of one pole to some point on the ground
between the poles, and then to the top of the other pole. What is the minimum possible
length of cable required? Justify your answer completely using calculus.
9. A company is designing propane tanks that are cylindrical with hemispherical ends.
Assume that the company wants tanks that will hold 1000 cubic feet of gas, and that the
ends are more expensive to make, costing $5 per square foot, while the cylindrical barrel
between the ends costs $2 per square foot. Use calculus to determine the minimum cost
to construct such a tank.
191
Chapter 3 Using Derivatives
3.5 Related Rates
Motivating Questions
• If two quantities that are related, such as the radius and volume of a spherical bal-
loon, are both changing as implicit functions of time, how are their rates of change
related? That is, how does the relationship between the values of the quantities affect
the relationship between their respective derivatives with respect to time?
In most of our applications of the derivative so far, we have been interested in the instanta-
neous rate at which one variable, say y, changes with respect to another, say x, leading us to
dy
compute and interpret dx . We next consider situations where several variable quantities are
related, but where each quantity is implicitly a function of time, which will be represented
by the variable t. Through knowing how the quantities are related, we will be interested in
determining how their respective rates of change with respect to time are related.
For example, suppose that air is being pumped into a spherical balloon so that its volume
increases at a constant rate of 20 cubic inches per second. Since the balloon’s volume and ra-
dius are related, by knowing how fast the volume is changing, we ought to be able to discover
how fast the radius is changing. We are interested in questions such as: can we determine
how fast the radius of the balloon is increasing at the moment the balloon’s diameter is 12
inches?
Preview Activity 3.5.1. A spherical balloon is being inflated at a constant rate of 20
cubic inches per second. How fast is the radius of the balloon changing at the instant
the balloon’s diameter is 12 inches? Is the radius changing more rapidly when d 12
or when d 16? Why?
a. Draw several spheres with different radii, and observe that as volume changes,
the radius, diameter, and surface area of the balloon also change.
b. Recall that the volume of a sphere of radius r is V 34 πr 3 . Note well that in
the setting of this problem, both V and r are changing as time t changes, and
thus both V and r may be viewed as implicit functions of t, with respective
dt and dt . Differentiate both sides of the equation V 3 πr with
derivatives dV dr 4 3
dV
respect to t (using the chain rule on the right) to find a formula for dt that
depends on both r and dr dt .
c. At this point in the problem, by differentiating we have “related the rates” of
change of V and r. Recall that we are given in the problem that the balloon is
being inflated at a constant rate of 20 cubic inches per second. Is this rate the
value of dr dV
dt or dt ? Why?
d. From part (c), we know the value of dV
dt at every value of t. Next, observe that
when the diameter of the balloon is 12, we know the value of the radius. In the
dt 4πr dt , substitute these values for the relevant quantities and
equation dV 2 dr
192
3.5 Related Rates
dr
solve for the remaining unknown quantity, which is dt . How fast is the radius
changing at the instant d 12?
e. How is the situation different when d 16? When is the radius changing more
rapidly, when d 12 or when d 16?
3.5.1 Related Rates Problems
In problems where two or more quantities can be related to one another, and all of the vari-
ables involved are implicitly functions of time, t, we are often interested in how their rates
are related; we call these related rates problems. Once we have an equation establishing the
relationship among the variables, we differentiate implicitly with respect to time to find
connections among the rates of change.
Example 3.5.1 Sand is being dumped by a conveyor belt onto a pile so that the sand forms a
right circular cone, as pictured in Figure 3.5.2.
h
r
Figure 3.5.2: A conical pile of sand.
Solution. As sand falls from the conveyor belt, several features of the sand pile will change:
the volume of the pile will grow, the height will increase, and the radius will get bigger, too.
All of these quantities are related to one another, and the rate at which each is changing is
related to the rate at which sand falls from the conveyor.
We begin by identifying which variables are changing and how they are related. In this
problem, we observe that the radius and height of the pile are related to its volume by the
standard equation for the volume of a cone,
1 2
V πr h.
3
Viewing each of V, r, and h as functions of t, we differentiate implicitly to arrive at an equa-
tion that relates their respective rates of change. Taking the derivative of each side of the
193
Chapter 3 Using Derivatives
equation with respect to t, we find
[ ]
d d 1 2
[V] πr h .
dt dt 3
d
On the left, dt [V] is simply dV
dt . On the right, the situation is more complicated, as both r
and h are implicit functions of t. Hence we need the product and chain rules. We find that
[ ]
dV d 1 2
πr h
dt dt 3
1 d 1 d
πr 2 [h] + πh [r 2 ]
3 dt 3 dt
1 dh 1 dr
πr 2 + πh2r
3 dt 3 dt
(Note particularly how we are using ideas from Section 2.7 on implicit differentiation. There
dy
we found that when y is an implicit function of x, dxd
[y 2 ] 2y dx . The same principles are
applied here when we compute dt d
[r 2 ] 2r dr
dt .)
The equation
dV 1 dh 2 dr
πr 2 + πrh ,
dt 3 dt 3 dt
relates the rates of change of V, h, and r.
If we are given sufficient additional information, we may then find the value of one or more
of these rates of change at a specific point in time.
Example 3.5.3 In the setting of Example 3.5.1, suppose we also know the following: (a) sand
falls from the conveyor in such a way that the height of the pile is always half the radius, and
(b) sand falls from the conveyor belt at a constant rate of 10 cubic feet per minute. How fast
is the height of the sandpile changing at the moment the radius is 4 feet?
Solution. The information that the height is always half the radius tells us that for all values
of t, h 12 r. Differentiating with respect to t, it follows that dh
dt 2 dt . These relationships
1 dr
enable us to relate dVdt to just one of r or h. Substituting the expressions involving r and dt
dr
dh
for h and dt , we now have that
dV 1 1 dr 2 1 dr
πr 2 · + πr · r · . (3.5.1)
dt 3 2 dt 3 2 dt
Since sand falls from the conveyor at the constant rate of 10 cubic feet per minute, the value
dt , the rate at which the volume of the sand pile changes, is dt 10 ft /min. We are
of dV dV 3
interested in how fast the height of the pile is changing at the instant when r 4, so we
substitute r 4 and dVdt 10 into Equation (3.5.1), to find
1 2 1 dr 2 1 dr 8 dr 16 dr
10 π4 · + π4 · 4 · π + π .
3 2 dt r4 3 2 dt r4 3 dt r4 3 dt r4
dr
Only the value of dt r4 remains unknown. We combine like terms on the right side of the
194
3.5 Related Rates
equation above to get 10 8π dr
dt r4 , and solve for dr
dt r4 to find
dr 10
≈ 0.39789
dt r4 8π
feet per second. Because we were interested in how fast the height of the pile was changing
dt when r 4. Since dt 2 dt for all values of t, it follows
at this instant, we want to know dh dh 1 dr
dh 5
≈ 0.19894 ft/min.
dt r4 8π
Note the difference between the notations dr dr
dt and dt r4 . The former represents the rate of
change of r with respect to t at an arbitrary value of t, while the latter is the rate of change
of r with respect to t at a particular moment, the moment when r 4.
Had we known that h 21 r at the beginning of Example 3.5.1, we could have immediately
simplified our work by writing V solely in terms of r to have
( )
1 2 1 1
V πr h πr 3 .
3 2 6
From this last equation, differentiating with respect to t implies
dV 1 dr
πr 2 ,
dt 2 dt
from which the same conclusions can be made.
Our work with the sandpile problem above is similar in many ways to our approach in Pre-
view Activity 3.5.1, and these steps are typical of most related rates problems. In certain
ways, they also resemble work we do in applied optimization problems, and here we sum-
marize the main approach for consideration in subsequent problems.
Note 3.5.4
• Identify the quantities in the problem that are changing and choose clearly defined
variable names for them. Draw one or more figures that clearly represent the situation.
• Determine all rates of change that are known or given and identify the rate(s) of change
to be found.
• Find an equation that relates the variables whose rates of change are known to those
variables whose rates of change are to be found.
• Differentiate implicitly with respect to t to relate the rates of change of the involved
quantities.
• Evaluate the derivatives and variables at the information relevant to the instant at
which a certain rate of change is sought. Use proper notation to identify when a de-
rivative is being evaluated at a particular instant, such as dr
dt r4 .
When identifying variables and drawing a picture, it is important to think about the dynamic
ways in which the quantities change. Sometimes a sequence of pictures can be helpful; for
195
Chapter 3 Using Derivatives
some pictures that can be easily modified as applets built in Geogebra, see the following
links,¹ which represent
• how a circular oil slick’s area grows as its radius increases http://gvsu.edu/s/9n;
• how the location of the base of a ladder and its height along a wall change as the ladder
slides http://gvsu.edu/s/9o;
• how the water level changes in a conical tank as it fills with water at a constant rate
http://gvsu.edu/s/9p (compare the problem in Activity 3.5.2);
• how a skateboarder’s shadow changes as he moves past a lamppost http://gvsu.edu/
s/9q.
Drawing well-labeled diagrams and envisioning how different parts of the figure change is
a key part of understanding related rates problems and being successful at solving them.
Activity 3.5.2. A water tank has the shape of an inverted circular cone (point down)
with a base of radius 6 feet and a depth of 8 feet. Suppose that water is being pumped
into the tank at a constant instantaneous rate of 4 cubic feet per minute.
a. Draw a picture of the conical tank, including a sketch of the water level at a
point in time when the tank is not yet full. Introduce variables that measure the
radius of the water’s surface and the water’s depth in the tank, and label them
on your figure.
b. Say that r is the radius and h the depth of the water at a given time, t. What
equation relates the radius and height of the water, and why?
c. Determine an equation that relates the volume of water in the tank at time t to
the depth h of the water at that time.
d. Through differentiation, find an equation that relates the instantaneous rate of
change of water volume with respect to time to the instantaneous rate of change
of water depth at time t.
e. Find the instantaneous rate at which the water level is rising when the water in
the tank is 3 feet deep.
f. When is the water rising most rapidly: at h 3, h 4, or h 5?
Recognizing which geometric relationships are relevant in a given problem is often the key
to finding the function to optimize. For instance, although the problem in Activity 3.5.2 is
about a conical tank, the most important fact is that there are two similar right triangles
involved. In another setting, we might use the Pythagorean Theorem to relate the legs of
the triangle. But in the conical tank, the fact that the water fills the tank so that that the ratio
of radius to depth is constant turns out to be the important relationship. In other situations
where a changing angle is involved, trigonometric functions may provide the means to find
relationships among various parts of the triangle.
¹We again refer to the work of Prof. Marc Renault of Shippensburg University, found at http://gvsu.edu/s/5p.
196
3.5 Related Rates
Activity 3.5.3. A television camera is positioned 4000 feet from the base of a rocket
launching pad. The angle of elevation of the camera has to change at the correct rate
in order to keep the rocket in sight. In addition, the auto-focus of the camera has
to take into account the increasing distance between the camera and the rocket. We
assume that the rocket rises vertically. (A similar problem is discussed and pictured
dynamically at http://gvsu.edu/s/9t. Exploring the applet at the link will be helpful
to you in answering the questions that follow.)
a. Draw a figure that summarizes the given situation. What parts of the picture
are changing? What parts are constant? Introduce appropriate variables to rep-
resent the quantities that are changing.
b. Find an equation that relates the camera’s angle of elevation to the height of the
rocket, and then find an equation that relates the instantaneous rate of change of
the camera’s elevation angle to the instantaneous rate of change of the rocket’s
height (where all rates of change are with respect to time).
c. Find an equation that relates the distance from the camera to the rocket to the
rocket’s height, as well as an equation that relates the instantaneous rate of
change of distance from the camera to the rocket to the instantaneous rate of
change of the rocket’s height (where all rates of change are with respect to time).
d. Suppose that the rocket’s speed is 600 ft/sec at the instant it has risen 3000 feet.
How fast is the distance from the television camera to the rocket changing at
that moment? If the camera is following the rocket, how fast is the camera’s
angle of elevation changing at that same moment?
e. If from an elevation of 3000 feet onward the rocket continues to rise at 600 feet/
sec, will the rate of change of distance with respect to time be greater when the
elevation is 4000 feet than it was at 3000 feet, or less? Why?
In addition to finding instantaneous rates of change at particular points in time, we can of-
ten make more general observations about how particular rates themselves will change over
time. For instance, when a conical tank is filling with water at a constant rate, it seems obvi-
ous that the depth of the water should increase more slowly over time. Note how carefully
we must phrase the relationship: we mean to say that while the depth, h, of the water is
increasing, its rate of change, dh
dt , is decreasing (both as a function of t and as a function of
h). We make this observation by solving the equation that relates the various rates for one
particular rate, without substituting any particular values for known variables or rates. For
instance, in the conical tank problem in Activity 3.5.2, we established that
dV 1 dh
πh 2 ,
dt 16 dt
and hence
dh 16 dV
.
dt πh 2 dt
Provided that dV dh
dt is constant, it is immediately apparent that as h gets larger, dt will get
smaller but remain positive. Hence, the depth of the water is increasing at a decreasing rate.
197
Chapter 3 Using Derivatives
Activity 3.5.4. As pictured in the applet at http://gvsu.edu/s/9q, a skateboarder
who is 6 feet tall rides under a 15 foot tall lamppost at a constant rate of 3 feet per sec-
ond. We are interested in understanding how fast his shadow is changing at various
points in time.
a. Draw an appropriate right triangle that represents a snapshot in time of the
skateboarder, lamppost, and his shadow. Let x denote the horizontal distance
from the base of the lamppost to the skateboarder and s represent the length of
his shadow. Label these quantities, as well as the skateboarder’s height and the
lamppost’s height on the diagram.
b. Observe that the skateboarder and the lamppost represent parallel line seg-
ments in the diagram, and thus similar triangles are present. Use similar tri-
angles to establish an equation that relates x and s.
dx ds
c. Use your work in (b) to find an equation that relates dt and dt .
d. At what rate is the length of the skateboarder’s shadow increasing at the instant
the skateboarder is 8 feet from the lamppost?
e. As the skateboarder’s distance from the lamppost increases, is his shadow’s
length increasing at an increasing rate, increasing at a decreasing rate, or in-
creasing at a constant rate?
f. Which is moving more rapidly: the skateboarder or the tip of his shadow? Ex-
plain, and justify your answer.
In the first three activities of this section, we provided guided instruction to build a solution
in a step by step way. For the closing activity and the following exercises, most of the detailed
work is left to the reader.
Activity 3.5.5. A baseball diamond is 90′ square. A batter hits a ball along the third
base line and runs to first base. At what rate is the distance between the ball and
first base changing when the ball is halfway to third base, if at that instant the ball is
traveling 100 feet/sec? At what rate is the distance between the ball and the runner
changing at the same instant, if at the same instant the runner is 1/8 of the way to first
base running at 30 feet/sec?
3.5.2 Summary
• When two or more related quantities are changing as implicit functions of time, their
rates of change can be related by implicitly differentiating the equation that relates the
quantities themselves. For instance, if the sides of a right triangle are all changing as
functions of time, say having lengths x, y, and z, then these quantities are related by
the Pythagorean Theorem: x 2 + y 2 z 2 . It follows by implicitly differentiating with
198
3.5 Related Rates
respect to t that their rates are related by the equation
dx dy dz
2x + 2y 2z ,
dt dt dt
so that if we know the values of x, y, and z at a particular time, as well as two of the
three rates, we can deduce the value of the third.
3.5.3 Exercises
1. Height of a conical pile of gravel. Gravel is being dumped from a conveyor belt at a
rate of 10 cubic feet per minute. It forms a pile in the shape of a right circular cone
whose base diameter and height are always the same. How fast is the height of the pile
increasing when the pile is 23 feet high? Recall that the volume of a right circular cone
with height h and radius of the base r is given by V 13 πr 2 h.
2. Movement of a shadow. A street light is at the top of a 13 foot tall pole. A 6 foot tall
woman walks away from the pole with a speed of 6 ft/sec along a straight path. How
fast is the tip of her shadow moving when she is 30 feet from the base of the pole?
3. A leaking conical tank. Water is leaking out of an inverted conical tank at a rate of
9600.0 cm3 /min at the same time that water is being pumped into the tank at a constant
rate. The tank has height 7.0 m and the the diameter at the top is 5.0 m. If the water
level is rising at a rate of 22.0 cm/min when the height of the water is 1.5 m, find the
rate at which water is being pumped into the tank in cubic centimeters per minute.
Hint. Let R be the unknown rate at which water is being pumped in. Then you know
dt R − 9600.0. Use geometry (similar triangles) to find
that if V is volume of water, dV
the relationship between the height of the water and the volume of the water at any
given time. Recall that the volume of a cone with base radius r and height h is given
by 31 πr 2 h
4. A sailboat is sitting at rest near its dock. A rope attached to the bow of the boat is drawn
in over a pulley that stands on a post on the end of the dock that is 5 feet higher than
the bow. If the rope is being pulled in at a rate of 2 feet per second, how fast is the boat
approaching the dock when the length of rope from bow to pulley is 13 feet?
5. A swimming pool is 60 feet long and 25 feet wide. Its depth varies uniformly from 3
feet at the shallow end to 15 feet at the deep end, as shown in the Figure 3.5.5.
Suppose the pool has been emptied and is now being filled with water at a rate of 800
cubic feet per minute. At what rate is the depth of water (measured at the deepest
point of the pool) increasing when it is 5 feet deep at that end? Over time, describe
how the depth of the water will increase: at an increasing rate, at a decreasing rate, or
at a constant rate. Explain.
199
Chapter 3 Using Derivatives
25
60
3
15
Figure 3.5.5: The swimming pool.
6. A baseball diamond is a square with sides 90 feet long. Suppose a baseball player is
advancing from second to third base at the rate of 24 feet per second, and an umpire is
standing on home plate. Let θ be the angle between the third baseline and the line of
sight from the umpire to the runner. How fast is θ changing when the runner is 30 feet
from third base?
7. Sand is being dumped off a conveyor belt onto a pile in such a way that the pile forms
in the shape of a cone whose radius is always equal to its height. Assuming that the
sand is being dumped at a rate of 10 cubic feet per minute, how fast is the height of the
pile changing when there are 1000 cubic feet on the pile?
200
CHAPTER 4
The Definite Integral
4.1 Determining distance traveled from velocity
Motivating Questions
• If we know the velocity of a moving body at every point in a given interval, can we
determine the distance the object has traveled on the time interval?
• How is the problem of finding distance traveled related to finding the area under a
certain curve?
• What does it mean to antidifferentiate a function and why is this process relevant to
finding distance traveled?
• If velocity is negative, how does this impact the problem of finding distance traveled?
In the first section of the text, we considered a moving object with known position at time
t, namely, a tennis ball tossed into the air with height s (in feet) at time t (in seconds) given
by s(t) 64 − 16(t − 1)2 . We investigated the average velocity of the ball on an interval
s(b)−s(a)
[a, b], computed by the difference quotient b−a . We found that we could determine the
instantaneous velocity of the ball at time t by taking the derivative of the position function,
s(t + h) − s(t)
s ′(t) lim .
h→0 h
Thus, if its position function is differentiable, we can find the velocity of a moving object at
any point in time.
From this study of position and velocity we have learned a great deal. We can use the de-
rivative to find a function’s instantaneous rate of change at any point in the domain, to find
where the function is increasing or decreasing, where it is concave up or concave down,
and to locate relative extremes. The vast majority of the problems and applications we have
considered have involved the situation where a particular function is known and we seek
information that relies on knowing the function’s instantaneous rate of change. For all these
tasks, we proceed from a function f to its derivative, f ′, and use the meaning of the deriv-
ative to help us answer important questions.
Chapter 4 The Definite Integral
We have also encountered the reverse situation, where we know the derivative of a function,
f ′, and try to deduce information about f . We will focus our attention in Chapter 4 on this
problem: if we know the instantaneous rate of change of a function, can we find the function
itself? We start with a more specific question: if we know the instantaneous velocity of an
object moving along a straight line path, can we find its corresponding position function?
Preview Activity 4.1.1. Suppose that a person is taking a walk along a long straight
path and walks at a constant rate of 3 miles per hour.
a. On the left-hand axes provided in Figure 4.1.1, sketch a labeled graph of the
velocity function v(t) 3.
mph miles
8 8
4 4
hrs hrs
1 2 1 2
Figure 4.1.1: At left, axes for plotting y v(t); at right, for plotting y s(t).
Note that while the scale on the two sets of axes is the same, the units on the
right-hand axes differ from those on the left. The right-hand axes will be used
in question (d).
b. How far did the person travel during the two hours? How is this distance related
to the area of a certain region under the graph of y v(t)?
c. Find an algebraic formula, s(t), for the position of the person at time t, assuming
that s(0) 0. Explain your thinking.
d. On the right-hand axes provided in Figure 4.1.1, sketch a labeled graph of the
position function y s(t).
e. For what values of t is the position function s increasing? Explain why this is
the case using relevant information about the velocity function v.
202
4.1 Determining distance traveled from velocity
4.1.1 Area under the graph of the velocity function
In Preview Activity 4.1.1, we learned that when the velocity of a moving object’s velocity is
constant (and positive), the area under the velocity curve over an interval of time tells us the
distance the object traveled.
mph mph
3 3
y = v(t)
v(t) = 2
A2
1 A1 1
hrs hrs
1 2 3 1 2 3
Figure 4.1.2: At left, a constant velocity function; at right, a non-constant velocity function.
The left-hand graph of Figure 4.1.2 shows the velocity of an object moving at 2 miles per
hour over the time interval [1, 1.5]. The area A1 of the shaded region under y v(t) on
[1, 1.5] is
miles 1
A1 2 · hours 1 mile.
hour 2
This result is simply the fact that distance equals rate times time, provided the rate is con-
stant. Thus, if v(t) is constant on the interval [a, b], the distance traveled on [a, b] is equal to
the area A given by
A v(a)(b − a) v(a)∆t,
where ∆t is the change in t over the interval. (Since the velocity is constant, we can use
any value of v(t) on the interval [a, b], we simply chose v(a), the value at the interval’s left
endpoint.) For several examples where the velocity function is piecewise constant, see http:/
/gvsu.edu/s/9T.¹
The situation is more complicated when the velocity function is not constant. But on rela-
tively small intervals where v(t) does not vary much, we can use the area principle to esti-
mate the distance traveled. The graph at right in Figure 4.1.2 shows a non-constant velocity
function. On the interval [1, 1.5], the velocity varies from v(1) 2.5 down to v(1.5) ≈ 2.1.
One estimate for the distance traveled is the area of the pictured rectangle,
miles 1
A2 v(1)∆t 2.5 · hours 1.25 miles.
hour 2
Note that because v is decreasing on [1, 1.5], A2 1.25 is an over-estimate of the actual
distance traveled.
¹Marc Renault, calculus applets.
203
Chapter 4 The Definite Integral
To estimate the area under this non-constant velocity function on a wider interval, say [0, 3],
one rectangle will not give a good approximation. Instead, we could use the six rectangles
pictured in Figure 4.1.3, find the area of each rectangle, and add up the total. Obviously
there are choices to make and issues to understand: How many rectangles should we use?
Where should we evaluate the function to decide the rectangle’s height? What happens
if the velocity is sometimes negative? Can we find the exact area under any non-constant
curve?
mph
3
y = v(t)
1
hrs
1 2 3
Figure 4.1.3: Using six rectangles to estimate the area under y v(t) on [0, 3].
We will study these questions and more in what follows; for now it suffices to observe that
the simple idea of the area of a rectangle gives us a powerful tool for estimating distance
traveled from a velocity function, as well as for estimating the area under an arbitrary curve.
To explore the use of multiple rectangles to approximate area under a non-constant velocity
function, see the applet found at http://gvsu.edu/s/9U.²
Activity 4.1.2. Suppose that a person is walking in such a way that her velocity varies
slightly according to the information given in Table 4.1.4 and graph given in Fig-
ure 4.1.5.
a. Using the grid, graph, and given data appropriately, estimate the distance trav-
eled by the walker during the two hour interval from t 0 to t 2. You should
use time intervals of width ∆t 0.5, choosing a way to use the function consis-
tently to determine the height of each rectangle in order to approximate distance
traveled.
b. How could you get a better approximation of the distance traveled on [0, 2]?
Explain, and then find this new estimate.
²Marc Renault, calculus applets.
204
4.1 Determining distance traveled from velocity
t v(t)
mph
0.00 1.500 3
y = v(t)
0.25 1.789
0.50 1.938
2
0.75 1.992
1.00 2.000
1.25 2.008 1
1.50 2.063
1.75 2.211
hrs
2.00 2.500
1 2
Table 4.1.4: Velocity data for the per-
son walking.
Figure 4.1.5: The graph of y v(t).
c. Now suppose that you know that v is given by v(t) 0.5t 3 − 1.5t 2 + 1.5t + 1.5.
Remember that v is the derivative of the walker’s position function, s. Find a
formula for s so that s ′ v.
d. Based on your work in (c), what is the value of s(2) − s(0)? What is the meaning
of this quantity?
4.1.2 Two approaches: area and antidifferentiation
When the velocity of a moving object is positive, the object’s position is always increasing.
(We will soon consider situations where velocity is negative; for now, we focus on the situa-
tion where velocity is always positive.) We have established that whenever v is constant on
an interval, the exact distance traveled is the area under the velocity curve. When v is not
constant, we can estimate the total distance traveled by finding the areas of rectangles that
approximate the area under the velocity curve.
Thus, we see that finding the area between a curve and the horizontal axis is an important
exercise: besides being an interesting geometric question, if the curve gives the velocity of
a moving object, the area under the curve tells us the exact distance traveled on an interval.
We can estimate this area if we have a graph or a table of values for the velocity function.
In Activity 4.1.2, we encountered an alternate approach to finding the distance traveled. If
y v(t) is a formula for the instantaneous velocity of a moving object, then v must be the
derivative of the object’s position function, s. If we can find a formula for s(t) from the
formula for v(t), we will know the position of the object at time t, and the change in position
over a particular time interval tells us the distance traveled on that interval.
For a simple example, consider the situation from Preview Activity 4.1.1, where a person is
walking along a straight line with velocity function v(t) 3 mph. On the left-hand graph
of the velocity function in Figure 4.1.6, we see the relationship between area and distance
205
Chapter 4 The Definite Integral
mph miles
8 8
s(t) = 3t
4 4
v(t) = 3
s(1.5) = 4.5
A = 3 · 1.25 = 3.75
hrs s(0.25) = 0.75 hrs
1 2 1 2
Figure 4.1.6: The velocity function v(t) 3 and corresponding position function s(t) 3t.
traveled,
miles
A3 · 1.25 hours 3.75 miles.
hour
In addition, we observe³ that if s(t) 3t, then s ′(t) 3, so s(t) 3t is the position function
whose derivative is the given velocity function, v(t) 3. The respective locations of the
person at times t 0.25 and t 1.5 are s(1.5) 4.5 and s(0.25) 0.75, and therefore
s(1.5) − s(0.25) 4.5 − 0.75 3.75 miles.
This is the person’s change in position on [0.25, 1.5], which is precisely the distance traveled.
In this example there are profound ideas and connections that we will study throughout
Chapter 4.
For now, observe that if we know a formula for a velocity function v, it can be very helpful to
find a function s that satisfies s ′ v. We say that s is an antiderivative of v. More generally,
we have the following formal definition.
Definition 4.1.7 If 1 and G are functions such that G′ 1, we say that G is an antiderivative
of 1.
For example, if 1(x) 3x 2 + 2x, G(x) x 3 + x 2 is an antiderivative of 1, because G′(x)
1(x). Note that we say “an” antiderivative of 1 rather than “the” antiderivative of 1, because
H(x) x 3 +x 2 +5 is also a function whose derivative is 1, and thus H is another antiderivative
of 1.
Activity 4.1.3. A ball is tossed vertically in such a way that its velocity function is
given by v(t) 32 − 32t, where t is measured in seconds and v in feet per second.
³Here we are making the implicit assumption that s(0) 0; we will discuss different possibilities for values of
s(0) in subsequent study.
206
4.1 Determining distance traveled from velocity
Assume that this function is valid for 0 ≤ t ≤ 2.
a. For what values of t is the velocity of the ball positive? What does this tell you
about the motion of the ball on this interval of time values?
b. Find an antiderivative, s, of v that satisfies s(0) 0.
c. Compute the value of s(1) − s( 12 ). What is the meaning of the value you find?
d. Using the graph of y v(t) provided in Figure 4.1.8, find the exact area of the
region under the velocity curve between t 21 and t 1. What is the meaning
of the value you find?
ft/sec
24
v(t) = 32 − 32t
12
sec
1 2
-12
-24
Figure 4.1.8: The graph of y v(t).
e. Answer the same questions as in (c) and (d) but instead using the interval [0, 1].
f. What is the value of s(2) − s(0)? What does this result tell you about the flight
of the ball? How is this value connected to the provided graph of y v(t)?
Explain.
4.1.3 When velocity is negative
The assumption that its velocity is positive on a given interval guarantees that the movement
of an object is always in a single direction, and hence ensures that its change in position is
the same as the distance it travels. As we saw in Activity 4.1.3, there are natural settings in
which an object’s velocity is negative, and we would like to understand this scenario as well.
Consider a simple example where a woman goes for a walk on the beach along a stretch of
very straight shoreline that runs east-west. We assume that her initial position is s(0) 0,
and that her position function increases as she moves east from her starting location. For
instance, s 1 mile represents one mile east of the start location, while s −1 tells us she is
207
Chapter 4 The Definite Integral
one mile west of where she began walking on the beach.
Now suppose she walks in the following manner. From the outset at t 0, she walks due
east at a constant rate of 3 mph for 1.5 hours. After 1.5 hours, she stops abruptly and begins
walking due west at a constant rate of 4 mph and does so for 0.5 hours. Then, after another
abrupt stop and start, she resumes walking at a constant rate of 3 mph to the east for one
more hour. What is the total distance she traveled on the time interval from t 0 to t 3?
What the total change in her position over that time?
These questions are possible to answer without calculus because the velocity is constant on
each interval. From t 0 to t 1.5, she traveled
D[0,1.5] 3 miles per hour · 1.5 hours 4.5 miles.
On t 1.5 to t 2, the distance traveled is
D[1.5,2] 4 miles per hour · 0.5 hours 2 miles.
Finally, in the last hour she walked
D[2,3] 3 miles per hour · 1 hours 3 miles,
so the total distance she traveled is
D D[0,1.5] + D[1.5,2] + D[2,3] 4.5 + 2 + 3 9.5 miles.
Since the velocity for 1.5 < t < 2 is v −4, indicating motion in the westward direction, the
woman first walked 4.5 miles east, then 2 miles west, followed by 3 more miles east. Thus,
the total change in her position is
change in position 4.5 − 2 + 3 5.5 miles.
We have been able to answer these questions fairly easily, and if we think about the problem
graphically, we can generalize our solution to the more complicated setting when velocity is
not constant, and possibly negative. In Figure 4.1.9, we see how the distances we computed
can be viewed as areas: A1 4.5 comes from multiplyimg rate times time (3 · 1.5), as do A2
and A3 . But while A2 is an area (and is therefore positive), because the velocity function is
negative for 1.5 < t < 2, this area has a negative sign associated with it. The negative area
distinguishes between distance traveled and change in position.
The distance traveled is the sum of the areas,
D A1 + A2 + A3 4.5 + 2 + 3 9.5 miles.
But the change in position has to account for travel in the negative direction. An area above
the t-axis is considered positive because it represents distance traveled in the positive direc-
tion, while one below the t-axis is viewed as negative because it represents travel in theneg-
ative direction. Thus, the change in the woman’s position is
s(3) − s(0) (+4.5) + (−2) + (+3) 5.5 miles.
208
4.1 Determining distance traveled from velocity
(3, 5.5)
mph miles (1.5, 4.5)
4.5 4.5
y = v(t) y = s(t)
3.0 3.0
1.5 1.5 (2, 2.5)
A1 = 4.5 A3 = 3
hrs hrs
1 3 1 3
-1.5 -1.5
A2 = 2
-3.0 -3.0
-4.5 -4.5
Figure 4.1.9: At left, the velocity function of the person walking; at right, the
corresponding position function.
In other words, the woman walks 4.5 miles in the positive direction, followed by two miles
in the negative direction, and then 3 more miles in the positive direction.
Negative velocity is also seen in the graph of the position function y s(t). Its slope is
negative (specifically, −4) on the interval 1.5 < t < 2 because the velocity is −4 on that
interval. The negative slope shows the position function is decreasing because the woman
is walking east, rather than west.
To summarize, we see that if velocity is sometimes negative, a moving object’s change in po-
sition different from its distance traveled. If we compute separately the distance traveled on
each interval where velocity is positive or negative, we can calculate either the total distance
traveled or the total change in position. We assign a negative value to distances traveled in
the negative direction when we calculate change in position, but a positive value when we
calculate the total distance traveled.
Activity 4.1.4. Suppose that an object moving along a straight line path has its velocity
v (in meters per second) at time t (in seconds) given by the piecewise linear function
whose graph is pictured at left in Figure 4.1.10. We view movement to the right as
being in the positive direction (with positive velocity), while movement to the left is
in the negative direction.
Suppose further that the object’s initial position at time t 0 is s(0) 1.
a. Determine the total distance traveled and the total change in position on the
time interval 0 ≤ t ≤ 2. What is the object’s position at t 2?
b. On what time intervals is the moving object’s position function increasing? Why?
On what intervals is the object’s position decreasing? Why?
209
Chapter 4 The Definite Integral
m/sec
4 8
y = v(t)
2 4
sec
2 4 6 8 2 4 6 8
-2 -4
-4 -8
Figure 4.1.10: The velocity function of a moving object.
c. What is the object’s position at t 8? How many total meters has it traveled to
get to this point (including distance in both directions)? Is this different from
the object’s total change in position on t 0 to t 8?
d. Find the exact position of the object at t 1, 2, 3, . . . , 8 and use this data to sketch
an accurate graph of y s(t) on the axes provided at right in Figure 4.1.10. How
can you use the provided information about y v(t) to determine the concavity
of s on each relevant interval?
4.1.4 Summary
• If we know the velocity of a moving body at every point in a given interval and the
velocity is positive throughout, we can estimate the object’s distance traveled and in
some circumstances determine this value exactly.
• In particular, when velocity is positive on an interval, we can find the total distance
traveled by finding the area under the velocity curve and above the t-axis on the given
time interval. We may only be able to estimate this area, depending on the shape of
the velocity curve.
• An antiderivative of a function f is a new function F whose derivative is f . That is, F
is an antiderivative of f provided that F′ f . In the context of velocity and position,
if we know a velocity function v, an antiderivative of v is a position function s that
satisfies s ′ v. If v is positive on a given interval, say [a, b], then the change in position,
s(b) − s(a), measures the distance the moving object traveled on [a, b].
• If its velocity is sometimes negative, a moving object is sometimes traveling in the
opposite direction or backtracking. To determine distance traveled, we have to think
210
4.1 Determining distance traveled from velocity
compute the distance separately on intervals where velocity is positive or negative,
and account for the change in position on each such interval.
4.1.5 Exercises
1. Estimating distance traveled from velocity data. A car comes to a stop six seconds
after the driver applies the brakes. While the brakes are on, the following velocities are
recorded:
Time since brakes applied (sec) 0 2 4 6
Velocity (ft/s) 90 46 17 0
Give lower and upper estimates (using all of the available data) for the distance the car
traveled after the brakes were applied.
On a sketch of velocity against time, show the lower and upper estimates you found above..
2. Distance from a linear veloity function. The velocity of a car is f (t) 11t meters/
second. Use a graph of f (t) to find the exact distance traveled by the car, in meters,
from t 0 to t 10 seconds.
3. Change in position from a linear velocity function. The velocity of a particle moving
along the x-axis is given by f (t) 12 − 4t cm/sec. Use a graph of f (t) to find the exact
change in position of the particle from time t 0 to t 4 seconds.
4. Comparing distance traveled from velocity graphs. Two cars start at the same time
and travel in the same direction along a straight road. The figure below gives the ve-
locity, v (in km/hr), of each car as a function of time (in hr).
The velocity of car A is given by the solid, blue curve, and the velocity of car B by
dashed, red curve.
(a) Which car attains the larger maximum velocity?
(b) Which stops first?
(c) Which travels farther?
5. Finding average acceleration from velocity data. Suppose that an accelerating car goes
from 0 mph to 68.2 mph in five seconds. Its velocity is given in the following table,
converted from miles per hour to feet per second, so that all time measurements are in
seconds. (Note: 1 mph is 22/15 feet per sec = 22/15 ft/s.) Find the average acceleration
of the car over each of the first two seconds.
211
Chapter 4 The Definite Integral
t 0 1 2 3 4 5
v(t) 0.00 34.09 59.09 77.27 90.91 100.00
6. Change in position from a quadratic velocity function. The velocity function is v(t)
t 2 −3t+2 for a particle moving along a line. Find the displacement (net distance covered)
of the particle during the time interval [−2, 5].
7. Along the eastern shore of Lake Michigan from Lake Macatawa (near Holland) to Grand
Haven, there is a bike path that runs almost directly north-south. For the purposes
of this problem, assume the road is completely straight, and that the function s(t)
tracks the position of the biker along this path in miles north of Pigeon Lake, which
lies roughly halfway between the ends of the bike path.
Suppose that the biker’s velocity function is given by the graph in Figure 4.1.11 on the
time interval 0 ≤ t ≤ 4 (where t is measured in hours), and that s(0) 1.
mph miles
10 y = v(t) 10
6 6
2 2
hrs hrs
-2 1 2 3 4 5 -2 1 2 3 4 5
-6 -6
-10 -10
Figure 4.1.11: The graph of the biker’s velocity, y v(t), at left. At right, axes to plot
an approximate sketch of y s(t).
a. Approximately how far north of Pigeon Lake was the cyclist when she was the
greatest distance away from Pigeon Lake? At what time did this occur?
b. What is the cyclist’s total change in position on the time interval 0 ≤ t ≤ 2? At
t 2, was she north or south of Pigeon Lake?
c. What is the total distance the biker traveled on 0 ≤ t ≤ 4? At the end of the ride,
how close was she to the point at which she started?
d. Sketch an approximate graph of y s(t), the position function of the cyclist, on
the interval 0 ≤ t ≤ 4. Label at least four important points on the graph of s.
8. A toy rocket is launched vertically from the ground on a day with no wind. The rocket’s
vertical velocity at time t (in seconds) is given by v(t) 500 − 32t feet/sec.
a. At what time after the rocket is launched does the rocket’s velocity equal zero?
Call this time value a. What happens to the rocket at t a?
b. Find the value of the total area enclosed by y v(t) and the t-axis on the interval
212
4.1 Determining distance traveled from velocity
0 ≤ t ≤ a. What does this area represent in terms of the physical setting of the
problem?
c. Find an antiderivative s of the function v. That is, find a function s such that
s ′(t) v(t).
d. Compute the value of s(a) − s(0). What does this number represent in terms of
the physical setting of the problem?
e. Compute s(5) − s(1). What does this number tell you about the rocket’s flight?
9. An object moving along a horizontal axis has its instantaneous velocity at time t in
seconds given by the function v pictured in Figure 4.1.12, where v is measured in feet/
sec. Assume that the curves that make up the parts of the graph of y v(t) are either
portions of straight lines or portions of circles.
y = v(t)
1
1 2 3 4 5 6 7
-1
Figure 4.1.12: The graph of y v(t), the velocity function of a moving object.
a. Determine the exact total distance the object traveled on 0 ≤ t ≤ 2.
b. What is the value and meaning of s(5) − s(2), where y s(t) is the position func-
tion of the moving object?
c. On which time interval did the object travel the greatest distance: [0, 2], [2, 4], or
[5, 7]?
d. On which time interval(s) is the position function s increasing? At which point(s)
does s achieve a relative maximum?
10. Filters at a water treatment plant become dirtier over time and thus become less ef-
fective; they are replaced every 30 days. During one 30-day period, the rate at which
pollution passes through the filters into a nearby lake (in units of particulate matter
per day) is measured every 6 days and is given in the following table. The time t is
measured in days since the filters were replaced.
Day, t 0 6 12 18 24 30
Rate of pollution in units per day, p(t) 7 8 10 13 18 35
Table 4.1.13: Pollution data for the water filters.
a. Plot the given data on a set of axes with time on the horizontal axis and the rate
213
Chapter 4 The Definite Integral
of pollution on the vertical axis.
b. Explain why the amount of pollution that entered the lake during this 30-day
period would be given exactly by the area bounded by y p(t) and the t-axis on
the time interval [0, 30].
c. Estimate the total amount of pollution entering the lake during this 30-day period.
Carefully explain how you determined your estimate.
214
4.2 Riemann Sums
4.2 Riemann Sums
Motivating Questions
• How can we use a Riemann sum to estimate the area between a given curve and the
horizontal axis over a particular interval?
• What are the differences among left, right, middle, and random Riemann sums?
• How can we write Riemann sums in an abbreviated form?
In Section 4.1, we learned that if an object moves with positive velocity v, the area between
y v(t) and the t-axis over a given time interval tells us the distance traveled by the object
over that time period. If v(t) is sometimes negative and we view the area of any region below
the t-axis as having an associated negative sign, then the sum of these signed areas tells us
the moving object’s change in position over a given time interval.
For instance, for the velocity function given
in Figure 4.2.1, if the areas of shaded re-
gions are A1 , A2 , and A3 as labeled, then y = v(t)
the total distance D traveled by the moving
object on [a, b] is
D A1 + A2 + A3 ,
while the total change in the object’s posi- A1 A3
tion on [a, b] is
s(b) − s(a) A1 − A2 + A3 . a A2 b
Because the motion is in the negative di-
rection on the interval where v(t) < 0, we
subtract A2 to determine the object’s total
change in position.
Figure 4.2.1: A velocity function that is
sometimes negative.
Of course, finding D and s(b) − s(a) for the graph in Figure 4.2.1 presumes that we can
actually find the areas A1 , A2 , and A3 . So far, we have worked with velocity functions that
were either constant or linear, so that the area bounded by the velocity function and the
horizontal axis is a combination of rectangles and triangles, and we can find the area exactly.
But when the curve bounds a region that is not a familiar geometric shape, we cannot find
its area exactly. Indeed, this is one of our biggest goals in Chapter 4: to learn how to find the
exact area bounded between a curve and the horizontal axis for as many different types of
functions as possible.
In Activity 4.1.2, we approximated the area under a nonlinear velocity function using rectan-
gles. In the following preview activity, we consider three different options for the heights
of the rectangles we will use.
215
Chapter 4 The Definite Integral
Preview Activity 4.2.1. A person walking along a straight path has her velocity in
miles per hour at time t given by the function v(t) 0.25t 3 − 1.5t 2 + 3t + 0.25, for
times in the interval 0 ≤ t ≤ 2. The graph of this function is also given in each of the
three diagrams in Figure 4.2.2.
mph mph mph
3 3 3
y = v(t) y = v(t) y = v(t)
2 2 2
1 A4 1 B4 1 C4
A3 B3 C3
A2 B2 C2
hrs B1 hrs C1 hrs
A1 1 2 1 2 1 2
Figure 4.2.2: Three approaches to estimating the area under y v(t) on the interval
[0, 2].
Note that in each diagram, we use four rectangles to estimate the area under y v(t)
on the interval [0, 2], but the method by which the four rectangles’ respective heights
are decided varies among the three individual graphs.
a. How are the heights of rectangles in the left-most diagram being chosen? Ex-
plain, and hence determine the value of
S A1 + A2 + A3 + A4
by evaluating the function y v(t) at appropriately chosen values and observ-
ing the width of each rectangle. Note, for example, that
1 1
A3 v(1) · 2 · 1.
2 2
b. Explain how the heights of rectangles are being chosen in the middle diagram
and find the value of
T B1 + B2 + B3 + B4 .
c. Likewise, determine the pattern of how heights of rectangles are chosen in the
right-most diagram and determine
U C1 + C2 + C3 + C4 .
d. Of the estimates S, T, and U, which do you think is the best approximation of
D, the total distance the person traveled on [0, 2]? Why?
216
4.2 Riemann Sums
4.2.1 Sigma Notation
We have used sums of areas of rectangles to approximate the area under a curve. Intuitively,
we expect that using a larger number of thinner rectangles will provide a better estimate for
the area. Consequently, we anticipate dealing with sums of a large number of terms. To do
so, we introduce sigma notation, named for the Greek letter Σ, which is the capital letter S in
the Greek alphabet.
For example, say we are interested in the sum
1 + 2 + 3 + · · · + 100,
the sum of the first 100 natural numbers. In sigma notation we write
∑
100
k 1 + 2 + 3 + · · · + 100.
k1
∑
We read the symbol 100k1 k as “the sum from k equals 1 to 100 of k.” The variable k is called
the index of summation, and any letter can be used for this variable. The pattern in the terms
of the sum is denoted by a function of the index; for example,
∑
10
(k 2 + 2k) (12 + 2 · 1) + (22 + 2 · 2) + (32 + 2 · 3) + · · · + (102 + 2 · 10),
k1
and more generally,
∑
n
f (k) f (1) + f (2) + · · · + f (n).
k1
Sigma notation allows us to vary easily the function being used to describe the terms in the
sum, and to adjust the number of terms in the sum simply by changing the value of n. We
test our understanding of this new notation in the following activity.
Activity 4.2.2. For each sum written in sigma notation, write the sum long-hand and
evaluate the sum to find its value. For each sum written in expanded form, write the
sum in
∑sigma notation.
a. 5k1 (k 2 + 2) d. 4 + 8 + 16 + 32 + · · · + 256
∑6
b. i3 (2i − 1)
∑6
c. 3 + 7 + 11 + 15 + · · · + 27 e. 1
i1 2i
4.2.2 Riemann Sums
When a moving body has a positive velocity function y v(t) on a given interval [a, b],
the area under the curve over the interval gives the total distance the body travels on [a, b].
We are also interested in finding the exact area bounded by y f (x) on an interval [a, b],
217
Chapter 4 The Definite Integral
a b
x0 x1 x2 ··· xi xi+1 ··· xn−1 xn
△x
Figure 4.2.3: Subdividing the interval [a, b] into n subintervals of equal length ∆x.
regardless of the meaning or context of the function f . For now, we continue to focus on
finding an accurate estimate of this area by using a sum of the areas of rectangles. Unless
otherwise indicated, we assume that f is continuous and non-negative on [a, b].
The first choice we make in such an approximation is the number of rectangles. If we desire
n rectangles of equal width to subdivide the interval [a, b], then each rectangle must have
width ∆x b−an . We let x 0 a, x n b, and define x i a + i∆x, so that x 1 x 0 + ∆x,
x2 x0 + 2∆x, and so on, as pictured in Figure 4.2.3.
We use each subinterval [x i , x i+1 ] as the base of a rectangle, and next choose the height of the
rectangle on that subinterval. There are three standard choices: we can use the left endpoint
of each subinterval, the right endpoint of each subinterval, or the midpoint of each. These
are precisely the options encountered in Preview Activity 4.2.1 and seen in Figure 4.2.2. We
next explore how these choices can be described in sigma notation.
Consider an arbitrary positive function f on [a, b] with the interval subdivided as shown in
Figure 4.2.3, and choose to use left endpoints. Then on each interval [x i , x i+1 ], the area of
the rectangle formed is given by
A i+1 f (x i ) · ∆x,
as seen in Figure 4.2.4.
y = f (x)
A1 A2 ··· Ai+1 ··· An
x0 x1 x2 xi xi+1 xn−1 xn
Figure 4.2.4: Subdividing the interval [a, b] into n subintervals of equal length ∆x and
approximating the area under y f (x) over [a, b] using left rectangles.
218
4.2 Riemann Sums
If we let L n denote the sum of the areas of these rectangles, we see that
L n A1 + A2 + · · · + A i+1 + · · · + A n
f (x0 ) · ∆x + f (x 1 ) · ∆x + · · · + f (x i ) · ∆x + · · · + f (x n−1 ) · ∆x.
In the more compact sigma notation, we have
∑
n−1
Ln f (x i )∆x.
i0
Note that since the index of summation begins at 0 and ends at n − 1, there are indeed n
terms in this sum. We call L n the left Riemann sum for the function f on the interval [a, b].
To see how the Riemann sums for right endpoints and midpoints are constructed, we con-
sider Figure 4.2.5. For the sum with right endpoints, we see that the area of the rectangle
on an arbitrary interval [x i , x i+1 ] is given by B i+1 f (x i+1 ) · ∆x, and that the sum of all such
areas of rectangles is given by
R n B1 + B2 + · · · + B i+1 + · · · + B n
f (x1 ) · ∆x + f (x2 ) · ∆x + · · · + f (x i+1 ) · ∆x + · · · + f (x n ) · ∆x
∑
n
f (x i )∆x.
i1
We call R n the right Riemann sum for the function f on the interval [a, b].
For the sum that uses midpoints, we introduce the notation
x i + x i+1
x i+1
2
so that x i+1 is the midpoint of the interval [x i , x i+1 ]. For instance, for the rectangle with area
C 1 in Figure 4.2.5, we now have
C 1 f (x 1 ) · ∆x.
y = f (x) y = f (x)
B1 B2 ··· Bi+1 ··· Bn C1 C2 ··· Ci+1 ··· Cn
x0 x1 x2 xi xi+1 xn−1 xn x0 x1 x2 xi xi+1 xn−1 xn
Figure 4.2.5: Riemann sums using right endpoints and midpoints.
219
Chapter 4 The Definite Integral
Figure 4.2.6: A snapshot of the applet found at http://gvsu.edu/s/a9.
Hence, the sum of all the areas of rectangles that use midpoints is
M n C1 + C 2 + · · · + C i+1 + · · · + C n
f (x1 ) · ∆x + f (x2 ) · ∆x + · · · + f (x i+1 ) · ∆x + · · · + f (x n ) · ∆x
∑
n
f (x i )∆x,
i1
and we say that M n is the middle Riemann sum for f on [a, b].
Thus, we have two variables to explore: the number of rectangles and the height of each
rectangle. We can explore these choices dynamically, and the applet¹ found at http://
gvsu.edu/s/a9 is a particularly useful one. There we see the image shown in Figure 4.2.6,
but with the opportunity to adjust the slider bars for the heights and the number of rec-
tangles. By moving the sliders, we can see how the heights of the rectangles change as we
consider left endpoints, midpoints, and right endpoints, as well as the impact that a larger
number of narrower rectangles has on the approximation of the exact area bounded by the
function and the horizontal axis.
When f (x) ≥ 0 on [a, b], each of the Riemann sums L n , R n , and M n provides an estimate of
the area under the curve y f (x) over the interval [a, b]. We also recall that in the context
of a nonnegative velocity function y v(t), the corresponding Riemann sums approximate
the distance traveled on [a, b] by a moving object with velocity function v.
There is a more general way to think of Riemann sums, and that is to allow any choice of
where the function is evaluated to determine the rectangle heights. Rather than saying we’ll
always choose left endpoints, or always choose midpoints, we simply say that a point x ∗i+1
will be selected at random in the interval [x i , x i+1 ] (so that x i ≤ x ∗i+1 ≤ x i+1 ). The Riemann
sum is then given by
∑
n
f (x1∗ ) · ∆x + f (x 2∗ ) · ∆x + · · · + f (x ∗i+1 ) · ∆x + · · · + f (x ∗n ) · ∆x f (x ∗i )∆x.
i1
¹Marc Renault, Geogebra Calculus Applets.
220
4.2 Riemann Sums
At http://gvsu.edu/s/a9, the applet noted earlier and referenced in Figure 4.2.6, by uncheck-
ing the “relative” box at the top left, and instead checking “random,” we can easily explore
the effect of using random point locations in subintervals on a Riemann sum. In computa-
tional practice, we most often use L n , R n , or M n , while the random Riemann sum is useful in
theoretical discussions. In the following activity, we investigate several different Riemann
sums for a particular velocity function.
Activity 4.2.3. Suppose that an object moving along a straight line path has its velocity
in feet per second at time t in seconds given by v(t) 29 (t − 3)2 + 2.
a. Carefully sketch the region whose exact area will tell you the value of the dis-
tance the object traveled on the time interval 2 ≤ t ≤ 5.
b. Estimate the distance traveled on [2, 5] by computing L4 , R 4 , and M4 .
c. Does averaging L4 and R 4 result in the same value as M4 ? If not, what do you
think the average of L4 and R 4 measures?
d. For this question, think about an arbitrary function f , rather than the particular
function v given above. If f is positive and increasing on [a, b], will L n over-
estimate or under-estimate the exact area under f on [a, b]? Will R n over- or
under-estimate the exact area under f on [a, b]? Explain.
4.2.3 When the function is sometimes negative
For a Riemann sum such as
∑
n−1
Ln f (x i )∆x,
i0
we can of course compute the sum even when f takes on negative values. We know that
when f is positive on [a, b], a Riemann sum estimates the area bounded between f and the
horizontal axis over the interval.
y = f (x) y = f (x) y = f (x)
A1 A3
A2
a b c d a b c d a b c d
Figure 4.2.7: At left and center, two left Riemann sums for a function f that is sometimes
negative; at right, the areas bounded by f on the interval [a, d].
221
Chapter 4 The Definite Integral
For the function pictured in the first graph of Figure 4.2.7, a left Riemann sum with 12 subin-
tervals over [a, d] is shown. The function is negative on the interval b ≤ x ≤ c, so at the
four left endpoints that fall in [b, c], the terms f (x i )∆x are negative. This means that those
four terms in the Riemann sum produce an estimate of the opposite of the area bounded by
y f (x) and the x-axis on [b, c].
In the middle graph of Figure 4.2.7, we see that by increasing the number of rectangles the
approximation of the area (or the opposite of the area) bounded by the curve appears to
improve.
In general, any Riemann sum of a continuous function f on an interval [a, b] approximates
the difference between the area that lies above the horizontal axis on [a, b] and under f
and the area that lies below the horizontal axis on [a, b] and above f . In the notation of
Figure 4.2.7, we may say that
L24 ≈ A1 − A2 + A3 ,
where L 24 is the left Riemann sum using 24 subintervals shown in the middle graph. A1 and
A3 are the areas of the regions where f is positive, and A2 is the area where f is negative.
We will call the quantity A1 − A2 + A3 the net signed area bounded by f over the interval
[a, d], where by the phrase “signed area” we indicate that we are attaching a minus sign to
the areas of regions that fall below the horizontal axis.
Finally, we recall that if the function f represents the velocity of a moving object, the sum
of the areas bounded by the curve tells us the total distance traveled over the relevant time
interval, while the net signed area bounded by the curve computes the object’s change in
position on the interval.
Activity 4.2.4. Suppose that an object moving along a straight line path has its velocity
v (in feet per second) at time t (in seconds) given by
1 2 7
v(t) t − 3t + .
2 2
a. Compute M5 , the middle Riemann sum, for v on the time interval [1, 5]. Be sure
to clearly identify the value of ∆t as well as the locations of t0 , t1 , · · ·, t5 . In addi-
tion, provide a careful sketch of the function and the corresponding rectangles
that are being used in the sum.
b. Building on your work in (a), estimate the total change in position of the object
on the interval [1, 5].
c. Building on your work in (a) and (b), estimate the total distance traveled by the
object on [1, 5].
d. Use appropriate computing technology² to compute M10 and M20 . What exact
value do you think the middle sum eventually approaches as n increases with-
out bound? What does that number represent in the physical context of the
overall problem?
²For instance, consider the applet at http://gvsu.edu/s/a9 and change the function and adjust the locations of
the blue points that represent the interval endpoints a and b.
222
4.2 Riemann Sums
4.2.4 Summary
• A Riemann sum is simply a sum of products of the form f (x ∗i )∆x that estimates the
area between a positive function and the horizontal axis over a given interval. If the
function is sometimes negative on the interval, the Riemann sum estimates the differ-
ence between the areas that lie above the horizontal axis and those that lie below the
axis.
• The three most common types of Riemann sums are left, right, and middle sums, but
we can also work with a more general Riemann sum. The only difference among these
sums is the location of the point at which the function is evaluated to determine the
height of the rectangle whose area is being computed. For a left Riemann sum, we
evaluate the function at the left endpoint of each subinterval, while for right and mid-
dle sums, we use right endpoints and midpoints, respectively.
• The left, right, and middle Riemann sums are denoted L n , R n , and M n , with formulas
∑
n−1
L n f (x0 )∆x + f (x1 )∆x + · · · + f (x n−1 )∆x f (x i )∆x,
i0
∑n
R n f (x1 )∆x + f (x2 )∆x + · · · + f (x n )∆x f (x i )∆x,
i1
∑n
M n f (x 1 )∆x + f (x 2 )∆x + · · · + f (x n )∆x f (x i )∆x,
i1
where x 0 a, x i a + i∆x, and x n b, using ∆x b−a
n . For the midpoint sum,
x i (x i−1 + x i )/2.
223
Chapter 4 The Definite Integral
4.2.5 Exercises
1. Evaluating Riemann sums for a quadratic function. The rectangles in the graph below
−x 2
illustrate a left endpoint Riemann sum for f (x) + 2x on the interval [3, 7].
4
The value of this left endpoint Riemann sum is , and this Riemann sum is (□ an
overestimate of □ equal to □ an underestimate of □ there is ambiguity) the area
of the region enclosed by y f (x), the x-axis, and the vertical lines x 3 and x 7.
Left endpoint Riemann sum for
y −x4 + 2x on [3, 7]
2
The rectangles in the graph below illustrate a right endpoint Riemann sum for f (x)
−x 2
+ 2x on the interval [3, 7].
4
The value of this right endpoint Riemann sum is , and this Riemann sum
is (□ an overestimate of □ equal to □ an underestimate of □ there is ambiguity)
the area of the region enclosed by y f (x), the x-axis, and the vertical lines x 3 and
x 7.
Right endpoint Riemann sum for
y −x4 + 2x on [3, 7]
2
224
4.2 Riemann Sums
2. Estimating distance traveled with a Riemann sum from data. Your task is to estimate
how far an object traveled during the time interval 0 ≤ t ≤ 8, but you only have the
following data about the velocity of the object.
time (sec) 0 1 2 3 4 5 6 7 8
velocity (feet/sec) -4 -2 -3 1 2 3 2 3 4
To get an idea of what the velocity function might look like, you pick up a black pen,
plot the data points, and connect them by curves. Your sketch looks something like the
black curve in the graph below.
Left endpoint approximation
You decide to use a left endpoint Riemann sum to estimate the total displacement. So,
you pick up a blue pen and draw rectangles whose height is determined by the velocity
measurement at the left endpoint of each one-second interval. By using the left end-
point Riemann sum as an approximation, you are assuming that the actual velocity is
approximately constant on each one-second interval (or, equivalently, that the actual
acceleration is approximately zero on each one-second interval), and that the velocity
and acceleration have discontinuous jumps every second. This assumption is proba-
bly incorrect because it is likely that the velocity and acceleration change continuously
over time. However, you decide to use this approximation anyway since it seems like
a reasonable approximation to the actual velocity given the limited amount of data.
(A) Using the left endpoint Riemann sum, find approximately how far the object trav-
eled.
Using the same data, you also decide to estimate how far the object traveled using a
right endpoint Riemann sum. So, you sketch the curve again with a black pen, and
draw rectangles whose height is determined by the velocity measurement at the right
endpoint of each one-second interval.
225
Chapter 4 The Definite Integral
Right endpoint approximation
(B) Using the right endpoint Riemann sum, find approximately how far the object trav-
eled.
3. Writing basic Riemann sums. On a sketch of y e x , represent the left Riemann sum
∫1
with n 2 approximating 0
e x dx. Write out the terms of the sum, but do not evaluate
it.
∫1
On another sketch, represent the right Riemann sum with n 2 approximating 0 e x dx.
Write out the terms of the sum, but do not evaluate it. Which sum is an overestimate?
Which sum is an underestimate?
4. Consider the function f (x) 3x + 4.
a. Compute M4 for y f (x) on the interval [2, 5]. Be sure to clearly identify the
value of ∆x, as well as the locations of x 0 , x1 , . . . , x4 . Include a careful sketch of
the function and the corresponding rectangles being used in the sum.
b. Use a familiar geometric formula to determine the exact value of the area of the
region bounded by y f (x) and the x-axis on [2, 5].
c. Explain why the values you computed in (a) and (b) turn out to be the same. Will
this be true if we use a number different than n 4 and compute M n ? Will L4 or
R 4 have the same value as the exact area of the region found in (b)?
d. Describe the collection of functions 1 for which it will always be the case that M n ,
regardless of the value of n, gives the exact net signed area bounded between the
function 1 and the x-axis on the interval [a, b].
5. Let S be the sum given by
S ((1.4)2 + 1) · 0.4 + ((1.8)2 + 1) · 0.4 + ((2.2)2 + 1) · 0.4 + ((2.6)2 + 1) · 0.4 + ((3.0)2 + 1) · 0.4.
a. Assume that S is a right Riemann sum. For what function f and what interval
[a, b] is S this function’s Riemann sum? Why?
b. How does your answer to (a) change if S is a left Riemann sum? a middle Riemann
sum?
226
4.2 Riemann Sums
c. Suppose that S really is a right Riemann sum. What is geometric quantity does S
approximate?
d. Use sigma notation to write a new sum R that is the right Riemann sum for the
same function, but that uses twice as many subintervals as S.
6. A car traveling along a straight road is braking and its velocity is measured at several
different points in time, as given in the following table.
seconds, t 0 0.3 0.6 0.9 1.2 1.5 1.8
Velocity in ft/sec, v(t) 100 88 74 59 40 19 0
Table 4.2.8: Data for the braking car.
a. Plot the given data on a set of axes with time on the horizontal axis and the ve-
locity on the vertical axis.
b. Estimate the total distance traveled during the car the time brakes using a middle
Riemann sum with 3 subintervals.
c. Estimate the total distance traveled on [0, 1.8] by computing L6 , R 6 , and 12 (L6 +R 6 ).
d. Assuming that v(t) is always decreasing on [0, 1.8], what is the maximum possi-
ble distance the car traveled before it stopped? Why?
7. The rate at which pollution escapes a scrubbing process at a manufacturing plant in-
creases over time as filters and other technologies become less effective. For this par-
ticular example, assume that the rate of pollution (in tons per week) is given by the
function r that is pictured in Figure 4.2.9.
a. Use the graph to estimate the value
of M4 on the interval [0, 4].
tons/week
b. What is the meaning of M4 in terms 4
of the pollution discharged by the y = r(t)
plant? 3
c. Suppose that r(t) 0.5e 0.5t . Use 2
this formula for r to compute L5 on
[0, 4].
1
d. Determine an upper bound on the weeks
total amount of pollution that can
escape the plant during the pic- 1 2 3 4
tured four week time period that is
accurate within an error of at most
one ton of pollution. Figure 4.2.9: The rate, r(t), of pollution
in tons per week.
227
Chapter 4 The Definite Integral
4.3 The Definite Integral
Motivating Questions
• How does increasing the number of subintervals affect the accuracy of the approxi-
mation generated by a Riemann sum?
• What is the definition of the definite integral of a function f over the interval [a, b]?
• What does the definite integral measure exactly, and what are some of the key prop-
erties of the definite integral?
In Figure 4.3.1, we see evidence that increasing the number of rectangles in a Riemann sum
improves the accuracy of the approximation of the net signed area bounded by the given
function.
y = f (x) y = f (x) y = f (x)
A1 A3
A2
a b c d a b c d a b c d
Figure 4.3.1: At left and center, two left Riemann sums for a function f that is sometimes
negative; at right, the exact areas bounded by f on the interval [a, d].
We therefore explore the natural idea of allowing the number of rectangles to increase with-
out bound. In an effort to compute the exact net signed area we also consider the differences
among left, right, and middle Riemann sums and the different results they generate as the
value of n increases. We begin with functions that are exclusively positive on the interval
under consideration.
Preview Activity 4.3.1. Consider the applet found at http://gvsu.edu/s/a9¹. There,
you will initially see the situation shown in Figure 4.3.2.
Note that the value of the chosen Riemann sum is displayed next to the word “rela-
tive,” and that you can change the type of Riemann sum being computed by dragging
the point on the slider bar below the phrase “sample point placement.”
228
4.3 The Definite Integral
Figure 4.3.2: A right Riemann sum with 10 subintervals for the function
x2
f (x) sin(2x) − 10 + 3 on the interval [1, 7]. The value of the sum is R 10 4.90595.
Explore to see how you can change the window in which the function is viewed, as
well as the function itself. You can set the minimum and maximum values of x by
clicking and dragging on the blue points that set the endpoints; you can change the
function by typing a new formula in the “f(x)” window at the bottom; and you can
adjust the overall window by “panning and zooming” by using the Shift key and the
scrolling feature of your mouse. More information on how to pan and zoom can be
found at http://gvsu.edu/s/Fl.
Work accordingly to adjust the applet so that it uses a left Riemann sum with n 5
subintervals for the function is f (x) 2x + 1. You should see the updated figure
shown in Figure 4.3.3. Then, answer the following questions.
a. Update the applet (and view window, as needed) so that the function being
considered is f (x) 2x + 1 on [1, 4], as directed above. For this function on this
interval, compute L n , M n , R n for n 5, n 25, and n 100. What appears to
be the exact area bounded by f (x) 2x + 1 and the x-axis on [1, 4]?
b. Use basic geometry to determine the exact area bounded by f (x) 2x + 1 and
the x-axis on [1, 4].
c. Based on your work in (a) and (b), what do you observe occurs when we increase
the number of subintervals used in the Riemann sum?
d. Update the applet to consider the function f (x) x 2 + 1 on the interval [1, 4]
(note that you need to enter “x ^ 2 + 1” for the function formula). Use the
229
Chapter 4 The Definite Integral
applet to compute L n , M n , R n for n 5, n 25, and n 100. What do you
conjecture is the exact area bounded by f (x) x 2 + 1 and the x-axis on [1, 4]?
e. Why can we not compute the exact value of the area bounded by f (x) x 2 + 1
and the x-axis on [1, 4] using a formula like we did in (b)?
Figure 4.3.3: A left Riemann sum with 5 subintervals for the function f (x) 2x + 1
on the interval [1, 4]. The value of the sum is L5 16.2.
4.3.1 The definition of the definite integral
In Preview Activity 4.3.1, we saw that as the number of rectangles got larger and larger, the
values of L n , M n , and R n all grew closer and closer to the same value. It turns out that this
occurs for any continuous function on an interval [a, b], and also for a Riemann sum using
any point x ∗i+1 in the interval [x i , x i+1 ]. Thus, as we let n → ∞, it doesn’t really matter where
we choose to evaluate the function within a given subinterval, because
∑
n
lim L n lim R n lim M n lim f (x ∗i )∆x.
n→∞ n→∞ n→∞ n→∞
i1
That these limits always exist (and share the same value) when f is continuous² allows us
to make the following definition.
¹Marc Renault, Shippensburg University, Geogebra Applets for Calclulus, http://gvsu.edu/s/5p.
²It turns out that a function need not be continuous in order to have a definite integral. For our purposes, we
assume that the functions we consider are continuous on the interval(s) of interest. It is straightforward to see that
any function that is piecewise continuous on an interval of interest will also have a well-defined definite integral.
230
4.3 The Definite Integral
y = f (x)
A1 A3
A2
a b c d
Figure 4.3.5: A continuous function f on the interval [a, d].
Definition 4.3.4 The definite integral of a continuous function f on the interval [a, b], de-
∫b
noted a
f (x) dx, is the real number given by
∫ b ∑
n
f (x) dx lim f (x ∗i )∆x,
a n→∞
i1
where ∆x b−a
n , x i a + i∆x (for i 0, . . . , n), and x ∗i satisfies x i−1 ≤ x ∗i ≤ x i (for i
1, . . . , n).
∫
We call the symbol the integral sign, the values a and b the limits of integration, and the
∫b
function f the integrand. The process of determining the real number a f (x) dx is called
evaluating the definite integral. While there are several different interpretations of the definite
∫b
integral, for now the most important is that a f (x) dx measures the net signed area bounded
by y f (x) and the x-axis on the interval [a, b].
For example, if f is the function pictured in Figure 4.3.5, and A1 , A2 , and A3 are the exact
areas bounded by f and the x-axis on the respective intervals [a, b], [b, c], and [c, d], then
∫ b ∫ c
f (x) dx A1 , f (x) dx −A2 ,
a b
∫ d
f (x) dx A3 ,
c
∫ d
and f (x) dx A1 − A2 + A3 .
a
We can also use definite integrals to express the change in position and the distance traveled
by a moving object. If v is a velocity function on an interval [a, b], then the change in position
231
Chapter 4 The Definite Integral
f (x) = 2x + 1
9
3 R4
1 (2x + 1) dx
1 4
Figure 4.3.6: The area bounded by f (x) 2x + 1 and the x-axis on the interval [1, 4].
of the object, s(b) − s(a), is given by
∫ b
s(b) − s(a) v(t) dt.
a
∫b
If the velocity function is nonnegative on [a, b], then a v(t) dt tells us the distance the object
traveled. If the velocity is sometimes negative on [a, b], we can use definite integrals to find
the areas bounded by the function on each interval where v does not change sign, and the
sum of these areas will tell us the distance the object traveled.
To compute the value of a definite integral from the definition, we have to take the limit
of a sum. While this is possible to do in select circumstances, it is also tedious and time-
consuming, and does not offer much additional insight into the meaning or interpretation
of the definite integral. Instead, in Section 4.4, we will learn the Fundamental Theorem of
Calculus, which provides a shortcut for evaluating a large class of definite integrals. This
will enable us to determine the exact net signed area bounded by a continuous function and
the x-axis in many circumstances.
For now, our goal is to understand the meaning and properties of the definite integral, rather
than to compute its value. To do this, we will rely on the net signed area interpretation of
the definite integral. So we will use as examples curves that produce regions whose areas
we can compute exactly through area formulas. We can thus compute the exact value of the
corresponding integral.
∫4
For instance, if we wish to evaluate the definite integral 1 (2x + 1) dx, we observe that the
region bounded by this function and the x-axis is the trapezoid shown in Figure 4.3.6. By
the formula for the area of a trapezoid, A 12 (3 + 9) · 3 18, so
∫ 4
(2x + 1) dx 18.
1
232
4.3 The Definite Integral
Activity 4.3.2. Use known geometric formulas and the net signed area interpretation
of the definite integral to evaluate each of the definite integrals below.
∫1
a. 0
3x dx
∫4
b. −1
(2 − 2x) dx
∫1 √
c. −1
1 − x 2 dx
∫4
d. −3
1(x) dx, where 1 is the function pictured in Figure 4.3.7. Assume that each
portion of 1 is either part of a line or part of a circle.
y = g(x)
1
-3 -2 -1 1 2 3 4
-1
Figure 4.3.7: A function 1 that is piecewise defined; each piece of the function is part
of a circle or part of a line.
4.3.2 Some properties of the definite integral
Regarding the definite integral of a function f over an interval [a, b] as the net signed area
bounded by f and the x-axis, we discover several standard properties of the definite integral.
It is helpful to remember that the definite integral is defined in terms of Riemann sums,
which consist of the areas of rectangles.
∫a
For any real number a and the definite integral a f (x) dx it is evident that no area is en-
closed, because the interval begins and ends with the same point. Hence,
∫a
If f is a continuous function and a is a real number, then a
f (x) dx 0.
Next, we consider the result of subdividing the interval of integration. In Figure 4.3.8, we
see that
∫ b ∫ c
f (x) dx A1 , f (x) dx A2 ,
a b
∫ c
and f (x) dx A1 + A2 ,
a
233
Chapter 4 The Definite Integral
which illustrates the following general rule.
y = f (x)
A1 A2
a b c
Figure 4.3.8: The area bounded by y f (x) on the interval [a, c].
If f is a continuous function and a, b, and c are real numbers, then
∫ c ∫ b ∫ c
f (x) dx f (x) dx + f (x) dx.
a a b
While this rule is easy to see if a < b < c, it in fact holds in general for any values of a, b, and
c. Another property of the definite integral states that if we reverse the order of the limits of
integration, we change the sign of the integral’s value.
If f is a continuous function and a and b are real numbers, then
∫ a ∫ b
f (x) dx − f (x) dx.
b a
This result makes sense because if we integrate from a to b, then in the defining Riemann
sum we set ∆x b−an , while if we integrate from b to a, we have ∆x n − n , and this
a−b b−a
is the only change in the sum used to define the integral.
There are two additional useful properties of the definite integral. When we worked with
derivative rules in Chapter 2, we formulated the Constant Multiple Rule and the Sum Rule.
Recall that the Constant Multiple Rule says that if f is a differentiable function and k is a
constant, then
d
[k f (x)] k f ′(x),
dx
234
4.3 The Definite Integral
and the Sum Rule says that if f and 1 are differentiable functions, then
d
[ f (x) + 1(x)] f ′(x) + 1 ′(x).
dx
These rules are useful because they allow to deal individually with the simplest parts of
certain functions by taking advantage of addition and multiplying by a constant. In other
words, the process of taking the derivative respects addition and multiplying by constants
in the simplest possible way.
It turns out that similar rules hold for the definite integral. First, let’s consider the functions
pictured in Figure 4.3.9.
B = 2 f (xi )△x
y = 2 f (x)
A = f (xi )△x
y = f (x)
B
A
a xi xi+1 b a xi xi+1 b
Figure 4.3.9: The areas bounded by y f (x) and y 2 f (x) on [a, b].
Because multiplying the function by 2 doubles its height at every x-value, we see that the
height of each rectangle in a left Riemann sum is doubled, f (x i ) for the original function,
versus 2 f (x i ) in the doubled function. For the areas A and B, it follows B 2A. As this is
true regardless of the value of n or the type of sum we use, we see that in the limit, the area
of the red region bounded by y 2 f (x) will be twice the area of the blue region bounded
by y f (x). As there is nothing special about the value 2 compared to an arbitrary constant
k, the following general principle holds.
Constant Multiple Rule.
If f is a continuous function and k is any real number, then
∫ b ∫ b
k · f (x) dx k f (x) dx.
a a
We see a similar situation with the sum of two functions f and 1.
235
Chapter 4 The Definite Integral
C = ( f (xi ) + g(xi ))△x
f +g
A = f (xi )△x
f
B = g(xi )△x
C
A g
B
a xi xi+1 b a xi xi+1 b a xi xi+1 b
Figure 4.3.10: The areas bounded by y f (x) and y 1(x) on [a, b], as well as the area
bounded by y f (x) + 1(x).
If we take the sum of two functions f and 1 at every point in the interval, the height of the
function f + 1 is given by ( f + 1)(x i ) f (x i ) + 1(x i ). Hence, for the pictured rectangles
with areas A, B, and C, it follows that C A + B. Because this will occur for every such
rectangle, in the limit the area of the gray region will be the sum of the areas of the blue and
red regions. In terms of definite integrals, we have the following general rule.
Sum Rule.
If f and 1 are continuous functions, then
∫ b ∫ b ∫ b
[ f (x) + 1(x)] dx f (x) dx + 1(x) dx.
a a a
The Constant Multiple and Sum Rules can be combined to say that for any continuous func-
tions f and 1 and any constants c and k,
∫ b ∫ b ∫ b
[c f (x) ± k1(x)] dx c f (x) dx ± k 1(x) dx.
a a a
Activity 4.3.3. Suppose that the following information is known about the functions
f , 1, x 2 , and x 3 :
∫2 ∫5
• 0
f (x) dx −3; 2
f (x) dx 2
∫2 ∫5
• 0
1(x) dx 4; 2
1(x) dx −1
∫2 ∫5
• 0
x 2 dx 83 ; 2
x 2 dx 117
3
∫2 ∫5
• 0
x 3 dx 4; 2
x 3 dx 609
4
Use the provided information and the rules discussed in the preceding section to
evaluate each of the following definite integrals.
236
4.3 The Definite Integral
∫2 ∫5
a. 5
f (x) dx d. 2
(3x 2 − 4x 3 ) dx
∫5
b. 0
1(x) dx
∫5 ∫0
c. 0
( f (x) + 1(x)) dx e. 5
(2x 3 − 71(x)) dx
4.3.3 How the definite integral is connected to a function’s average value
One of the most valuable applications of the definite integral is that it provides a way to
discuss the average value of a function, even for a function that takes on infinitely many
values. Recall that if we wish to take the average of n numbers y1 , y2 , . . ., y n , we compute
y1 + y2 + · · · + y n
AV G .
n
Since integrals arise from Riemann sums in which we add n values of a function, it should
not be surprising that evaluating an integral is similar to averaging the output values of a
function. Consider, for instance, the right Riemann sum R n of a function f , which is given
by
R n f (x1 )∆x + f (x2 )∆x + · · · + f (x n )∆x ( f (x 1 ) + f (x2 ) + · · · + f (x n ))∆x.
Since ∆x b−a
n , we can thus write
b−a
R n ( f (x1 ) + f (x 2 ) + · · · + f (x n )) ·
n
f (x 1 ) + f (x2 ) + · · · + f (x n )
(b − a) . (4.3.1)
n
We see that the right Riemann sum with n subintervals is just the length of the interval (b −a)
times the average of the n function values found at the right endpoints. And just as with
our efforts to compute area, the larger the value of n we use, the more accurate our average
will be. Indeed, we will define the average value of f on [a, b] to be
f (x1 ) + f (x2 ) + · · · + f (x n )
fAVG[a,b] lim .
n→∞ n
But we also know that for any continuous function f on [a, b], taking the limit of a Riemann
∫b
sum leads precisely to the definite integral. That is, limn→∞ R n a
f (x) dx, and thus taking
the limit as n → ∞ in Equation (4.3.1), we have that
∫ b
f (x) dx (b − a) · fAVG[a,b] . (4.3.2)
a
Solving Equation (4.3.2) for fAVG[a,b] , we have the following general principle.
237
Chapter 4 The Definite Integral
Average value of a function.
If f is a continuous function on [a, b], then its average value on [a, b] is given by the
formula ∫ b
1
fAVG[a,b] · f (x) dx.
b−a a
Equation (4.3.2) tells us another way to interpret the definite integral: the definite integral
of a function f from a to b is the length of the interval (b − a) times the average value of the
function on the interval. In addition, when the function f is nonnegative on [a, b], Equa-
tion (4.3.2) has a natural visual interpretation.
y = f (x) y = f (x) y = f (x)
fAVG[a,b] A2
A1
Rb
a f (x) dx (b − a) · fAVG[a,b]
a b a b a b
Figure 4.3.11: A function y f (x), the area it bounds, and its average value on [a, b].
∫b
Consider Figure 4.3.11, where we see at left the shaded region whose area is a f (x) dx, at
center the shaded rectangle whose dimensions are (b − a) by fAVG[a,b] , and at right these two
figures superimposed. Note that in dark green we show the horizontal line y fAVG[a,b] .
Thus, the area of the green rectangle is given by (b − a) · fAVG[a,b] , which is precisely the
∫b
value of a f (x) dx. The area of the blue region in the left figure is the same as the area of
the green rectangle in the center figure. We can also observe that the areas A1 and A2 in the
rightmost figure appear to be equal. Thus, knowing the average value of a function enables
us to construct a rectangle whose area is the same as the value of the definite integral of the
function on the interval. The java applet³ at http://gvsu.edu/s/az provides an opportunity
to explore how the average value of the function changes as the interval changes, through
an image similar to that found in Figure 4.3.11.
√
Activity 4.3.4. Suppose that v(t) 4 − (t − 2)2 tells us the instantaneous velocity of
a moving object on the interval 0 ≤ t ≤ 4, where t is measured in minutes and v is
measured in meters per minute.
√
a. Sketch an accurate graph of y v(t). What kind of curve is y 4 − (t − 2)2 ?
∫4
b. Evaluate 0
v(t) dt exactly.
³David Austin, http://gvsu.edu/s/5r.
238
4.3 The Definite Integral
c. In terms of the physical problem of the moving object with velocity v(t), what
∫4
is the meaning of 0
v(t) dt? Include units on your answer.
d. Determine the exact average value of v(t) on [0, 4]. Include units on your answer.
e. Sketch a rectangle whose base is the line segment from t 0 to t 4 on the
∫4
t-axis such that the rectangle’s area is equal to the value of 0
v(t) dt. What is
the rectangle’s exact height?
f. How can you use the average value you found in (d) to compute the total dis-
tance traveled by the moving object over [0, 4]?
4.3.4 Summary
• Any Riemann sum of a continuous function f on an interval [a, b] provides an estimate
of the net signed area bounded by the function and the horizontal axis on the interval.
Increasing the number of subintervals in the Riemann sum improves the accuracy of
this estimate, and letting the number of subintervals increase without bound results
in the values of the corresponding Riemann sums approaching the exact value of the
enclosed net signed area.
• When we take the limit of Riemann sums, we arrive at what we call the definite integral
∫b
of f over the interval [a, b]. In particular, the symbol a f (x) dx denotes the definite
integral of f over [a, b], and this quantity is defined by the equation
∫ b ∑
n
f (x) dx lim f (x ∗i )∆x,
a n→∞
i1
where ∆x b−a
n , x i a + i∆x (for i 0, . . . , n), and x ∗i satisfies x i−1 ≤ x ∗i ≤ x i (for
i 1, . . . , n).
∫b
• The definite integral a f (x) dx measures the exact net signed area bounded by f and
the horizontal axis on [a, b]; in addition, the value of the definite integral is related to
∫b
what we call the average value of the function on [a, b]: fAVG[a,b] b−a · a f (x) dx. In
1
∫b
the setting where we consider the integral of a velocity function v, a
v(t) dt measures
the exact change in position of the moving object on [a, b]; when v is nonnegative,
∫b
a
v(t) dt is the object’s distance traveled on [a, b].
• The definite integral is a sophisticated sum, and thus has some of the same natural
properties that finite sums have. Perhaps most important of these is how the definite
integral respects sums and constant multiples of functions, which can be summarized
by the rule
∫ b ∫ b ∫ b
[c f (x) ± k1(x)] dx c f (x) dx ± k 1(x) dx
a a a
where f and 1 are continuous functions on [a, b] and c and k are arbitrary constants.
239
Chapter 4 The Definite Integral
4.3.5 Exercises
1. Evaluating definite integrals from graphical information. Use the following figure,
which shows a graph of f (x) to find each of the indicated integrals.
Note that the first area (with vertical,
red shading) is 18 and the second (with
oblique, black shading) is 6.
∫b
A. f (x)dx
∫ ac
B. b f (x)dx
∫c
C. a f (x)dx
∫c
D. a
| f (x)|dx
2. Estimating definite integrals from a graph. Use the graph of f (x) shown below to find
the following integrals.
∫0
A. −4 f (x)dx
B. If the vertical red shaded∫ area in the
6
graph has area A, estimate: −4 f (x)dx
(Your estimate may be written in terms of A.)
3. Finding the average value of a linear function. Find the average value of f (x) 7x + 1
over [3, 8].
4. Finding the average value of a function given graphically. The figure below to the left
is a graph of f (x), and below to the right is 1(x).
240
4.3 The Definite Integral
f (x) 1(x)
(a) What is the average value of f (x) on 0 ≤ x ≤ 2?
(b) What is the average value of 1(x) on 0 ≤ x ≤ 2?
(c) What is the average value of f (x) · 1(x) on 0 ≤ x ≤ 2?
(d) Is the following statement true?
Average( f ) · Average(g) Average( f · 1)
5. Estimating a definite integral and average value from a graph. Use the figure below,
which shows the graph of y f (x), to answer the following questions.
∫3
A. Estimate the integral: −3 f (x) dx
B. Which of the following average values
of f is larger?
⊙ Between x −3 and x 3
⊙ Between x 0 and x 3
6. Using rules to combine known integral values. Suppose
∫ −4.5 ∫ −7.5 ∫ −4.5
f (x) dx 10, f (x) dx 8, and f (x) dx 10.
−9 −9 −6
Find ∫ ∫
−6 −7.5
f (x) dx and (10 f (x) − 8) dx.
−7.5 −6
241
Chapter 4 The Definite Integral
7. The velocity of an object moving along an axis is given by the piecewise linear function
v that is pictured in Figure 4.3.12. Assume that the object is moving to the right when
its velocity is positive, and moving to the left when its velocity is negative. Assume that
the given velocity function is valid for t 0 to t 4.
a. Write an expression involving defi- ft/sec
nite integrals whose value is the to- 2
tal change in position of the object
on the interval [0, 4]. 1
b. Use the provided graph of v to
sec
determine the value of the total
change in position on [0, 4]. 1 2 3 4
c. Write an expression involving defi- -1
nite integrals whose value is the to- y = v(t)
tal distance traveled by the object
on [0, 4]. What is the exact value of -2
the total distance traveled on [0, 4]?
d. What is the object’s exact average
Figure 4.3.12: The velocity function of a
velocity on [0, 4]?
moving object.
e. Find an algebraic formula for the
object’s position function on [0, 1.5]
that satisfies s(0) 0.
8. Suppose that the velocity of a moving object is given by v(t) t(t − 1)(t − 3), measured
in feet per second, and that this function is valid for 0 ≤ t ≤ 4.
a. Write an expression involving definite integrals whose value is the total change
in position of the object on the interval [0, 4].
b. Use appropriate technology (such as http://gvsu.edu/s/a9⁴) to compute Rie-
mann sums to estimate the object’s total change in position on [0, 4]. Work to
ensure that your estimate is accurate to two decimal places, and explain how you
know this to be the case.
c. Write an expression involving definite integrals whose value is the total distance
traveled by the object on [0, 4].
d. Use appropriate technology to compute Riemann sums to estimate the object’s
total distance travelled on [0, 4]. Work to ensure that your estimate is accurate to
two decimal places, and explain how you know this to be the case.
e. What is the object’s average velocity on [0, 4], accurate to two decimal places?
9. Consider the graphs of two functions f and 1 that are provided in Figure 4.3.13. Each
piece of f and 1 is either part of a straight line or part of a circle.
⁴Marc Renault, Shippensburg University.
242
4.3 The Definite Integral
y = g(x)
2 2
y = f (x)
1 1
1 2 3 4 1 2 3 4
-1 -1
-2 -2
Figure 4.3.13: Two functions f and 1.
∫1
a. Determine the exact value of 0
[ f (x) + 1(x)] dx.
∫4
b. Determine the exact value of 1
[2 f (x) − 31(x)] dx.
c. Find the exact average value of h(x) 1(x) − f (x) on [0, 4].
d. For what constant c does the following equation hold?
∫ 4 ∫ 4
c dx [ f (x) + 1(x)] dx
0 0
10. Let f (x) 3 − x 2 and 1(x) 2x 2 .
a. On the interval [−1, 1], sketch a labeled graph of y f (x) and write a definite
integral whose value is the exact area bounded by y f (x) on [−1, 1].
b. On the interval [−1, 1], sketch a labeled graph of y 1(x) and write a definite
integral whose value is the exact area bounded by y 1(x) on [−1, 1].
c. Write an expression involving a difference of definite integrals whose value is the
exact area that lies between y f (x) and y 1(x) on [−1, 1].
d. Explain why your expression in (c) has the same value as the single integral
∫1
−1
[ f (x) − 1(x)] dx.
e. Explain why, in general, if p(x) ≥ q(x) for all x in [a, b], the exact area between
y p(x) and y q(x) is given by
∫ b
[p(x) − q(x)] dx.
a
243
Chapter 4 The Definite Integral
4.4 The Fundamental Theorem of Calculus
Motivating Questions
• How can we find the exact value of a definite integral without taking the limit of a
Riemann sum?
• What is the statement of the Fundamental Theorem of Calculus, and how do anti-
derivatives of functions play a key role in applying the theorem?
• What is the meaning of the definite integral of a rate of change in contexts other than
when the rate of change represents velocity?
Much of our work in Chapter 4 has been motivated by the velocity-distance problem: if we
know the instantaneous velocity function, v(t), for a moving object on a given time interval
[a, b], can we determine the distance it traveled on [a, b]? If the velocity function is nonneg-
ative on [a, b], the area bounded by y v(t) and the t-axis on [a, b] is equal to the distance
∫b
traveled. This area is also the value of the definite integral a v(t) dt. If the velocity is some-
times negative, the total area bounded by the velocity function still tells us distance traveled,
while the net signed area tells us the object’s change in position.
For instance, for the velocity function in Figure 4.4.1, the total distance D traveled by the
moving object on [a, b] is
D A1 + A2 + A3 ,
and the total change in the object’s position is
s(b) − s(a) A1 − A2 + A3 .
The areas A1 , A2 , and A3 are each given by definite integrals, which may be computed by
limits of Riemann sums (and in special circumstances by geometric formulas).
y = v(t)
A1 A3
a A2 b
Figure 4.4.1: A velocity function that is sometimes negative.
244
4.4 The Fundamental Theorem of Calculus
We turn our attention to an alternate approach.
Preview Activity 4.4.1. A student with a third floor dormitory window 32 feet off the
ground tosses a water balloon straight up in the air with an initial velocity of 16 feet
per second. It turns out that the instantaneous velocity of the water balloon is given
by v(t) −32t + 16, where v is measured in feet per second and t is measured in
seconds.
a. Let s(t) represent the height of the water balloon above ground at time t, and
note that s is an antiderivative of v. That is, v is the derivative of s: s ′(t) v(t).
Find a formula for s(t) that satisfies the initial condition that the balloon is tossed
from 32 feet above ground. In other words, make your formula for s satisfy
s(0) 32.
b. When does the water balloon reach its maximum height? When does it land?
c. Compute s( 12 ) − s(0), s(2) − s( 12 ), and s(2) − s(0). What do these represent?
d. What is the total vertical distance traveled by the water balloon from the time it
is tossed until the time it lands?
e. Sketch a graph of the velocity function y v(t) on the time interval [0, 2]. What
is the total net signed area bounded by y v(t) and the t-axis on [0, 2]? Answer
this question in two ways: first by using your work above, and then by using a
familiar geometric formula to compute areas of certain relevant regions.
4.4.1 The Fundamental Theorem of Calculus
Suppose we know the position function s(t) and the velocity function v(t) of an object mov-
ing in a straight line, and for the moment let us assume that v(t) is positive on [a, b]. Then,
as shown in Figure 4.4.2, we know two different ways to compute the distance, D, the object
travels: one is that D s(b) − s(a), the object’s change in position. The other is the area
∫b
under the velocity curve, which is given by the definite integral, so D a v(t) dt. Since
both of these expressions tell us the distance traveled, it follows that they are equal, so
∫ b
s(b) − s(a) v(t) dt. (4.4.1)
a
Equation (4.4.1) holds even when velocity is sometimes negative, because s(b) − s(a),the
object’s change in position, is also measured by the net signed area on [a, b] which is given
∫b
by a
v(t) dt.
Perhaps the most powerful fact Equation (4.4.1) reveals is that we can compute the integral’s
value if we can find a formula for s. Remember, s and v are related by the fact that v is the
derivative of s, or equivalently that s is an antiderivative of v.
Example 4.4.3 Determine the exact distance traveled on [1, 5] by an object with velocity func-
245
Chapter 4 The Definite Integral
y = v(t)
Rb
D= a v(t) dt
= s(b) − s(a)
a b
Figure 4.4.2: Finding distance traveled when we know a velocity function v.
tion v(t) 3t 2 + 40 feet per second. The distance traveled on the interval [1, 5] is given by
∫ 5 ∫ 5
D v(t) dt (3t 2 + 40) dt s(5) − s(1),
1 1
where s is an antiderivative of v. Now, the derivative of t 3 is 3t 2 and the derivative of 40t is
40, so it follows that s(t) t 3 + 40t is an antiderivative of v. Therefore,
∫ 5
D 3t 2 + 40 dt s(5) − s(1)
1
(5 + 40 · 5) − (13 + 40 · 1) 284 feet.
3
Note the key lesson of Example 4.4.3: to find the distance traveled, we need to compute the
area under a curve, which is given by the definite integral. But to evaluate the integral, we
can find an antiderivative, s, of the velocity function, and then compute the total change in
s on the interval. In particular, we can evaluate the integral without computing the limit
of a Riemann sum. It will be convenient to have a shorthand symbol for a function’s anti-
derivative. For a continuous function f , we will often denote an antiderivative of f by F,
so that F′(x) f (x) for all relevant x. Using the notation V in place of s (so that V is an
antiderivative of v) in Equation (4.4.1), we can write
∫ b
V(b) − V(a) v(t) dt. (4.4.2)
a
∫b
Now, to evaluate the definite integral a f (x) dx for an arbitrary continuous function f , we
could certainly think of f as representing the velocity of some moving object, and x as the
variable that represents time. But Equations (4.4.1) and (4.4.2) hold for any continuous ve-
locity function, even when v is sometimes negative. So Equation (4.4.2) offers a shortcut
route to evaluating any definite integral, provided that we can find an antiderivative of the
246
4.4 The Fundamental Theorem of Calculus
140
120 y = v(t)
100
80
60
40 D=
R5
v(t) dt
1
20
= 284
1 3 5
Figure 4.4.4: The exact area of the region enclosed by v(t) 3t 2 + 40 on [1, 5].
integrand. The Fundamental Theorem of Calculus (FTC) summarizes these observations.
Fundamental Theorem of Calculus.
∫b
If f is a continuous function on [a, b], and F is any antiderivative of f , then a
f (x) dx
F(b) − F(a).
A common alternate notation for F(b) − F(a) is
F(b) − F(a) F(x)| ba ,
where we read the righthand side as “the function F evaluated from a to b.” In this notation,
the FTC says that
∫ b
f (x) dx F(x)| ba .
a
The FTC opens the door to evaluating a wide range of integrals if we can find an antideriv-
[ 3 x ] x 2 , the FTC tells us that
d 1 3
ative F for the integrand f . For instance since dx
∫ 1 1
1 3
x 2 dx x
0 3 0
1 3 1 3
(1) − (0)
3 3
1
.
3
But finding an antiderivative can be far from simple; it is often difficult or even impossible.
While we can differentiate just about any function, even some relatively simple functions
don’t have an elementary antiderivative. A significant portion of integral calculus (which
247
Chapter 4 The Definite Integral
is the main focus of second semester college calculus) is devoted to the problem of finding
antiderivatives.
Activity 4.4.2. Use the Fundamental Theorem of Calculus to evaluate each of the
following integrals exactly. For each, sketch a graph of the integrand on the relevant
interval and write one sentence that explains the meaning of the value of the integral
in terms
∫ 4 of the (net signed) area bounded by the curve.
∫1
a. −1 (2 − 2x) dx d. −1 x 5 dx
∫ π
2
b. 0
sin(x) dx
∫1 ∫2
c. 0
e x dx e. 0
(3x 3 − 2x 2 − e x ) dx
4.4.2 Basic antiderivatives
The general problem of finding an antiderivative is difficult. In part, this is due to the fact
that we are trying to undo the process of differentiating, and the undoing is much more
difficult than the doing. For example, while it is evident that an antiderivative of f (x)
sin(x) is F(x) − cos(x) and that an antiderivative of 1(x) x 2 is G(x) 13 x 3 , combinations
of f and 1 can be far more complicated. Consider the functions
sin(x)
5 sin(x) − 4x 2 , x 2 sin(x), , and sin(x 2 ).
x2
What is involved in trying to find an antiderivative for each? From our experience with de-
rivative rules, we know that derivatives of sums and constant multiples of basic functions
are simple to execute, but derivatives involving products, quotients, and composites of fa-
miliar functions are more complicated. Therefore, it stands to reason that antidifferentiating
products, quotients, and composites of basic functions may be even more challenging. We
defer our study of all but the most elementary antiderivatives to later in the text.
We do note that whenever we know the derivative of a function, we have a function-derivative
pair, so we also know the antiderivative of a function. For instance, since we know that
d
[− cos(x)] sin(x),
dx
we also know that F(x) − cos(x) is an antiderivative of f (x) sin(x). F and f together
form a function-derivative pair. Clearly, every basic derivative rule leads us to such a pair,
and thus to a known antiderivative.
In Activity 4.4.3, we will construct a list of the basic antiderivatives we know at this time.
Those rules will help us antidifferentiate sums and constant multiples of basic functions.
For example, since − cos(x) is an antiderivative of sin(x) and 13 x 3 is an antiderivative of x 2 ,
it follows that
4
F(x) −5 cos(x) − x 3
3
is an antiderivative of f (x) 5 sin(x) − 4x 2 , by the sum and constant multiple rules for
differentiation.
248
4.4 The Fundamental Theorem of Calculus
Finally, before proceeding to build a list of common functions whose antiderivatives we
know, we recall that each function has more than one antiderivative. Because the derivative
of any constant is zero, we may add a constant of our choice to any antiderivative. For
instance, we know that G(x) 31 x 3 is an antiderivative of 1(x) x 2 . But we could also have
chosen G(x) 13 x 3 + 7, since in this case as well, G′(x) x 2 . If 1(x) x 2 , we say that the
general antiderivative of 1 is
1 3
G(x) x + C,
3
where C represents an arbitrary real number constant. Regardless of the formula for 1,
including +C in the formula for its antiderivative G results in the most general possible
antiderivative.
Our current interest in antiderivatives is so that we can evaluate definite integrals by the
Fundamental Theorem of Calculus. For that task, the constant C is irrelevant, and we usually
omit it. To see why, consider the definite integral
∫ 1
x 2 dx.
0
For the integrand 1(x) x 2 , suppose we find and use the general antiderivative G(x)
3 x + C. Then, by the FTC,
1 3
∫ 1 1
1 3
x 2 dx x +C
0 3 0
( ) ( )
1 3 1
(1) + C − (0)3 + C
3 3
1
+C−0−C
3
1
.
3
Observe that the C-values appear as opposites in the evaluation of the integral and thus do
not affect the definite integral’s value.
In the following activity, we work to build a list of basic functions whose antiderivatives we
already know.
Activity 4.4.3. Use your knowledge of derivatives of basic functions to complete Ta-
ble 4.4.5 of antiderivatives. For each entry, your task is to find a function F whose
derivative is the given function f . When finished, use the FTC and the results in the
table to evaluate the three given definite integrals.
249
Chapter 4 The Definite Integral
given function, f (x) antiderivative, F(x)
k, (k is constant)
x n , n , −1
x, x > 0
1
sin(x)
cos(x)
sec(x) tan(x)
csc(x) cot(x)
sec2 (x)
csc2 (x)
ex
a x (a > 1)
1
1+x 2
√ 1
1−x 2
Table 4.4.5: Familiar basic functions and their antiderivatives.
∫ 1 ( )
a. x 3 − x − e x + 2 dx
0
∫ π/3
b. (2 sin(t) − 4 cos(t) + sec2 (t) − π) dt
0
∫ 1 √
c. ( x − x 2 ) dx
0
4.4.3 The total change theorem
Let us review three interpretations of the definite integral.
• For a moving object with instantaneous velocity v(t), the object’s change in position on
∫b ∫b
the time interval [a, b] is given by a v(t) dt, and whenever v(t) ≥ 0 on [a, b], a
v(t) dt
tells us the total distance traveled by the object on [a, b].
∫b
• For any continuous function f , its definite integral a f (x) dx represents the net signed
area bounded by y f (x) and the x-axis on [a, b], where regions that lie below the
x-axis have a minus sign associated with their area.
• The value of a definite integral is linked to the average value of a function: for a con-
tinuous function f on [a, b], its average value fAVG[a,b] is given by
∫ b
1
fAVG[a,b] f (x) dx.
b−a a
250
4.4 The Fundamental Theorem of Calculus
The Fundamental Theorem of Calculus now enables us to evaluate exactly (without taking a
limit of Riemann sums) any definite integral for which we are able to find an antiderivative
of the integrand.
A slight change in perspective allows us to gain even more insight into the meaning of the
definite integral. Recall Equation (4.4.2), where we wrote the Fundamental Theorem of Cal-
culus for a velocity function v with antiderivative V as
∫ b
V(b) − V(a) v(t) dt.
a
If we instead replace V with s (which represents position) and replace v with s ′ (since ve-
locity is the derivative of position), Equation (4.4.2) then reads as
∫ b
s(b) − s(a) s ′(t) dt. (4.4.3)
a
In words, this version of the FTC tells us that the total change in an object’s position function
on a particular interval is given by the definite integral of the position function’s derivative
over that interval.
Of course, this result is not limited to only the setting of position and velocity. Writing the
result in terms of a more general function f , we have the Total Change Theorem.
Total Change Theorem.
If f is a continuously differentiable function on [a, b] with derivative f ′, then f (b) −
∫b
f (a) a f ′(x) dx. That is, the definite integral of the rate of change of a function on
[a, b] is the total change of the function itself on [a, b].
The Total Change Theorem tells us more about the relationship between the graph of a func-
tion and that of its derivative. Recall that heights on the graph of the derivative function are
equal to slopes on the graph of the function itself. If instead we know f ′ and are seeking
information about f , we can say the following:
differences in heights on f correspond to net signed areas bounded by f ′.
To see why this is so, consider the difference f (1) − f (0). This value is 3, because f (1) 3
and f (0) 0, but also because the net signed area bounded by y f ′(x) on [0, 1] is 3. That
is, ∫ 1
f (1) − f (0) f ′(x) dx.
0
In addition to this observation about area, the Total Change Theorem enables us to answer
questions about a function whose rate of change we know.
Example 4.4.7 Suppose that pollutants are leaking out of an underground storage tank at a
rate of r(t) gallons/day, where t is measured in days. It is conjectured that r(t) is given by the
formula r(t) 0.0069t 3 − 0.125t 2 + 11.079 over a certain 12-day period. The graph of y r(t)
251
Chapter 4 The Definite Integral
(2, 4)
4 4
(3, 3)
3 3
(1, 3)
2 2
3
1 1
1 3 4 (0, 0) (4, 0)
1 2 1 1 2 3 4
-1 -1
3
-2 -2
y = f ′ (x) y = f (x)
-3 -3
-4 -4
Figure 4.4.6: The graphs of f ′(x) 4 − 2x (at left) and an antiderivative f (x) 4x − x 2 at
right. Differences in heights on f correspond to net signed areas bounded by f ′.
∫ 10
is given in Figure 4.4.8. What is the meaning of 4 r(t) dt and what is its value? What is the
average rate at which pollutants are leaving the tank on the time interval 4 ≤ t ≤ 10?
gal/day
12
10
8 y = r(t)
6
4
2
days
2 4 6 8 10 12
Figure 4.4.8: The rate r(t) of pollution leaking from a tank, measured in gallons per day.
∫ 10
Solution. Since r(t) ≥ 0, the value of 4
r(t) dt is the area under the curve on the interval
252
4.4 The Fundamental Theorem of Calculus
[4, 10]. A Riemann sum for this area will have rectangles with heights measured in gallons
per day and widths measured in days, so the area of each rectangle will have units of
gallons
· days gallons.
day
Thus, the definite integral tells us the total number of gallons of pollutant that leak from the
tank from day 4 to day 10. The Total Change Theorem tells us the same thing: if we let R(t)
denote the total number of gallons of pollutant that have leaked from the tank up to day t,
then R′(t) r(t), and
∫ 10
r(t) dt R(10) − R(4),
4
the number of gallons that have leaked from day 4 to day 10.
To compute the exact value of the integral, we use the Fundamental Theorem of Calculus.
Antidifferentiating r(t) 0.0069t 3 − 0.125t 2 + 11.079, we find that
∫ 10 10
1 4 1
0.0069t 3 − 0.125t 2 + 11.079 dt 0.0069 · t − 0.125 · t 3 + 11.079t
4 4 3 4
≈ 44.282.
Thus, approximately 44.282 gallons of pollutant leaked over the six day time period.
To find the average rate at which pollutant leaked from the tank over 4 ≤ t ≤ 10, we compute
the average value of r on [4, 10]. Thus,
∫ 10
1 44.282
rAVG[4,10] r(t) dt ≈ 7.380
10 − 4 4 6
gallons per day.
Activity 4.4.4. During a 40-minute workout, a person riding an exercise machine
burns calories at a rate of c calories per minute, where the function y c(t) is given
in Figure 4.4.9. On the interval 0 ≤ t ≤ 10, the formula for c is c(t) −0.05t 2 + t + 10,
while on 30 ≤ t ≤ 40, its formula is c(t) −0.05t 2 + 3t − 30.
a. What is the exact total number of calories the person burns during the first 10
minutes of her workout?
b. Let C(t) be an antiderivative of c(t). What is the meaning of C(40) − C(0) in the
context of the person exercising? Include units on your answer.
c. Determine the exact average rate at which the person burned calories during
the 40-minute workout.
d. At what time(s), if any, is the instantaneous rate at which the person is burning
calories equal to the average rate at which she burns calories, on the time interval
0 ≤ t ≤ 40?
253
Chapter 4 The Definite Integral
cal/min
15 y = c(t)
10
5
min
10 20 30 40
Figure 4.4.9: The rate c(t) at which a person exercising burns calories, measured in
calories per minute.
4.4.4 Summary
• We can find the exact value of a definite integral without taking the limit of a Riemann
sum or using a familiar area formula by finding the antiderivative of the integrand,
and hence applying the Fundamental Theorem of Calculus.
• The Fundamental Theorem of Calculus says that if f is a continuous function on [a, b]
and F is an antiderivative of f , then
∫ b
f (x) dx F(b) − F(a).
a
Hence, if we can find an antiderivative for the integrand f , evaluating the definite
integral comes from simply computing the change in F on [a, b].
• A slightly different perspective on the FTC allows us to restate it as the Total Change
Theorem, which says that
∫ b
f ′(x) dx f (b) − f (a),
a
for any continuously differentiable function f . This means that the definite integral
of the instantaneous rate of change of a function f on an interval [a, b] is equal to the
total change in the function f on [a, b].
254
4.4 The Fundamental Theorem of Calculus
4.4.5 Exercises
1. Finding exact displacement. The velocity function is v(t) −t 2 + 4t − 3 for a particle
moving along a line. Find the displacement (net distance covered) of the particle during
the time interval [−1, 5].
∫ 4
1
2. Evaluating the definite integral of a rational function. Find the value of dx.
2 x2
3. Evaluating the definite integral of a linear function. Evaluate the definite integral
∫ 9
(4x + 10) dx.
2
4. Evaluating the definite integral of a quadratic function. Evaluate the definite integral
∫ 6
(36 − x 2 ) dx.
−6
5. Simplifying an integrand before integrating. Evaluate the definite integral
∫ 8
8x 2 + 3
√ dx.
3 x
6. Evaluating the definite integral of a trigonometric function. Evaluate the definite
integral
∫ π
8 sin(x) dx.
0
7. The instantaneous velocity (in meters per minute) of a moving object is given by the
function v as pictured in Figure 4.4.10. Assume that on the interval 0 ≤ t ≤ 4, v(t) is
given by v(t) − 14 t 3 + 32 t 2 + 1, and that on every other interval v is piecewise linear, as
shown.
a. Determine the exact distance traveled by the object on the time interval 0 ≤ t ≤ 4.
b. What is the object’s average velocity on [12, 24]?
c. At what time is the object’s acceleration greatest?
d. Suppose that the velocity of the object is increased by a constant value c for all val-
ues of t. What value of c will make the object’s total distance traveled on [12, 24]
be 210 meters?
255
Chapter 4 The Definite Integral
m/min y = v(t)
15
12
9
6
3
min
4 8 12 16 20 24
Figure 4.4.10: The velocity function of a moving body.
8. A function f is given piecewise by the formula
−x 2 + 2x + 1, if 0 ≤ x < 2
f (x) −x + 3, if 2 ≤ x < 3 .
x 2 − 8x + 15, if 3 ≤ x ≤ 5
a. Determine the exact value of the net signed area enclosed by f and the x-axis on
the interval [2, 5].
b. Compute the exact average value of f on [0, 5].
c. Find a formula for a function 1 on 5 ≤ x ≤ 7 so that if we extend the above
∫7
definition of f so that f (x) 1(x) if 5 ≤ x ≤ 7, it follows that 0
f (x) dx 0.
9. When an aircraft attempts to climb as rapidly as possible, its climb rate (in feet per
minute) decreases as altitude increases, because the air is less dense at higher altitudes.
Given below is a table showing performance data for a certain single engine aircraft,
giving its climb rate at various altitudes, where c(h) denotes the climb rate of the air-
plane at an altitude h.
h (feet) 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10,000
c (ft/min) 925 875 830 780 730 685 635 585 535 490 440
Let a new function called m(h) measure the number of minutes required for a plane at
altitude h to climb the next foot of altitude.
a. Determine a similar table of values for m(h) and explain how it is related to the
table above. Be sure to explain the units.
b. Give a careful interpretation of a function whose derivative is m(h). Describe
what the input is and what the output is. Also, explain in plain English what the
function tells us.
c. Determine a definite integral whose value tells us exactly the number of minutes
required for the airplane to ascend to 10,000 feet of altitude. Clearly explain why
256
4.4 The Fundamental Theorem of Calculus
the value of this integral has the required meaning.
d. Use the Riemann sum M5 to estimate the value of the integral you found in (c).
Include units on your result.
10. In Chapter 1, we showed that for an object moving along a straight line with position
function s(t), the object’s “average velocity on the interval [a, b]” is given by
s(b) − s(a)
AV[a,b] .
b−a
More recently in Chapter 4, we found that for an object moving along a straight line
with velocity function v(t), the object’s “average value of its velocity function on [a, b]”
is ∫ b
1
v AVG[a,b] v(t) dt.
b−a a
Are the “average velocity on the interval [a, b]” and the “average value of the velocity
function on [a, b]” the same thing? Why or why not? Explain.
11. In Table 4.4.5 in Activity 4.4.3, we noted that for x > 0, the antiderivative of f (x) x1
is F(x) ln(x). Here we observe that a key difference between f (x) and F(x) is that f
is defined for all x , 0, while F is only defined for x > 0, and see how we can actually
define the antiderivative of f for all values of x.
a. Suppose that x < 0, and let G(x) ln(−x). Compute G′(x).
b. Explain why G is an antiderivative of f for x < 0.
c. Let H(x) ln(|x|), and recall that
{
−x, if x < 0
|x| .
x, if x ≥ 0
Explain why H(x) G(x) for x < 0 and H(x) F(x) for x > 0.
d. Now discuss why we say that the antiderivative of 1
x is ln(|x|) for all x , 0.
257
Chapter 4 The Definite Integral
258
CHAPTER 5
Evaluating Integrals
5.1 Constructing Accurate Graphs of Antiderivatives
Motivating Questions
• Given the graph of a function’s derivative, how can we construct a completely accu-
rate graph of the original function?
• How many antiderivatives does a given function have? What do those antiderivatives
all have in common?
∫x
• Given a function f , how does the rule A(x) 0
f (t) dt define a new function A?
A recurring theme in our discussion of differential calculus has been the question “Given
information about the derivative of an unknown function f , how much information can we
obtain about f itself?” In Activity 1.8.3, the graph of y f ′(x) was known (along with the
value of f at a single point) and we endeavored to sketch a possible graph of f near the
known point. In Example 3.1.7 — we investigated how the first derivative test enables us
to use information about f ′ to determine where the original function f is increasing and
decreasing, as well as where f has relative extreme values. If we know a formula or graph
of f ′, by computing f ′′ we can find where the original function f is concave up and concave
down. Thus, knowing f ′ and f ′′ enables us to understand the shape of the graph of f .
We returned to this question in even more detail in Section 4.1. In that setting, we knew
the instantaneous velocity of a moving object and worked to determine as much as possible
about the object’s position function. We found connections between the net signed area
under the velocity function and the corresponding change in position of the function, and
the Total Change Theorem further illuminated these connections between f ′ and f , showing
that the total change in the value of f over an interval [a, b] is determined by the net signed
area bounded by f ′ and the x-axis on the same interval.
In what follows, we explore the situation where we possess an accurate graph of the deriv-
ative function along with a single value of the function f . From that information, we’d like
to determine a graph of f that shows where f is increasing, decreasing, concave up, and
concave down, and also provides an accurate function value at any point.
Chapter 5 Evaluating Integrals
Preview Activity 5.1.1. Suppose that the following information is known about a
function f : the graph of its derivative, y f ′(x), is given in Figure 5.1.1. Further,
assume that f ′ is piecewise linear (as pictured) and that for x ≤ 0 and x ≥ 6, f ′(x) 0.
Finally, it is given that f (0) 1.
y = f ′ (x)
3 3
1 1
1 3 5 1 3 5
-1 -1
-3 -3
Figure 5.1.1: At left, the graph of y f ′(x); at right, axes for plotting y f (x).
a. On what interval(s) is f an increasing function? On what intervals is f decreas-
ing?
b. On what interval(s) is f concave up? concave down?
c. At what point(s) does f have a relative minimum? a relative maximum?
d. Recall that the Total Change Theorem tells us that
∫ 1
f (1) − f (0) f ′(x) dx.
0
What is the exact value of f (1)?
e. Use the given information and similar reasoning to that in (d) to determine the
exact value of f (2), f (3), f (4), f (5), and f (6).
f. Based on your responses to all of the preceding questions, sketch a complete
and accurate graph of y f (x) on the axes provided, being sure to indicate the
behavior of f for x < 0 and x > 6.
260
5.1 Constructing Accurate Graphs of Antiderivatives
5.1.1 Constructing the graph of an antiderivative
Preview Activity 5.1.1 demonstrates that when we can find the exact area under the graph
of a function on any given interval, it is possible to construct a graph of the function’s anti-
derivative. That is, we can find a function whose derivative is given. We can now determine
not only the overall shape of the antiderivative graph, but also the actual height of the graph
at any point of interest.
This is a consequence of the Fundamental Theorem of Calculus: if we know a function f
and the value of the antiderivative F at some starting point a, we can determine the value of
∫b
F(b) via the definite integral. Since F(b) − F(a) a
f (x) dx, it follows that
∫ b
F(b) F(a) + f (x) dx. (5.1.1)
a
∫b
We can also interpret the equation F(b) − F(a) a
f (x) dx in terms of the graphs of f and
F as follows. On an interval [a, b],
differences in heights on the graph of the antiderivative given by F(b) − F(a) correspond
to the net signed area bounded by the original function on the interval [a, b], which is
∫b
given by a
f (x) dx.
Activity 5.1.2. Suppose that the function y f (x) is given by the graph shown in
Figure 5.1.2, and that the pieces of f are either portions of lines or portions of circles.
In addition, let F be an antiderivative of f and say that F(0) −1. Finally, assume that
for x ≤ 0 and x ≥ 7, f (x) 0.
y = f (x)
1
1 2 3 4 5 6 7
-1
Figure 5.1.2: At left, the graph of y f (x).
a. On what interval(s) is F an increasing function? On what intervals is F decreas-
ing?
b. On what interval(s) is F concave up? concave down? neither?
c. At what point(s) does F have a relative minimum? a relative maximum?
d. Use the given information to determine the exact value of F(x) for x 1, 2, . . . , 7.
In addition, what are the values of F(−1) and F(8)?
261
Chapter 5 Evaluating Integrals
e. Based on your responses to all of the preceding questions, sketch a complete
and accurate graph of y F(x) on the axes provided, being sure to indicate the
behavior of F for x < 0 and x > 7. Clearly indicate the scale on the vertical and
horizontal axes of your graph.
f. What happens if we change one key piece of information: in particular, say that
G is an antiderivative of f and G(0) 0. How (if at all) would your answers to
the preceding questions change? Sketch a graph of G on the same axes as the
graph of F you constructed in (e).
5.1.2 Multiple antiderivatives of a single function
In the final question of Activity 5.1.2, we encountered a very important idea: a function f has
more than one antiderivative. Each antiderivative of f is determined uniquely by its value
at a single point. For example, suppose that f is the function given at left in Figure 5.1.3,
and suppose further that F is an antiderivative of f that satisfies F(0) 1.
f G
3
2
F
1
1 3 5 2 4 H
-1
-2
-3
Figure 5.1.3: At left, the graph of y f (x). At right, three different antiderivatives of f .
Then, using Equation (5.1.1), we can compute
∫ 1
F(1) F(0) + f (x) dx
0
1 + 0.5
1.5.
Similarly, F(2) 1.5, F(3) −0.5, F(4) −2, F(5) −0.5, and F(6) 1. In addition, we can
use the fact that F′ f to ascertain where F is increasing and decreasing, concave up and
concave down, and has relative extremes and inflection points. We ultimately find that the
graph of F is the one given in blue in Figure 5.1.3.
If we want an antiderivative G for which G(0) 3, then G will have the exact same shape
as F (since both share the derivative f ), but G will be shifted vertically from the graph of
262
5.1 Constructing Accurate Graphs of Antiderivatives
∫1
F, as pictured in red in Figure 5.1.3. Note that G(1) − G(0) 0 f (x) dx 0.5, just as
F(1) − F(0) 0.5, but since G(0) 3, G(1) G(0) + 0.5 3.5, whereas F(1) 1.5. In
the same way, if we assigned a different initial value to the antiderivative, say H(0) −1, we
would get still another antiderivative, as shown in magenta in Figure 5.1.3.
This example demonstrates an important fact that holds more generally:
If G and H are both antiderivatives of a function f , then the function G − H must be
constant.
To see why this result holds, observe that if G and H are both antiderivatives of f , then
G′ f and H ′ f . Hence,
d
[G(x) − H(x)] G′(x) − H ′(x) f (x) − f (x) 0.
dx
Since the only way a function can have derivative zero is by being a constant function, it
follows that the function G − H must be constant.
We now see that if a function has at least one antiderivative, it must have infinitely many:
we can add any constant of our choice to the antiderivative and get another antiderivative.
For this reason, we sometimes refer to the general antiderivative of a function f .
To identify a particular antiderivative of f , we must know a single value of the antideriv-
ative F (this value is often called an initial condition). For example, if f (x) x 2 , its general
antiderivative is F(x) 31 x 3 + C, where we include the “+C” to indicate that F includes
all of the possible antiderivatives of f . If we know that F(2) 3, we substitute 2 for x in
F(x) 31 x 3 + C, and find that
1
3 (2)3 + C,
3
or C 3 − 8
3 13 . Therefore, the particular antiderivative in this case is F(x) 13 x 3 + 31 .
Activity 5.1.3. For each of the following functions, sketch an accurate graph of the
antiderivative that satisfies the given initial condition. In addition, sketch the graph
of two additional antiderivatives of the given function, and state the corresponding
initial conditions that each of them satisfy. If possible, find an algebraic formula for
the antiderivative that satisfies the initial condition.
a. original function: 1(x) |x| − 1; initial condition: G(−1) 0; interval for sketch:
[−2, 2]
b. original function: h(x) sin(x); initial condition: H(0) 1; interval for sketch:
[0, 4π]
x2 , if 0 < x < 1
c. original function: p(x) −(x − 2)2 , if 1 < x < 2 ; initial condition: P(0) 1;
0 otherwise
interval for sketch: [−1, 3]
263
Chapter 5 Evaluating Integrals
5.1.3 Functions defined by integrals
Equation (5.1.1) allows us to compute the value of the antiderivative F at a point b, provided
that we know F(a) and can evaluate the definite integral from a to b of f . That is,
∫ b
F(b) F(a) + f (x) dx.
a
In several situations, we have used this formula to compute F(b) for several different values
of b, and then plotted the points (b, F(b)) to help us draw an accurate graph of F. This
suggests that we may want to think of b, the upper limit of integration, as a variable itself.
To that end, we introduce the idea of an integral function, a function whose formula involves
a definite integral.
Definition 5.1.4 If f is a continuous function, we define the corresponding integral function
A according to the rule ∫ x
A(x) f (t) dt. (5.1.2)
a
Note that because x is the independent variable in the function A, and determines the end-
point of the interval of integration, we need to use a different variable as the variable of
integration. A standard choice is t, but any variable other than x is acceptable.
One way to think of the function A is as the “net signed area from a up to x” function, where
we consider the region bounded by y f (t). For example, in Figure 5.1.5, we see a∫ function
x
f pictured at left, and its corresponding area function (choosing a 0), A(x) 0 f (t) dt
shown at right.
y = f (t)
3
1
π 2π
x
1
A(x)
-1
π 2π
x
Figure 5.1.5:
∫ x At left, the graph of the given function f . At right, the area function
A(x) 0 f (t) dt.
264
5.1 Constructing Accurate Graphs of Antiderivatives
The function A measures the net signed area from t 0 to t x bounded by the curve
y f (t); this value is then reported as the corresponding height on the graph of y A(x). At
http://gvsu.edu/s/cz, we find a java applet¹ that brings the static picture in Figure 5.1.5 to
life. There, the user can move the red point on the function f and see how the corresponding
height changes at the light blue point on the graph of A.
The choice of a is somewhat arbitrary. In the activity that follows, we explore how the value
of a affects the graph of the integral function.
Activity 5.1.4. Suppose that 1 is given by the graph at left
∫ xin Figure 5.1.6 and that A
is the corresponding integral function defined by A(x) 1 1(t) dt.
g
3 3
1 1
1 3 5 1 3 5
-1 -1
-3 -3
Figure 5.1.6: At left, the graph of y
∫ x1(t); at right, axes for plotting y A(x), where
A is defined by the formula A(x) 1 1(t) dt.
a. On what interval(s) is A an increasing function? On what intervals is A decreas-
ing? Why?
b. On what interval(s) do you think A is concave up? concave down? Why?
c. At what point(s) does A have a relative minimum? a relative maximum?
d. Use the given information to determine the exact values of A(0), A(1), A(2), A(3),
A(4), A(5), and A(6).
e. Based on your responses to all of the preceding questions, sketch a complete
and accurate graph of y A(x) on the axes provided, being sure to indicate the
behavior of A for x < 0 and x > 6.
∫x
f. How does the graph of B compare to A if B is instead defined by B(x) 0
1(t) dt?
¹David Austin, Grand Valley State University
265
Chapter 5 Evaluating Integrals
5.1.4 Summary
• Given the graph of a function f , we can construct the graph of its antiderivative F
provided that (a) we know a starting value of F, say F(a), and (b) we can evaluate the
∫b
integral f (x) dx exactly for relevant choices of a and b. For instance, if we wish to
a ∫3
know F(3), we can compute F(3) F(a) + a f (x) dx. When we combine this infor-
mation about the function values of F together with our understanding of how the
behavior of F′ f affects the overall shape of F, we can develop a completely accurate
graph of the antiderivative F.
• Because the derivative of a constant is zero, if F is an antiderivative of f , it follows that
G(x) F(x) + C will also be an antiderivative of f . Moreover, any two antiderivatives
of a function f differ precisely by a constant. Thus, any function with at least one
antiderivative in fact has infinitely many, and the graphs of any two antiderivatives
will differ only by a vertical translation.
∫x
• Given a function f , the rule A(x) a f (t) dt defines a new function A that measures
the net-signed area bounded by f on the interval [a, x]. We call the function A the
integral function corresponding to f .
5.1.5 Exercises
1. Definite integral of a piecewise linear function. Use the graph of f (x) shown below
to find the following integrals.
∫0
A. −5 f (x)dx
B. If the vertical red shaded area in the graph
∫7
has area A, estimate: −5 f (x)dx
(Your estimate may be written in terms of A.)
266
5.1 Constructing Accurate Graphs of Antiderivatives
2. A smooth function that starts out at 0. Consider the graph of the function f (x) shown
below.
A. Estimate the integral
B. If F is an antiderivative of the same function
f and F(0) 30, estimate F(7).
3. A piecewise constant function. Assume f ′ is given by the graph below. Suppose f is
continuous and that f (3) 0.
Sketch, on a sheet of work paper, an accurate
graph of f , and use it to find f (0) and f (7. Then
∫7
find the value of the integral: 0 f ′(x) dx.
(Note that you can do this in two different ways!)
4. Another piecewise linear function. The figure below shows f .
If F′ f and F(0) 0, find F(b) for b 1, 2, 3, 4, 5, 6.
267
Chapter 5 Evaluating Integrals
5. A moving particle has its velocity given by the quadratic function v pictured in Fig-
ure 5.1.7. In addition, it is given that A1 76 and A2 83 , as well as that for the corre-
sponding position function s, s(0) 0.5.
a. Use the given information to determine s(1), s(3), s(5), and s(6).
b. On what interval(s) is s increasing? On what interval(s) is s decreasing?
c. On what interval(s) is s concave up? On what interval(s) is s concave down?
d. Sketch an accurate, labeled graph of s on the axes at right in Figure 5.1.7.
e. Note that v(t) −2 + 12 (t − 3)2 . Find a formula for s.
3 3 s
v
1
A1
t t
1 6 2 4 6
A2 -1
-3 -3
Figure 5.1.7: At left, the given graph of v. At right, axes for plotting s.
6. A person exercising on a treadmill experiences different levels of resistance and thus
burns calories at different rates, depending on the treadmill’s setting. In a particu-
lar workout, the rate at which a person is burning calories is given by the piecewise
constant function c pictured in Figure 5.1.8. Note that the units on c are “calories per
minute.”
a. Let C be an antiderivative of c. What does the function C measure? What are its
units?
b. Assume that C(0) 0. Determine the exact value of C(t) at the values
t 5, 10, 15, 20, 25, 30.
c. Sketch an accurate graph of C on the axes provided at right in Figure 5.1.8. Be
certain to label the scale on the vertical axis.
d. Determine a formula for C that does not involve an integral and is valid for 5 ≤
t ≤ 10.
268
5.1 Constructing Accurate Graphs of Antiderivatives
cal/min
15
c
10
5
min
10 20 30 10 20 30
Figure 5.1.8: At left, the given graph of c. At right, axes for plotting C.
7. Consider the piecewise linear function f given
∫ x in Figure 5.1.9.∫ xLet the functions A,
B, and C be defined by the rules A(x) −1 f (t) dt, B(x) 0 f (t) dt, and C(x)
∫x
1
f (t) dt.
a. For the values x −1, 0, 1, . . . , 6, make a table that lists corresponding values of
A(x), B(x), and C(x).
b. On the axes provided in Figure 5.1.9, sketch the graphs of A, B, and C.
c. How are the graphs of A, B, and C related?
d. How would you best describe the relationship between the function A and the
function f ?
3 3
f
1 1
1 3 5 1 3 5
-1 -1
-3 -3
Figure 5.1.9: At left, the given graph of f . At right, axes for plotting A, B, and C.
269
Chapter 5 Evaluating Integrals
5.2 The Second Fundamental Theorem of Calculus
Motivating Questions
∫x
• How does the integral function A(x) 1
f (t) dt define an antiderivative of f ?
• What is the statement of the Second Fundamental Theorem of Calculus?
• How do the First and Second Fundamental Theorems of Calculus enable us to for-
mally see how differentiation and integration are almost inverse processes?
In Section 4.4, we learned the Fundamental Theorem of Calculus (FTC), which from here
forward will be referred to as the First Fundamental Theorem of Calculus, as in this section
we develop a corresponding result that follows it. Recall that the First FTC tells us that if f
is a continuous function on [a, b] and F is any antiderivative of f (that is, F′ f ), then
∫ b
f (x) dx F(b) − F(a).
a
We have used this result in two settings:
1 If we have a graph of f and we can compute the exact area bounded by f on an interval
[a, b], we can compute the change in an antiderivative F over the interval.
2 If we can find an algebraic formula for an antiderivative of f , we can evaluate the
integral to find the net signed area bounded by the function on the interval.
For the former, see Preview Activity 5.1.1 or Activity 5.1.2. For the latter, we can easily
evaluate exactly integrals such as
∫ 4
x 2 dx,
1
since we know that the function F(x) 13 x 3 is an antiderivative of f (x) x 2 . Thus,
∫ 4 4
1
x dx x 3
2
1 3 1
1 1
(4)3 − (1)3
3 3
21.
Thus, the First FTC can used in two ways. First, to find the difference F(b) − F(a) for an anti-
derivative F of the integrand f , even if we may not have a formula for F itself. To do this, we
∫b
must know the value of the integral a f (x) dx exactly, perhaps through known geometric
formulas for area. In addition, the First FTC provides a way to find the exact value of a defi-
nite integral, and hence a certain net signed area exactly, by finding an antiderivative of the
integrand and evaluating its total change over the interval. In this case, we need to know a
formula for the antiderivative F. Both of these perspectives are reflected in Figure 5.2.1.
270
5.2 The Second Fundamental Theorem of Calculus
20 20 (4, 64
3 )
F(4) − F(1) = 21
10 f (x) = x2 10 F(x) = 31 x3
(1, 13 )
R4 2
1 x dx = 21
1 2 3 4 1 2 3 4
Figure 5.2.1: At left, the graph of f (x) x 2 on the interval [1, 4] and the area it bounds. At
right, the antiderivative function F(x) 31 x 3 , whose total change on [1, 4] is the value of the
definite integral at left.
The value of a definite integral may have additional meaning depending on context: as the
change in position when the integrand is a velocity function, the total amount of pollutant
leaked from a tank when the integrand is the rate at which pollution is leaking, or other total
changes if the integrand is a rate function. Also, the value of the definite integral is connected
∫b
to the average value of a continuous function on a given interval: fAVG[a,b] 1
b−a a
f (x) dx.
∫x
In the last part of Section 5.1, we studied integral functions of the form A(x) c f (t) dt.
Figure 5.1.5 is a particularly important image to keep in mind as we work with integral
functions, and the corresponding java applet at gvsu.edu/s/cz can help us understand the
function A. In what∫ x follows, we use the First FTC to gain additional understanding of the
function A(x) c f (t) dt, where the integrand f is given (either through a graph or a
formula), and c is a constant.
Preview Activity 5.2.1. Consider the function A defined by the rule
∫ x
A(x) f (t) dt,
1
where f (t) 4 − 2t.
a. Compute A(1) and A(2) exactly.
b. Use the First Fundamental Theorem of Calculus to find a formula ∫ x for A(x) that
does not involve integrals. That is, use the first FTC to evaluate 1 (4 − 2t) dt.
c. Observe that f is a linear function; what kind of function is A?
271
Chapter 5 Evaluating Integrals
d. Using the formula you found in (b) that does not involve integrals, compute
A′(x).
e. While we have defined f by the rule f (t) 4 − 2t, it is equivalent to say that f
is given by the rule f (x) 4 − 2x. What do you observe about the relationship
between A and f ?
5.2.1 The Second Fundamental Theorem of Calculus
The result of Preview Activity 5.2.1 is not particular to the function f (t) 4 − 2t, nor to the
choice of “1” as the lower bound in the∫integral that defines the function A. For instance, if
x
we let f (t) cos(t) − t and set A(x) 2 f (t) dt, we can determine a formula for A by the
First FTC. Specifically,
∫ x
A(x) (cos(t) − t) dt
2
x
1
sin(t) − t 2
2 2
1
sin(x) − x 2 − (sin(2) − 2) .
2
Differentiating A(x), since (sin(2) − 2) is constant, it follows that
A′(x) cos(x) − x,
and thus we see that A′(x) f (x), so A is an antiderivative of f . And since
∫ 2
A(2) f (t) dt 0,
2
A is the only antiderivative of f for which A(2) 0.
In general, if f is any continuous function, and we define the function A by the rule
∫ x
A(x) f (t) dt,
c
where c is an arbitrary constant, then we can show that A is an antiderivative of f . To see
why, let’s demonstrate that A′(x) f (x) by using the limit definition of the derivative. Doing
so, we observe that
A(x + h) − A(x)
A′(x) lim
h→0 h
∫ x+h ∫x
c
f (t) dt − c
f (t) dt
lim
h→0 h
∫ x+h
x
f (t) dt
lim , (5.2.1)
h→0 h
272
5.2 The Second Fundamental Theorem of Calculus
∫x ∫ x+h ∫ x+h
where Equation (5.2.1) follows from the fact that c
f (t) dt + x
f (t) dt c
f (t) dt.
Now, observe that for small values of h,
∫ x+h
f (t) dt ≈ f (x) · h,
x
by a simple left-hand approximation of the integral. Thus, as we take the limit in Equa-
tion (5.2.1), it follows that
∫ x+h
f (t) dt f (x) · h
A′(x) lim x
lim f (x).
h→0 h h→0 h
∫c
Hence, A is indeed an antiderivative of f . In addition, A(c) c f (t) dt 0. The preceding
argument demonstrates the truth of the Second Fundamental Theorem of Calculus, which
we state as follows.
The Second Fundamental Theorem of Calculus.
If f is a continuous function and c is any constant, then f has a unique antiderivative
∫x
A that satisfies A(c) 0, and that antiderivative is given by the rule A(x) c f (t) dt.
Activity 5.2.2. Suppose that f is the function given in Figure 5.2.2 and that f is a
piecewise function whose parts are either portions of lines or portions of circles, as
pictured.
y = f (x)
1
1 2 3 4 5 6 7
-1
Figure 5.2.2: At left, the graph of y f (x). At right, axes for sketching y A(x).
∫x
In addition, let A be the function defined by the rule A(x) 2
f (t) dt.
a. What does the Second FTC tell us about the relationship between A and f ?
b. Compute A(1) and A(3) exactly.
c. Sketch a precise graph of y A(x) on the axes at right that accurately reflects
where A is increasing and decreasing, where A is concave up and concave down,
and the exact values of A at x 0, 1, . . . , 7.
d. How is A similar to, but different from, the function F that you found in Activ-
ity 5.1.2?
273
Chapter 5 Evaluating Integrals
e. With as ∫little additional work as∫ possible, sketch precise graphs of the functions
x x
B(x) 3 f (t) dt and C(x) 1 f (t) dt. Justify your results with at least one
sentence of explanation.
5.2.2 Understanding Integral Functions
The Second FTC provides us with a way to construct an antiderivative of any continuous
function. In particular, if we are given a continuous function 1 and wish to find an anti-
derivative of G, we can now say that
∫ x
G(x) 1(t) dt
c
provides the rule for such an antiderivative, and moreover that G(c) 0. Note especially
that we know that G′(x) 1(x), or
[∫ x ]
d
1(t) dt 1(x). (5.2.2)
dx c
This result is useful for understanding the graph of G.
Example 5.2.3 Investigate the behavior of the integral function
∫ x
e −t dt.
2
E(x)
0
Solution. E is closely related to the well known error function ¹ in probability and statistics.
It turns out that the function e −t does not have an elementary antiderivative.
2
While we cannot evaluate E exactly for any value other than x 0, we still can gain a tremen-
dous amount of information about the function E. By applying the rule in Equation (5.2.2)
to E, it follows that [∫ x ]
′ d
e dt e −x ,
−t 2 2
E (x)
dx 0
so we know a formula for the derivative of E, and we know that E(0) 0. This information
is precisely the type we were given in Activity 3.1.2, where we were given information about
the derivative of a function, but lacked a formula for the function itself.
Using the first and second derivatives of E, along with the fact that E(0) 0, we can de-
termine more information about the behavior of E. First, we note that for all real numbers
x, e −x > 0, and thus E′(x) > 0 for all x. Thus E is an always increasing function. Further,
2
as x → ∞, E′(x) e −x → 0, so the slope of the function E tends to zero as x → ∞ (and
2
similarly as x → −∞). Indeed, it turns out that E has horizontal asymptotes as x increases
or decreases without bound.
In addition, we can observe that E′′(x) −2xe −x , and that E′′(0) 0, while E′′(x) < 0 for
2
x > 0 and E′′(x) > 0 for x < 0. This information tells us that E is concave up for x < 0 and
concave down for x > 0 with a point of inflection at x 0.
274
5.2 The Second Fundamental Theorem of Calculus
The only thing we lack at this point is a sense of how big E can get as x increases. If we use
a midpoint Riemann sum with 10 subintervals to estimate E(2), we see that E(2) ≈ 0.8822; a
similar calculation to estimate E(3) shows little change (E(3) ≈ 0.8862), so it appears that as
x increases without bound, E approaches a value just larger than 0.886, which aligns with
the fact that E has horizontal asymptotes. Putting all of this information together (and using
the symmetry of f (t) e −t ), we see the results shown in Figure 5.2.4.
2
R x −t 2
f (t) = e−t
2 E(x) = 0e dt
1 1
-2 2 -2 2
-1 -1
Figure ∫5.2.4: At left, the graph of f (t) e −t . At right, the integral function
2
x
E(x) 0 e −t dt, which is the unique antiderivative of f that satisfies E(0) 0.
2
Because E is the antiderivative of f (t) e −t that satisfies E(0) 0, values on the graph of
2
y E(x) represent the net signed area of the region bounded by f (t) e −t from 0 up to x.
2
We see that the value of E increases rapidly near zero but then levels off as x increases, since
there is less and less additional accumulated area bounded by f (t) e −t as x increases.
2
∫x
Activity 5.2.3. Suppose that f (t) t
1+t 2
and F(x) 0
f (t) dt.
a. On the axes at left in Figure 5.2.5, plot a graph of f (t) 1+t
t
2 on the interval
−10 ≤ t ≤ 10. Clearly label the vertical axes with appropriate scale.
b. What is the key relationship between F and f , according to the Second FTC?
c. Use the first derivative test to determine the intervals on which F is increasing
and decreasing.
d. Use the second derivative test to determine the intervals on which F is concave
up and concave down. Note that f ′(t) can be simplified to be written in the form
∫x 2
¹The error function is defined by the rule erf(x) √2
π 0
e −t dt and has the key property that 0 ≤ erf(x) < 1
for all x ≥ 0 and moreover that limx→∞ erf(x) 1.
275
Chapter 5 Evaluating Integrals
1−t 2
f ′(t) (1+t 2 )2
.
e. Using technology appropriately, estimate the values of F(5) and F(10) through
appropriate Riemann sums.
f. Sketch an accurate graph of y F(x) on the righthand axes provided, and
clearly label the vertical axes with appropriate scale.
Figure 5.2.5: Axes for plotting f and F.
5.2.3 Differentiating an Integral Function
We have seen that the Second FTC enables us to construct
∫x an antiderivative F for any con-
tinuous function f as the integral function F(x) c f (t) dt. If we have a function of the
∫x [∫ x ]
form F(x) c
f (t) dt, then we know that F′(x) d
dx c
f (t) dt f (x). This shows that
integral functions, while perhaps having the most complicated formulas of any functions
we have encountered, are nonetheless particularly simple to differentiate. For instance, if
∫ x
F(x) sin(t 2 ) dt,
π
then by the Second FTC, we know immediately that
F′(x) sin(x 2 ).
In general, we know by the Second FTC that
[∫ x ]
d
f (t) dt f (x).
dx a
276
5.2 The Second Fundamental Theorem of Calculus
This equation says that “the derivative of the integral function whose integrand is f , is f .”
We see that if we first integrate the function f from t a to t x, and then differentiate
with respect to x, these two processes “undo” each other.
What happens if we differentiate a function f (t) and then integrate the result from t a to
t x? That is, what can we say about the quantity
∫ x
d [ ]
f (t) dt?
a dt
[ ]
We note that f (t) is an antiderivative of d
dt f (t) and apply the First FTC. We see that
∫
d [ ]
x x
f (t) dt f (t)
a dt a
f (x) − f (a).
Thus, we see that if we first differentiate f and then integrate the result from a to x, we
return to the function f , minus the constant value f (a). So the two processes almost undo
each other, up to the constant f (a).
The observations made in the preceding two paragraphs demonstrate that differentiating
and integrating (where we integrate from a constant up to a variable) are almost inverse
processes. This should not be surprising: integrating involves antidifferentiating, which re-
verses the process of differentiating. On the other hand, we see that there is some subtlety
involved, because integrating the derivative of a function does not quite produce the func-
tion itself. This is because every function has an entire family of antiderivatives, and any
two of those antiderivatives differ only by a constant.
Activity 5.2.4. Evaluate each of the following derivatives and definite integrals. Clearly
cite whether
[∫ x you use
] the First or Second FTC in so∫ doing.
x d [ ]
d
a. dx 4 e dtt 2
d. 3 dt ln(1 + t 2 ) dt
∫x [ ] [∫ x 3 ]
b. d t4
dt e. d
dx 4
sin(t 2 ) dt .
−2 dt 1+t 4
[∫ 1 ]
c. d
dx x
cos(t 3 ) dt
5.2.4 Summary
∫x
• For a continuous function f , the integral function A(x) 1
f (t) dt defines an anti-
derivative of f .
• The Second Fundamental Theorem of Calculus is the formal, more general statement
∫ofx the preceding fact: if f is a continuous function and c is any constant, then A(x)
c
f (t) dt is the unique antiderivative of f that satisfies A(c) 0.
277
Chapter 5 Evaluating Integrals
• Together, the First and Second FTC enable us to formally see how differentiation and
integration are almost inverse processes through the observations that
∫ x
d [ ]
f (t) dt f (x) − f (c)
c dt
and
[∫ x ]
d
f (t) dt f (x).
dx c
5.2.5 Exercises
∫x
1. A definite integral starting at 3. Let 1(x) 0
f (t) dt, where f (t) is given in the figure
below.
Find each of the following:
A. 1(0)
B. 1 ′(1)
C. The interval (with endpoints given to
the nearest 0.25) where 1 is concave up:
D. The value of x where 1 takes its max-
imum on the interval 0 ≤ x ≤ 8.
∫ a
d
2. Variable in the lower limit. Find the derivative: ln(ln(t)) dt.
dx x
Approximating a function with derivative e −x /5 . Find a good numerical approxima-
2
3.
tion to F(4) for the function with the properties that F′(x) e −x /5 and F(0) 3.
2
4. ∫ x 1 be the function pictured at left in Figure 5.2.6, and let F be defined by F(x)
Let
2
1(t) dt. Assume that the shaded areas have values A1 4.29, A2 12.75, A3 0.36,
and A4 1.79. Assume further that the portion of A2 that lies between x 0.5 and
x 2 is 6.06.
Sketch a carefully labeled graph of F on the axes provided, and include a written analy-
sis of how you know where F is zero, increasing, decreasing, CCU, and CCD.
278
5.2 The Second Fundamental Theorem of Calculus
6 15
4 10
y = g(t)
A2
2 5
A4
A1 1 2 3 4 5 6 -1 1 2 3 4 5 6
-2 -5
A3
-4 -10
Figure 5.2.6: At left, the graph of 1. At right, axes for plotting F.
5. The tide removes sand from the beach at a small ocean park at a rate modeled by the
function ( )
4πt
R(t) 2 + 5 sin
25
A pumping station adds sand to the beach at rate modeled by the function
15t
S(t)
1 + 3t
Both R(t) and S(t) are measured in cubic yards of sand per hour, t is measured in hours,
and the valid times are 0 ≤ t ≤ 6. At time t 0, the beach holds 2500 cubic yards of
sand.
a. What definite integral measures how much sand the tide will remove during the
time period 0 ≤ t ≤ 6? Why?
b. Write an expression for Y(x), the total number of cubic yards of sand on the beach
at time x. Carefully explain your thinking and reasoning.
c. At what instantaneous rate is the total number of cubic yards of sand on the beach
at time t 4 changing?
d. Over the time interval 0 ≤ t ≤ 6, at what time t is the amount of sand on the beach
least? What is this minimum value? Explain and justify your answers fully.
6. When an aircraft attempts to climb as rapidly as possible, its climb rate (in feet per
minute) decreases as altitude increases, because the air is less dense at higher altitudes.
Given below is a table showing performance data for a certain single engine aircraft,
giving its climb rate at various altitudes, where c(h) denotes the climb rate of the air-
plane at an altitude h.
279
Chapter 5 Evaluating Integrals
h (feet) 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10,000
c (ft/min) 925 875 830 780 730 685 635 585 535 490 440
Table 5.2.7: Data for the climbing aircraft.
Let a new function m, that also depends on h, (say y m(h)) measure the number of
minutes required for a plane at altitude h to climb the next foot of altitude.
a. Determine a similar table of values for m(h) and explain how it is related to the
table above. Be sure to discuss the units on m.
b. Give a careful interpretation of a function whose derivative is m(h). Describe
what the input is and what the output is. Also, explain in plain English what the
function tells us.
c. Determine a definite integral whose value tells us exactly the number of minutes
required for the airplane to ascend to 10,000 feet of altitude. Clearly explain why
the value of this integral has the required meaning.
d. Determine a formula for a function M(h) whose value tells us the exact number
of minutes required for the airplane to ascend to h feet of altitude.
e. Estimate the values of M(6000) and M(10000) as accurately as you can. Include
units on your results.
280
5.3 Integration by Substitution
5.3 Integration by Substitution
Motivating Questions
• How can we begin to find algebraic formulas for antiderivatives of more complicated
algebraic functions?
• What is an indefinite integral and how is its notation used in discussing antideriva-
tives?
• How does the technique of u-substitution work to help us evaluate certain indefinite
integrals, and how does this process rely on identifying function-derivative pairs?
In Section 4.4, we learned the key role that antiderivatives play in the process of evaluating
definite integrals exactly. The Fundamental Theorem of Calculus tells us that if F is any
antiderivative of f , then
∫ b
f (x) dx F(b) − F(a).
a
Furthermore, we realized that each elementary derivative rule developed in Chapter 2 leads
to a corresponding elementary antiderivative, as summarized in Table 4.4.5. Thus, if we
wish to evaluate an integral such as
∫ ( )
1 √
x 3 − x + 5x dx,
0
√
it is straightforward to do so, since we can easily antidifferentiate f (x) x 3 − x+5x . Because
one antiderivative of f is F(x) 14 x 4 − 23 x 3/2 + ln(5)
1
5x , the Fundamental Theorem of Calculus
tells us that
∫ ( ) 1
1 √ 1 2 1 x
x 3 − x + 5x dx x 4 − x 3/2 + 5
0 4 3 ln(5) 0
( ) ( )
1 4 2 3/2 1 1 1 0
(1) − (1) + 5 − 0−0+ 5
4 3 ln(5) ln(5)
5 4
− + .
12 ln(5)
We see that we have a natural interest in being able to find such algebraic antiderivatives. We
emphasize algebraic antiderivatives, as opposed to any antiderivative,
∫x since we know by the
Second Fundamental Theorem of Calculus that G(x) a f (t) dt is indeed an antiderivative
of the given function f , but one that still involves a definite integral. Our goal in this section
is to “undo” the process of differentiation to find an algebraic antiderivative for a given
function.
281
Chapter 5 Evaluating Integrals
Preview Activity 5.3.1. In Section 2.5, we learned the Chain Rule and how it can be
applied to find the derivative of a composite function. In particular, if u is a differen-
tiable function of x, and f is a differentiable function of u(x), then
d [ ]
f (u(x)) f ′(u(x)) · u ′(x).
dx
In words, we say that the derivative of a composite function c(x) f (u(x)), where
f is considered the “outer” function and u the “inner” function, is “the derivative of
the outer function, evaluated at the inner function, times the derivative of the inner
function.”
a. For each of the following functions, use the Chain Rule to find the function’s
derivative. Be sure to label each derivative by name (e.g., the derivative of 1(x)
should be labeled 1 ′(x)).
i. 1(x) e 3x iv. q(x) (2 − 7x)4
ii. h(x) sin(5x + 1)
iii. p(x) arctan(2x) v. r(x) 34−11x
b. For each of the following functions, use your work in (a) to help you determine
the general antiderivative¹ of the function. Label each antiderivative by name
(e.g., the antiderivative of m should be called M). In addition, check your work
by computing the derivative of each proposed antiderivative.
i. m(x) e 3x iv. v(x) (2 − 7x)3
ii. n(x) cos(5x + 1)
iii. s(x) 1+4x
1
2 v. w(x) 34−11x
c. Based on your experience in parts (a) and (b), conjecture an antiderivative for
each of the following functions. Test your conjectures by computing the deriv-
ative of each proposed antiderivative.
2
i. a(x) cos(πx) iii. c(x) xe x
ii. b(x) (4x + 7)11
5.3.1 Reversing the Chain Rule: First Steps
Whenever f is a familiar function whose antiderivative is known and u(x) is a linear func-
tion, it is straightforward to antidifferentiate a function of the form
h(x) f (u(x)).
¹Recall that the general antiderivative of a function includes “+C” to reflect the entire family of functions that
share the same derivative.
282
5.3 Integration by Substitution
Example 5.3.1 Determine the general antiderivative of
h(x) (5x − 3)6 .
Check the result by differentiating.
For this composite function, the outer function f is f (u) u 6 , while the inner function is
u(x) 5x − 3. Since the antiderivative of f is F(u) 71 u 7 + C, we see that the antiderivative
of h is
1 1 1
H(x) (5x − 3)7 · + C (5x − 3)7 + C.
7 5 35
The inclusion of the constant 15 is essential precisely because the derivative of the inner func-
tion is u ′(x) 5. Indeed, if we now compute H ′(x), we find by the Chain Rule (and Constant
Multiple Rule) that
1
H ′(x) · 7(5x − 3)6 · 5 (5x − 3)6 h(x),
35
and thus H is indeed the general antiderivative of h.
Hence, in the special case where the outer function is familiar and the inner function is linear,
we can antidifferentiate composite functions according to the following rule.
If h(x) f (ax + b) and F is a known algebraic antiderivative of f , then the general
antiderivative of h is given by
1
H(x) F(ax + b) + C.
a
It is useful to have shorthand notation that indicates the instruction to find an antiderivative.
Thus, in a similar way to how the notation
d [ ]
f (x)
dx
represents the derivative of f (x) with respect to x, we use the notation of the indefinite inte-
gral,
∫
f (x) dx
to represent the general antiderivative of f with respect to x. Returning to the earlier exam-
ple with h(x) (5x − 3)6 , we can rephrase the relationship between h and its antiderivative
H through the notation
∫
1
(5x − 3)6 dx (5x − 6)7 + C.
35
When we find an antiderivative, we will often say that we evaluate an indefinite integral.
∫ Just
d
as the notation dx [□] means “find the derivative with respect to x of □,” the notation □ dx
means “find a function of x whose derivative is □.”
283
Chapter 5 Evaluating Integrals
Activity 5.3.2. Evaluate each of the following indefinite integrals. Check each anti-
derivative
∫ that you find by differentiating. ∫
a. sin(8 − 3x) dx d. csc(2x + 1) cot(2x + 1) dx
∫ ∫
b. sec2 (4x) dx e. √ 1 dx
1−16x 2
∫ ∫
c. 1
11x−9 dx f. 5−x dx
5.3.2 Reversing the Chain Rule: u-substitution
A natural question arises from our recent work: what happens when the inner function is
not linear? For example, can we find antiderivatives of such functions as
2 2
1(x) xe x and h(x) e x ?
It is important to remember that differentiation and antidifferentiation are almost inverse
processes (that they are not is due to the +C that arises when antidifferentiating). This
almost-inverse relationship enables us to take any known derivative rule and rewrite it as a
corresponding rule for an indefinite integral. For example, since
d [ 5]
x 5x 4 ,
dx
we can equivalently write ∫
5x 4 dx x 5 + C.
Recall that the Chain Rule states that
d [ ]
f (1(x)) f ′(1(x)) · 1 ′(x).
dx
Restating this relationship in terms of an indefinite integral,
∫
f ′(1(x))1 ′(x) dx f (1(x)) + C. (5.3.1)
Equation (5.3.1) tells us that if we can view a given function as f ′(1(x))1 ′(x) for some appro-
priate choices of f and 1, then we can antidifferentiate the function by reversing the Chain
Rule. Note that both 1(x) and 1 ′(x) appear in the form of f ′(1(x))1 ′(x); we will sometimes
say that we seek to identify a function-derivative pair (1(x) and 1 ′(x)) when trying to apply the
rule in Equation (5.3.1).
If we can identify a function-derivative pair, we will introduce a new variable u to represent
the function 1(x). With u 1(x), it follows in Leibniz notation that du ′
dx 1 (x), so that in
′
terms of differentials², du 1 (x) dx. Now converting the indefinite integral to a new one in
²If we recall from the definition of the derivative that du du ′
dx ≈ ∆x and use the fact that dx 1 (x), then we see that
∆u
1 ′ (x) ′
≈ ∆x . Solving for ∆u, ∆u ≈ 1 (x)∆x. It is this last relationship that, when expressed in “differential” notation
∆u
enables us to write du 1 ′ (x) dx in the change of variable formula.
284
5.3 Integration by Substitution
terms of u, we have ∫ ∫
′ ′
f (1(x))1 (x) dx f ′(u) du.
Provided that f ′ is an elementary function whose antiderivative is known, we can easily
evaluate the indefinite integral in u, and then go on to determine the desired overall anti-
derivative of f ′(1(x))1 ′(x). We call this process u-substitution, and summarize the rule as
follows:
With the substitution u 1(x),
∫ ∫
f ′(1(x))1 ′(x) dx f ′(u) du f (u) + C f (1(x)) + C.
To see u-substitution at work, we consider the following example.
Example 5.3.2 Evaluate the indefinite integral
∫
x 3 · sin(7x 4 + 3) dx
and check the result by differentiating.
Solution. We can make two algebraic observations regarding the integrand, x 3 ·sin(7x 4 +3).
First, sin(7x 4 + 3) is a composite function; as such, we know we’ll need a more sophisticated
approach to antidifferentiating. Second, x 3 is almost the derivative of (7x 4 + 3); the only
issue is a missing constant. Thus, x 3 and (7x 4 + 3) are nearly a function-derivative pair.
Furthermore, we know the antiderivative of f (u) sin(u). The combination of these ob-
servations suggests that we can evaluate the given indefinite integral by reversing the chain
rule through u-substitution.
Letting u represent the inner function of the composite function sin(7x 4 + 3), we have u
dx 28x . In differential notation, it follows that du 28x dx, and thus
7x 4 + 3, and thus du 3 3
x dx 28 du. The original indefinite integral may be slightly rewritten as
3 1
∫
sin(7x 4 + 3) · x 3 dx,
and so by substituting u for 7x 4 + 3 and 1
28 du for x 3 dx, it follows that
∫ ∫
1
sin(7x + 3) · x dx
4 3
sin(u) · du.
28
Now we may evaluate the easier integral in u, and then replace u by the expression 7x 4 + 3.
Doing so, we find
∫ ∫
1
sin(7x 4 + 3) · x 3 dx sin(u) · du
28
∫
1
sin(u) du
28
285
Chapter 5 Evaluating Integrals
1
(− cos(u)) + C
28
1
− cos(7x 4 + 3) + C.
28
To check our work, we observe by the Chain Rule that
[ ]
d 1 1
− cos(7x 4 + 3) − · (−1) sin(7x 4 + 3) · 28x 3 sin(7x 4 + 3) · x 3 ,
dx 28 28
which is indeed the original integrand.
The u-substitution worked because the function multiplying sin(7x 4 + 3) was x 3 . If instead
that function was x 2 or x 4 , the substitution process would not have worked. This is one of the
primary challenges of antidifferentiation: slight changes in the integrand make tremendous
differences. For instance, we can use u-substitution with u x 2 and du 2xdx to find that
∫ ∫
x2 1
xe dx eu · du
2
∫
1
e u du
2
1
eu + C
2
1 2
e x + C.
2
However, for the similar indefinite integral
∫
2
e x dx,
the u-substitution u x 2 is no longer possible because the factor of x is missing. Hence,
part of the lesson of u-substitution is just how specialized the process is: it only applies to
situations where, up to a missing constant, the integrand is the result of applying the Chain
Rule to a different, related function.
Activity 5.3.3. Evaluate each of the following indefinite integrals by using these steps:
• Find two functions within the integrand that form (up to a possible missing
constant) a function-derivative pair;
• Make a substitution and convert the integral to one involving u and du;
• Evaluate the new integral in u;
• Convert the resulting function of u back to a function of x by using your earlier
substitution;
• Check your work by differentiating the function of x. You should come up with
the integrand originally given.
286
5.3 Integration by Substitution
∫ ∫ √
x2 cos( x)
a. 5x 3 +1
dx c. √ dx
x
∫
b. e x sin(e x ) dx
5.3.3 Evaluating Definite Integrals via u-substitution
We have introduced u-substitution as a means to evaluate indefinite integrals of functions
that can be written, up to a constant multiple, in the form f (1(x))1 ′(x). This same technique
can be used to evaluate definite integrals involving such functions, though we need to be
careful with the corresponding limits of integration. Consider, for instance, the definite
integral
∫ 5
2
xe x dx.
2
Whenever we write a definite integral, it is implicit that the limits of integration correspond
to the variable of integration. To be more explicit, observe that
∫ 5 ∫ x5
x2 2
xe dx xe x dx.
2 x2
When we execute a u-substitution, we change the variable of integration; it is essential to note
that this also changes the limits of integration. For instance, with the substitution u x 2 and
du 2x dx, it also follows that when x 2, u 22 4, and when x 5, u 52 25. Thus,
under the change of variables of u-substitution, we now have
∫ x5 ∫ u25
2 1
xe x dx eu · du
x2 u4 2
u25
1
eu
2 u4
1 25 1 4
e − e .
2 2
∫ 2
Alternatively, we could consider the related indefinite integral xe x dx, find the antideriv-
2
ative 21 e x through u-substitution, and then evaluate the original definite integral. With that
method, we’d have
∫ 5 5
2 1 x2
xe x dx e
2 2 2
1 25 1 4
e − e ,
2 2
which is, of course, the same result.
287
Chapter 5 Evaluating Integrals
Activity 5.3.4. Evaluate each of the following definite integrals exactly through an
appropriate
∫2 u-substitution.
x
∫ 4/π cos( 1 )
a. 1 1+4x 2 dx c. 2/π x 2 x dx
∫1
b. 0
e −x (2e −x + 3)9 dx
5.3.4 Summary
• To find algebraic formulas for antiderivatives of more complicated algebraic functions,
we need to think carefully about how we can reverse known differentiation rules. To
that end, it is essential that we understand and recall known derivatives of basic func-
tions, as well as the standard derivative rules.
∫
• The indefinite integral provides notation for antiderivatives. When we write “ f (x) dx,”
we mean “the general antiderivative of f .” In particular, if we have functions f and F
such that F′ f , the following two statements say the exact thing:
∫
d
[F(x)] f (x) and f (x) dx F(x) + C.
dx
That is, f is the derivative of F, and F is an antiderivative of f .
∫ technique of u-substitution helps us to evaluate indefinite integrals of the form
• The
f (1(x))1 ′(x) dx through the substitutions u 1(x) and du 1 ′(x) dx, so that
∫ ∫
f (1(x))1 ′(x) dx f (u) du.
A key part of choosing the expression in x to be represented by u is the identification of
a function-derivative pair. To do so, we often look for an “inner” function 1(x) that is
part of a composite function, while investigating whether 1 ′(x) (or a constant multiple
of 1 ′(x)) is present as a multiplying factor of the integrand.
5.3.5 Exercises
1. Product involving 4th power of a polynomial. Find the following integral.
∫
( )2
t3 t4 − 4 dt.
2. Product involving sin(x 6 ). Find the the general antiderivative F(x) of the function f (x)
given below.
f (x) 7x 5 sin(x 6 ).
3. Fraction involving ln9 . Find the following integral.
∫
ln8 (z)
dz
z
288
5.3 Integration by Substitution
4. Fraction involving e 5x . Find the following integral.
∫
e 5x
dx
1 + e 5x
√
y
5. Fraction involving e 5 . Find the following integral.
∫ √
2e 4 y
√ dy
y
6. Definite integral involving e −cos(q) . Use the Fundamental Theorem of Calculus to find
∫ π ( )
e sin(q ) · cos q dq
π/2
7. This problem centers on finding antiderivatives for the basic trigonometric functions
other than sin(x) and cos(x).
∫
a. Consider the indefinite integral tan(x) dx. By rewriting the integrand as
sin(x)
tan(x)
cos(x)
and identifying an ∫appropriate function-derivative pair, make a u-substitution
and hence evaluate tan(x) dx.
∫
b. In a similar way, evaluate cot(x) dx.
c. Consider the indefinite integral
∫
sec2 (x) + sec(x) tan(x)
dx.
sec(x) + tan(x)
Evaluate this integral using the substitution u sec(x) + tan(x).
d. Simplify the integrand in (c) by factoring the numerator. What is a far simpler
way to write the integrand?
∫
e. Combine your work in (c) and (d) to determine sec(x) dx.
∫
f. Using (c)-(e) as a guide, evaluate csc(x) dx.
∫ √
8. Consider the indefinite integral x x − 1 dx.
a. At first glance, this integrand may not seem suited to substitution due to the pres-
ence of x in separate locations in the integrand. Nonetheless, using the composite
√
function x − 1 as a guide, let u x − 1. Determine expressions for both x and
dx in terms of u.
b. Convert the given integral in x to a new integral in u.
√
c. Evaluate the integral in (b) by noting that u u 1/2 and observing that it is now
possible to rewrite the integrand in u by expanding through multiplication.
289
Chapter 5 Evaluating Integrals
∫ √ ∫ √
d. Evaluate each of the integrals x 2 x − 1 dx and x x 2 − 1 dx. Write a para-
graph to discuss the similarities among the three indefinite integrals in this prob-
lem and the role of substitution and algebraic rearrangement in each.
∫
9. Consider the indefinite integral sin3 (x) dx.
a. Explain why the substitution u sin(x) will not work to help evaluate the given
integral.
b. Recall the Fundamental Trigonometric Identity, which states that
sin2 (x) + cos2 (x) 1.
By observing that sin3 (x) sin(x) · sin2 (x), use the Fundamental Trigonometric
Identity to rewrite the integrand as the product of sin(x) with another function.
c. Explain why the substitution u cos(x) now provides a possible way to evaluate
the integral in (b).
∫
d. Use your work in (a)-(c) to evaluate the indefinite integral sin3 (x) dx.
∫
e. Use a similar approach to evaluate cos3 (x) dx.
10. For the town of Mathland, MI, residential power consumption has shown certain trends
over recent years. Based on data reflecting average usage, engineers at the power com-
pany have modeled the town’s rate of energy consumption by the function
r(t) 4 + sin(0.263t + 4.7) + cos(0.526t + 9.4).
Here, t measures time in hours after midnight on a typical weekday, and r is the rate
of consumption in megawatts³ at time t. Units are critical throughout this problem.
a. Sketch a carefully labeled graph of r(t) on the interval [0,24] and explain its mean-
ing. Why is this a reasonable model of power consumption?
∫ 24
b. Without calculating its value, explain the meaning of 0
r(t) dt. Include appro-
priate units on your answer.
c. Determine the exact amount of energy Mathland consumes in a typical day.
d. What is Mathland’s average rate of power consumption in a given 24-hour period?
What are the units on this quantity?
³The unit megawatt is itself a rate, which measures energy consumption per unit time. A megawatt-hour is the
total amount of energy that is equivalent to a constant stream of 1 megawatt of power being sustained for 1 hour.
290
5.4 Integration by Parts
5.4 Integration by Parts
Motivating Questions
• How
∫ do we evaluate∫indefinite integrals that involve products of basic functions such
as x sin(x) dx and xe x dx?
• What is the method of integration by parts and how can we consistently apply it to
integrate products of basic functions?
• How does the algebraic structure of functions guide us in identifying u and dv in
using integration by parts?
∫ of u-substitution for evaluating indefinite integrals.
In Section 5.3, we learned the technique
For example, the indefinite integral x 3 sin(x 4 ) dx is perfectly suited to u-substitution, be-
cause one factor is a composite function and the other factor is the derivative (up to a con-
stant) of the inner function. Recognizing the algebraic structure of a function can help us to
find its antiderivative.
Next we consider integrands with a different elementary algebraic structure: a product of
basic functions. For instance, suppose we are interested in evaluating the indefinite integral
∫
x sin(x) dx.
The integrand is the product of the basic functions f (x) x and 1(x) sin(x). We know
that it is relatively complicated to compute the derivative of the product of two functions, so
we should expect that antidifferentiating
∫ a product should be similarly involved. Intuitively,
we expect that evaluating x sin(x) dx will involve somehow reversing the Product Rule.
To that end, in Preview Activity 5.4.1 we refresh our understanding of the Product Rule and
then investigate some indefinite integrals that involve products of basic functions.
Preview Activity 5.4.1. In Section 2.3, we developed the Product Rule and studied
how it is employed to differentiate a product of two functions. In particular, recall
that if f and 1 are differentiable functions of x, then
d [ ]
f (x) · 1(x) f (x) · 1 ′(x) + 1(x) · f ′(x).
dx
a. For each of the following functions, use the Product Rule to find the function’s
derivative. Be sure to label each derivative by name (e.g., the derivative of 1(x)
should be labeled 1 ′(x)).
i. 1(x) x sin(x) iv. q(x) x 2 cos(x)
ii. h(x) xe x
iii. p(x) x ln(x) v. r(x) e x sin(x)
291
Chapter 5 Evaluating Integrals
b. Use your work in (a) to help you evaluate the following indefinite integrals. Use
differentiation to check your work.
∫ ∫
i. xe x + e x dx iv. x cos(x) + sin(x) dx
∫
ii. e x (sin(x) + cos(x)) dx
∫ ∫
iii. 2x cos(x) − x 2 sin(x) dx v. 1 + ln(x) dx
c. Observe that the examples in (b) work nicely because of the derivatives you
were asked to calculate in (a). Each integrand in (b) is precisely the result of
differentiating one of the products of basic functions found in (a). To see what
happens when an integrand is still a product but not necessarily the result of
differentiating an elementary product, we consider how to evaluate
∫
x cos(x) dx.
i. First, observe that
d
[x sin(x)] x cos(x) + sin(x).
dx
Integrating both sides indefinitely and using the fact that the integral of a
sum is the sum of the integrals, we find that
∫ ( ) ∫ ∫
d
[x sin(x)] dx x cos(x) dx + sin(x) dx.
dx
In this last equation, evaluate the indefinite integral on the left side as well
as the rightmost indefinite integral on the right.
ii. ∫In the most recent equation from (i.), solve the equation for the expression
x cos(x) dx.
iii. For which product of basic functions have you now found the antideriva-
tive?
5.4.1 Reversing the Product Rule: Integration by Parts
Problem (c) in Preview Activity 5.4.1 provides a clue to the general technique known as
Integration by Parts, which comes from reversing the Product Rule. Recall that the Product
Rule states that
d [ ]
f (x)1(x) f (x)1 ′(x) + 1(x) f ′(x).
dx
Integrating both sides of this equation indefinitely with respect to x, we find
∫ ∫ ∫
d [ ]
f (x)1(x) dx f (x)1 ′(x) dx + 1(x) f ′(x) dx. (5.4.1)
dx
292
5.4 Integration by Parts
On the left side of Equation (5.4.1), we have the indefinite integral of the derivative of a
function. Temporarily omitting the constant that may arise, we have
∫ ∫
f (x)1(x) f (x)1 ′(x) dx + 1(x) f ′(x) dx. (5.4.2)
We solve for the first indefinite integral on the left to generate the rule
∫ ∫
f (x)1 ′(x) dx f (x)1(x) − 1(x) f ′(x) dx. (5.4.3)
Often we express Equation (5.4.3) in terms of the variables u and v, where u f (x) and
v 1(x). In differential notation, du f ′(x) dx and dv 1 ′(x) dx, so we can state the rule
for Integration by Parts in its most common form as follows:
∫ ∫
u dv uv − v du.
To apply integration by parts, we look for a product of basic functions
∫ that we can identify as
u and dv. If we can
∫ antidifferentiate dv to find v, and evaluating v du is not more difficult
than evaluating u dv, then this substitution usually proves to be fruitful. To demonstrate,
we consider the following example.
Example 5.4.1 Evaluate the indefinite integral
∫
x cos(x) dx
using integration by parts.
Solution. When we use integration by parts, we have a choice for u and dv. In this problem,
we can either let u x and dv cos(x) dx, or let u cos(x) and dv x dx. While there is
∫ a universal rule for how to choose u and dv, a good
not ∫ guideline is this: do so in a way that
v du is at least as simple as the original problem u dv.
This leads us to choose¹ u x and dv cos(x) dx, from which it follows that du 1 dx and
v sin(x). With this substitution, the rule for integration by parts tells us that
∫ ∫
x cos(x) dx x sin(x) − sin(x) · 1 dx.
∫
All that remains to do is evaluate the (simpler) integral sin(x) · 1 dx. Doing so, we find
∫
x cos(x) dx x sin(x) − (− cos(x)) + C x sin(x) + cos(x) + C.
Observe that when we get to the final stage of evaluating the last remaining antiderivative,
it is at this step that we include the integration constant, +C.
¹Observe that if we considered the alternate choice, and let u cos(x) and dv x dx, then du − sin(x) dx
293
Chapter 5 Evaluating Integrals
The general technique of integration by parts involves trading the problem of integrating
the product of two functions for the problem of integrating
∫ the product of two related
∫ func-
tions. That is, we convert the problem of evaluating u dv to that of evaluating v du. This
clearly
∫ shapes our choice of u and v. In Example 5.4.1, the original integral to evaluate was
x cos(x) dx, and through
∫ the substitution provided by integration by parts, we were in-
stead able to evaluate sin(x) · 1 dx. Note that the original function x was replaced by its
derivative, while cos(x) was replaced by its antiderivative.
Activity 5.4.2. Evaluate each of the following indefinite integrals. Check each anti-
derivative
∫ that you find by differentiating. ∫
a. te −t dt c. z sec2 (z) dz
∫ ∫
b. 4x sin(3x) dx d. x ln(x) dx
5.4.2 Some Subtleties with Integration by Parts
Sometimes integration by parts is not an obvious choice, but the technique is appropriate
nonetheless. Integration by parts allows us to replace one function in a product with its
derivative while replacing the other with its antiderivative. For instance, consider evaluating
∫
arctan(x) dx.
Initially, this problem seems ill-suited to integration by parts, since there does not appear to
be a product of functions present. But if we note that arctan(x) arctan(x)·1, and realize that
we know the derivative of arctan(x) as well as the antiderivative of 1, we see the possibility
for the substitution u arctan(x) and dv 1 dx. We explore this substitution further in
Activity 5.4.3.
∫
In a related problem, consider t 3 sin(t 2 ) dt. Observe that there is a composite function
present in sin(t 2 ), but there is not an obvious function-derivative pair, as we have t 3 (rather
than simply t) multiplying sin(t 2 ). In this problem we use both u-substitution and integra-
tion by parts. First we write t 3 t · t 2 and consider the indefinite integral
∫
t · t 2 · sin(t 2 ) dt.
We let z t 2 so that dz 2t dt, and thus t dt 12 dz. (We are using the variable z to
perform a “z-substitution” first so that we may then apply integration by parts.) Under this
z-substitution, we now have
∫ ∫
1
t · t · sin(t ) dt
2 2
z · sin(z) · dz.
2
∫ ∫
and v 12 x 2 , from which we would write x cos(x) dx 12 x 2 cos(x) − 12 x 2 (− sin(x)) dx. Thus we have replaced
the problem of integrating x cos(x) with that of integrating 12 x 2 sin(x); the latter is clearly more complicated, which
shows that this alternate choice is not as helpful as the first choice.
294
5.4 Integration by Parts
The resulting integral can be evaluated by parts. This, too, is explored further in Activ-
ity 5.4.3.
These problems show that we sometimes must think creatively in choosing the variables
for substitution in integration by parts, and that we may need to use substitution for an
additional change of variables.
Activity 5.4.3. Evaluate each of the following indefinite integrals, using the provided
hints.
∫
a. Evaluate arctan(x) dx by using Integration by Parts with the substitution u
arctan(x) and dv 1 dx.
∫
b. Evaluate ln(z) dz. Consider a similar substitution to the one in (a).
∫
c. Use the substitution z t 2 to transform the integral t 3 sin(t 2 ) dt to a new
integral in the variable z, and evaluate that new integral by parts.
∫ 3
d. Evaluate s 5 e s ds using an approach similar to that described in (c).
∫
e. Evaluate e 2t cos(e t ) dt. You will find it helpful to note that e 2t e t · e t .
5.4.3 Using Integration by Parts Multiple Times
Integration by parts is well suited to integrating the product of basic functions, allowing us
to trade a given integrand for a new one where one function in the product is replaced ∫ by its
derivative,
∫ and the other is replaced by its antiderivative. The goal in this trade of u dv for
v du is that the new integral be simpler to evaluate than the original one. Sometimes it is
necessary to apply integration by parts more than once in order to evaluate a given integral.
∫
Example 5.4.2 Evaluate t 2 e t dt.
Solution. Let u t 2 and dv e t dt. Then du 2t dt and v e t , and thus
∫ ∫
t e dt t e −
2 t 2 t
2te t dt.
The integral on the right side is simpler to evaluate than the one on the left, but it still requires
integration by parts. Now letting u 2t and dv e t dt, we have du 2 dt and v e t , so
that ∫ ( ∫ )
t 2 e t dt t 2 e t − 2te t − 2e t dt .
(Note the parentheses,
∫ which remind us to distribute the minus sign to the entire value of
the integral 2te t dt.) The final integral on the right is a basic one; evaluating that integral
and distributing the minus sign, we find
∫
t 2 e t dt t 2 e t − 2te t + 2e t + C.
295
Chapter 5 Evaluating Integrals
Of course, even more than two applications of integration by parts may be necessary. In the
preceding example, if the integrand had been t 3 e t , we would have had to use integration by
parts three times.
Next, we consider the slightly different scenario.
∫
Example 5.4.3 Evaluate e t cos(t) dt.
Solution. We can choose to let u be either e t or cos(t); we pick u cos(t), and thus dv
e t dt. With du − sin(t) dt and v e t , integration by parts tells us that
∫ ∫
e cos(t) dt e cos(t) −
t t
e t (− sin(t)) dt,
or equivalently that ∫ ∫
e t cos(t) dt e t cos(t) + e t sin(t) dt. (5.4.4)
The new integral has the same algebraic structure as the original one. While the overall
situation isn’t necessarily better than what we started with, it hasn’t gotten worse. Thus,
we proceed to integrate by parts again. This time we let u sin(t) and dv e t dt, so that
du cos(t) dt and v e t , which implies
∫ ( ∫ )
e cos(t) dt e cos(t) + e sin(t) −
t t t t
e cos(t) dt . (5.4.5)
We seem to be back where we∫started, as two applications of integration by parts has led us
back to the original problem, e t cos(t) dt. But if we look closely at Equation (5.4.5),
∫ we see
that we can use algebra to solve for the value of the desired integral. Adding e t cos(t) dt
to both sides of the equation, we have
∫
2 e t cos(t) dt e t cos(t) + e t sin(t),
and therefore ∫
1( t )
e t cos(t) dt e cos(t) + e t sin(t) + C.
2
Note that since we never actually encountered an integral we could evaluate directly, we
didn’t have the opportunity to add the integration constant C until the final step.
Activity 5.4.4. Evaluate each of the following indefinite integrals.
∫
a. x 2 sin(x) dx
∫
b. t 3 ln(t) dt
∫
c. e z sin(z) dz
∫
d. s 2 e 3s ds
296
5.4 Integration by Parts
∫
e. t arctan(t) dt (Hint: At a certain point in this problem, it is very helpful to note
t2
that 1+t 2
1− 1
1+t 2
.)
5.4.4 Evaluating Definite Integrals Using Integration by Parts
We can use the technique of integration by parts to evaluate a definite integral.
Example 5.4.4 Evaluate
∫ π/2
t sin(t) dt.
0
Solution. One option is to find an antiderivative (using indefinite integral notation) and
then apply the Fundamental Theorem of Calculus to find that
∫ π/2 π/2
t sin(t) dt (−t cos(t) + sin(t))
0 0
( π π π )
− cos( ) + sin( ) − (−0 cos(0) + sin(0))
2 2 2
1.
Alternatively, we can apply integration by parts and work with definite integrals throughout.
With this method, we must remember to evaluate the product uv over the given limits of
integration. Using the substitution u t and dv sin(t) dt, so that du dt and v − cos(t),
we write
∫ π/2 π/2 ∫ π/2
t sin(t) dt − t cos(t) − (− cos(t)) dt
0 0 0
π/2 π/2
− t cos(t) + sin(t)
0 0
( π π π )
− cos( ) + sin( ) − (−0 cos(0) + sin(0))
2 2 2
1.
As with any substitution technique, it is important to use notation carefully and completely,
and to ensure that the end result makes sense.
5.4.5 When u-substitution and Integration by Parts Fail to Help
Both integration techniques we have discussed apply in relatively limited circumstances. It
is not hard to find examples of functions for which neither technique produces an antideriv-
ative; indeed, there are many, many functions that appear elementary but that do not have
an elementary algebraic antiderivative. For instance, neither u-substitution nor integration
297
Chapter 5 Evaluating Integrals
by parts proves fruitful for the indefinite integrals
∫ ∫
2
e x dx and x tan(x) dx.
While there are other integration techniques, some of which we will consider briefly, none of
2
them enables us to find an algebraic antiderivative for e x or x tan(x). We do know from the
Second Fundamental Theorem∫ x 2 of Calculus that we can construct2 an integral antiderivative
∫x
for each function; F(x) 0 e t dt is an antiderivative of f (x) e x , and G(x) 0 t tan(t) dt
is an antiderivative of 1(x) x tan(x). But finding an elementary algebraic formula that
doesn’t involve integrals for either F or G turns out not only to be impossible through u-
substitution or integration by parts, but indeed impossible altogether. Antidifferentiation is
much harder in general than differentiation.
5.4.6 Summary
• Through the method of integration by parts, ∫ we can evaluate∫ indefinite integrals that
involve products of basic functions such as x sin(x) dx and x ln(x) dx. Using a sub-
stitution enables us to trade one of the functions in the product for its derivative, and
the other for its antiderivative, in an effort to find a different product of functions that
is easier to integrate.
• If
∫ the algebraic structure of an integrand is a product of basic functions in the form
f (x)1 ′(x) dx, we can use the substitution u f (x) and dv 1 ′(x) dx and apply the
rule ∫ ∫
u dv uv − v du
∫
to evaluate the original integral f (x)1 ′(x) dx by instead evaluating
∫ ∫
v du f ′(x)1(x) dx.
• When deciding to integrate by parts, we have to select∫ both u and dv. That selection is
guided by the overall∫principle that the new integral v du not be more difficult than
the original integral u dv. In addition, it is often helpful to recognize if one of the
functions present is much easier to differentiate than antidifferentiate (such as ln(x)),
in which case that function often is best assigned the variable u. In addition, dv must
be a function that we can antidifferentiate.
5.4.7 Exercises
1. Choose which method to use. For each of the following integrals, indicate whether
integration by substitution or integration by parts is more appropriate, or if neither
method is appropriate. Do not evaluate the integrals.
∫
1. x sin x dx
298
5.4 Integration by Parts
∫
x4
2. 1+x 5
dx
∫ 5
3. x 4 e x dx
∫
4. x 4 cos(x 5 ) dx
∫
5. √ 1 dx
5x+1
2. Product involving cos(5x). Use integration by parts to evaluate the integral.
∫
3x cos(4x) dx.
3. Product involving e 8z . Find the integral
∫
(z + 1) e 5z dz.
4. Definite integral of te −t . Evaluate the definite integral.
∫ 5
te −t dt.
0
∫x
5. Let f (t) te −2t and F(x) 0
f (t) dt.
a. Determine F′(x).
b. Use the First FTC to find a formula for F that does not involve an integral.
c. Is F an increasing or decreasing function for x > 0? Why?
∫
6. Consider the indefinite integral given by e 2x cos(e x ) dx.
a. Noting that e 2x e x ·e x , use the substitution z e x to determine a new, equivalent
integral in the variable z.
b. Evaluate the integral you found in (a) using an appropriate technique.
∫
c. How is the problem of evaluating e 2x cos(e 2x ) dx different from evaluating the
integral in (a)? Do so.
d. Evaluate each of the following integrals as well, keeping in mind the approach(es)
used earlier in this problem:
∫
• e 2x sin(e x ) dx
∫
• e 3x sin(e 3x ) dx
∫ 2 2 2
• xe x cos(e x ) sin(e x ) dx
7. For each of the following indefinite integrals, determine whether you would use u-
substitution, integration by parts, neither∗, or both to evaluate the integral. In each
case, write one sentence to explain your reasoning, and include a statement of any sub-
stitutions used. (That is, if you decide in a problem to let u e 3x , you should state
that, as well as that du 3e 3x dx.) Finally, use your chosen approach to evaluate each
299
Chapter 5 Evaluating Integrals
integral. (∗ one of the following problems does not have an elementary antiderivative
and you are not expected to actually evaluate this integral; this will correspond with a
choice of “neither” among those given.)
∫
a. x 2 cos(x 3 ) dx
∫
b. x 5 cos(x 3 ) dx (Hint: x 5 x 2 · x 3 )
∫
c. x ln(x 2 ) dx
∫
d. sin(x 4 ) dx
∫
e. x 3 sin(x 4 ) dx
∫
f. x 7 sin(x 4 ) dx
300
5.5 Other Options for Finding Algebraic Antiderivatives
5.5 Other Options for Finding Algebraic Antiderivatives
Motivating Questions
• How does the method of partial fractions enable any rational function to be antidif-
ferentiated?
• What role have integral tables historically played in the study of calculus and how
∫ √
can a table be used to evaluate integrals such as a 2 + u 2 du?
• What role can a computer algebra system play in the process of finding antideriva-
tives?
We have learned two antidifferentiation techniques: u-substitution and integration by parts.
The former is used to reverse the chain rule, while the latter to reverse the product rule.
But
∫ we have seen that each works only in specialized
∫ circumstances. For example, while
x 2 x
xe dx may be evaluated by u-substitution and xe dx by integration by parts, neither
∫ 2
method provides a route to evaluate e x dx, and in fact an elementary algebraic antideriv-
2
ative for e x does not exist. No antidifferentiation method will provide us with a simple
algebraic formula for a function F(x) that satisfies F′(x) e x .
2
In this section of the text, our main goals are to identify some classes of functions that can be
antidifferentiated, and to learn some methods to do so. We should also recognize that there
are many functions for which an algebraic formula for an antiderivative does not exist, and
appreciate the role that computing technology can play in finding antiderivatives of other
complicated functions.
Preview Activity 5.5.1. For each of the indefinite integrals below, the main question
is to decide whether the integral can be evaluated using u-substitution, integration
by parts, a combination of the two, or neither. For integrals for which your answer
is affirmative, state the substitution(s) you would use. It is not necessary to actually
evaluate any of the integrals completely, unless the integral can be evaluated imme-
diately using a familiar basic antiderivative.
∫ ∫ ∫ ∫
a. x 2 sin(x 3 ) dx, x 2 sin(x) dx, sin(x 3 ) dx, x 5 sin(x 3 ) dx
∫ ∫ ∫ ∫
1 x 2x+3 ex
b. 1+x 2
dx, 1+x 2
dx, 1+x 2
dx, 1+(e x )2
dx,
∫ ∫ ln(x)
∫ ∫
c. x ln(x) dx, x dx, ln(1 + x 2 ) dx, x ln(1 + x 2 ) dx,
∫ √ ∫ ∫ ∫
d. x 1 − x 2 dx, √ 1
2
dx, √ x
2
dx, √1
2
dx,
1−x 1−x x 1−x
301
Chapter 5 Evaluating Integrals
5.5.1 The Method of Partial Fractions
The method of partial fractions is used to integrate rational functions. It involves reversing
the process of finding a common denominator.
Example 5.5.1 Evaluate ∫
5x
dx.
x2 −x−2
Solution. If we factor the denominator, we can see how R might be the sum of two fractions
A
of the form x−2 + x+1
B
, so we suppose that
5x A B
+
(x − 2)(x + 1) x − 2 x + 1
and look for the constants A and B.
Multiplying both sides of this equation by (x − 2)(x + 1), we find that
5x A(x + 1) + B(x − 2).
Since we want this equation to hold for every value of x, we can use insightful choices of
specific x-values to help us find A and B. Taking x −1, we have
5(−1) A(0) + B(−3),
so B 53 . Choosing x 2, it follows
5(2) A(3) + B(0),
so A 10
3 . Thus, ∫ ∫
5x 10/3 5/3
dx + dx.
x −x−2
2 x−2 x+1
This integral is straightforward to evaluate, and hence we find that
∫
5x 10 5
dx ln |x − 2| + ln |x + 1| + C.
x2 −x−2 3 3
It turns out that we can use the method of partial fractions to rewrite any ratinal function
P(x)
R(x) Q(x) where the degree of the polynomial P is less than¹ the degree of Q as a sum of
simpler rational functions of one of the following forms:
A A Ax + B Ax + B
, , , or
x − c (x − c)n x 2 + k (x 2 + k)
n
where A, B, and c are real numbers, and k is a positive real number. Because we can an-
tidifferentiate each of these basic forms, partial fractions enables us to antidifferentiate any
rational function.
¹If the degree of P is greater than or equal to the degree of Q, long division may be used to write R as the sum
of a polynomial plus a rational function where the numerator’s degree is less than the denominator’s.
302
5.5 Other Options for Finding Algebraic Antiderivatives
A computer algebra system such as Maple, Mathematica, or WolframAlpha can be used to find
the partial fraction decomposition of any rational function. In WolframAlpha, entering
partial fraction 5x/(x^2-x-2)
results in the output
5x 10 5
+ .
x 2 − x − 2 3(x − 2) 3(x + 1)
We will use technology to generate partial fraction decompositions of rational functions, and
then evaluate the integrals using established methods.
Activity 5.5.2. For each of the following problems, evaluate the integral by using the
partial fraction decomposition provided.
∫ 1/4 1/4
a. 1
x 2 −2x−3
dx, given that 1
x 2 −2x−3
x−3 − x+1
∫
x 2 +1 x 2 +1
b. x 3 −x 2
dx, given that x 3 −x 2
− x1 − 1
x2
+ 2
x−1
∫
−x+2
c. x−2
x 4 +x 2
dx, given that x−2
x 4 +x 2
1
x − 2
x2
+ 1+x 2
5.5.2 Using an Integral Table
Calculus has a long history, going back to Greek mathematicians in 400-300 BC. Its main
foundations were first investigated and understood independently by Isaac Newton and
Gottfried Wilhelm Leibniz in the late 1600s, making the modern ideas of calculus well over
300 years old. It is instructive to realize that until the late 1980s, the personal computer did
not exist, so calculus (and other mathematics) had to be done by hand for roughly 300 years.
In the 21st century, however, computers have revolutionized many aspects of the world we
live in, including mathematics. In this section we take a short historical tour to precede
discussing the role computer algebra systems can play in evaluating indefinite integrals. In
particular, we consider a class of integrals involving certain radical expressions.
As seen in the short table of integrals found in Appendix A, there are many forms of integrals
√ √
that involve a 2 ± w 2 and w 2 − a 2 . These integral rules can be developed using a technique
known as trigonometric substitution that we choose to omit; instead, we will simply accept
the results presented in the table. To see how these rules are used, consider the differences
among ∫ ∫ ∫ √
1 x
√ dx, √ dx, and 1 − x 2 dx.
1−x 2 1−x 2
The first integral is a familiar basic one, and results in arcsin(x) + C. The second integral
can be evaluated using a standard u-substitution with u 1 − x 2 . The third, however, is not
familiar and does not lend itself to u-substitution.
In Appendix A, we find the rule
∫ √ (u)
u√ 2 a2
(h) a 2 − u 2 du a − u2 + arcsin + C.
2 2 a
303
Chapter 5 Evaluating Integrals
Using the substitutions a 1 and u x (so that du dx), it follows that
∫ √
x√ 1
1 − x 2 dx 1 − x 2 − arcsin(x) + C.
2 2
Whenever we are applying a rule in the table, we are doing a u-substitution, especially when
the substitution is more complicated than setting u x as in the last example.
Example 5.5.2 Evaluate the integral
∫ √
9 + 64x 2 dx.
Solution. Here, we want to use Rule (c) from the table, but we now set a 3 and u 8x.
We also choose the “+” option in the rule. With this substitution, it follows that du 8dx,
so dx 18 du. Applying the substitution,
∫ √ ∫ √ ∫ √
1 1
9 + 64x 2 dx 9 + u2 · du 9 + u 2 du.
8 8
By Rule (c), we now find that
∫ √ ( )
1 u√ 2 9 √
9+ 64x 2 dx u + 9 + ln |u + u 2 + 9| + C
8 2 2
( √ √
)
1 8x 9
64x 2 + 9 + ln |8x + 64x 2 + 9| + C .
8 2 2
Whenever we use a u-subsitution in conjunction with Appendix A, it’s important that we
not forget to address any constants that arise and include them in our computations, such
as the 18 that appeared in Example 5.5.2.
Activity 5.5.3. For each of the following integrals, evaluate the integral using u-
∫ √ and/or an entry from the table found∫ in Appendix
substitution A.
a. x 2 + 4 dx c. √ 2 dx
2 16+25x
∫ ∫
b. √ x dx d. √ 1 dx
x 2 +4 x 2 49−36x 2
5.5.3 Using Computer Algebra Systems
A computer algebra system (CAS) is a computer program that is capable of executing sym-
bolic mathematics. For example, if we ask a CAS to solve the equation ax 2 + bx + c 0√for the
b −4ac
variable x, where a, b, and c are arbitrary constants, the program will return x −b± 2a
2
.
Research to develop the first CAS dates to the 1960s, and these programs became publicly
available in the early 1990s. Two prominent examples are the programs Maple and Mathemat-
ica, which were among the first computer algebra systems to offer a graphical user interface.
Today, Maple and Mathematica are exceptionally powerful professional software packages
304
5.5 Other Options for Finding Algebraic Antiderivatives
that can execute an amazing array of sophisticated mathematical computations. They are
also very expensive, as each is a proprietary program. The CAS SAGE is an open-source,
free alternative to Maple and Mathematica.
For the purposes of this text, when we need to use a CAS, we are going to turn instead to a
similar, but somewhat different computational tool, the web-based “computational knowl-
edge engine” called WolframAlpha. There are two features of WolframAlpha that make it stand
out from the CAS options mentioned above: (1) unlike Maple and Mathematica, WolframAlpha
is free (provided we are willing to navigate some pop-up advertising); and (2) unlike any
of the three, the syntax in WolframAlpha is flexible. Think of WolframAlpha as being a little
bit like doing a Google search: the program will interpret what is input, and then provide a
summary of options.
If we want to have WolframAlpha evaluate an integral for us, we can provide it syntax such
as
integrate x^2 dx
to which the program responds with
∫
x3
x 2 dx + constant.
3
While there is much to be enthusiastic about regarding CAS programs such as WolframAlpha,
there are several things we should be cautious about: (1) a CAS only responds to exactly what
is input; (2) a CAS can answer using powerful functions from very advanced mathematics;
and (3) there are problems that even a CAS cannot do without additional human insight.
Although (1) likely goes without saying, we have to be careful with our input: if we enter
syntax that defines the wrong function, the CAS will work with precisely the function we
define. For example, if we are interested in evaluating the integral
∫
1
dx,
16 − 5x 2
and we mistakenly enter
integrate 1/16 - 5x^2 dx
a CAS will (correctly) reply with
1 5
x − x3.
16 3
But if we are sufficiently well-versed in antidifferentiation, we will recognize that this func-
1
tion cannot be the one that we seek: integrating a rational function such as 16−5x 2 , we expect
the logarithm function to be present in the result.
∫
1
Regarding (2), even for a relatively simple integral such as 16−5x 2 dx, some CASs will in-
voke advanced functions rather than simple ones. For instance, if we use Maple to execute
the command
305
Chapter 5 Evaluating Integrals
int(1/(16-5*x^2), x);
the program responds with
∫ √ √
1 5 5
dx arctanh( x).
16 − 5x 2 20 4
While this is correct (save for the missing arbitrary constant, which Maple never reports),
the inverse hyperbolic tangent function is not a common nor familiar one; a simpler way to
express this function can be found by using the partial fractions method, and happens to be
the result reported by WolframAlpha:
∫
1 1 ( √ √ )
dx √ log(4 5 + 5x) − log(4 5 − 5x) + constant.
16 − 5x 2 8 5
Using sophisticated functions from more advanced mathematics is sometimes the way a
CAS says to the user “I don’t know how to do this problem.” For example, if we want to
evaluate ∫
e −x dx,
2
and we ask WolframAlpha to do so, the input
integrate exp(-x^2) dx
results in the output
∫ √
−x 2 π
e dx erf(x) + constant.
2
The function “erf(x)” is the error function, which is actually defined by an integral:
∫ x
2
e −t dt.
2
erf(x) √
π 0
So, in producing output involving an integral, the CAS has basically reported back to us the
very question we asked.
Finally, as remarked at (3) above, there are times that a CAS will actually fail without some
additional human insight. If we consider the integral
∫ √
(1 + x)e x 1 + x 2 e 2x dx
and ask WolframAlpha to evaluate
int (1+x) * exp(x) * sqrt(1+x^2 * exp(2x)) dx,
the program thinks for a moment and then reports
306
5.5 Other Options for Finding Algebraic Antiderivatives
(no result found in terms of standard mathematical functions)
But in fact this integral is not that difficult to evaluate. If we let u xe x , then du (1 +
x)e x dx, which means that the preceding integral has form
∫ √ ∫ √
(1 + x)e x 1 + x 2 e 2x dx 1 + u 2 du,
which is a straightforward one for any CAS to evaluate.
So, we should proceed with some caution: while any CAS is capable of evaluating a wide
range of integrals (both definite and indefinite), there are times when the result can mislead
us. We must think carefully about the meaning of the output, whether it is consistent with
what we expect, and whether or not it makes sense to proceed.
5.5.4 Summary
• We can antidifferentiate any rational function with the method of partial fractions. Any
polynomial function can be factored into a product of linear and irreducible quadratic
terms, so any rational function may be written as the sum of a polynomial plus rational
A Bx+C
terms of the form (x−c) n (where n is a natural number) and x 2 +k (where k is a positive
real number).
• Until the development of compute algebra
∫ √ systems, integral tables enabled students of
calculus to evaluate integrals such as a 2 + u 2 du, where a is a positive real number.
A short table of integrals may be found in Appendix A.
• Computer algebra systems can play an important role in finding antiderivatives, though
we must be cautious to use correct input, to watch for unusual or unfamiliar advanced
functions that the CAS may cite in its result, and to consider the possibility that a CAS
may need further assistance or insight from us in order to answer a particular question.
5.5.5 Exercises
1. Partial fractions: linear over difference of squares. Calculate the integral below by
partial fractions and by using the indicated substitution. Be sure that you can show
how the results you obtain are the same.
∫
2x
dx
x 2 − 36
First, rewrite this with partial fractions:
∫ ∫ ∫
2x
x 2 −36
dx dx + dx
+ +C.
Next, use the substitution w x 2 − 36 to find the integral.
307
Chapter 5 Evaluating Integrals
2. Partial fractions: constant over product. Calculate the integral:
∫
1
dx.
(x + 6)(x + 8)
3. Partial fractions: linear over quadratic. Calculate the integral
∫
8x + 6
dx.
x 2 − 3x + 2
4. Partial fractions: cubic over 4th degree. Consider the following indefinite integral.
∫
9x 3 + 6x 2 + 100x + 75
dx
x 4 + 25x 2
The integrand has partial fractions decomposition:
a b cx + d
+ +
x 2 x x 2 + 25
where
a
b
c
d
Now integrate term by term to evaluate the integral.
5. Partial fractions: quadratic over factored cubic. The form of the partial fraction de-
composition of a rational function is given below.
( )
− 4x 2 + x + 32 A Bx + C
+ 2
(x − 4)(x 2 + 9) x−4 x +9
A B C
Now evaluate the indefinite integral.
∫ ( )
− 4x 2 + x + 32
dx.
(x − 4)(x 2 + 9)
6. For each of the following integrals involving rational functions, (1) use a CAS to find the
partial fraction decomposition of the integrand; (2) evaluate the integral of the resulting
function without the assistance of technology; (3) use a CAS to evaluate the original
integral to test and compare your result in (2).
∫
x 3 +x+1
a. x 4 −1
dx
∫
x 5 +x 2 +3
b. x 3 −6x 2 +11x−6
dx
308
5.5 Other Options for Finding Algebraic Antiderivatives
∫
x 2 −x−1
c. (x−3)3
dx
7. For each of the following integrals involving radical functions, (1) use an appropriate
u-substitution along with Appendix A to evaluate the integral without the assistance
of technology, and (2) use a CAS to evaluate the original integral to test and compare
your result in (1).
∫
a. √ 1 dx
x 9x 2 +25
∫ √
b. x 1 + x 4 dx
∫ √
c. e x 4 + e 2x dx
∫
d. √ tan(x) dx
9−cos2 (x)
8. Consider the indefinite integral given by
∫ √ √
x + 1 + x2
dx.
x
a. Explain why u-substitution does not offer a way to simplify this integral by dis-
cussing at least two different options you might try for u.
b. Explain why integration by parts does not seem to be a reasonable way to proceed,
either, by considering one option for u and dv.
c. Is there any line in the integral table in Appendix A that is helpful for this integral?
d. Evaluate the given integral using WolframAlpha. What do you observe?
309
Chapter 5 Evaluating Integrals
5.6 Numerical Integration
Motivating Questions
∫1
• How do we accurately evaluate a definite integral such as 0 e −x dx when we can-
2
not use the First Fundamental Theorem of Calculus because the integrand lacks an
elementary algebraic antiderivative? Are there ways to generate accurate estimates
without using extremely large values of n in Riemann sums?
• What is the Trapezoid Rule, and how is it related to left, right, and middle Riemann
sums?
• How are the errors in the Trapezoid Rule and Midpoint Rule related, and how can
they be used to develop an even more accurate rule?
When we first explored finding the net signed area bounded by a curve, we developed the
concept of a Riemann sum as a helpful estimation tool and a key step in the definition of the
definite integral. Recall that the left, right, and middle Riemann sums of a function f on an
interval [a, b] are given by
∑
n−1
L n f (x 0 )∆x + f (x 1 )∆x + · · · + f (x n−1 )∆x f (x i )∆x, (5.6.1)
i0
∑n
R n f (x1 )∆x + f (x2 )∆x + · · · + f (x n )∆x f (x i )∆x, (5.6.2)
i1
∑n
M n f (x 1 )∆x + f (x 2 )∆x + · · · + f (x n )∆x f (x i )∆x, (5.6.3)
i1
where x0 a, x i a + i∆x, x n b, and ∆x b−a
n . For the middle sum, we defined
x i (x i−1 + x i )/2.
A Riemann sum is a sum of (possibly signed) areas of rectangles. The value of n determines
the number of rectangles, and our choice of left endpoints, right endpoints, or midpoints
determines the heights of the rectangles. We can see the similarities and differences among
these three options in Figure 5.6.1, where we consider the function f (x) 20 1
(x−4)3 +7 on the
interval [1, 8], and use 5 rectangles for each of the Riemann sums. While it is a good exercise
to compute a few Riemann sums by hand, just to ensure that we understand how they work
and how varying the function, the number of subintervals, and the choice of endpoints or
midpoints affects the result, using computing technology is the best way to determine L n ,
R n , and M n . Any computer algebra system will offer this capability; as we saw in Preview
Activity 4.3.1, a straightforward option that is freely available online is the applet¹ at http://
gvsu.edu/s/a9. Note that we can adjust the formula for f (x), the window of x- and y-values
of interest, the number of subintervals, and the method. (See Preview Activity 4.3.1 for any
needed reminders on how the applet works.)
In this section we explore several different alternatives for estimating definite integrals. Our
¹Marc Renault, Shippensburg University
310
5.6 Numerical Integration
y = f (x) y = f (x) y = f (x)
1 LEFT 8 1 RIGHT 8 1 MID 8
Figure 5.6.1: Left, right, and middle Riemann sums for y f (x) on [1, 8] with 5
subintervals.
main goal is to develop formulas to estimate definite integrals accurately without using a
large numbers of rectangles.
Preview Activity 5.6.1. As we begin to investigate ways to approximate definite inte-
grals, it will be insightful to compare results to integrals whose exact values we know.
∫3
To that end, the following sequence of questions centers on 0
x 2 dx.
a. Use the applet at http://gvsu.edu/s/a9 with the function f (x) x 2 on the
window of x values from 0 to 3 to compute L3 , the left Riemann sum with three
subintervals.
b. Likewise, use the applet to compute R 3 and M3 , the right and middle Riemann
sums with three subintervals, respectively.
c. Use the Fundamental Theorem of Calculus to compute the exact value of I
∫3
0
x 2 dx.
d. We define the error that results from an approximation of a definite integral to
be the approximation’s value minus the integral’s exact value. What is the error
that results from using L 3 ? From R 3 ? From M3 ?
e. In what follows in this section, we will learn a new approach to estimating the
value of a definite integral known as the Trapezoid Rule. The basic idea is to use
trapezoids, rather than rectangles, to estimate the area under a curve. What is
the formula for the area of a trapezoid with bases of length b 1 and b 2 and height
h?
f. Working by hand, estimate the area under f (x) x 2 on [0, 3] using three subin-
tervals and three corresponding trapezoids. What is the error in this approxi-
mation? How does it compare to the errors you calculated in (d)?
311
Chapter 5 Evaluating Integrals
5.6.1 The Trapezoid Rule
So far, we have used the simplest possible quadrilaterals (that is, rectangles) to estimate
areas. It is natural, however, to wonder if other familiar shapes might serve us even better.
An alternative to L n , R n , and M n is called the Trapezoid Rule. Rather than using a rectangle
to estimate the (signed) area bounded by y f (x) on a small interval, we use a trapezoid.
For example, in Figure 5.6.2, we estimate the area under the curve using three subintervals
and the trapezoids that result from connecting the corresponding points on the curve with
straight lines.
y = f (x)
D1 D2 D3
x0 x1 x2 x3
∫b
Figure 5.6.2: Estimating a f (x) dx using three subintervals and trapezoids, rather than
rectangles, where a x 0 and b x 3 .
The biggest difference between the Trapezoid Rule and a Riemann sum is that on each subin-
terval, the Trapezoid Rule uses two function values, rather than one, to estimate the (signed)
area bounded by the curve. For instance, to compute D1 , the area of the trapezoid on [x0 , x 1 ],
we observe that the left base has length f (x0 ), while the right base has length f (x 1 ). The
height of the trapezoid is x1 − x0 ∆x b−a
3 . The area of a trapezoid is the average of the
bases times the height, so we have
1
D1 ( f (x0 ) + f (x1 )) · ∆x.
2
Using similar computations for D2 and D3 , we find that T3 , the trapezoidal approximation
∫b
to a
f (x) dx is given by
T3 D1 + D2 + D3
1 1 1
( f (x 0 ) + f (x1 )) · ∆x + ( f (x1 ) + f (x 2 )) · ∆x + ( f (x 2 ) + f (x3 )) · ∆x.
2 2 2
312
5.6 Numerical Integration
Because both left and right endpoints are being used, we recognize within the trapezoidal
approximation the use of both left and right Riemann sums. Rearranging the expression for
T3 by removing factors of 12 and ∆x, grouping the left endpoint and right endpoint evalua-
tions of f , we see that
1[ ] 1[ ]
T3 f (x0 ) + f (x 1 ) + f (x2 ) ∆x + f (x1 ) + f (x 2 ) + f (x3 ) ∆x. (5.6.4)
2 2
We now observe that two familiar sums have arisen. The left Riemann sum L3 is L3
f (x 0 )∆x + f (x1 )∆x + f (x2 )∆x, and the right Riemann sum is R 3 f (x1 )∆x + f (x2 )∆x +
f (x 3 )∆x. Substituting L3 and R 3 for the corresponding expressions in Equation (5.6.4), it
follows that T3 12 [L3 + R 3 ]. We have thus seen a very important result: using trapezoids
to estimate the (signed) area bounded by a curve is the same as averaging the estimates
generated by using left and right endpoints.
The Trapezoid Rule.
∫b
The trapezoidal approximation, Tn , of the definite integral a
f (x) dx using n subin-
tervals is given by the rule
[ ]
1 1 1
Tn ( f (x0 ) + f (x 1 )) + ( f (x 1 ) + f (x2 )) + · · · + ( f (x n−1 ) + f (x n )) ∆x.
2 2 2
∑
n−1
1
( f (x i ) + f (x i+1 ))∆x.
2
i0
Moreover, Tn 1
2 [L n + R n ].
Activity 5.6.2. In this activity, we explore the relationships among the errors gener-
ated by left, right, midpoint, and trapezoid approximations to the definite integral
∫2
1
1 x2
dx.
∫2
1
a. Use the First FTC to evaluate 1 x2
dx exactly.
b. Use appropriate computing technology to compute the following approxima-
∫2
1
tions for 1 x2
dx: T4 , M4 , T8 , and M8 .
c. Let the error that results from an approximation be the approximation’s value
minus the exact value of the definite integral. For instance, if we let ET,4 repre-
sent the error that results from using the trapezoid rule with 4 subintervals to
estimate the integral, we have
∫ 2
1
ET,4 T4 − dx.
1 x2
Similarly, we compute the error of the midpoint rule approximation with 8 subin-
tervals by the formula
∫ 2
1
E M,8 M8 − 2
dx.
1 x
313
Chapter 5 Evaluating Integrals
Based on your work in (a) and (b) above, compute ET,4 , ET,8 , E M,4 , E M,8 .
d. Which rule consistently over-estimates the exact value of the definite integral?
Which rule consistently under-estimates the definite integral?
e. What behavior(s) of the function f (x) 1
x2
lead to your observations in (d)?
5.6.2 Comparing the Midpoint and Trapezoid Rules
We know from the definition of the definite integral that if we let n be large enough, we
can make any of the approximations L n , R n , and M n as close as we’d like (in theory) to the
∫b
exact value of a f (x) dx. Thus, it may be natural to wonder why we ever use any rule other
than L n or R n (with a sufficiently large n value) to estimate a definite integral. One of the
primary reasons is that as n → ∞, ∆x b−a n → 0, and thus in a Riemann sum calculation
with a large n value, we end up multiplying by a number that is very close to zero. Doing
so often generates roundoff error, because representing numbers close to zero accurately is
a persistent challenge for computers.
Hence, we explore ways to estimate definite integrals to high levels of precision, but without
using extremely large values of n. Paying close attention to patterns in errors, such as those
observed in Activity 5.6.2, is one way to begin to see some alternate approaches.
To begin, we compare the errors in the Midpoint and Trapezoid rules. First, consider a
function that is concave up on a given interval, and picture approximating the area bounded
on that interval by both the Midpoint and Trapezoid rules using a single subinterval.
T1 M1 M1
∫b
Figure 5.6.3: Estimating a f (x) dx using a single subinterval: at left, the trapezoid rule; in
the middle, the midpoint rule; at right, a modified way to think about the midpoint rule.
As seen in Figure 5.6.3, it is evident that whenever the function is concave up on an inter-
val, the Trapezoid Rule with one subinterval, T1 , will overestimate the exact value of the
definite integral on that interval. From a careful analysis of the line that bounds the top of
the rectangle for the Midpoint Rule (shown in magenta), we see that if we rotate this line
segment until it is tangent to the curve at the midpoint of the interval (as shown at right in
314
5.6 Numerical Integration
Figure 5.6.3), the resulting trapezoid has the same area as M1 , and this value is less than the
exact value of the definite integral. Thus, when the function is concave up on the interval,
M1 underestimates the integral’s true value.
M1
∫b
Figure 5.6.4: Comparing the error in estimating a f (x) dx using a single subinterval: in
red, the error from the Trapezoid rule; in light red, the error from the Midpoint rule.
These observations extend easily to the situation where the function’s concavity remains
consistent but we use larger values of n in the Midpoint and Trapezoid Rules. Hence, when-
∫b
ever f is concave up on [a, b], Tn will overestimate the value of f (x) dx, while M n will
∫b a
underestimate a
f (x) dx. The reverse observations are true in the situation where f is con-
cave down.
Next, we compare the size of the errors between M n and Tn . Again, we focus on M1 and T1
on an interval where the concavity of f is consistent. In Figure 5.6.4, where the error of the
Trapezoid Rule is shaded in red, while the error of the Midpoint Rule is shaded lighter red,
it is visually apparent that the error in the Trapezoid Rule is more significant. To see how
much more significant, let’s consider two examples and some particular computations.
∫1
If we let f (x) 1 − x 2 and consider 0
f (x) dx, we know by the First FTC that the exact value
of the integral is
∫ 1 1
x3 2
(1 − x 2 ) dx x − .
0 3 0 3
Using appropriate technology to compute M4 , M8 , T4 , and T8 , as well as the corresponding
errors E M,4 , E M,8 , ET,4 , and ET,8 , as we did in Activity 5.6.2, we find the results summarized
∫2
1
in Table 5.6.5. We also include the approximations and their errors for the example 1 x2
dx
from Activity 5.6.2.
315
Chapter 5 Evaluating Integrals
∫1 ∫2
Rule 0
(1 − x 2 ) dx 0.6 error dx 0.5
1
1 x2
error
T4 0.65625 −0.0104166667 0.5089937642 0.0089937642
M4 0.671875 0.0052083333 0.4955479365 −0.0044520635
T8 0.6640625 −0.0026041667 0.5022708502 0.0022708502
M8 0.66796875 0.0013020833 0.4988674899 −0.0011325101
Table 5.6.5: Calculations of T4 , M4 , T8 , and M8 , along with corresponding errors, for the
∫1 ∫2
definite integrals 0
(1 − x 2 ) dx and 1
1 x2
dx.
∫b
For a given function f and interval [a, b], ET,4 T4 − a f (x) dx calculates the difference
between the approximation generated by the Trapezoid Rule with n 4 and the exact value
of the definite integral. If we look at not only ET,4 , but also the other errors generated by
using Tn and M n with n 4 and n 8 in the two examples noted in Table 5.6.5, we see
an evident pattern. Not only is the sign of the error (which measures whether the rule
generates an over- or under-estimate) tied to the rule used and the function’s concavity, but
the magnitude of the errors generated by Tn and M n seems closely connected. In particular,
the errors generated by the Midpoint Rule seem to be about half the size (in absolute value)
of those generated by the Trapezoid Rule.
That is, we can observe in both examples that E M,4 ≈ − 12 ET,4 and E M,8 ≈ − 21 ET,8 . This
property of the Midpoint and Trapezoid Rules turns out to hold in general: for a function of
consistent concavity, the error in the Midpoint Rule has the opposite sign and approximately
half the magnitude of the error of the Trapezoid Rule. Written symbolically,
1
E M,n ≈ − ET,n .
2
This important relationship suggests a way to combine the Midpoint and Trapezoid Rules
to create an even more accurate approximation to a definite integral.
5.6.3 Simpson’s Rule
When we first developed the Trapezoid Rule, we observed that it is an average of the Left
and Right Riemann sums:
1
Tn (L n + R n ).
2
If a function is always increasing or always decreasing on the interval [a, b], one of L n and
∫b
R n will over-estimate the true value of a f (x) dx, while the other will under-estimate the
integral. Thus, the errors that result from L n and R n will have opposite signs; so averaging
L n and R n eliminates a considerable amount of the error present in the respective approx-
imations. In a similar way, it makes sense to think about averaging M n and Tn in order to
generate a still more accurate approximation.
We’ve just observed that M n is typically about twice as accurate as Tn . So we use the
weighted average
2M n + Tn
S2n . (5.6.5)
3
316
5.6 Numerical Integration
The rule for S2n giving by Equation (5.6.5) is usually known as Simpson’s Rule.² Note that
we use “S2n ” rather that “S n ” since the n points the Midpoint Rule uses are different from
the n points the Trapezoid Rule uses, and thus Simpson’s Rule is using 2n points at which
to evaluate the function. We build upon the results in Table 5.6.5 to see the approximations
generated by Simpson’s Rule. In particular, in Table 5.6.6, we include all of the results in
Table 5.6.5, but include additional results for S8 2M43+T4 and S16 2M83+T8 .
∫1 ∫2
Rule 0
(1 − x 2 ) dx 0.6 error 1
1 x2
dx 0.5 error
T4 0.65625 −0.0104166667 0.5089937642 0.0089937642
M4 0.671875 0.0052083333 0.4955479365 −0.0044520635
S8 0.6666666667 0 0.5000298792 0.0000298792
T8 0.6640625 −0.0026041667 0.5022708502 0.0022708502
M8 0.66796875 0.0013020833 0.4988674899 −0.0011325101
S16 0.6666666667 0 0.5000019434 0.0000019434
Table 5.6.6: Table 5.6.5 updated to include S8 , S16 , and the corresponding errors.
∫2
The results seen in Table 5.6.6 are striking. If we consider the S16 approximation of 1 x12 dx,
the error is only ES,16 0.0000019434. By contrast, L8 0.5491458502, so the error of that
estimate is EL,8 0.0491458502. Moreover, we observe that generating the approximations
for Simpson’s Rule is almost no additional work: once we have L n , R n , and M n for a given
value of n, it is a simple exercise to generate Tn , and from there to calculate S2n . Finally, note
∫1
that the error in the Simpson’s Rule approximations of 0
(1 − x 2 ) dx is zero!³
∫1
e −x dx, for
2
These rules are not only useful for approximating definite integrals such as 0
which we cannot find an elementary antiderivative of e −x , but also for approximating def-
2
inite integrals when we are given a function through a table of data.
Activity 5.6.3. A car traveling along a straight road is braking and its velocity is mea-
sured at several different points in time, as given in the following table. Assume that
v is continuous, always decreasing, and always decreasing at a decreasing rate, as is
suggested by the data.
²Thomas Simpson was an 18th century mathematician; his idea was to extend the Trapezoid rule, but rather
than using straight lines to build trapezoids, to use quadratic functions to build regions whose area was bounded
by parabolas (whose areas he could find exactly). Simpson’s Rule is often developed from the more sophisticated
perspective of using interpolation by quadratic functions.
³Similar to how the Midpoint and Trapezoid approximations are exact for linear functions, Simpson’s Rule
approximations are exact for quadratic and cubic functions. See additional discussion on this issue later in the
section and in the exercises.
317
Chapter 5 Evaluating Integrals
v
seconds, t Velocity in ft/sec, v(t)
0 100
0.3 99
0.6 96
0.9 90
1.2 80
1.5 50
t
1.8 0
0.3 0.6 0.9 1.2 1.5 1.8
Table 5.6.7: Data for the braking car.
Figure 5.6.8: Axes for plotting the data
in Activity 5.6.3.
a. Plot the given data on the set of axes provided in Figure 5.6.8 with time on the
horizontal axis and the velocity on the vertical axis.
b. What definite integral will give you the exact distance the car traveled on [0, 1.8]?
c. Estimate the total distance traveled on [0, 1.8] by computing L3 , R 3 , and T3 .
Which of these under-estimates the true distance traveled?
d. Estimate the total distance traveled on [0, 1.8] by computing M3 . Is this an over-
or under-estimate? Why?
e. Using your results from (c) and (d), improve your estimate further by using
Simpson’s Rule.
f. What is your best estimate of the average velocity of the car on [0, 1.8]? Why?
What are the units on this quantity?
5.6.4 Overall observations regarding L n , R n , Tn , M n , and S2n .
As we conclude our discussion of numerical approximation of definite integrals, it is impor-
tant to summarize general trends in how the various rules over- or under-estimate the true
value of a definite integral, and by how much. To revisit some past observations and see
some new ones, we consider the following activity.
Activity 5.6.4. Consider the functions f (x) 2 − x 2 , 1(x) 2 − x 3 , and h(x) 2 − x 4 ,
all on the interval [0, 1]. For each of the questions that require a numerical answer in
318
5.6 Numerical Integration
what follows, write your answer exactly in fraction form.
a. On the three sets of axes provided in Figure 5.6.9, sketch a graph of each function
on the interval [0, 1], and compute L1 and R 1 for each. What do you observe?
∫ 1 ∫ 1
b. Compute M1 for each function to approximate f (x) dx, 1(x) dx, and
∫ 1
0 0
h(x) dx, respectively.
0
c. Compute T1 for each of the three functions, and hence compute S2 for each of
the three functions.
∫1 ∫1 ∫1
d. Evaluate each of the integrals 0
f (x) dx, 0
1(x) dx, and 0
h(x) dx exactly us-
ing the First FTC.
e. For each of the three functions f , 1, and h, compare the results of L1 , R 1 , M1 , T1 ,
and S2 to the true value of the corresponding definite integral. What patterns
do you observe?
2 2 2
1 1 1
Figure 5.6.9: Axes for plotting the functions in Activity 5.6.4.
The results seen in Activity 5.6.4 generalize nicely. For instance, if f is decreasing on [a, b],
∫b
L n will over-estimate the exact value of a f (x) dx, and if f is concave down on [a, b], M n
will over-estimate the exact value of the integral. An excellent exercise is to write a collection
of scenarios of possible function behavior, and then categorize whether each of L n , R n , Tn ,
and M n is an over- or under-estimate.
Finally, we make two important notes about Simpson’s Rule. When T. Simpson first devel-
oped this rule, his idea was to replace the function f on a given interval with a quadratic
function that shared three values with the function f . In so doing, he guaranteed that this
new approximation rule would be exact for the definite integral of any quadratic polyno-
mial. In one of the pleasant surprises of numerical analysis, it turns out that even though it
was designed to be exact for quadratic polynomials, Simpson’s Rule is exact for any cubic
∫5
polynomial: that is, if we are interested in an integral such as 2 (5x 3 − 2x 2 + 7x − 4) dx, S2n
will always be exact, regardless of the value of n. This is just one more piece of evidence
that shows how effective Simpson’s Rule is as an approximation tool for estimating definite
319
Chapter 5 Evaluating Integrals
integrals.⁴
5.6.5 Summary
∫1
• For a definite integral such as 0 e −x dx when we cannot use the First Fundamental
2
Theorem of Calculus because the integrand lacks an elementary algebraic antideriva-
tive, we can estimate the integral’s value by using a sequence of Riemann sum approx-
imations. Typically, we start by computing L n , R n , and M n for one or more chosen
values of n.
∫b
• The Trapezoid Rule, which estimates a f (x) dx by using trapezoids, rather than rec-
tangles, can also be viewed as the average of Left and Right Riemann sums. That is,
Tn 21 (L n + R n ).
• The Midpoint Rule is typically twice as accurate as the Trapezoid Rule, and the signs
of the respective errors of these rules are opposites. Hence, by taking the weighted
∫b
average S2n 2Mn3+Tn , we can build a much more accurate approximation to a f (x) dx
by using approximations we have already computed. The rule for S2n is known as
Simpson’s Rule, which can also be developed by approximating a given continuous
function with pieces of quadratic polynomials.
5.6.6 Exercises
Two notes about how Exercise 5.6.6.1 is coded: (i) as explained in the problem header, you
need responses to every single entry before you can get individual parts marked as correct;
if you enter only an answer for (a) and submit, (a) will be marked wrong regardless. And
(ii), in this problem, the notation ”SIMP(2)” is actually what we have called ”SIMP(4)” in our
previous work. Different authors use different notation, and the author of this WeBWorK
exercise chooses to write ”SIMP(n)” where we have written ”SIMP(2n)” in Section 5.6.
∫4
1. Various methods for e x numerically. (a) What is the exact value of 0
e x dx?
(b) Find LEFT(2), RIGHT(2), TRAP(2), MID(2), and SIMP(2); compute the error for each.
(c) Repeat part (b) with n 4 (instead of n 2).
(d) For each rule in part (b), as n goes from n 2 to n 4, does the error go down
approximately as you would expect? Explain by calculating the ratios of the errors:
Error LEFT(2)/Error LEFT(4) =
Error RIGHT(2)/Error RIGHT(4) =
Error TRAP(2)/Error TRAP(4) =
Error MID(2)/Error MID(4) =
⁴One reason that Simpson’s Rule is so effective is that S2n benefits from using 2n + 1 points of data. Because
it combines M n , which uses n midpoints, and Tn , which uses the n + 1 endpoints of the chosen subintervals, S2n
takes advantage of the maximum amount of information we have when we know function values at the endpoints
and midpoints of n subintervals.
320
5.6 Numerical Integration
Error SIMP(2)/Error SIMP(4) =
(Be sure that you can explain in words why these do (or don’t) make sense.)
2. Comparison of methods for increasing concave down function. Using the figure
∫3
showing f (x) below, order the following approximations to the integral 0
f (x) dx and
its exact value from smallest to largest.
Enter each of ”LEFT(n)”, ”RIGHT(n)”, ”TRAP(n)”, ”MID(n)” and ”Exact” in one of the
following answer blanks to indicate the correct ordering:
< < < <
3. Comparing accuracy for two similar functions. Using a fixed number of subdivisions,
we approximate the integrals of f and 1 on the interval shown in the figure below.
(The function f (x) is shown in blue, and
1(x) in black).
For which function, f or 1 is LEFT
more accurate?
For which function, f or 1 is RIGHT
more accurate?
For which function, f or 1 is MID
more accurate?
For which function, f or 1 is TRAP
more accurate?
4. Identifying and comparing methods. Consider the four functions shown below. On
∫b
the first two, an approximation for a
f (x) dx is shown.
321
Chapter 5 Evaluating Integrals
1. 2.
3. 4.
1. For graph number 1, Which integration method is shown?
⊙ midpoint rule
⊙ left rule
⊙ right rule
⊙ trapezoid rule
Is this method an over- or underestimate?
2. For graph number 2, Which integration method is shown?
⊙ right rule
⊙ midpoint rule
⊙ left rule
⊙ trapezoid rule
Is this method an over- or underestimate?
3. On a copy of graph number 3, sketch an estimate with n 2 subdivisions using the
midpoint rule.
Is this method an over- or underestimate?
322
5.6 Numerical Integration
4. On a copy of graph number 4, sketch an estimate with n 2 subdivisions using the
trapezoid rule.
Is this method an over- or underestimate?
∫1
5. Consider the definite integral 0
x tan(x) dx.
a. Explain why this integral cannot be evaluated exactly by using either u-substitution
or by integrating by parts.
b. Using appropriate subintervals, compute L4 , R 4 , M4 , T4 , and S8 .
c. Which of the approximations in (b) is an over-estimate to the true value of
∫1
0
x tan(x) dx? Which is an under-estimate? How do you know?
6. For an unknown function f (x), the following information is known.
• f is continuous on [3, 6];
• f is either always increasing or always decreasing on [3, 6];
• f has the same concavity throughout the interval [3, 6];
∫6
• As approximations to 3
f (x) dx, L4 7.23, R 4 6.75, and M4 7.05.
a. Is f increasing or decreasing on [3, 6]? What data tells you?
b. Is f concave up or concave down on [3, 6]? Why?
∫6
c. Determine the best possible estimate you can for 3 f (x) dx, based on the given
information.
7. The rate at which water flows through Table Rock Dam on the White River in Branson,
MO, is measured in cubic feet per second (CFS). As engineers open the floodgates, flow
rates are recorded according to the following chart.
seconds, t 0 10 20 30 40 50 60
flow in CFS, r(t) 2000 2100 2400 3000 3900 5100 6500
Table 5.6.10: Water flow data.
a. What definite integral measures the total volume of water to flow through the
dam in the 60 second time period provided by the table above?
b. Use the given data to calculate M n for the largest possible value of n to approxi-
mate the integral you stated in (a). Do you think M n over- or under-estimates the
exact value of the integral? Why?
c. Approximate the integral stated in (a) by calculating S n for the largest possible
value of n, based on the given data.
1
d. Compute 60 S n and 2000+2100+2400+3000+3900+5100+6500
7 . What quantity do both of
these values estimate? Which is a more accurate approximation?
323
Chapter 5 Evaluating Integrals
324
CHAPTER 6
Using Definite Integrals
6.1 Using Definite Integrals to Find Area and Length
Motivating Questions
• How can we use definite integrals to measure the area between two curves?
• How do we decide whether to integrate with respect to x or with respect to y when
we try to find the area of a region?
• How can a definite integral be used to measure the length of a curve?
Early on in our work with the definite integral, we learned that for an object moving along an
axis, the area under a non-negative velocity function v between a and b tells us the distance
the object traveled on that time interval, and that area is given precisely by the definite in-
∫b ∫b
tegral a v(t) dt. In general, for any nonnegative function f on an interval [a, b], a f (x) dx
measures the area bounded by the curve and the x-axis between x a and x b.
Next, we will explore how definite integrals can be used to represent other physically im-
portant properties. In Preview Activity 6.1.1, we investigate how a single definite integral
may be used to represent the area between two curves.
Preview Activity 6.1.1. Consider the functions given by f (x) 5 − (x − 1)2 and 1(x)
4 − x.
a. Use algebra to find the points where the graphs of f and 1 intersect.
b. Sketch an accurate graph of f and 1 on the axes provided, labeling the curves
by name and the intersection points with ordered pairs.
c. Find and evaluate exactly an integral expression that represents the area be-
tween y f (x) and the x-axis on the interval between the intersection points of
f and 1.
d. Find and evaluate exactly an integral expression that represents the area be-
tween y 1(x) and the x-axis on the interval between the intersection points of
Chapter 6 Using Definite Integrals
f and 1.
e. What is the exact area between f and 1 between their intersection points? Why?
6
4
2
1 2 3
Figure 6.1.1: Axes for plotting f and 1 in Preview Activity 6.1.1
6.1.1 The Area Between Two Curves
In Preview Activity 6.1.1, we saw a natural way to think about the area between two curves:
it is the area beneath the upper curve minus the area below the lower curve.
Example 6.1.2 Find the area bounded between the graphs of f (x) (x − 1)2 + 1 and 1(x)
x + 2.
6 6 6
4 4 4
g g g
2 2 2
f f
1 2 3 1 2 3 1 2 3
Figure 6.1.3: The areas bounded by the functions f (x) (x − 1)2 + 1 and 1(x) x + 2 on the
interval [0, 3].
Solution. In Figure 6.1.3, we see that the graphs intersect at (0, 2) and (3, 5). We can find
326
6.1 Using Definite Integrals to Find Area and Length
these intersection points algebraically by solving the system of equations given by y x + 2
and y (x − 1)2 + 1: substituting x + 2 for y in the second equation yields x + 2 (x − 1)2 + 1,
so x + 2 x 2 − 2x + 1 + 1, and thus
x 2 − 3x x(x − 3) 0,
from which it follows that x 0 or x 3. Using y x + 2, we find the corresponding
y-values of the intersection points.
On the interval [0, 3], the area beneath 1 is
∫ 3
21
(x + 2) dx ,
0 2
while the area under f on the same interval is
∫ 3
[(x − 1)2 + 1] dx 6.
0
Thus, the area between the curves is
∫ 3 ∫ 3
21 9
A (x + 2) dx − [(x − 1)2 + 1] dx −6 . (6.1.1)
0 0 2 2
We can also think of the area this way: if we slice up the region between two curves into
thin vertical rectangles (in the same spirit as we originally sliced the region between a single
curve and the x-axis in Section 4.2), we see (as shown in Figure 6.1.4) that the height of a
typical rectangle is given by the difference between the two functions, 1(x) − f (x), and its
width is ∆x. Thus the area of the rectangle is
Arect (1(x) − f (x))∆x.
6
4
g g
g(x) − f (x) 2
f f
△x
x 1 2 3
Figure 6.1.4: The area bounded by the functions f (x) (x − 1)2 + 1 and 1(x) x + 2 on the
interval [0, 3].
327
Chapter 6 Using Definite Integrals
The area between the two curves on [0, 3] is thus approximated by the Riemann sum
∑
n
A≈ (1(x i ) − f (x i ))∆x,
i1
and as we let n → ∞, it follows that the area is given by the single definite integral
∫ 3
A (1(x) − f (x)) dx. (6.1.2)
0
In many applications of the definite integral, we will find it helpful to think of a “representa-
tive slice” and use the definite integral to add these slices. Here, the integral sums the areas
of thin rectangles.
Finally, it doesn’t matter whether we think of the area between two curves as the difference
between the area bounded by the individual curves (as in (6.1.1)) or as the limit of a Riemann
sum of the areas of thin rectangles between the curves (as in (6.1.2)). These two results are
the same, since the difference of two integrals is the integral of the difference:
∫ 3 ∫ 3 ∫ 3
1(x) dx − f (x) dx (1(x) − f (x)) dx.
0 0 0
Our work so far in this section illustrates the following general principle.
If two curves y 1(x) and y f (x) intersect at (a, 1(a)) and (b, 1(b)), and for all x
∫b
such that a ≤ x ≤ b, 1(x) ≥ f (x), then the area between the curves is A a
(1(x) −
f (x)) dx.
Activity 6.1.2. In each of the following problems, our goal is to determine the area
of the region described. For each region, (i) determine the intersection points of the
curves, (ii) sketch the region whose area is being found, (iii) draw and label a repre-
sentative slice, and (iv) state the area of the representative slice. Then, state a definite
integral whose value is the exact area of the region, and evaluate the integral to find
the numeric value of the region’s area.
√
a. The finite region bounded by y x and y 14 x.
b. The finite region bounded by y 12 − 2x 2 and y x 2 − 8.
c. The area bounded by the y-axis, f (x) cos(x), and 1(x) sin(x), where we
consider the region formed by the first positive value of x for which f and 1
intersect.
d. The finite regions between the curves y x 3 − x and y x 2 .
328
6.1 Using Definite Integrals to Find Area and Length
6.1.2 Finding Area with Horizontal Slices
At times, the shape of a region may dictate that we use horizontal rectangular slices, instead
of vertical ones.
Example 6.1.5 Find the area of the region bounded by the parabola x y 2 − 1 and the line
y x − 1, shown at left in Figure 6.1.6.
x = y2 − 1 x = y2 − 1 x = y2 − 1
2 2 2
1 1 1
△y
1 2 3 1 2 3 1 2 3
-1 y = x−1 -1 y = x−1 -1 x = y+1
Figure 6.1.6: The area bounded by the functions x y 2 − 1 and y x − 1 (at left), with the
region sliced vertically (center) and horizontally (at right).
Solution. By solving the second equation for x and writing x y + 1, we find that y + 1
y 2 − 1. Hence the curves intersect where y 2 − y − 2 0. Thus, we find y −1 or y 2, so
the intersection points of the two curves are (0, −1) and (3, 2).
If we attempt to use vertical rectangles to slice up the area (as in the center graph of Fig-
ure 6.1.6), we see that from x −1 to x 0 the curves that bound the top and bottom of
the rectangle are one and the same. This suggests, as shown in the rightmost graph in the
figure, that we try using horizontal rectangles.
Note that the width of a horizontal rectangle depends on y. Between y −1 and y 2, the
right end of a representative rectangle is determined by the line x y + 1, and the left end
is determined by the parabola, x y 2 − 1. The thickness of the rectangle is ∆y.
Therefore, the area of the rectangle is
Arect [(y + 1) − (y 2 − 1)]∆y,
and the area between the two curves on the y-interval [−1, 2] is approximated by the Rie-
mann sum
∑
n
A≈ [(y i + 1) − (y i2 − 1)]∆y.
i1
Taking the limit of the Riemann sum, it follows that the area of the region is
∫ y2
A [(y + 1) − (y 2 − 1)] dy. (6.1.3)
y−1
329
Chapter 6 Using Definite Integrals
We emphasize that we are integrating with respect to y; this is because we chose to use
horizontal rectangles whose widths depend on y and whose thickness is denoted ∆y. It is
a straightforward exercise to evaluate the integral in Equation (6.1.3) and find that A 92 .
Just as with the use of vertical rectangles of thickness ∆x, we have a general principle for
finding the area between two curves, which we state as follows.
If two curves x 1(y) and x f (y) intersect at (1(c), c) and (1(d), d), and for all y
such that c ≤ y ≤ d, 1(y) ≥ f (y), then the area between the curves is
∫ yd
A (1(y) − f (y)) dy.
yc
Activity 6.1.3. In each of the following problems, our goal is to determine the area
of the region described. For each region, (i) determine the intersection points of the
curves, (ii) sketch the region whose area is being found, (iii) draw and label a repre-
sentative slice, and (iv) state the area of the representative slice. Then, state a defi-
nite integral whose value is the exact area of the region, and evaluate the integral to
find the numeric value of the region’s area. Note well: At the step where you draw
a representative slice, you need to make a choice about whether to slice vertically or
horizontally.
a. The finite region bounded by x y 2 and x 6 − 2y 2 .
b. The finite region bounded by x 1 − y 2 and x 2 − 2y 2 .
c. The area bounded by the x-axis, y x 2 , and y 2 − x.
d. The finite regions between the curves x y 2 − 2y and y x.
6.1.3 Finding the length of a curve
We can also use the definite integral to find the length of a portion of a curve. We use
the same fundamental principle: we slice the curve up into small pieces whose lengths we
can easily approximate. Specifically, we subdivide the curve into small approximating line
segments, as shown at left in Figure 6.1.7. We estimate the length L slice of each portion of
the curve on a small interval of length ∆x. We use the right triangle with legs parallel to
the coordinate axes and hypotenuse connecting the endpoints of the slice, as seen at right in
Figure 6.1.7. The length, h, of the hypotenuse approximates the length, Lslice , of the curve
between the two selected points. Thus,
√
Lslice ≈ h (∆x)2 + (∆y)2 .
Next we use algebra to rearrange the expression for the length of the hypotenuse into a form
that we can integrate. By removing a factor of (∆x)2 , we find
√
Lslice ≈ (∆x)2 + (∆y)2
330
6.1 Using Definite Integrals to Find Area and Length
y
f
h
△y
Lslice
x
x0 x1 x2 x3 △x
Figure 6.1.7: At left, a continuous function y f (x) whose length we seek on the interval
a x0 to b x3 . At right, a close up view of a portion of the curve.
√ ( )
(∆y)2
(∆x)2 1 +
(∆x)2
√
(∆y)2
1+ · ∆x.
(∆x)2
∆y dy
Then, as n → ∞ and ∆x → 0, we have that ∆x → dx f ′(x). Thus, we can say that
√
Lslice ≈ 1 + f ′(x)2 ∆x.
Taking a Riemann sum of all of these slices and letting n → ∞, we arrive at the following
fact.
Given a differentiable function f on an interval [a, b], the total arc length, L, along
the curve y f (x) from x a to x b is given by
∫ b √
L 1 + f ′(x)2 dx.
a
Activity 6.1.4. Each of the following questions somehow involves the arc length along
a curve.
a. Use the definition and appropriate computational technology to determine the
arc length along y x 2 from x −1 to x 1.
331
Chapter 6 Using Definite Integrals
√
b. Find the arc length of y 4 − x 2 on the interval −2 ≤ x ≤ 2. Find this value in
two different ways: (a) by using a definite integral, and (b) by using a familiar
property of the curve.
c. Determine the arc length of y xe 3x on the interval [0, 1].
d. Will the integrals that arise calculating arc length typically be ones that we can
evaluate exactly using the First FTC, or ones that we need to approximate? Why?
e. A moving particle is traveling along the curve given by y f (x) 0.1x 2 +1, and
does so at a constant rate of 7 cm/sec, where both x and y are measured in cm
(that is, the curve y f (x) is the path along which the object actually travels;
the curve is not a “position function”). Find the position of the particle when
t 4 sec, assuming that when t 0, the particle’s location is (0, f (0)).
6.1.4 Summary
• To find the area between two curves, we think about slicing the region into thin rec-
tangles. If, for instance, the area of a typical rectangle on the interval x a to x b
is given by Arect (1(x) − f (x))∆x, then the exact area of the region is given by the
definite integral
∫ b
A (1(x) − f (x)) dx.
a
• The shape of the region usually dictates whether we should use vertical rectangles
of thickness ∆x or horizontal rectangles of thickness ∆y. We want the height of the
rectangle given by the difference between two curves: if those curves are best thought
of as functions of y, we use horizontal rectangles, whereas if those curves are best
viewed as functions of x, we use vertical rectangles.
• The arc length, L, along the curve y f (x) from x a to x b is given by
∫ b √
L 1 + f ′(x)2 dx.
a
6.1.5 Exercises
1. Area between two power functions. Find the area of the region between y x 1/2 and
y x 1/5 for 0 ≤ x ≤ 1.
2. Area between two trigonometric functions. Find the area between y 8 sin x and
y 9 cos x over the interval [0, π]. Sketch the curves if necessary.
¹This integral is actually ”improper” because the integrand is undefined at the endpoints, x ±2. We will learn
how to evaluate such integrals in Section 6.5.
332
6.1 Using Definite Integrals to Find Area and Length
3. Area between two curves. Sketch the region enclosed by x + y 2 56 and x + y 0.
Decide whether to integrate with respect to x or y, and then find the area of the region.
√
4. Arc length of a curve. Find the arc length of the graph of the function f (x) 2 x 3
from x 2 to x 5.
5. Find the exact area of each described region.
a. The finite region between the curves x y(y − 2) and x −(y − 1)(y − 3).
b. The region between the sine and cosine functions on the interval [ π4 , 4 ].
3π
c. The finite region between x y 2 − y − 2 and y 2x − 1.
d. The finite region between y mx and y x 2 − 1, where m is a positive constant.
6. Let f (x) 1 − x 2 and 1(x) ax 2 − a, where a is an unknown positive real number. For
what value(s) of a is the area between the curves f and 1 equal to 2?
7. Let f (x) 2 − x 2 . Recall that the average value of any continuous function f on an
∫b
interval [a, b] is given by 1
b−a a
f (x) dx.
√
a. Find the average value of f (x) 2 − x 2 on the interval [0, 2]. Call this value r.
b. Sketch a graph of y f (x) and y r. Find their intersection point(s).
√
c. Show that on the interval [0, 2], the amount of area that lies below y f (x)
and above y r is equal to the amount of area that lies below y r and above
y f (x).
d. Will the result of (c) be true for any continuous function and its average value on
any interval? Why?
333
Chapter 6 Using Definite Integrals
6.2 Using Definite Integrals to Find Volume
Motivating Questions
• How can we use a definite integral to find the volume of a three-dimensional solid
of revolution that results from revolving a two-dimensional region about a particular
axis?
• In what circumstances do we integrate with respect to y instead of integrating with
respect to x?
• What adjustments do we need to make if we revolve about a line other than the x- or
y-axis?
Just as we can use definite integrals to add the areas of rectangular slices to find the exact
area that lies between two curves, we can also use integrals to find the volume of regions
whose cross-sections have a particular shape.
In particular, we can determine the volume of solids whose cross-sections are all thin cylin-
ders (or washers) by adding up the volumes of these individual slices. We first consider a
familiar shape in Preview Activity 6.2.1: a circular cone.
Preview Activity 6.2.1. Consider a circular cone of radius 3 and height 5, which we
view horizontally as pictured in Figure 6.2.1. Our goal in this activity is to use a
definite integral to determine the volume of the cone.
y
3
x
x 5
∆x
Figure 6.2.1: The circular cone described in Preview Activity 6.2.1
a. Find a formula for the linear function y f (x) that is pictured in Figure 6.2.1.
b. For the representative slice of thickness ∆x that is located horizontally at a lo-
334
6.2 Using Definite Integrals to Find Volume
cation x (somewhere between x 0 and x 5), what is the radius of the repre-
sentative slice? Note that the radius depends on the value of x.
c. What is the volume of the representative slice you found in (b)?
d. What definite integral will sum the volumes of the thin slices across the full
horizontal span of the cone? What is the exact value of this definite integral?
e. Compare the result of your work in (d) to the volume of the cone that comes
from using the formula Vcone 31 πr 2 h.
6.2.1 The Volume of a Solid of Revolution
A solid of revolution is a three dimensional solid that can be generated by revolving one or
more curves around a fixed axis. For example, the circular cone in Figure 6.2.1 is the solid
of revolution generated by revolving the portion of the line y 3 − 35 x from x 0 to x 5
about the x-axis. Notice that if we slice a solid of revolution perpendicular to the axis of
revolution, the resulting cross-section is a circle.
We first consider solids whose slices are thin cylinders. Recall that the volume of a cylinder
is given by V πr 2 h.
Example 6.2.2 Find the volume of the solid of revolution generated when the region R
bounded by y 4 − x 2 and the x-axis is revolved about the x-axis.
Solution. First, we observe that y 4 − x 2 intersects the x-axis at the points (−2, 0) and
(2, 0). When we revolve the region R about the x-axis, we get the three-dimensional solid
pictured in Figure 6.2.3.
y
y = 4 − x2
x
∆x
Figure 6.2.3: The solid of revolution in Example 6.2.2.
335
Chapter 6 Using Definite Integrals
We slice the solid into vertical slices of thickness ∆x between x −2 and x 2. A repre-
sentative slice is a cylinder of height ∆x and radius 4 − x 2 . Hence, the volume of the slice
is
Vslice π(4 − x 2 )2 ∆x.
Using a definite integral to sum the volumes of the representative slices, it follows that
∫ 2
V π(4 − x 2 )2 dx.
−2
It is straightforward to evaluate the integral and find that the volume is V 15 π.
512
For a solid such as the one in Example 6.2.2, where each slice is a cylindrical disk, we first
find the volume of a typical slice (noting particularly how this volume depends on x), and
then integrate over the range of x-values that bound the solid. Often, we will be content
with simply finding the integral that represents the volume; if we desire a numeric value for
the integral, we typically use a calculator or computer algebra system to find that value.
This method for finding the volume of a solid of revolution is often called the disk method.
The Disk Method.
If y r(x) is a nonnegative continuous function on [a, b], then the volume of the solid
of revolution generated by revolving the curve about the x-axis over this interval is
given by
∫ b
V πr(x)2 dx.
a
A different type of solid can emerge when two curves are involved, as we see in the following
example.
Example 6.2.4 Find the volume of the solid of revolution generated when the finite region R
that lies between y 4 − x 2 and y x + 2 is revolved about the x-axis.
Solution. First, we must determine where the curves y 4 − x 2 and y x + 2 intersect.
Substituting the expression for y from the second equation into the first equation, we find
that x + 2 4 − x 2 . Rearranging, it follows that
x 2 + x − 2 0,
and the solutions to this equation are x −2 and x 1. The curves therefore cross at (−2, 0)
and (1, 1).
When we revolve the region R about the x-axis, we get the three-dimensional solid pictured
at left in Figure 6.2.5.
Immediately we see a major difference between the solid in this example and the one in
Example 6.2.2: here, the three-dimensional solid of revolution isn’t “solid” because it has
open space in its center along the axis of revolution. If we slice the solid perpendicular to
the axis of revolution, we observe that the resulting slice is not a solid disk, but rather a
washer, as pictured at right in Figure 6.2.5. At a given location x between x −2 and x 1,
336
6.2 Using Definite Integrals to Find Volume
the small radius r(x) of the inner circle is determined by the curve y x + 2, so r(x) x + 2.
Similarly, the big radius R(x) comes from the function y 4 − x 2 , and thus R(x) 4 − x 2 .
y
R(x)
x
r(x)
Figure 6.2.5: At left, the solid of revolution in Example 6.2.4. At right, a typical slice with
inner radius r(x) and outer radius R(x).
To find the volume of a representative slice, we compute the volume of the outer disk and
subtract the volume of the inner disk. Since
πR(x)2 ∆x − πr(x)2 ∆x π[R(x)2 − r(x)2 ]∆x,
it follows that the volume of a typical slice is
Vslice π[(4 − x 2 )2 − (x + 2)2 ]∆x.
Using a definite integral to sum the volumes of the respective slices across the integral, we
find that ∫ 1
V π[(4 − x 2 )2 − (x + 2)2 ] dx.
−2
Evaluating the integral, we find that the volume of the solid of revolution is V 5 π.
108
This method for finding the volume of a solid of revolution generated by two curves is often
called the washer method.
337
Chapter 6 Using Definite Integrals
The Washer Method.
If y R(x) and y r(x) are nonnegative continuous functions on [a, b] that satisfy
R(x) ≥ r(x) for all x in [a, b], then the volume of the solid of revolution generated by
revolving the region between them about the x-axis over this interval is given by
∫ b
V π[R(x)2 − r(x)2 ] dx.
a
Activity 6.2.2. In each of the following questions, draw a careful, labeled sketch of the
region described, as well as the resulting solid that results from revolving the region
about the stated axis. In addition, draw a representative slice and state the volume of
that slice, along with a definite integral whose value is the volume of the entire solid.
It is not necessary to evaluate the integrals you find.
√
a. The region S bounded by the x-axis, the curve y x, and the line x 4;
revolve S about the x-axis.
√
b. The region S bounded by the y-axis, the curve y x, and the line y 2;
revolve S about the x-axis.
√
c. The finite region S bounded by the curves y x and y x 3 ; revolve S about
the x-axis.
d. The finite region S bounded by the curves y 2x 2 + 1 and y x 2 + 4; revolve S
about the x-axis.
√
e. The region S bounded by the y-axis, the curve y x, and the line y 2;
revolve S about the y-axis. How is this problem different from the one posed in
part (b)?
6.2.2 Revolving about the y-axis
When we revolve a given region about the y-axis, the representative slices now have thick-
ness ∆y, which means that we must integrate with respect to y.
Example 6.2.6 Find
√ the volume of the solid of revolution generated when the region R that
lies between y x and y x 4 is revolved about the y-axis.
Solution. These two curves intersect when x 1, hence at the point (1, 1). When we re-
volve the region R about the y-axis, we get the three-dimensional solid pictured at left in
Figure 6.2.7.
Note that the slices are cylindrical washers only if taken perpendicular to the y-axis. We
slice the solid horizontally, starting at y 0 and proceeding up to y 1. The thickness of a
representative slice is ∆y, so we√must express the integrand in terms of y. The inner radius
is determined by the curve y x, so we solve for x and get x y 2 r(y). In the same way,
we solve the curve y x 4 (which governs the outer radius) for x in terms of y, and hence
338
6.2 Using Definite Integrals to Find Volume
√4
x y. Therefore, the volume of a typical slice is
√
Vslice π[R(y)2 − r(y)2 ] π[( 4 y)2 − (y 2 )2 ]∆y.
y
R(y)
r(y)
x
Figure 6.2.7: At left, the solid of revolution in Example 6.2.6. At right, a typical slice with
inner radius r(y) and outer radius R(y).
We use a definite integral to sum the volumes of all the slices from y 0 to y 1. The total
volume is ∫ y1
[√ ]
V π ( 4 y)2 − (y 2 )2 dy.
y0
It is straightforward to evaluate the integral and find that V 15 π.
7
Activity 6.2.3. In each of the following questions, draw a careful, labeled sketch of the
region described, as well as the resulting solid that results from revolving the region
about the stated axis. In addition, draw a representative slice and state the volume of
that slice, along with a definite integral whose value is the volume of the entire solid.
It is not necessary to evaluate the integrals you find.
√
a. The region S bounded by the y-axis, the curve y x, and the line y 2;
revolve S about the y-axis.
√
b. The region S bounded by the x-axis, the curve y x, and the line x 4;
revolve S about the y-axis.
c. The finite region S in the first quadrant bounded by the curves y 2x and
y x 3 ; revolve S about the x-axis.
339
Chapter 6 Using Definite Integrals
d. The finite region S in the first quadrant bounded by the curves y 2x and
y x 3 ; revolve S about the y-axis.
e. The finite region S bounded by the curves x (y − 1)2 and y x − 1; revolve S
about the y-axis
6.2.3 Revolving about horizontal and vertical lines other than the coordi-
nate axes
It is possible to revolve a region around any horizontal or vertical line. Doing so adjusts the
radii of the cylinders or washers involved by a constant value. A careful, well-labeled plot
of the solid of revolution will usually reveal how the different axis of revolution affects the
definite integral.
Example 6.2.8 Find the volume of the solid of revolution generated when the finite region S
that lies between y x 2 and y x is revolved about the line y −1.
Solution. Graphing the region between the two curves in the first quadrant between their
points of intersection ((0, 0) and (1, 1)) and then revolving the region about the line y −1,
we see the solid shown in Figure 6.2.9. Each slice of the solid perpendicular to the axis of
revolution is a washer, and the radii of each washer are governed by the curves y x 2 and
y x. But we also see that there is one added change: the axis of revolution adds a fixed
length to each radius. The inner radius of a typical slice, r(x), is given by r(x) x 2 + 1, while
the outer radius is R(x) x + 1.
y
x
Figure 6.2.9: The solid of revolution described in Example 6.2.8.
Therefore, the volume of a typical slice is
[ ]
Vslice π[R(x)2 − r(x)2 ]∆x π (x + 1)2 − (x 2 + 1)2 ∆x.
340
6.2 Using Definite Integrals to Find Volume
Finally, we integrate to find the total volume, and
∫ 1 [ ] 7
V π (x + 1)2 − (x 2 + 1)2 dx π.
0 15
Activity 6.2.4. In each of the following questions, draw a careful, labeled sketch of the
region described, as well as the resulting solid that results from revolving the region
about the stated axis. In addition, draw a representative slice and state the volume of
that slice, along with a definite integral whose value is the volume of the entire solid.
It is not necessary to evaluate the integrals you find. For each prompt, use the finite
region S in the first quadrant bounded by the curves y 2x and y x 3 .
a. Revolve S about the line y −2. c. Revolve S about the line x −1.
b. Revolve S about the line y 4. d. Revolve S about the line x 5.
6.2.4 Summary
• We can use a definite integral to find the volume of a three-dimensional solid of revo-
lution that results from revolving a two-dimensional region about a particular axis by
taking slices perpendicular to the axis of revolution which will then be circular disks
or washers.
• If we revolve about a vertical line and slice perpendicular to that line, then our slices
are horizontal and of thickness ∆y. This leads us to integrate with respect to y, as
opposed to with respect to x when we slice a solid vertically.
• If we revolve about a line other than the x- or y-axis, we need to carefully account for
the shift that occurs in the radius of a typical slice. Normally, this shift involves taking
a sum or difference of the function along with the constant connected to the equation
for the horizontal or vertical line; a well-labeled diagram is usually the best way to
decide the new expression for the radius.
6.2.5 Exercises
1. Solid of revolution from one function about the x-axis. The region bounded by y
e x , y 0, x −2, x −1 is rotated around the x-axis. Find the volume.
2. Solid of revolution from one function about the y-axis. Find the volume of the solid
obtained by rotating the region in the first quadrant bounded by y x 6 , y 1, and the
y-axis around the y-axis.
3. Solid of revolution from two functions about the x-axis. Find the volume of the solid
obtained by rotating the region in the first quadrant bounded by y x 6 , y 1, and the
y-axis around the x-axis.
4. Solid of revolution from two functions about a horizontal line. Find the volume of
the solid obtained by rotating the region in the first quadrant bounded by y x 6 , y 1,
341
Chapter 6 Using Definite Integrals
and the y-axis about the line y −5.
5. Solid of revolution from two functions about a different horizontal line. Find the
volume of the solid obtained by rotating the region bounded by the curves
y x6 , y1
about the line y 5 .
6. Solid of revolution from two functions about a vertical line. Find the volume of the
solid obtained by rotating the region bounded by the given curves about the line x −6
y x2 , x y2
3
7. Consider the curve f (x) 3 cos( x4 ) and the portion of its graph that lies in the first
quadrant between the y-axis and the first positive value of x for which f (x) 0. Let R
denote the region bounded by this portion of f , the x-axis, and the y-axis.
a. Set up a definite integral whose value is the exact arc length of f that lies along
the upper boundary of R. Use technology appropriately to evaluate the integral
you find.
b. Set up a definite integral whose value is the exact area of R. Use technology
appropriately to evaluate the integral you find.
c. Suppose that the region R is revolved around the x-axis. Set up a definite integral
whose value is the exact volume of the solid of revolution that is generated. Use
technology appropriately to evaluate the integral you find.
d. Suppose instead that R is revolved around the y-axis. If possible, set up an in-
tegral expression whose value is the exact volume of the solid of revolution and
evaluate the integral using appropriate technology. If not possible, explain why.
8. Consider the curves given by y sin(x) and y cos(x). For each of the following
problems, you should include a sketch of the region/solid being considered, as well as
a labeled representative slice.
a. Sketch the region R bounded by the y-axis and the curves y sin(x) and y
cos(x) up to the first positive value of x at which they intersect. What is the exact
intersection point of the curves?
b. Set up a definite integral whose value is the exact area of R.
c. Set up a definite integral whose value is the exact volume of the solid of revolution
generated by revolving R about the x-axis.
d. Set up a definite integral whose value is the exact volume of the solid of revolution
generated by revolving R about the y-axis.
e. Set up a definite integral whose value is the exact volume of the solid of revolution
generated by revolving R about the line y 2.
f. Set up a definite integral whose value is the exact volume of the solid of revolution
generated by revolving R about the line x −1.
342
6.2 Using Definite Integrals to Find Volume
9. Consider the finite region R that is bounded by the curves y 1 + 12 (x − 2)2 , y 21 x 2 ,
and x 0.
a. Determine a definite integral whose value is the area of the region enclosed by
the two curves.
b. Find an expression involving one or more definite integrals whose value is the
volume of the solid of revolution generated by revolving the region R about the
line y −1.
c. Determine an expression involving one or more definite integrals whose value is
the volume of the solid of revolution generated by revolving the region R about
the y-axis.
d. Find an expression involving one or more definite integrals whose value is the
perimeter of the region R.
343
Chapter 6 Using Definite Integrals
6.3 Density, Mass, and Center of Mass
Motivating Questions
• How are mass, density, and volume related?
• How is the mass of an object with varying density computed?
• What is is the center of mass of an object, and how are definite integrals used to
compute it?
Studying the units on the integrand and variable of integration helps us understand the
meaning of a definite integral. For instance, if v(t) is the velocity of an object moving along
an axis, measured in feet per second, and t measures time in seconds, then both the definite
integral and its Riemann sum approximation,
∫ b ∑
n
v(t) dt ≈ v(t i )∆t,
a i1
have units given by the product of the units of v(t) and t:
(feet/sec) · (sec) feet.
∫b
Thus, a
v(t) dt measures the total change in position of the moving object in feet.
Unit analysis will be particularly helpful to us in what follows.
Preview Activity 6.3.1. In each of the following scenarios, we consider the distribu-
tion of a quantity along an axis.
a. Suppose that the function c(x) 200 + 100e −0.1x models the density of traffic on
a straight road, measured in cars per mile, where x is number of miles east of a
∫2
major interchange, and consider the definite integral 0
(200 + 100e −0.1x ) dx.
i. What are the units on the product c(x) · ∆x?
ii. What are the units on the definite integral and its Riemann sum approxi-
mation given by
∫ 2 ∑
n
c(x) dx ≈ c(x i )∆x?
0 i1
∫2 ∫2( )
iii. Evaluate the definite integral 0 c(x) dx 0 200 + 100e −0.1x dx and write
one sentence to explain the meaning of the value you find.
b. On a 6 foot long shelf filled with books, the function B models the distribution
of the weight of the books, in pounds per inch, where x is the number of inches
from the left end of the bookshelf. Let B(x) be given by the rule B(x) 0.5 +
1
(x+1)2
.
344
6.3 Density, Mass, and Center of Mass
i. What are the units on the product B(x) · ∆x?
ii. What are the units on the definite integral and its Riemann sum approxi-
mation given by
∫ 36 ∑
n
B(x) dx ≈ B(x i )∆x?
12 i1
∫ 72 ∫ 72 ( )
iii. Evaluate the definite integral 0
B(x) dx 0
0.5 + 1
(x+1)2
dx and write
one sentence to explain the meaning of the value you find.
6.3.1 Density
The mass of a quantity, typically measured in metric units such as grams or kilograms, is
a measure of the amount of the quantity. In a corresponding way, the density of an object
measures the distribution of mass per unit volume. For instance, if a brick has mass 3 kg
and volume 0.002 m3 , then the density of the brick is
3kg kg
3
1500 3 .
0.002m m
As another example, the mass density of water is 1000 kg/m3 . Each of these relationships
demonstrate the following general principle.
For an object of constant density d, with mass m and volume V,
m
d , or m d · V.
V
But what happens when the density is not constant?
The formula m d · V is reminiscent of two other equations that we have used in our work:
for a body moving in a fixed direction, distance = rate · time, and, for a rectangle, its area is
given by A l · w. These formulas hold when the principal quantities involved, such as the
rate the body moves and the height of the rectangle, are constant. When these quantities are
not constant, we have turned to the definite integral for assistance. By working with small
slices on which the quantity of interest (such as velocity) is approximately constant, we can
use a definite integral to add up the values on the pieces.
For example, if we have a nonnegative velocity function that is not constant, over a short
time interval ∆t we know that the distance traveled is approximately v(t)∆t, since v(t) is
almost constant on a small interval. Similarly, if we are thinking about the area under a
nonnegative function f whose value is changing, on a short interval ∆x the area under the
curve is approximately the area of the rectangle whose height is f (x) and whose width is
∆x: f (x)∆x. Both of these principles are represented visually in Figure 6.3.1. In a similar
way, if the density of some object is not constant, we can use a definite integral to compute
345
Chapter 6 Using Definite Integrals
ft/sec y
y = v(t)
y = f (x)
v(t)
f (x)
sec x
△t △x
Figure 6.3.1: At left, estimating a small amount of distance traveled, v(t)∆t, and at right, a
small amount of area under the curve, f (x)∆x.
the overall mass of the object. We will focus on problems where the density varies in only
one dimension, say along a single axis.
Let’s consider a thin bar of length b whose left end is at the origin, where x 0, and assume
that the bar has constant cross-sectional area of 1 cm2 . We let ρ(x) represent the mass density
function of the bar, measured in grams per cubic centimeter. That is, given a location x, ρ(x)
tells us approximately how much mass will be found in a one-centimeter wide slice of the
bar at x.
x
∆x
Figure 6.3.2: A thin bar of constant cross-sectional area 1 cm2 with density function ρ(x)
g/cm3 .
The volume of a thin slice of the bar of width ∆x, as pictured in Figure 6.3.2, is the cross-
sectional area times ∆x. Since the cross-sections each have constant area 1 cm2 , it follows
that the volume of the slice is 1∆x cm3 . And because mass is the product of density and
volume, we see that the mass of this slice is approximately
g
massslice ≈ ρ(x) · 1∆x cm3 ρ(x) · ∆x g.
cm3
346
6.3 Density, Mass, and Center of Mass
The corresponding Riemann sum (and the integral that it approximates),
∑
n ∫ b
ρ(x i )∆x ≈ ρ(x) dx,
i1 0
therefore measure the mass of the bar between 0 and b. (The Riemann sum is an approxi-
mation, while the integral will be the exact mass.)
For objects whose cross-sectional area is constant and whose mass is distributed relative to
horizontal location, x, it makes sense to think of the density function ρ(x) with units “mass
per unit length,” such as g/cm. Thus, when we compute ρ(x) · ∆x on a small slice ∆x, the
resulting units are g/cm · cm = g, which thus measures the mass of the slice. The general
principle follows.
For an object of constant cross-sectional area whose mass is distributed along a single
axis according to the function ρ(x) (whose units are units of mass per unit of length),
the total mass, M, of the object between x a and x b is given by
∫ b
M ρ(x) dx.
a
Activity 6.3.2. Consider the following situations in which mass is distributed in a
non-constant manner.
a. Suppose that a thin rod with constant cross-sectional area of 1 cm2 has its mass
distributed according to the density function ρ(x) 2e −0.2x , where x is the dis-
tance in cm from the left end of the rod, and the units on ρ(x) are g/cm. If the
rod is 10 cm long, determine the exact mass of the rod.
b. Consider the cone that has a base of radius 4 m and a height of 5 m. Picture the
cone lying horizontally with the center of its base at the origin and think of the
cone as a solid of revolution.
i. Write and evaluate a definite integral whose value is the volume of the
cone.
ii. Next, suppose that the cone has uniform density of 800 kg/m3 . What is
the mass of the solid cone?
iii. Now suppose that the cone’s density is not uniform, but rather that the
cone is most dense at its base. In particular, assume that the density of the
cone is uniform across cross sections parallel to its base, but that in each
such cross section that is a distance x units from the origin, the density of
the cross section is given by the function ρ(x) 400 + 1+x200
2 , measured in
3
kg/m . Determine and evaluate a definite integral whose value is the mass
of this cone of non-uniform density. Do so by first thinking about the mass
of a given slice of the cone x units away from the base; remember that in
such a slice, the density will be essentially constant.
347
Chapter 6 Using Definite Integrals
c. Let a thin rod of constant cross-sectional area 1 cm2 and length 12 cm have its
mass be distributed according to the density function ρ(x) 251
(x − 15)2 , mea-
sured in g/cm. Find the exact location z at which to cut the bar so that the two
pieces will each have identical mass.
6.3.2 Weighted Averages
The concept of an average is a natural one, and one that we have used repeatedly as part of
our understanding of the meaning of the definite integral. If we have n values a1 , a2 , . . ., a n ,
we know that their average is given by
a1 + a2 + · · · + a n
,
n
and for a quantity being measured by a function f on an interval [a, b], the average value of
the quantity on [a, b] is
∫ b
1
f (x) dx.
b−a a
As we continue to think about problems involving the distribution of mass, it is natural to
consider the idea of a weighted average, where certain quantities involved are counted more
in the average.
A common use of weighted averages class grade grade points credits
is in the computation of a student’s chemistry B+ 3.3 5
GPA, where grades are weighted ac-
calculus A- 3.7 4
cording to credit hours. Let’s consider
history B- 2.7 3
the scenario in Table 6.3.3.
psychology B- 2.7 3
Table 6.3.3: A college student’s semester grades.
If all of the classes were of the same weight (i.e., the same number of credits), the student’s
GPA would simply be calculated by taking the average
3.3 + 3.7 + 2.7 + 2.7
3.1.
4
But since the chemistry and calculus courses have higher weights (of 5 and 4 credits respec-
tively), we actually compute the GPA according to the weighted average
3.3 · 5 + 3.7 · 4 + 2.7 · 3 + 2.7 · 3
3.16.
5+4+3+3
The weighted average reflects the fact that chemistry and calculus, as courses with higher
credits, have a greater impact on the students’ grade point average. Note particularly that in
the weighted average, each grade gets multiplied by its weight, and we divide by the sum
of the weights.
In the following activity, we explore further how weighted averages can be used to find the
balancing point of a physical system.
348
6.3 Density, Mass, and Center of Mass
Activity 6.3.3. For quantities of equal weight, such as two children on a teeter-totter,
the balancing point is found by taking the average of their locations. When the weights
of the quantities differ, we use a weighted average of their respective locations to find
the balancing point.
a. Suppose that a shelf is 6 feet long, with its left end situated at x 0. If one
book of weight 1 lb is placed at x1 0, and another book of weight 1 lb is
placed at x2 6, what is the location of x, the point at which the shelf would
(theoretically) balance on a fulcrum?
b. Now, say that we place four books on the shelf, each weighing 1 lb: at x 1 0, at
x2 2, at x3 4, and at x4 6. Find x, the balancing point of the shelf.
c. How does x change if we change the location of the third book? Say the locations
of the 1-lb books are x1 0, x2 2, x 3 3, and x4 6.
d. Next, suppose that we place four books on the shelf, but of varying weights: at
x1 0 a 2-lb book, at x2 2 a 3-lb book, at x3 4 a 1-lb book, and at x4 6 a 1-lb
book. Use a weighted average of the locations to find x, the balancing point of
the shelf. How does the balancing point in this scenario compare to that found
in (b)?
e. What happens if we change the location of one of the books? Say that we keep
everything the same in (d), except that x3 5. How does x change?
f. What happens if we change the weight of one of the books? Say that we keep
everything the same in (d), except that the book at x3 4 now weighs 2 lbs.
How does x change?
g. Experiment with a couple of different scenarios of your choosing where you
move one of the books to the left, or you decrease the weight of one of the books.
h. Write a couple of sentences to explain how adjusting the location of one of the
books or the weight of one of the books affects the location of the balancing point
of the shelf. Think carefully here about how your changes should be considered
relative to the location of the balancing point x of the current scenario.
6.3.3 Center of Mass
In Activity 6.3.3, we saw that the balancing point of a system of point-masses¹ (such as books
on a shelf) is found by taking a weighted average of their respective locations. In the activity,
we were computing the center of mass of a system of masses distributed along an axis, which
is the balancing point of the axis on which the masses rest.
¹In the activity, we actually used weight rather than mass. Since weight is proportional to mass, the computations
for the balancing point result in the same location regardless of whether we use weight or mass. The gravitational
constant is present in both the numerator and denominator of the weighted average.
349
Chapter 6 Using Definite Integrals
Center of Mass (point-masses).
For a collection of n masses m 1 , . . ., m n that are distributed along a single axis at the
locations x1 , . . ., x n , the center of mass is given by
x1 m1 + x2 m2 + · · · + x n m n
x .
m1 + m2 + · · · + m n
Now consider a thin bar over which density is distributed continuously. If the density is
constant, it is obvious that the balancing point of the bar is its midpoint. But if density is not
constant, we must compute a weighted average. Let’s say that the function ρ(x) tells us the
density distribution along the bar, measured in g/cm. If we slice the bar into small sections,
we can think of the bar as holding a collection of adjacent point-masses. The mass m i of a
slice of thickness ∆x at location x i , is m i ≈ ρ(x i )∆x.
If we slice the bar into n pieces, we can approximate its center of mass by
x 1 · ρ(x 1 )∆x + x2 · ρ(x2 )∆x + · · · + x n · ρ(x n )∆x
x≈ .
ρ(x 1 )∆x + ρ(x2 )∆x + · · · + ρ(x n )∆x
Rewriting the sums in sigma notation, we have
∑n
i1 x i · ρ(x i )∆x
x≈ ∑ n . (6.3.1)
i1 ρ(x i )∆x
The greater the number of slices, the more accurate our estimate of the balancing point will
be. The sums in Equation (6.3.1) can be viewed as Riemann sums, so in the limit as n → ∞,
we find that the center of mass is given by the quotient of two integrals.
Center of Mass (continuous mass distribution).
For a thin rod of density ρ(x) distributed along an axis from x a to x b, the center
of mass of the rod is given by
∫b
xρ(x) dx
x ∫a b .
a
ρ(x) dx
Note that the denominator of x is the mass of the bar, and that this quotient of integrals is
simply the continuous version of the weighted average of locations, x, along the bar.
Activity 6.3.4. Consider a thin bar of length 20 cm whose density is distributed ac-
cording to the function ρ(x) 4 + 0.1x, where x 0 represents the left end of the bar.
Assume that ρ is measured in g/cm and x is measured in cm.
a. Find the total mass, M, of the bar.
b. Without doing any calculations, do you expect the center of mass of the bar to
be equal to 10, less than 10, or greater than 10? Why?
350
6.3 Density, Mass, and Center of Mass
c. Compute x, the exact center of mass of the bar.
d. What is the average density of the bar?
e. Now consider a different density function, given by p(x) 4e 0.020732x , also for a
bar of length 20 cm whose left end is at x 0. Plot both ρ(x) and p(x) on the
same axes. Without doing any calculations, which bar do you expect to have the
greater center of mass? Why?
f. Compute the exact center of mass of the bar described in (e) whose density func-
tion is p(x) 4e 0.020732x . Check the result against the prediction you made in
(e).
6.3.4 Summary
• For an object of constant density D, with volume V and mass m, we know that m
D · V.
• If an object with constant cross-sectional area (such as a thin bar) has its density dis-
tributed along an axis according to the function ρ(x), then we can find the mass of the
object between x a and x b by
∫ b
m ρ(x) dx.
a
• For a system of point-masses distributed along an axis, say m 1 , . . . , m n at locations
x1 , . . . , x n , the center of mass, x, is given by the weighted average
∑n
xi mi
x ∑i1
n .
i1 mi
If instead we have mass continuously distributed along an axis, such as by a density
function ρ(x) for a thin bar of constant cross-sectional area, the center of mass of the
portion of the bar between x a and x b is given by
∫b
xρ(x) dx
x ∫a b .
a
ρ(x) dx
In each situation, x represents the balancing point of the system of masses or of the
portion of the bar.
6.3.5 Exercises
1. Center of mass for a linear density function. A rod has length 4 meters. At a distance
x meters from its left end, the density of the rod is given by δ(x) 5 + 2x g/m.
351
Chapter 6 Using Definite Integrals
(a) Complete the Riemann sum for the total mass of the rod.
(b) Convert the Riemann sum to an integral and find the exact mass.
2. Center of mass for a nonlinear density function. A rod with uniform density (mass/
unit length) δ(x) 8 + sin(x) lies on the x-axis between x 0 and x π. Find the mass
and center of mass of the rod.
3. Interpreting the density of cars on a road. Suppose that the density of cars (in cars
per mile) down
( a 20-mile
( √ stretch))of the Pennsylvania Turnpike is approximated by
δ(x) 250 2 + sin 4 x + 0.125 , at a distance x miles from the Breezewood toll
plaza. Sketch a graph of this function for 0 ≤ x ≤ 20.
(a) Complete the Riemann sum that approximates the total number of cars on this 20-
mile stretch.
(b) Find the total number of cars on the 20-mile stretch.
Number =
4. Center of mass in a point-mass system. A point mass of 1 grams located 7 centimeters
to the left of the origin and a point mass of 4 grams located 8 centimeters to the right
of the origin are connected by a thin, light rod. Find the center of mass of the system.
5. Let a thin rod of length a have density distribution function ρ(x) 10e −0.1x , where x is
measured in cm and ρ in grams per centimeter.
a. If the mass of the rod is 30 g, what is the value of a?
b. For the 30g rod, will the center of mass lie at its midpoint, to the left of the mid-
point, or to the right of the midpoint? Why?
c. For the 30g rod, find the center of mass, and compare your prediction in (b).
d. At what value of x should the 30g rod be cut in order to form two pieces of equal
mass?
6. Consider two thin bars of constant cross-sectional area, each of length 10 cm, with re-
spective mass density functions ρ(x) 1+x −0.1x .
2 and p(x) e
1
a. Find the mass of each bar.
b. Find the center of mass of each bar.
c. Now consider a new 10 cm bar whose mass density function is f (x) ρ(x) + p(x).
(i) Explain how you can easily find the mass of this new bar with little to no
additional work.
∫ 10
(ii) Similarly, compute 0
x f (x) dx as simply as possible, in light of earlier com-
putations.
(iii) True or false: the center of mass of this new bar is the average of the centers
of mass of the two earlier bars. Write at least one sentence to say why your
conclusion makes sense.
7. Consider the curve given by y f (x) 2xe −1.25x + (30 − x)e −0.25(30−x) .
a. Plot this curve in the window x 0 . . . 30, y 0 . . . 3 (with constrained scaling so
352
6.3 Density, Mass, and Center of Mass
the units on the x and y axis are equal), and use it to generate a solid of revolution
about the x-axis. Explain why this curve could generate a reasonable model of a
baseball bat.
b. Let x and y be measured in inches. Find the total volume of the baseball bat
generated by revolving the given curve about the x-axis. Include units on your
answer.
c. Suppose that the baseball bat has constant weight density, and that the weight
density is 0.6 ounces per cubic inch. Find the total weight of the bat whose volume
you found in (b).
d. Because the baseball bat does not have constant cross-sectional area, we see that
the amount of weight concentrated at a location x along the bat is determined by
the volume of a slice at location x. Explain why we can think about the function
ρ(x) 0.6π f (x)2 (where f is the function given at the start of the problem) as
being the weight density function for how the weight of the baseball bat is dis-
tributed from x 0 to x 30.
e. Compute the center of mass of the baseball bat.
353
Chapter 6 Using Definite Integrals
6.4 Physics Applications: Work, Force, and Pressure
Motivating Questions
• How do we measure the work accomplished by a varying force that moves an object
a certain distance?
• What is the total force exerted by water against a dam?
• How are both of the above concepts and their corresponding use of definite integrals
similar to problems we have encountered in the past involving formulas such as “dis-
tance equals rate times time” and “mass equals density times volume”?
y
y = f (x)
y = v(t)
ρ(x)
△x
f (x) v(t)
t
a △x b a △t b
Figure 6.4.1: Three settings where we compute the accumulation of a varying quantity: the
area under y f (x), the distance traveled by an object with velocity y v(t), and the mass
of a bar with density function y ρ(x).
We have seen several different circumstances where the definite integral enables us to mea-
sure the accumulation of a quantity that varies, provided the quantity is approximately con-
stant over small intervals. For instance, to find the area bounded by a nonnegative curve
y f (x) and the x-axis on an interval [a, b], we take a representative slice of width ∆x that
has area Aslice f (x)∆x. As we let the width of the representative slice tend to zero, we find
that the exact area of the region is
∫ b
A f (x) dx.
a
In a similar way, if we know the velocity v(t) of a moving object and we wish to know the
distance the object travels on an interval [a, b] where v(t) is nonnegative, we can use a def-
inite integral to generalize the fact that d r · t when the rate, r, is constant. On a short
time interval ∆t, v(t) is roughly constant, so for a small slice of time, dslice v(t)∆t. As
the width of the time interval ∆t tends to zero, the exact distance traveled is given by the
354
6.4 Physics Applications: Work, Force, and Pressure
definite integral
∫ b
d v(t) dt.
a
Finally, if we want to determine the mass of an object of non-constant density, because M
D ·V (mass equals density times volume, provided that density is constant), we can consider
a small slice of an object on which the density is approximately constant, and a definite
integral may be used to determine the exact mass of the object. For instance, if we have a
thin rod whose cross sections have constant density, but whose density is distributed along
the x axis according to the function y ρ(x), it follows that for a small slice of the rod that
is ∆x thick, Mslice ρ(x)∆x. In the limit as ∆x → 0, we then find that the total mass is given
by
∫ b
M ρ(x) dx.
a
All three of these situations are similar in that we have a basic rule (A l · w, d r · t,
M D · V) where one of the two quantities being multiplied is no longer constant; in each,
we consider a small interval for the other variable in the formula, calculate the approximate
value of the desired quantity (area, distance, or mass) over the small interval, and then use a
definite integral to sum the results as the length of the small intervals is allowed to approach
zero. It should be apparent that this approach will work effectively for other situations where
we have a quantity that varies.
We next turn to the notion of work: from physics, a basic principle is that work is the product
of force and distance. For example, if a person exerts a force of 20 pounds to lift a 20-pound
weight 4 feet off the ground, the total work accomplished is
W F · d 20 · 4 80 foot-pounds.
If force and distance are measured in English units (pounds and feet), then the units of
work are foot-pounds. If we work in metric units, where forces are measured in Newtons and
distances in meters, the units of work are Newton-meters.
Of course, the formula W F · d only applies when the force is constant over the distance d.
In Preview Activity 6.4.1, we explore one way that we can use a definite integral to compute
the total work accomplished when the force exerted varies.
Preview Activity 6.4.1. A bucket is being lifted from the bottom of a 50-foot deep well;
its weight (including the water), B, in pounds at a height h feet above the water is given
by the function B(h). When the bucket leaves the water, the bucket and water together
weigh B(0) 20 pounds, and when the bucket reaches the top of the well, B(50) 12
pounds. Assume that the bucket loses water at a constant rate (as a function of height,
h) throughout its journey from the bottom to the top of the well.
a. Find a formula for B(h).
b. Compute the value of the product B(5)∆h, where ∆h 2 feet. Include units on
your answer. Explain why this product represents the approximate work it took
to move the bucket of water from h 5 to h 7.
355
Chapter 6 Using Definite Integrals
c. Is the value in (b) an over- or under-estimate of the actual amount of work it
took to move the bucket from h 5 to h 7? Why?
d. Compute the value of the product B(22)∆h, where ∆h 0.25 feet. Include units
on your answer. What is the meaning of the value you found?
e. More generally, what does the quantity Wslice B(h)∆h measure for a given
value of h and a small positive value of ∆h?
∫ 50
f. Evaluate the definite integral 0
B(h) dh. What is the meaning of the value you
find? Why?
6.4.1 Work
Because work is calculated by the rule W F · d whenever the force F is constant, it follows
that we can use a definite integral to compute the work accomplished by a varying force.
For example, suppose that a bucket whose weight at height h is given by B(h) 12 + 8e −0.1h
is being lifted in a 50-foot well.
In contrast to the problem in the preview activity, this bucket is not leaking at a constant
rate; but because the weight of the bucket and water is not constant, we have to use a definite
integral to determine the total work done in lifting the bucket. At a height h above the water,
the approximate work to move the bucket a small distance ∆h is
Wslice B(h)∆h (12 + 8e −0.1h )∆h.
Hence, if we let ∆h tend to 0 and take the sum of all of the slices of work accomplished on
these small intervals, it follows that the total work is given by
∫ 50 ∫ 50
W B(h) dh (12 + 8e −0.1h ) dh.
0 0
While it is a straightforward exercise to evaluate this integral exactly using the First Fun-
damental Theorem of Calculus, in applied settings such as this one we will typically use
∫ 50
computing technology. Here, it turns out that W 0
(12 + 8e −0.1h ) dh ≈ 679.461 foot-
pounds.
Our work in Preview Activity 6.4.1 and in the most recent discussion above employs the
following important general principle.
For an object being moved in the positive direction along an axis with location x by
a force F(x), the total work to move the object from a to b is given by
∫ b
W F(x) dx.
a
356
6.4 Physics Applications: Work, Force, and Pressure
Activity 6.4.2. Consider the following situations in which a varying force accom-
plishes work.
a. Suppose that a heavy rope hangs over the side of a cliff. The rope is 200 feet
long and weighs 0.3 pounds per foot; initially the rope is fully extended. How
much work is required to haul in the entire length of the rope? (Hint: set up
a function F(h) whose value is the weight of the rope remaining over the cliff
after h feet have been hauled in.)
b. A leaky bucket is being hauled up from a 100 foot deep well. When lifted from
the water, the bucket and water together weigh 40 pounds. As the bucket is
being hauled upward at a constant rate, the bucket leaks water at a constant
rate so that it is losing weight at a rate of 0.1 pounds per foot. What function
B(h) tells the weight of the bucket after the bucket has been lifted h feet? What
is the total amount of work accomplished in lifting the bucket to the top of the
well?
c. Now suppose that the bucket in (b) does not leak at a constant rate, but rather
that its weight at a height h feet above the water is given by B(h) 25+15e −0.05h .
What is the total work required to lift the bucket 100 feet? What is the average
force exerted on the bucket on the interval h 0 to h 100?
d. From physics, Hooke’s Law for springs states that the amount of force required to
hold a spring that is compressed (or extended) to a particular length is propor-
tionate to the distance the spring is compressed (or extended) from its natural
length. That is, the force to compress (or extend) a spring x units from its natural
length is F(x) kx for some constant k (which is called the spring constant.) For
springs, we choose to measure the force in pounds and the distance the spring
is compressed in feet. Suppose that a force of 5 pounds extends a particular
spring 4 inches (1/3 foot) beyond its natural length.
i. Use the given fact that F(1/3) 5 to find the spring constant k.
ii. Find the work done to extend the spring from its natural length to 1 foot
beyond its natural length.
iii. Find the work required to extend the spring from 1 foot beyond its natural
length to 1.5 feet beyond its natural length.
6.4.2 Work: Pumping Liquid from a Tank
In certain geographic locations where the water table is high, residential homes with base-
ments have a peculiar feature: in the basement, one finds a large hole in the floor, and in the
hole, there is water. For example, in Figure 6.4.2 we see a sump crock¹. A sump crock pro-
vides an outlet for water that may build up beneath the basement floor to prevent flooding
the basement.
¹Image credit to www.warreninspect.com/basement-moisture.
357
Chapter 6 Using Definite Integrals
In the crock we see a floating pump. This
pump is activated by elevation, so when the
water level reaches a particular height, the
pump turns on and pumps water out of the
crock, hence relieving the water buildup
beneath the foundation. One of the ques-
tions we’d like to answer is: how much
work does a sump pump accomplish?
Figure 6.4.2: A sump crock.
Example 6.4.3 Suppose that a sump crock has the shape of a frustum of a cone, as pictured
in Figure 6.4.4. The crock has a diameter of 3 feet at its surface, a diameter of 1.5 feet at its
base, and a depth of 4 feet. In addition, suppose that the sump pump is set up so that it
pumps the water vertically up a pipe to a drain that is located at ground level just outside
a basement window. To accomplish this, the pump must send the water to a location 9 feet
above the surface of the sump crock. How much work is required to empty the sump crock
if it is initially completely full?
y+
(0, 1.5)
∆x
(4, 0.75)
x+
Figure 6.4.4: A sump crock with approximately cylindrical cross-sections.
Solution. It is helpful to think of the depth below the surface of the crock as being the
independent variable, so we let the positive x-axis point down, and the positive y-axis to
the right, as pictured in the figure. Because the pump sits on the surface of the water, it
makes sense to think about the pump moving the water one “slice” at a time, where it takes
a thin slice from the surface, pumps it out of the tank, and then proceeds to pump the next
slice below.
Each slice of water is cylindrical in shape. We see that the radius of each slice varies according
to the linear function y f (x) that passes through the points (0, 1.5) and (4, 0.75), where
x is the depth of the particular slice in the tank; it is a straightforward exercise to find that
358
6.4 Physics Applications: Work, Force, and Pressure
f (x) 1.5 − 0.1875x. Now we think about the problem in several steps:
a. determining the volume of a typical slice;
b. finding the weight² of a typical slice (and thus the force that must be exerted on it);
c. deciding the distance that a typical slice moves;
d. and computing the work to move a representative slice.
Once we know the work it takes to move one slice, we use a definite integral over an appro-
priate interval to find the total work.
Consider a representative cylindrical slice at a depth of x feet below the top of the crock. The
approximate volume of that slice is given by
Vslice π f (x)2 ∆x π(1.5 − 0.1875x)2 ∆x.
Since water weighs 62.4 lb/ft3 , the approximate weight of a representative slice is
Fslice 62.4 · Vslice 62.4π(1.5 − 0.1875x)2 ∆x.
This is also the approximate force the pump must exert to move the slice.
Because the slice is located at a depth of x feet below the top of the crock, the slice being
moved by the pump must move x feet to get to the level of the basement floor, and then, as
stated in the problem description, another 9 feet to reach the drain at ground level. Hence,
the total distance a representative slice travels is
dslice x + 9.
Finally, the work to move a representative slice is given by
Wslice Fslice · dslice 62.4π(1.5 − 0.1875x)2 ∆x · (x + 9).
We sum the work required to move slices throughout the tank (from x 0 to x 4), let
∆x → 0, and hence ∫ 4
W 62.4π(1.5 − 0.1875x)2 (x + 9) dx.
0
When evaluated using appropriate technology, the integral shows that the total work is W
3463.2π foot-pounds.
The preceding example demonstrates the standard approach to finding the work required
to empty a tank filled with liquid. The main task in each such problem is to determine
the volume of a representative slice, followed by the force exerted on the slice, as well as
the distance such a slice moves. In the case where the units are metric, there is one key
difference: in the metric setting, rather than weight, we normally first find the mass of a
slice. For instance, if distance is measured in meters, the mass density of water is 1000 kg/
²We assume that the weight density of water is 62.4 pounds per cubic foot.
359
Chapter 6 Using Definite Integrals
m3 . In that setting, we can find the mass of a typical slice (in kg). To determine the force
required to move it, we use F ma, where m is the object’s mass and a is the gravitational
constant a 9.81 N/kg. That is, in metric units, the weight density of water is 9810 N/m3 .
Activity 6.4.3. In each of the following problems, determine the total work required
to accomplish the described task. In parts (b) and (c), a key step is to find a formula
for a function that describes the curve that forms the side boundary of the tank.
y+
x+
Figure 6.4.5: A trough with triangular ends, as described in Activity 6.4.3, part (c).
a. Consider a vertical cylindrical tank of radius 2 meters and depth 6 meters. Sup-
pose the tank is filled with 4 meters of water of mass density 1000 kg/m3 , and
the top 1 meter of water is pumped over the top of the tank.
b. Consider a hemispherical tank with a radius of 10 feet. Suppose that the tank is
full to a depth of 7 feet with water of weight density 62.4 pounds/ft3 , and the
top 5 feet of water are pumped out of the tank to a tanker truck whose height is
5 feet above the top of the tank.
c. Consider a trough with triangular ends, as pictured in Figure 6.4.5, where the
tank is 10 feet long, the top is 5 feet wide, and the tank is 4 feet deep. Say that
the trough is full to within 1 foot of the top with water of weight density 62.4
pounds/ft3 , and a pump is used to empty the tank until the water remaining in
the tank is 1 foot deep.
6.4.3 Force due to Hydrostatic Pressure
When building a dam, engineers need to know how much force water will exert against the
face of the dam. This force comes from water pressure. The pressure a force exerts on a
region is measured in units of force per unit of area: for example, the air pressure in a tire is
often measured in pounds per square inch (PSI). Hence, we see that the general relationship
is given by
F
P , or F P · A,
A
360
6.4 Physics Applications: Work, Force, and Pressure
where P represents pressure, F represents force, and A the area of the region being consid-
ered. Of course, in the equation F PA, we assume that the pressure is constant over the
entire region A.
We know from experience that the deeper one dives underwater while swimming, the greater
the pressure exerted by the water. This is because at a greater depth, there is more water
right on top of the swimmer: it is the force that “column” of water exerts that determines
the pressure the swimmer experiences. The total water pressure is found by computing the
total weight of the column of water that lies above a region of area 1 square foot at a fixed
depth. At a depth of d feet, a rectangular column has volume V 1 · 1 · d ft3 , so the corre-
sponding weight of the water overhead is 62.4d. This is the amount of force being exerted
on a 1 square foot region at a depth d feet underwater, so the pressure exerted by water at
depth d is P 62.4d (lbs/ft2 ).
Because pressure is force per unit area, or P F
A, we can compute the total force from a
variable pressure by integrating F PA.
Example 6.4.6 Consider a trapezoid-shaped dam that is 60 feet wide at its base and 90 feet
wide at its top, and assume the dam is 25 feet tall with water that rises to within 5 feet of the
top of its face. Water weighs 62.4 pounds per cubic foot. How much force does the water
exert against the dam?
Solution. First, we sketch a picture of the dam, as shown in Figure 6.4.7. Note that, as in
problems involving the work to pump out a tank, we let the positive x-axis point down.
45
y+
x−5 y = f (x)
x △x
x+ (25, 30)
Figure 6.4.7: A trapezoidal dam that is 25 feet tall, 60 feet wide at its base, 90 feet wide at
its top, with the water line 5 feet down from the top of its face.
Pressure is constant at a fixed depth, so we consider a slice of water at constant depth on the
face, as shown in the figure. The area of this slice is approximately the area of the rectangle
pictured. Since the width of that rectangle depends on the variable x, we find a formula for
the line that represents one side of the dam. It is straightforward to find that y 45 − 35 x.
361
Chapter 6 Using Definite Integrals
Hence, the approximate area of a representative slice is
3
Aslice 2 f (x)∆x 2(45 − x)∆x.
5
At any point on this slice, the depth is approximately constant, so the pressure can be con-
sidered constant. Because the water rises to within 5 feet of the top of the dam, the depth of
any point on the representative slice is approximately (x − 5). Now, since pressure is given
by P 62.4d, we have that at any point on the slice
Pslice 62.4(x − 5).
Knowing both the pressure and area, we can find the force the water exerts on the slice.
Using F PA, it follows that
3
Fslice Pslice · Aslice 62.4(x − 5) · 2(45 − x)∆x.
5
Finally, we use a definite integral to sum the forces over the appropriate range of x-values.
Since the water rises to within 5 feet of the top of the dam, we start at x 5 and take slices
all the way to the bottom of the dam, where x 25. Hence,
∫ x25
3
F 62.4(x − 5) · 2(45 − x) dx.
x5 5
Using technology to evaluate the integral, we find F 848640 pounds.
Activity 6.4.4. In each of the following problems, determine the total force exerted by
water against the surface that is described.
y+
x+
Figure 6.4.8: A trough with triangular ends, as described in Activity 6.4.4, part (c).
a. Consider a rectangular dam that is 100 feet wide and 50 feet tall, and suppose
that water presses against the dam all the way to the top.
b. Consider a semicircular dam with a radius of 30 feet. Suppose that the water
rises to within 10 feet of the top of the dam.
362
6.4 Physics Applications: Work, Force, and Pressure
c. Consider a trough with triangular ends, as pictured in Figure 6.4.8, where the
tank is 10 feet long, the top is 5 feet wide, and the tank is 4 feet deep. Say that
the trough is full to within 1 foot of the top with water of weight density 62.4
pounds/ft3 . How much force does the water exert against one of the triangular
ends?
Although there are many different formulas involving work, force, and pressure, the funda-
mental ideas behind these problems are similar to others we’ve encountered in applications
of the definite integral. We slice the quantity of interest into more manageable pieces and
then use a definite integral to add them up.
6.4.4 Summary
• To measure the work done by a varying force in moving an object, we divide the prob-
lem into pieces on which we can use the formula W F · d, and then use a definite
integral to sum the work done on each piece.
• To find the total force exerted by water against a dam, we use the formula F P · A to
measure the force exerted on a slice that lies at a fixed depth, and then use a definite
integral to sum the forces across the appropriate range of depths.
• Because work is computed as the product of force and distance (provided force is con-
stant), and the force water exerts on a dam can be computed as the product of pressure
and area (provided pressure is constant), problems involving these concepts are sim-
ilar to earlier problems we did using definite integrals to find distance (via “distance
equals rate times time”) and mass (“mass equals density times volume”).
6.4.5 Exercises
1. Work to empty a conical tank. A tank in the shape of an inverted right circular cone
has height 5 meters and radius 4 meters. It is filled with 3 meters of hot chocolate. Find
the work required to empty the tank by pumping the hot chocolate over the top of the
tank. The density of hot chocolate is δ 1070 kg/m3 . Your answer must include the
correct units.
2. Work to empty a cylindrical tank. A fuel oil tank is an upright cylinder, buried so
that its circular top is 10 feet beneath ground level. The tank has a radius of 4 feet and
is 12 feet high, although the current oil level is only 10 feet deep. Calculate the work
required to pump all of the oil to the surface. Oil weighs 50 lb/ft3 .
3. Work to empty a rectangular pool. A rectangular swimming pool 50 ft long, 30 ft
wide, and 8 ft deep is filled with water to a depth of 6 ft. Use an integral to find the
work required to pump all the water out over the top. (Take as the density of water
δ 62.4lb/ft3 .)
4. Work to empty a cylindrical tank to differing heights. Water in a cylinder of height
11 ft and radius 4 ft is to be pumped out. The density of water is 62.4 lb/ft3 . Find the
363
Chapter 6 Using Definite Integrals
work required if
(a) The tank is full of water and the water is to be pumped over the top of the tank.
(b) The tank is full of water and the water must be pumped to a height 6 ft above the
top of the tank.
(c) The depth of water in the tank is 6 ft and the water must be pumped over the top of
the tank.
5. Force due to hydrostatic pressure. A lobster tank in a restaurant is 1.25 m long by 0.5
m wide by 90 cm deep. Taking the density of water to be 1000kg/m3 , find the water
forces
on the bottom of the tank.
on each of the larger sides of the tank.
on each of the smaller sides of the tank.
3
6. Consider the curve f (x) 3 cos( x4 ) and the portion of its graph that lies in the first
quadrant between the y-axis and the first positive value of x for which f (x) 0. Let
R denote the region bounded by this portion of f , the x-axis, and the y-axis. Assume
that x and y are each measured in feet.
a. Picture the coordinate axes rotated 90 degrees clockwise so that the positive x-
axis points straight down, and the positive y-axis points to the right. Suppose
that R is rotated about the x axis to form a solid of revolution, and we consider
this solid as a storage tank. Suppose that the resulting tank is filled to a depth
of 1.5 feet with water weighing 62.4 pounds per cubic foot. Find the amount of
work required to lower the water in the tank until it is 0.5 feet deep, by pumping
the water to the top of the tank.
b. Again picture the coordinate axes rotated 90 degrees clockwise so that the positive
x-axis points straight down, and the positive y-axis points to the right. Suppose
that R, together with its reflection across the x-axis, forms one end of a storage
tank that is 10 feet long. Suppose that the resulting tank is filled completely with
water weighing 62.4 pounds per cubic foot. Find a formula for a function that
tells the amount of work required to lower the water by h feet.
c. Suppose that the tank described in (b) is completely filled with water. Find the
total force due to hydrostatic pressure exerted by the water on one end of the tank.
7. A cylindrical tank, buried on its side, has radius 3 feet and length 10 feet. It is filled
completely with water whose weight density is 62.4 lbs/ft3 , and the top of the tank is
two feet underground.
a. Set up, but do not evaluate, an integral expression that represents the amount of
work required to empty the top half of the water in the tank to a truck whose tank
lies 4.5 feet above ground.
b. With the tank now only half-full, set up, but do not evaluate an integral expression
that represents the total force due to hydrostatic pressure against one end of the
tank.
364
6.5 Improper Integrals
6.5 Improper Integrals
Motivating Questions
• What are improper integrals and why are they important?
• What does it mean to say that an improper integral converges or diverges?
• What are some typical improper integrals that we can classify as convergent or di-
vergent?
Another important application of the definite integral measures the likelihood of certain
events. For instance, consider a company that manufactures incandescent light bulbs. Based
on a large volume of test results, they have determined that the fraction of light bulbs that
fail between times t a and t b of use (where t is measured in months) is given by
∫ b
0.3e −0.3t dt.
a
For example, the fraction of light bulbs that fail during their third month of use is given by
∫ 3 3
0.3e −0.3t dt −e −0.3t
2 2
−e −0.9 + e −0.6
≈ 0.1422.
Thus about 14.22% of all lightbulbs fail between t 2 and t 3. Clearly we could adjust the
limits of integration to measure the fraction of light bulbs that fail during any time period
of interest.
Preview Activity 6.5.1. A company with a large customer base has a call center that
receives thousands of calls a day. After studying the data that represents how long
callers wait for assistance, they find that the function p(t) 0.25e −0.25t models the
time customers wait in the following way: the fraction of customers who wait between
t a and t b minutes is given by
∫ b
p(t) dt.
a
Use this information to answer the following questions.
a. Determine the fraction of callers who wait between 5 and 10 minutes.
b. Determine the fraction of callers who wait between 10 and 20 minutes.
c. Next, let’s study the fraction who wait up to a certain number of minutes:
365
Chapter 6 Using Definite Integrals
i. What is the fraction of callers who wait between 0 and 5 minutes?
ii. What is the fraction of callers who wait between 0 and 10 minutes?
iii. Between 0 and 15 minutes? Between 0 and 20?
d. Let F(b) represent the fraction of callers who wait between 0 and b minutes.
Find a formula for F(b) that involves a definite integral, and then use the First
FTC to find a formula for F(b) that does not involve a definite integral.
e. What is the value of the limit limb→∞ F(b)? What is its meaning in the context
of the problem?
6.5.1 Improper Integrals Involving Unbounded Intervals
In view of the above examples, we see that we may want to integrate over an interval whose
upper limit grows without bound. For example, to find the fraction of light bulbs that fail
eventually, we wish to find
∫ b
lim 0.3e −0.3t dt,
b→∞ 0
for which we will also use the notation
∫ ∞
0.3e −0.3t dt. (6.5.1)
0
Such an integral can be interpreted as the area of an unbounded region, as pictured at right
in Figure 6.5.1.
y y
···
t
b t
Figure 6.5.1: At left, the area bounded by p(t) 0.3e −0.3t on the finite interval [0, b]; at
right, the result of letting b → ∞. By “· · ·” in the righthand figure, we mean that the region
extends to the right without bound.
366
6.5 Improper Integrals
We call an integral for which the interval of integration is unbounded improper. For instance,
the integrals
∫ ∞ ∫ 0 ∫ ∞
1 1
e −x dx
2
dx, dx, and
1 x 2
−∞ 1 + x 2
−∞
are all improper because they have limits of integration that involve ∞. To evaluate an im-
proper integral we replace it with a limit of proper integrals. That is,
∫ ∞ ∫ b
f (x) dx lim f (x) dx.
0 b→∞ 0
∫b
We first attempt to evaluate 0 f (x) dx using the First FTC, and then evaluate the limit. Is
it even possible for the area of an unbounded region to be finite? The following activity
explores this issue and others in more detail.
∫∞ ∫∞
1 1
Activity 6.5.2. In this activity we explore the improper integrals 1 x dx and 1 x 3/2
dx.
∫∞
1
a. First we investigate 1 x dx.
∫ 10 ∫ 1000
1 1
i. Use the First FTC to determine the exact values of dx, dx, and
∫ 100000 1 x 1 x
1
1
dx. Then, use your computational device to compute a decimal
x
approximation of each result.
∫b
1
ii. Use the First FTC to evaluate the definite integral 1 x
dx (which results in
an expression that depends on b).
iii. Now, use your work from (ii.) to evaluate the limit given by
∫ b
1
lim dx.
b→∞ 1 x
∫∞
1
b. Next, we investigate 1 x 3/2
dx.
∫ 10 ∫ 1000
1 1
i. Use the First FTC to determine the exact values of dx, dx,
∫ 100000 1 x 3/2 1 x 3/2
1
and 1 x 3/2
dx. Then, use your calculator to compute a decimal approx-
imation of each result.
∫b
1
ii. Use the First FTC to evaluate the definite integral 1 x 3/2
dx (which results
in an expression that depends on b).
iii. Now, use your work from (ii.) to evaluate the limit given by
∫ b
1
lim dx.
b→∞ 1 x 3/2
c. Plot the functions y and y x 3/2 on the same coordinate axes for the values
1
x
1
x 0 . . . 10. How would you compare their behavior as x increases without
bound? What is similar? What is different?
∫∞ ∫∞
d. How would you characterize the value of 1 x1 dx? of 1 x 3/2 1
dx? What does
this tell us about the respective areas bounded by these two curves for x ≥ 1?
367
Chapter 6 Using Definite Integrals
6.5.2 Convergence and Divergence
∫b
Activity 6.5.2 suggests that limb→∞ 1 f (x) dx is either finite or infinite (or it doesn’t exist).
With these possibilities in mind, we introduce the following terminology.
∫∞
If f (x) is nonnegative for x ≥ a, then we say that the improper integral a
f (x) dx
converges provided that
∫ b
lim f (x) dx
b→∞ a
∫∞
exists and is finite. Otherwise, we say that a
f (x) dx diverges.
We will restrict our interest to improper integrals for which the integrand is nonnegative.
∫ ∞that limx→∞ f (x) 0, for if f does not approach 0 as x → ∞, then it is
Also, we require
impossible for a f (x) dx to converge.
Activity 6.5.3. Determine whether each of the following improper integrals converges
or diverges.
∫∞ For each integral that converges, find∫its exact value.
∞
a. 1 x12 dx d. 4 (x+2)3
5/4 dx
∫∞ ∫∞
b. e −x/4 dx e. 0
xe −x/4 dx
0
∫∞
1
∫∞ f. dx, where p is a positive real
1 xp
9
c. 2 (x+5)2/3
dx number
6.5.3 Improper Integrals Involving Unbounded Integrands
An integral is also called improper if the integrand is unbounded on the interval of integra-
tion. For example, consider
∫ 1
1
√ dx.
0 x
Because f (x) √1x has a vertical asymptote at x 0, f is not continuous on [0, 1], and the
integral represents the area of the unbounded region shown at right in Figure 6.5.2. We
address the problem of the integrand being unbounded by replacing the improper integral
∫1
with a limit of proper integrals. For example, to evaluate √1 dx, we replace 0 with a and
0 x
let a approach 0 from the right. Thus,
∫ 1 ∫ 1
1 1
√ dx lim+ √ dx.
0 x a→0 a x
∫1
We evaluate the proper integral a √1x dx, and then take the limit. We will say that the im-
proper integral converges if this limit exists, and diverges otherwise. In this example, we
368
6.5 Improper Integrals
y y
f (x) = √1 f (x) = √1
x x
x x
a 1 1
Figure 6.5.2: At left, the area bounded by f (x) √1x on the finite interval [a, 1]; at right, the
result of letting a → 0+ , where we see that the shaded region will extend vertically without
bound.
observe that
∫ 1 ∫ 1
1 1
√ dx lim+ √ dx
0 x a→0 a x
√ 1
lim+ 2 x a
a→0
√ √
lim+ 2 1 − 2 a
a→0
2,
∫1
so the improper integral √1 dx converges (to the value 2).
0 x
We have to be particularly careful with unbounded integrands, for they may arise in ways
that may not initially be obvious. Consider, for instance, the integral
∫ 3
1
dx.
1 (x − 2)2
At first glance we might think that we can simply apply the Fundamental Theorem of Cal-
2 to get − x−2 and then evaluating from 1 to 3. Were we to do
1 1
culus by antidifferentiating (x−2)
so, we would be erroneously applying the FTC because f (x) (x−2)1
2 fails to be continuous
throughout the interval, as seen in Figure 6.5.3. Such an incorrect application of the FTC
leads to an impossible result (−2), which would itself suggest that something we did must
be wrong. Instead, we must address the vertical asymptote at x 2 by writing
∫ 3 ∫ a ∫ 3
1 1 1
dx lim− dx + lim dx.
1 (x − 2)2 a→2 1 (x − 2)2 b→2+ b (x − 2)2
We then evaluate two separate limits of proper integrals. For instance, doing so for the
369
Chapter 6 Using Definite Integrals
y
1
y= (x−2)2
1 2 3 x
Figure 6.5.3: The function f (x) 1
(x−2)2
on an interval including x 2.
integral with a approaching 2 from the left, we find
∫ 2 ∫ a
1 1
dx lim− dx
1 (x − 2)2 a→2 1 (x − 2)2
a
1
lim− −
a→2 (x − 2) 1
1 1
lim− − +
a→2 (a − 2) 1 − 2
∞,
∫2
since 1
a−2 → −∞ as a approaches 2 from the left. Thus, the improper integral 1
1 (x−2)2
dx
∫3
1
diverges; similar work shows that 2 (x−2)2
dx also diverges. From either of these two results,
∫3
1
we can conclude that that the original integral, 1 (x−2)2
dx diverges, too.
Activity 6.5.4. For each of the following definite integrals, decide whether the integral
is improper or not. If the integral is proper, evaluate it using the First FTC. If the
integral is improper, determine whether or not the integral converges or diverges; if
the integral
∫1 converges, find its exact value. ∫2
1
a. 0 x 1/3 dx d. −2 x12 dx
∫2 ∫ π/2
b. 0
e −x dx e. 0
tan(x) dx
∫4 ∫1
c. √1 dx f. √ 1 dx
1 4−x 0 1−x 2
370
6.5 Improper Integrals
6.5.4 Summary
∫b
• An integral a f (x) dx can be improper if at least one of a or b is ±∞, making the
interval unbounded, or if f has a vertical asymptote at x c for some value of c that
satisfies a ≤ c ≤ b. One reason that improper integrals are important is that certain
probabilities can be represented by integrals that involve infinite limits.
• When we encounter an improper integral, we work to understand it by replacing the
improper integral with a limit of proper integrals. For instance, we write
∫ ∞ ∫ b
f (x) dx lim f (x) dx,
a b→∞ a
and then work to determine whether the limit exists and is finite. For any improper in-
tegral, if the resulting limit of proper integrals exists and is finite, we say the improper
integral converges. Otherwise, the improper integral diverges.
• An important class of improper integrals is given by
∫ ∞
1
dx
1 xp
where p is a positive real number. We can show that this improper integral converges
whenever p > 1, and diverges whenever 0 < p ≤ 1. A related class of improper
∫1
integrals is 1
0 xp
dx, which converges for 0 < p < 1, and diverges for p ≥ 1.
6.5.5 Exercises
1. An improper integral on a finite interval. Calculate the integral below or explain why
it diverges.
∫ 3
9
√ dx
0 x x
2. An improper integral on an infinite interval. Calculate the integral below, if it con-
verges. ∫ ∞
3x 2 e −x dx
3
2
3. An improper integral involving a ratio of exponential functions. Calculate the inte-
gral, if it converges.
∫ 3
e 2x
dx
−∞ 1 + e
2x
4. A subtle improper integral. Calculate the integral, if it converges.
∫ 3
1
dv
−3 v
371
Chapter 6 Using Definite Integrals
5. An improper integral involving a ratio of trigonometric functions. Find the area un-
der the curve y tan(t) between t 0 and t π/2.
6. Determine, with justification, whether each of the following improper integrals con-
verges or diverges.
∫∞ ln(x)
a. e x dx
∫∞
1
b. e x ln(x)
dx
∫∞
1
c. e x(ln(x))2
dx
∫∞
1
d. e x(ln(x))p
dx, where p is a positive real number
∫1 ln(x)
e. 0 x dx
∫1
f. 0
ln(x) dx
7. Sometimes we may encounter an improper integral for which we cannot easily ∫ ∞ evalu-
1
ate the limit of the corresponding proper integrals. For instance, consider 1 1+x 3 dx.
1
While it is hard (or perhaps impossible) to find an antiderivative for 1+x 3 , we can still
determine whether or not the improper integral converges or diverges by comparison
to a simpler one. Observe that for all x > 0, 1 + x 3 > x 3 , and therefore
1 1
< 3.
1 + x3 x
It therefore follows that ∫ ∫
b b
1 1
dx < dx
1 1 + x3 1 x3
∫∞
for every b > 1. If we let b → ∞ so as to consider the two improper integrals 1
dx
∫∞ 1 1+x 3
1
and 1 dx, we know that the larger of the two improper integrals converges. And
x3
thus, since the smaller one lies below∫a convergent integral, it follows that the smaller
∞ 1
one must converge, too. In particular, 1 1+x 3 dx must converge, even though we never
explicitly evaluated the corresponding limit of proper integrals. We use this idea and
similar ones in the exercises that follow.
∫∞
a. Explain why x 2 + x + 1 > x 2 for all x ≥ 1, and hence show that 1
dx
∫∞ 1 x 2 +x+1
1
converges by comparison to 1 x2
dx.
b. Observe that for each x > 1, ln(x) < x. Explain why
∫ b ∫ b
1 1
dx < dx
2 x 2 ln(x)
∫b
for each b > 2. Why must it be true that 1
2 ln(x)
dx diverges?
372
6.5 Improper Integrals
√
c. Explain why x x+1
4
4 > 1 for all x > 1. Then, determine whether or not the im-
proper integral
∫ √
∞
1 x4 + 1
· dx
1 x x4
converges or diverges.
373
Chapter 6 Using Definite Integrals
374
CHAPTER 7
Differential Equations
7.1 An Introduction to Differential Equations
Motivating Questions
• What is a differential equation and what kinds of information can it tell us?
• How do differential equations arise in the world around us?
• What do we mean by a solution to a differential equation?
In previous chapters, we have seen that a function’s derivative tells us the rate at which
the function is changing. The Fundamental Theorem of Calculus helped us determine the
total change of a function over an interval from the function’s rate of change. For instance,
an object’s velocity tells us the rate of change of that object’s position. By integrating the
velocity over a time interval, we can determine how much the position changes over that
time interval. If we know where the object is at the beginning of that interval, we have
enough information to predict where it will be at the end of the interval.
In this chapter, we introduce the concept of differential equations. A differential equation is
an equation that provides a description of a function’s derivative, which means that it tells
us the function’s rate of change. Using this information, we would like to learn as much as
possible about the function itself. Ideally we would like to have an algebraic description of
the function. As we’ll see, this may be too much to ask in some situations, but we will still
be able to make accurate approximations.
Preview Activity 7.1.1. The position of a moving object is given by the function s(t),
where s is measured in feet and t in seconds. We determine that the velocity is v(t)
4t + 1 feet per second.
a. How much does the position change over the time interval [0, 4]?
b. Does this give you enough information to determine s(4), the position at time
t 4? If so, what is s(4)? If not, what additional information would you need
to know to determine s(4)?
Chapter 7 Differential Equations
c. Suppose you are told that the object’s initial position s(0) 7. Determine s(2),
the object’s position 2 seconds later.
d. If you are told instead that the object’s initial position is s(0) 3, what is s(2)?
e. If we only know the velocity v(t) 4t + 1, is it possible that the object’s position
at all times is s(t) 2t 2 + t − 4? Explain how you know.
f. Are there other possibilities for s(t)? If so, what are they?
g. If, in addition to knowing the velocity function is v(t) 4t + 1, we know the
initial position s(0), how many possibilities are there for s(t)?
7.1.1 What is a differential equation?
A differential equation is an equation that describes the derivative, or derivatives, of a func-
tion that is unknown to us. For instance, the equation
dy
x sin x
dx
describes the derivative of a function y(x) that is unknown to us.
As many important examples of differential equations involve quantities that change in time,
the independent variable in our discussion will frequently be time t. In the preview activity,
we considered the differential equation
ds
4t + 1.
dt
Knowing the velocity and the starting position of a moving object, we were able to find its
position at any later time.
Because differential equations describe the derivative of a function, they give us information
about how that function changes. Our goal will be to use this information to predict the value
of the function in the future; in this way, differential equations provide us with something
like a crystal ball.
Differential equations arise frequently in our every day world. For instance, you may hear a
bank advertising:
Your money will grow at a 3% annual interest rate with us.
This innocuous statement is really a differential equation. Let’s translate: A(t) will be amount
of money you have in your account at time t. The rate at which your money grows is the de-
rivative dA/dt, and we are told that this rate is 0.03A. This leads to the differential equation
dA
0.03A.
dt
376
7.1 An Introduction to Differential Equations
dt 4t + 1.
This differential equation has a slightly different feel than the previous equation ds
In the earlier example, the rate of change depends only on the independent variable t, and
we may find s(t) by integrating the velocity 4t + 1. In the banking example, however, the
rate of change depends on the dependent variable A, so we’ll need some new techniques in
order to find A(t).
Activity 7.1.2. Express the following statements as differential equations. In each
case, you will need to introduce notation to describe the important quantities in the
statement so be sure to clearly state what your notation means.
a. The population of a town grows continuously at an annual rate of 1.25%.
b. A radioactive sample loses mass at a rate of 5.6% of its mass every day.
c. You have a bank account that continuously earns 4% interest every year. At the
same time, you withdraw money continually from the account at the rate of
$1000 per year.
d. A cup of hot chocolate is sitting in a 70◦ room. The temperature of the hot choco-
late cools continuously by 10% of the difference between the hot chocolate’s
temperature and the room temperature every minute.
e. A can of cold soda is sitting in a 70◦ room. The temperature of the soda warms
continuously at the rate of 10% of the difference between the soda’s temperature
and the room’s temperature every minute.
7.1.2 Differential equations in the world around us
Differential equations give a natural way to describe phenomena we see in the real world.
For instance, physical principles are frequently expressed as a description of how a quantity
changes. A good example is Newton’s Second Law, which says:
The product of an object’s mass and acceleration equals the force applied to it.
For instance, when gravity acts on an object near the earth’s surface, it exerts a force equal
to m1, the mass of the object times the gravitational constant 1. We therefore have
ma m1, or
dv
1,
dt
where v is the velocity of the object, and 1 9.8 meters per second squared. Notice that this
physical principle does not tell us what the object’s velocity is, but rather how the object’s
velocity changes.
Activity 7.1.3. Shown below are two graphs depicting the velocity of falling objects.
On the left is the velocity of a skydiver, while on the right is the velocity of a meteorite
entering the Earth’s atmosphere.
377
Chapter 7 Differential Equations
6 6
v v
5 5
4 4
3 3
2 2
1 1
t t
1 2 3 1 2 3
Figure 7.1.1: A skydiver’s velocity. Figure 7.1.2: A meteorite’s velocity.
a. Begin with the skydiver’s velocity and use the given graph to measure the rate
of change dv/dt when the velocity is v 0.5, 1.0, 1.5, 2.0, and 2.5. Plot your
values on the graph below. You will want to think carefully about this: you are
plotting the derivative dv/dt as a function of velocity.
378
7.1 An Introduction to Differential Equations
1.5
dv
dt
1.0
0.5
v
1 2 3 4 5
-0.5
-1.0
-1.5
b. Now do the same thing with the meteorite’s velocity: use the given graph to
measure the rate of change dv/dt when the velocity is v 3.5, 4.0, 4.5, and 5.0.
Plot your values on the graph above.
c. You should find that all your points lie on a line. Write the equation of this line
being careful to use proper notation for the quantities on the horizontal and
vertical axes.
d. The relationship you just found is a differential equation. Write a complete sen-
tence that explains its meaning.
e. By looking at the differential equation, determine the values of the velocity for
which the velocity increases.
f. By looking at the differential equation, determine the values of the velocity for
which the velocity decreases.
g. By looking at the differential equation, determine the values of the velocity for
which the velocity remains constant.
The point of this activity is to demonstrate how differential equations model processes in
the real world. In this example, two factors influence the velocities: gravity and wind resis-
tance. The differential equation describes how these factors influence the rate of change of
the velocities.
7.1.3 Solving a differential equation
A differential equation describes the derivative, or derivatives, of a function that is unknown
to us. By a solution to a differential equation, we mean simply a function that satisies this
379
Chapter 7 Differential Equations
description.
For instance, the first differential equation we looked at is
ds
4t + 1,
dt
which describes an unknown function s(t). We may check that s(t) 2t 2 + t is a solution
because it satisfies this description. Notice that s(t) 2t 2 + t + 4 is also a solution.
If we have a candidate for a solution, it is straightforward to check whether it is a solution
or not. Before we demonstrate, however, let’s consider the same issue in a simpler context.
Suppose we are given the equation 2x 2 − 2x 2x + 6 and asked whether x 3 is a solution.
To answer this question, we could rewrite the variable x in the equation with the symbol □:
2□2 − 2□ 2□ + 6.
To determine whether x 3 is a solution, we can investigate the value of each side of the
equation separately when the value 3 is placed in □ and see if indeed the two resulting values
are equal. Doing so, we observe that
2□2 − 2□ 2 · 32 − 2 · 3 12,
and
2□ + 6 2 · 3 + 6 12.
Therefore, x 3 is indeed a solution.
We will do the same thing with differential equations. Consider the differential equation
dv
1.5 − 0.5v, or
dt
d□
1.5 − 0.5□.
dt
Let’s ask whether v(t) 3 − 2e −0.5t is a solution¹. Using this formula for v, observe first that
dv d□ d
[3 − 2e −0.5t ] −2e −0.5t · (−0.5) e −0.5t
dt dt dt
and
1.5 − 0.5v 1.5 − 0.5□ 1.5 − 0.5(3 − 2e −0.5t ) 1.5 − 1.5 + e −0.5t e −0.5t .
−0.5t , we have indeed found
dt and 1.5 − 0.5v agree for all values of t when v 3 − 2e
Since dv
a solution to the differential equation.
¹At this time, don’t worry about why we chose this function; we will learn techniques for finding solutions to
differential equations soon enough.
380
7.1 An Introduction to Differential Equations
Activity 7.1.4. Consider the differential equation
dv
1.5 − 0.5v.
dt
Which of the following functions are solutions of this differential equation?
a. v(t) 1.5t − 0.25t 2 . c. v(t) 3.
d. v(t) 3 + Ce −0.5t where C is any
b. v(t) 3 + 2e −0.5t . constant.
This activity shows us something interesting. Notice that the differential equation has in-
finitely many solutions, which are parametrized by the constant C in v(t) 3 + Ce −0.5t . In
Figure 7.1.3, we see the graphs of these solutions for a few values of C, as labeled.
6
v
5
3
4 2
1
3 0
−1
2
−2
1 −3
t
1 2 3
Figure 7.1.3: The family of solutions to the differential equation dv
dt 1.5 − 0.5v.
Notice that the value of C is connected to the initial value of the velocity v(0), since v(0)
3 + C. In other words, while the differential equation describes how the velocity changes
as a function of the velocity itself, this is not enough information to determine the velocity
uniquely: we also need to know the initial velocity. For this reason, differential equations
will typically have infinitely many solutions, one corresponding to each initial value. We
have seen this phenomenon before: given the velocity of a moving object v(t), we cannot
uniquely determine the object’s position function unless we also know its initial position.
If we are given a differential equation and an initial value for the unknown function, we say
that we have an initial value problem. For instance,
dv
1.5 − 0.5v, v(0) 0.5
dt
is an initial value problem. In this problem, we know the value of v at one time and we know
381
Chapter 7 Differential Equations
how v is changing. Consequently, there should be exactly one function v that satisfies the
initial value problem.
This demonstrates the following important general property of initial value problems.
Initial value problems that are “well behaved” have exactly one solution, which exists
in some interval around the initial point.
We won’t worry about what “well behaved” means—it is a technical condition that will be
satisfied by all the differential equations we consider.
To close this section, we note that differential equations may be classified based on certain
characteristics they may possess. You may see many different types of differential equations
in a later course in differential equations. For now, we would like to introduce a few terms
that are used to describe differential equations.
A first-order differential equation is one in which only the first derivative of the function
occurs. For this reason,
dv
1.5 − 0.5v
dt
is a first-order equation while
d2 y
−10y
dt 2
is a second-order equation.
A differential equation is autonomous if the independent variable does not appear in the
description of the derivative. For instance,
dv
1.5 − 0.5v
dt
is autonomous because the description of the derivative dv/dt does not depend on time.
The equation
dy
1.5t − 0.5y,
dt
however, is not autonomous.
7.1.4 Summary
• A differential equation is simply an equation that describes the derivative(s) of an un-
known function.
• Physical principles, as well as some everyday situations, often describe how a quantity
changes, which lead to differential equations.
• A solution to a differential equation is a function whose derivatives satisfy the equa-
tion’s description. Differential equations typically have infinitely many solutions, pa-
rametrized by the initial values.
382
7.1 An Introduction to Differential Equations
7.1.5 Exercises
1. Matching solutions with equations. Match the solutions to the differential equations.
If there is more than one solution to an equation, select the answer that includes all
solutions.
(a)
dy
4y A. y sin(4x) or y 4 sin(x)
dx
d2 y B. y sin(4x)
(b) dx 2
16y
C. y e −4x or y e 4x
dy
(c) dx −4y
D. y e 4x
d2 y
(d) dx 2
−16y E. y e −4x
F. y 4 sin(x)
2. Finding constant to complete solution. Find a positive value of k for which y sin(kt)
satisfies
d2 y
+ 9y 0.
dt 2
3. Choosing solution of dy/dt k(1 − Ay). Let A and k be positive constants.
dy
Which of the given functions is a solution to dt −k(y + A)?
⊙ y −A + Ce −kt
⊙ y A + Ce −kt
⊙ y −A + Ce kt
⊙ y A−1 + Ce Akt
⊙ y A + Ce kt
⊙ y A−1 + Ce −Akt
4. Suppose that T(t) represents the temperature of a cup of coffee set out in a room, where
T is expressed in degrees Fahrenheit and t in minutes. A physical principle known as
Newton’s Law of Cooling tells us that
dT 1
− T + 5.
dt 15
a. Supposes that T(0) 105. What does the differential equation give us for the
dt |T105 ? Explain in a complete sentence the meaning of these two facts.
value of dT
b. Is T increasing or decreasing at t 0?
c. What is the approximate temperature at t 1?
d. On the graph below, make a plot of dT/dt as a function of T.
383
Chapter 7 Differential Equations
5
dT
4
dt
3
2
1
T
30 60 90 120
-1
-2
-3
e. For which values of T does T increase? For which values of T does T decrease?
f. What do you think is the temperature of the room? Explain your thinking.
g. Verify that T(t) 75 + 30e −t/15 is the solution to the differential equation with
initial value T(0) 105. What happens to this solution after a long time?
5. Suppose that the population of a particular species is described by the function P(t),
where P is expressed in millions. Suppose further that the population’s rate of change
is governed by the differential equation
dP
f (P)
dt
where f (P) is the function graphed below.
dP
dt
P
1 2 3 4
a. For which values of the population P does the population increase?
b. For which values of the population P does the population decrease?
384
7.1 An Introduction to Differential Equations
c. If P(0) 3, how will the population change in time?
d. If the initial population satisfies 0 < P(0) < 1, what will happen to the population
after a very long time?
e. If the initial population satisfies 1 < P(0) < 3, what will happen to the population
after a very long time?
f. If the initial population satisfies 3 < P(0), what will happen to the population
after a very long time?
g. This model for a population’s growth is sometimes called “growth with a thresh-
old.” Explain why this is an appropriate name.
6. In this problem, we test further what it means for a function to be a solution to a given
differential equation.
a. Consider the differential equation
dy
y − t.
dt
Determine whether the following functions are solutions to the given differential
equation.
i. y(t) t + 1 + 2e t
ii. y(t) t + 1
iii. y(t) t + 2
b. When you weigh bananas in a scale at the grocery store, the height h of the ba-
nanas is described by the differential equation
d2 h
−kh
dt 2
where k is the spring constant, a constant that depends on the properties of the
spring in the scale. After you put the bananas in the scale, you (cleverly) observe
that the height of the bananas is given by h(t) 4 sin(3t). What is the value of
the spring constant?
385
Chapter 7 Differential Equations
7.2 Qualitative behavior of solutions to DEs
Motivating Questions
• What is a slope field?
• How can we use a slope field to obtain qualitative information about the solutions of
a differential equation?
• What are stable and unstable equilibrium solutions of an autonomous differential
equation?
In earlier work, we have used the tangent line to the graph of a function f at a point a to
approximate the values of f near a. The usefulness of this approximation is that we need to
know very little about the function; armed with only the value f (a) and the derivative f ′(a),
we may find the equation of the tangent line and the approximation
f (x) ≈ f (a) + f ′(a)(x − a).
Remember that a first-order differential equation gives us information about the derivative
of an unknown function. Since the derivative at a point tells us the slope of the tangent
line at this point, a differential equation gives us crucial information about the tangent lines
to the graph of a solution. We will use this information about the tangent lines to create a
slope field for the differential equation, which enables us to sketch solutions to initial value
problems. Our aim will be to understand the solutions qualitatively. That is, we would
like to understand the basic nature of solutions, such as their long-range behavior, without
precisely determining the value of a solution at a particular point.
Preview Activity 7.2.1. Let’s consider the initial value problem
dy
t − 2, y(0) 1.
dt
a. Use the differential equation to find the slope of the tangent line to the solution
y(t) at t 0. Then use the initial value to find the equation of the tangent line
at t 0. Sketch this tangent line over the interval −0.25 ≤ t ≤ 0.25 on the axes
provided in Figure 7.2.1.
b. Also shown in Figure 7.2.1 are the tangent lines to the solution y(t) at the points
t 1, 2, and 3 (we will see how to find these later). Use the graph to measure the
slope of each tangent line and verify that each agrees with the value specified
by the differential equation.
c. Using these tangent lines as a guide, sketch a graph of the solution y(t) over the
interval 0 ≤ t ≤ 3 so that the lines are tangent to the graph of y(t).
386
7.2 Qualitative behavior of solutions to DEs
d. Graph the solution you found in (d) 3
on the axes provided, and compare it y
to the sketch you made using the tan-
gent lines.
2
1
t
1 2 3
-1
-2
Figure 7.2.1: Grid for plotting
partial tangent lines.
7.2.1 Slope fields
Preview Activity 7.2.1 shows that we can sketch the solution to an initial value problem if
we know an appropriate collection of tangent lines. We can use the differential equation to
find the slope of the tangent line at any point of interest, and hence plot such a collection.
dy
Let’s continue looking at the differential equation dt t − 2. If t 0, this equation says
that dy/dt 0 − 2 −2. Note that this value holds regardless of the value of y. We will
therefore sketch tangent lines for several values of y and t 0 with a slope of −2, as shown
in Figure 7.2.2.
3 3
y y
2 2
1 1
t t
1 2 3 1 2 3
-1 -1
-2 -2
Figure 7.2.2: Tangent lines Figure 7.2.3: Adding
at points with t 0. tangent lines at points
with t 1.
Let’s continue in the same way: if t 1, the differential equation tells us that dy/dt 1 − 2
−1, and this holds regardless of the value of y. We now sketch tangent lines for several
values of y and t 1 with a slope of −1 in Figure 7.2.3.
387
Chapter 7 Differential Equations
Similarly, we see that when t 2, dy/dt 0 and when t 3, dy/dt 1. We may therefore
add to our growing collection of tangent line plots to achieve Figure 7.2.4.
3 3
y y
2 2
1 1
t t
1 2 3 1 2 3
-1 -1
-2 -2
Figure 7.2.4: Adding Figure 7.2.5: A completed
tangent lines at points slope field.
with t 2 and t 3.
In Figure 7.2.4, we begin to see the solutions to the differential equation emerge. For the
sake of even greater clarity, we add more tangent lines to provide the more complete picture
shown at right in Figure 7.2.5.
Figure 7.2.5 is called a slope field for the differential equation. It allows us to sketch solutions
just as we did in the preview activity. We can begin with the initial value y(0) 1 and start
sketching the solution by following the tangent line. Whenever the solution passes through
a point at which a tangent line is drawn, that line is tangent to the solution. This principle
leads us to the sequence of images in Figure 7.2.6.
3 3 3
y y y
2 2 2
1 1 1
t t t
1 2 3 1 2 3 1 2 3
-1 -1 -1
-2 -2 -2
Figure 7.2.6: A sequence of images that show how to sketch the IVP solution that satisfies
y(0) 1.
In fact, we can draw solutions for any initial value. Figure 7.2.7 shows solutions for several
dy
different initial values for y(0). Just as we did for the equation dt t − 2, we can construct
a slope field for any differential equation of interest. The slope field provides us with visual
information about how we expect solutions to the differential equation to behave.
388
7.2 Qualitative behavior of solutions to DEs
3
y
2
1
t
1 2 3
-1
-2
dy
Figure 7.2.7: Different solutions to dt t − 2 that correspond to different initial values.
Activity 7.2.2. Consider the autonomous differential equation
dy 1
− (y − 4).
dt 2
dy
a. Make a plot of dt versus y on the axes provided in Figure 7.2.8. Looking at the
graph, for what values of y does y increase and for what values of y does y
decrease?
b. Next, sketch the slope field for this differential equation on the axes provided
in Figure 7.2.9.
c. Use your work in (b) to sketch (on the same axes in Figure 7.2.9.) solutions that
satisfy y(0) 0, y(0) 2, y(0) 4 and y(0) 6.
d. Verify that y(t) 4 + 2e −t/2 is a solution to the given differential equation with
the initial value y(0) 6. Compare its graph to the one you sketched in (c).
e. What is special about the solution where y(0) 4?
7
y
6
3
5
dy
dt
4
2
3
1 2
y
1
-1 1 2 3 4 5 6 7 t
-1 -1 1 2 3 4 5 6 7
-1
-2
Figure 7.2.9: Axes for plotting the
dy dy
Figure 7.2.8: Axes for plotting dt versus y. slope field for dt − 12 (y − 4).
389
Chapter 7 Differential Equations
7.2.2 Equilibrium solutions and stability
As our work in Activity 7.2.2 demonstrates, first-order autonomous equations may have
solutions that are constant. These are simple to detect by inspecting the differential equation
dy/dt f (y): constant solutions necessarily have a zero derivative, so dy/dt 0 f (y).
dy
For example, in Activity 7.2.2, we considered the equation dt f (y) − 12 (y − 4). Constant
solutions are found by setting f (y) − 21 (y − 4) 0, which we immediately see implies that
y 4.
dy
Values of y for which f (y) 0 in an autonomous differential equation dt f (y) are called
equilibrium solutions of the differential equation.
Activity 7.2.3. Consider the autonomous differential equation
dy 1
− y(y − 4).
dt 2
dy
a. Make a plot of dt versus y on the axes provided in Figure 7.2.10. Looking at
the graph, for what values of y does y increase and for what values of y does y
decrease?
6
y
5
3 4
dy
dt 3
2
2
1
y
1
t
-1 1 2 3 4 5 6 7
-1 1 2 3 4 5 6 7
-1 -1
-2 -2
Figure 7.2.10: Axes for plotting dy/dt Figure 7.2.11: Axes for plotting the
dy dy
vs y for dt − 12 y(y − 4). slope field for dt − 12 y(y − 4).
b. Identify any equilibrium solutions of the given differential equation.
c. Now sketch the slope field for the given differential equation on the axes pro-
vided in Figure 7.2.11.
d. Sketch the solutions to the given differential equation that correspond to initial
values y(0) −1, 0, 1, . . . , 5.
e. An equilibrium solution y is called stable if nearby solutions converge to y. This
means that if the initial condition varies slightly from y, then limt→∞ y(t) y.
Conversely, an equilibrium solution y is called unstable if nearby solutions are
390
7.2 Qualitative behavior of solutions to DEs
pushed away from y. Using your work above, classify the equilibrium solutions
you found in (b) as either stable or unstable.
f. Suppose that y(t) describes the population of a species of living organisms and
that the initial value y(0) is positive. What can you say about the eventual fate
of this population?
g. Now consider a general autonomous differential equation of the form dy/dt
f (y). Remember that an equilibrium solution y satisfies f (y) 0. If we graph
dy/dt f (y) as a function of y, for which of the differential equations repre-
sented in Figure 7.2.12 and Figure 7.2.13 is y a stable equilibrium and for which
is y unstable? Why?
dy dy
dt = f (y) dt = f (y)
y y
y y
dy dy
Figure 7.2.12: Plot of dt as a function of Figure 7.2.13: Plot of dt as a different
y. function of y.
7.2.3 Summary
• A slope field is a plot created by graphing the tangent lines of many different solutions
to a differential equation.
• Once we have a slope field, we may sketch the graph of solutions by drawing a curve
that is always tangent to the lines in the slope field.
• Autonomous differential equations sometimes have constant solutions that we call
equilibrium solutions. These may be classified as stable or unstable, depending on
the behavior of nearby solutions.
391
Chapter 7 Differential Equations
7.2.4 Exercises
1. Graphing equilibrium solutions. Consider the direction field below for a differential
equation. Use the graph to find the equilibrium solutions.
2. Sketching solution curves. Consider the two slope fields shown, in figures 1 and 2
below.
figure 1 figure 2
On a print-out of these slope fields, sketch for each three solution curves to the differ-
ential equations that generated them. Then complete the following statements:
For the slope field in figure 1, a solution passing through the point (4,-3) has a (□ pos-
itive □ negative □ zero □ undefined) slope.
For the slope field in figure 1, a solution passing through the point (-2,-3) has a (□ pos-
itive □ negative □ zero □ undefined) slope.
For the slope field in figure 2, a solution passing through the point (2,-1) has a (□ pos-
itive □ negative □ zero □ undefined) slope.
For the slope field in figure 2, a solution passing through the point (0,3) has a (□ pos-
itive □ negative □ zero □ undefined) slope.
392
7.2 Qualitative behavior of solutions to DEs
3. Matching equations with direction fields. Match the following equations with their
direction field. While you can probably solve this problem by guessing, it is useful to
try to predict characteristics of the direction field and then match them to the picture.
Here are some handy characteristics to start with -- you will develop more as you prac-
tice.
A. Set y equal to zero and look at how the derivative behaves along the x-axis.
B. Do the same for the y-axis by setting x equal to 0
C. Consider the curve in the plane defined by setting y ′ 0 -- this should correspond
to the points in the picture where the slope is zero.
D. Setting y ′ equal to a constant other than zero gives the curve of points where the
slope is that constant. These are called isoclines, and can be used to construct the
direction field picture by hand.
(a) y ′ 2x y + 2xe −x
2
y
(b) y ′ + 3 cos(2x)
x
(c) y ′ y + xe −x + 1
(d) y ′ 2 sin(3x) + 1 + y
A B
393
Chapter 7 Differential Equations
C D
4. Describing equilibrium solutions. Given the differential equation x ′(t) −x 4 − 9x 3 −
19x 2 + 9x + 20.
List the constant (or equilibrium) solutions to this differential equation in increasing
order and indicate whether or not these equations are stable, semi-stable, or unstable.
(It helps to sketch the graph.
(□ □ stable □ unstable □ semi-stable)
(□ □ stable □ unstable □ semi-stable)
(□ □ stable □ unstable □ semi-stable)
(□ □ stable □ unstable □ semi-stable)
5. Consider the differential equation
dy
t − y.
dt
4
a. Sketch a slope field on the axes at y
right. 3
b. Sketch the solutions whose initial 2
values are y(0) −4, −3, . . . , 4.
1
c. What do your sketches suggest is t
the solution whose initial value is
y(0) −1? Verify that this is -4 -3 -2 -1 1 2 3 4
-1
indeed the solution to this initial
value problem. -2
d. By considering the differential -3
equation and the graphs you have
sketched, what is the relationship -4
between t and y at a point where a
solution has a local minimum?
394
7.2 Qualitative behavior of solutions to DEs
6. Consider the situation from problem 2 of Section 7.1: Suppose that the population of
a particular species is described by the function P(t), where P is expressed in millions.
Suppose further that the population’s rate of change is governed by the differential
equation
dP
f (P)
dt
where f (P) is the function graphed below.
dP
dt
P
1 2 3 4
a. Sketch a slope field for this differential equation. You do not have enough infor-
mation to determine the actual slopes, but you should have enough information
to determine where slopes are positive, negative, zero, large, or small, and hence
determine the qualitative behavior of solutions.
b. Sketch some solutions to this differential equation when the initial population
P(0) > 0.
c. Identify any equilibrium solutions to the differential equation and classify them
as stable or unstable.
d. If P(0) > 1, what is the eventual fate of the species? if P(0) < 1?
e. Remember that we referred to this model for population growth as “growth with
a threshold.” Explain why this characterization makes sense by considering so-
lutions whose inital value is close to 1.
7. The population of a species of fish in a lake is P(t) where P is measured in thousands
of fish and t is measured in months. The growth of the population is described by the
differential equation
dP
f (P) P(6 − P).
dt
a. Sketch a graph of f (P) P(6−P) and use it to determine the equilibrium solutions
and whether they are stable or unstable. Write a complete sentence that describes
the long-term behavior of the fish population.
b. Suppose now that the owners of the lake allow fishers to remove 1000 fish from
395
Chapter 7 Differential Equations
the lake every month (remember that P(t) is measured in thousands of fish). Mod-
ify the differential equation to take this into account. Sketch the new graph of
dP/dt versus P. Determine the new equilibrium solutions and decide whether
they are stable or unstable.
c. Given the situation in part (b), give a description of the long-term behavior of the
fish population.
d. Suppose that fishermen remove h thousand fish per month. How is the differen-
tial equation modified?
e. What is the largest number of fish that can be removed per month without elimi-
nating the fish population? If fish are removed at this maximum rate, what is the
eventual population of fish?
8. Let y(t) be the number of thousands of mice that live on a farm; assume time t is mea-
sured in years.¹
a. The population of the mice grows at a yearly rate that is twenty times the number
of mice. Express this as a differential equation.
b. At some point, the farmer brings C cats to the farm. The number of mice that the
cats can eat in a year is
y
M(y) C
2+ y
thousand mice per year. Explain how this modifies the differential equation that
you found in part a).
c. Sketch a graph of the function M(y) for a single cat C 1 and explain its features
by looking, for instance, at the behavior of M(y) when y is small and when y is
large.
d. Suppose that C 1. Find the equilibrium solutions and determine whether they
are stable or unstable. Use this to explain the long-term behavior of the mice
population depending on the initial population of the mice.
e. Suppose that C 60. Find the equilibrium solutions and determine whether
they are stable or unstable. Use this to explain the long-term behavior of the mice
population depending on the initial population of the mice.
f. What is the smallest number of cats you would need to keep the mice population
from growing arbitrarily large?
¹This problem is based on an ecological analysis presented in a research paper by C.S. Hollings: The Com-
ponents of Predation as Revealed by a Study of Small Mammal Predation of the European Pine Sawfly, Canadian
Entomology 91: 283-320.
396
7.3 Euler’s method
7.3 Euler’s method
Motivating Questions
• What is Euler’s method and how can we use it to approximate the solution to an
initial value problem?
• How accurate is Euler’s method?
In Section 7.2, we saw how a slope field can be used to sketch solutions to a differential
equation. In particular, the slope field is a plot of a large collection of tangent lines to a
large number of solutions of the differential equation, and we sketch a single solution by
simply following these tangent lines. With a little more thought, we can use this same idea
to approximate numerically the solutions of a differential equation.
Preview Activity 7.3.1. Consider the initial value problem
dy 1
(y + 1), y(0) 0.
dt 2
a. Use the differential equation to find the slope of the tangent line to the solution
y(t) at t 0. Then use the given initial value to find the equation of the tangent
line at t 0.
b. Sketch the tangent line on the axes provided in Figure 7.3.1 on the interval 0 ≤
t ≤ 2 and use it to approximate y(2), the value of the solution at t 2.
7
y
6
5
4
3
2
1
t
1 2 3 4 5 6 7
Figure 7.3.1: Grid for plotting the tangent line.
c. Assuming that your approximation for y(2) is the actual value of y(2), use the
differential equation to find the slope of the tangent line to y(t) at t 2. Then,
write the equation of the tangent line at t 2.
397
Chapter 7 Differential Equations
d. Add a sketch of this tangent line on the interval 2 ≤ t ≤ 4 to your plot Fig-
ure 7.3.1; use this new tangent line to approximate y(4), the value of the solution
at t 4.
e. Repeat the same step to find an approximation for y(6).
7.3.1 Euler’s Method
Preview Activity 7.3.1 demonstrates an algorithm known as Euler’s¹ Method, which gener-
ates a numerical approximation to the solution of an initial value problem. In this algorithm,
we will approximate the solution by taking horizontal steps of a fixed size that we denote
by ∆t.
Before explaining the algorithm in detail,
let’s remember how we compute the slope y
of a line: the slope is the ratio of the vertical
change to the horizontal change, as shown
in Figure 7.3.2.
∆y
In other words, m ∆t . Solving for ∆y, we
see that the vertical change is the product
of the slope and the horizontal change, or ∆y
∆y m∆t.
Now, suppose that we would like to solve
∆t
the initial value problem
dy
t − y, y(0) 1. t
dt
Figure 7.3.2: The role of slope in Euler’s
Method.
There is an algorithm by which we can find an algebraic formula for the solution to this
initial value problem, and we can check that this solution is y(t) t − 1 + 2e −t . But we are
instead interested in generating an approximate solution by creating a sequence of points
(t i , y i ), where y i ≈ y(t i ). For this first example, we choose ∆t 0.2.
Since we know that y(0) 1, we will take the initial point to be (t0 , y0 ) (0, 1) and move
horizontally by ∆t 0.2 to the point (t1 , y1 ). Thus, t1 t0 + ∆t 0.2. Now, the differential
equation tells us that the slope of the tangent line at this point is
dy
m 0 − 1 −1,
dt (0,1)
¹“Euler” is pronounced “Oy-ler.” Among other things, Euler is the mathematician credited with the famous
number e; if you incorrectly pronounce his name “You-ler,” you fail to appreciate his genius and legacy.
398
7.3 Euler’s method
so to move along the tangent line by taking a
horizontal step of size ∆t 0.2, we must also 1.2
y
move vertically by
(t0 , y0 )
∆y m∆t −1 · 0.2 −0.2. 0.8 (t1 , y1 )
We then have the approximation y(0.2) ≈ y1
y0 + ∆y 1 − 0.2 0.8. At this point, we have
executed one step of Euler’s method, as seen 0.4
graphically in Figure 7.3.3.
t
0.4 0.8 1.2
Figure 7.3.3: One step of Euler’s
method.
Now we repeat this process: at (t1 , y1 ) (0.2, 0.8), the differential equation tells us that the
slope is
dy
m 0.2 − 0.8 −0.6.
dt (0.2,0.8)
If we move forward horizontally by ∆t to t2
t1 + ∆ 0.4, we must move vertically by
1.2
y
∆y −0.6 · 0.2 −0.12. (t0 , y0 )
0.8 (t1 , y1 )
We consequently arrive at y2 y1 +∆y 0.8−
0.12 0.68, which gives y(0.2) ≈ 0.68. Now
(t2 , y2 )
we have completed the second step of Euler’s
method, as shown in Figure 7.3.4. 0.4
t
0.4 0.8 1.2
Figure 7.3.4: Two steps of Euler’s
method.
If we continue in this way, we may generate the points (t i , y i ) shown in Figure 7.3.5. Because
we can find a formula for the actual solution y(t) to this differential equation, we can graph
y(t) and compare it to the points generated by Euler’s method, as shown in Figure 7.3.6.
399
Chapter 7 Differential Equations
1.2 1.2
y y
0.8 0.8
0.4 0.4
t t
0.4 0.8 1.2 0.4 0.8 1.2
Figure 7.3.5: The points and piecewise Figure 7.3.6: The approximate solution
linear approximate solution generated by compared to the exact solution (shown in
Euler’s method. blue).
Because we need to generate a large number of points (t i , y i ), it is convenient to organize the
implementation of Euler’s method in a table as shown. We begin with the given initial data.
ti yi dy/dt ∆y
0.0000 1.0000
From here, we compute the slope of the tangent line m dy/dt using the formula for dy/dt
from the differential equation, and then we find ∆y, the change in y, using the rule ∆y
m∆t.
ti yi dy/dt ∆y
0.0000 1.0000 −1.0000 −0.2000
Next, we increase t i by ∆t and y i by ∆y to get
ti yi dy/dt ∆y
0.0000 1.0000 −1.0000 −0.2000
0.2000 0.8000
We continue the process for however many steps we decide, eventually generating a table
like Table 7.3.7.
ti yi dy/dt ∆y
0.0000 1.0000 −1.0000 −0.2000
0.2000 0.8000 −0.6000 −0.1200
0.4000 0.6800 −0.2800 −0.0560
0.6000 0.6240 −0.0240 −0.0048
0.8000 0.6192 0.1808 0.0362
1.0000 0.6554 0.3446 0.0689
1.2000 0.7243 0.4757 0.0951
Table 7.3.7: Euler’s method for 6 steps with ∆t 0.2.
400
7.3 Euler’s method
Activity 7.3.2. Consider the initial value problem
dy
2t − 1, y(0) 0
dt
a. Use Euler’s method with ∆t 0.2 to approximate the solution at t i 0.2, 0.4, 0.6,
0.8, and 1.0. Record your work in the following table, and sketch the points
(t i , y i ) on the axes provided.
ti yi dy/dt ∆y 0.2
0.0000 0.0000 y t
0.2000 0.4 0.8 1.2
0.4000 -0.2
0.6000
0.8000
1.0000 -0.6
Table 7.3.8: Table for recording re-
sults of Euler’s method. -1.0
Figure 7.3.9: Grid for plotting points
generated by Euler’s method.
b. Find the exact solution to the original initial value problem and use this function
to find the error in your approximation at each one of the points t i .
c. Explain why the value y5 generated by Euler’s method for this initial value prob-
lem produces the same value as a left Riemann sum for the definite integral
∫1
0
(2t − 1) dt.
d. How would your computations differ if the initial value was y(0) 1? What
does this mean about different solutions to this differential equation?
dy
Activity 7.3.3. Consider the differential equation dt 6y − y 2 .
a. Sketch the slope field for this differential equation on the axes provided in Fig-
ure 7.3.10.
b. Identify any equilibrium solutions and determine whether they are stable or
unstable.
c. What is the long-term behavior of the solution that satisfies the initial value
y(0) 1?
d. Using the initial value y(0) 1, use Euler’s method with ∆t 0.2 to approx-
imate the solution at t i 0.2, 0.4, 0.6, 0.8, and 1.0. Record your results in Ta-
ble 7.3.11 and sketch the corresponding points (t i , y i ) on the axes provided in
401
Chapter 7 Differential Equations
Figure 7.3.12. Note the different horizontal scale on the axes in Figure 7.3.12
compared to Figure 7.3.10.
8
y
6
4
2
t
2 4 6 8
Figure 7.3.10: Grid for plotting the slope field of the given differential equation.
ti yi dy/dt ∆y
8
0.0 1.0000 y
0.2
0.4 6
0.6
0.8 4
1.0
Table 7.3.11: Table for recording re- 2
sults of Euler’s method with ∆t 0.2.
t
0.4 0.8 1.2
Figure 7.3.12: Axes for plotting the
results of Euler’s method.
e. What happens if we apply Euler’s method to approximate the solution with
y(0) 6?
7.3.2 The error in Euler’s method
Since we are approximating the solutions to an initial value problem using tangent lines,
we should expect that the error in the approximation will be smaller when the step size is
402
7.3 Euler’s method
smaller. Consider the initial value problem
dy
y, y(0) 1,
dt
whose solution we can easily find.
The question posed by this initial value problem is “what function do we know that is the
same as its own derivative and has value 1 when t 0?” It is not hard to see that the solution
is y(t) e t . We now apply Euler’s method to approximate y(1) e using several values of
∆t. These approximations will be denoted by E∆t , and we’ll use them to see how accurate
Euler’s Method is.
To begin, we apply Euler’s method with a step size of ∆t 0.2. In that case, we find that
y(1) ≈ E0.2 2.4883. The error is therefore
y(1) − E0.2 e − 2.4883 ≈ 0.2300.
Repeatedly halving ∆t gives the following results, expressed in both tabular and graphical
form.
∆t E∆t Error
0.200 2.4883 0.2300 Error
0.100 2.5937 0.1245
0.2
0.050 2.6533 0.0650
0.025 2.6851 0.0332
Table 7.3.13: Errors that correspond to dif-
ferent ∆t values. 0.1
∆t
0.1 0.2
Figure 7.3.14: A plot of the error as a
function of ∆t.
Notice, both numerically and graphically, that the error is roughly halved when ∆t is halved.
This example illustrates the following general principle.
If Euler’s method is used to approximate the solution to an initial value problem at a point
t, then the error is proportional to ∆t. That is,
y(t) − E∆t ≈ K∆t
for some constant of proportionality K.
403
Chapter 7 Differential Equations
7.3.3 Summary
• Euler’s method is an algorithm for approximating the solution to an initial value prob-
lem by following the tangent lines while we take horizontal steps across the t-axis.
• If we wish to approximate y(t) for some fixed t by taking horizontal steps of size ∆t,
then the error in our approximation is proportional to ∆t.
7.3.4 Exercises
1. A few steps of Euler’s method. Consider the differential equation y ′ −x − y.
Use Euler’s method with ∆x 0.1 to estimate y when x 1.4 for the solution curve
satisfying
Use Euler’s method with ∆x 0.1 to estimate y when x 2.4 for the solution curve
satisfying y(1) 0.
2. Using Euler’s method for a solution of y ′ 4y. Consider the solution of the differen-
tial equation y ′ −2y passing through y(0) 1.
A. Sketch the slope field for this differential equation, and sketch the solution passing
through the point (0,1).
B. Use Euler’s method with step size ∆x 0.2 to estimate the solution at x 0.2, 0.4, . . . , 1.
(Be sure not to round your answers at each step!)
C. Plot your estimated solution on your slope field. Compare the solution and the slope
field. Is the estimated solution an over or under estimate for the actual solution?
D. Check that y e −2x is a solution to y ′ −2y with y(0) 1.
3. Using Euler’s method with different time steps. Use Euler’s method to solve
dB
0.05B
dt
with initial value B 800 when t 0 .
A. ∆t 1 and 1 step: B(1) ≈
B. ∆t 0.5 and 2 steps: B(1) ≈
C. ∆t 0.25 and 4 steps: B(1) ≈
D. Suppose B is the balance in a bank account earning interest. Be sure that you can
explain why the result of your calculation in part (a) is equivalent to compounding the
interest once a year instead of continuously. Then interpret the result of your calcula-
tions in parts (b) and (c) in terms of compound interest.
4. Newton’s Law of Cooling says that the rate at which an object, such as a cup of cof-
fee, cools is proportional to the difference in the object’s temperature and room tem-
perature. If T(t) is the object’s temperature and Tr is room temperature, this law is
404
7.3 Euler’s method
expressed at
dT
−k(T − Tr ),
dt
where k is a constant of proportionality. In this problem, temperature is measured in
degrees Fahrenheit and time in minutes.
a. Two calculus students, Alice and Bob, enter a 70◦ classroom at the same time.
Each has a cup of coffee that is 100◦ . The differential equation for Alice has a
constant of proportionality k 0.5, while the constant of proportionality for Bob
is k 0.1. What is the initial rate of change for Alice’s coffee? What is the initial
rate of change for Bob’s coffee?
b. What feature of Alice’s and Bob’s cups of coffee could explain this difference?
c. As the heating unit turns on and off in the room, the temperature in the room is
Tr 70 + 10 sin t.
Implement Euler’s method with a step size of ∆t 0.1 to approximate the tem-
perature of Alice’s coffee over the time interval 0 ≤ t ≤ 50. This will most easily
be performed using a spreadsheet such as Excel. Graph the temperature of her
coffee and room temperature over this interval.
d. In the same way, implement Euler’s method to approximate the temperature of
Bob’s coffee over the same time interval. Graph the temperature of his coffee and
room temperature over the interval.
e. Explain the similarities and differences that you see in the behavior of Alice’s and
Bob’s cups of coffee.
5. We have seen that the error in approximating the solution to an initial value problem is
proportional to ∆t. That is, if E∆t is the Euler’s method approximation to the solution
to an initial value problem at t, then
y(t) − E∆t ≈ K∆t
for some constant of proportionality K.
In this problem, we will see how to use this fact to improve our estimates, using an idea
called accelerated convergence.
a. We will create a new approximation by assuming the error is exactly proportional
to ∆t, according to the formula
y(t) − E∆t K∆t.
Using our earlier results from the initial value problem dy/dt y and y(0) 1
with ∆t 0.2 and ∆t 0.1, we have
y(1) − 2.4883 0.2K
y(1) − 2.5937 0.1K.
This is a system of two linear equations in the unknowns y(1) and K. Solve this
system to find a new approximation for y(1). (You may remember that the exact
value is y(1) e 2.71828 . . ..)
405
Chapter 7 Differential Equations
b. Use the other data, E0.05 2.6533 and E0.025 2.6851 to do similar work as in (a)
to obtain another approximation. Which gives the better approximation? Why
do you think this is?
c. Let’s now study the initial value problem
dy
t − y, y(0) 0.
dt
Approximate y(0.3) by applying Euler’s method to find approximations E0.1 and
E0.05 . Now use the idea of accelerated convergence to obtain a better approxi-
mation. (For the sake of comparison, you want to note that the actual value is
y(0.3) 0.0408.)
6. In this problem, we’ll modify Euler’s method to obtain better approximations to solu-
tions of initial value problems. This method is called the Improved Euler’s method.
In Euler’s method, we walk across an interval of width ∆t using the slope obtained from
the differential equation at the left endpoint of the interval. Of course, the slope of the
solution will most likely change over this interval. We can improve our approximation
by trying to incorporate the change in the slope over the interval.
Let’s again consider the initial value problem dy/dt y and y(0) 1, which we will
approximate using steps of width ∆t 0.2. Our first interval is therefore 0 ≤ t ≤ 0.2.
At t 0, the differential equation tells us that the slope is 1, and the approximation we
obtain from Euler’s method is that y(0.2) ≈ y1 1 + 1(0.2) 1.2.
This gives us some idea for how the slope has changed over the interval 0 ≤ t ≤ 0.2.
We know the slope at t 0 is 1, while the slope at t 0.2 is 1.2, trusting in the Euler’s
method approximation. We will therefore refine our estimate of the initial slope to be
the average of these two slopes; that is, we will estimate the slope to be (1 + 1.2)/2 1.1.
This gives the new approximation y(1) y1 1 + 1.1(0.2) 1.22.
The first few steps look like what is found in Table 7.3.15.
ti yi Slope at (t i+1 , y i+1 ) Average slope
0.0 1.0000 1.2000 1.1000
0.2 1.2200 1.4640 1.3420
0.4 1.4884 1.7861 1.6372
.. .. .. ..
. . . .
Table 7.3.15: The first several steps of the improved Euler’s method
a. Continue with this method to obtain an approximation for y(1) e.
b. Repeat this method with ∆t 0.1 to obtain a better approximation for y(1).
c. We saw that the error in Euler’s method is proportional to ∆t. Using your results
from parts (a) and (b), what power of ∆t appears to be proportional to the error
in the Improved Euler’s Method?
406
7.4 Separable differential equations
7.4 Separable differential equations
Motivating Questions
• What is a separable differential equation?
• How can we find solutions to a separable differential equation?
• Are some of the differential equations that arise in applications separable?
In Sections 7.2 and 7.3, we have seen several ways to approximate the solution to an initial
value problem. Given the frequency with which differential equations arise in the world
around us, we would like to have some techniques for finding explicit algebraic solutions of
certain initial value problems. In this section, we focus on a particular class of differential
equations (called separable) and develop a method for finding algebraic formulas for their
solutions.
A separable differential equation is a differential equation whose algebraic structure allows the
variables to be separated in a particular way. For instance, consider the equation
dy
t y.
dt
We would like to separate the variables t and y so that all occurrences of t appear on the
right-hand side, and all occurrences of y appear on the left, multiplied by dy/dt. For this
example, we divide both sides by y so that
1 dy
t.
y dt
Note that when we attempt to separate the variables in a differential equation, we require
that one side is a product in which the derivative dy/dt is one factor and the other factor is
solely an expression involving y.
Not every differential equation is separable. For example, if we consider the equation
dy
t − y,
dt
it may seem natural to separate it by writing
dy
y+ t.
dt
As we will see, this will not be helpful, since the left-hand side is not a product of a function
dy
of y with dt .
407
Chapter 7 Differential Equations
Preview Activity 7.4.1. In this preview activity, we explore whether certain differen-
tial equations are separable or not, and then revisit some key ideas from earlier work
in integral calculus.
a. Which of the following differential equations are separable? If the equation is
dy
separable, write the equation in the revised form 1(y) dt h(t).
dy
i. −3y.
dt
dy
ii. t y − y.
dt
dy
iii. t + 1.
dt
dy
iv. t2 − y2.
dt
b. Explain why any autonomous differential equation is guaranteed to be separa-
ble.
c. Why do we include the term “+C” in the expression
∫
x2
x dx + C?
2
d. Suppose we know that a certain function f satisfies the equation
∫ ∫
′
f (x) dx x dx.
What can you conclude about f ?
7.4.1 Solving separable differential equations
Before we discuss a general approach to solving a separable differential equation, it is in-
structive to consider an example.
Example 7.4.1 Find all functions y that are solutions to the differential equation
dy t
2.
dt y
Solution. We begin by separating the variables and writing
dy
y2 t.
dt
408
7.4 Separable differential equations
Integrating both sides of the equation with respect to the independent variable t shows that
∫ ∫
2 dy
y dt t dt.
dt
Next, we notice that the left-hand side allows us to change the variable of antidifferentiation¹
dy
from t to y. In particular, dy dt dt, so we now have
∫ ∫
y dy
2
t dt.
This equation says that two families of antiderivatives are equal to each other. Therefore,
when we find representative antiderivatives of both sides, we know they must differ by an
arbitrary constant C. Antidifferentiating and including the integration constant C on the
y3 t2
right, we find that + C.
3 2
It is not necessary to include an arbitrary constant on both sides of the equation; we know
that y 3 /3 and t 2 /2 are in the same family of antiderivatives and must therefore differ by a
single constant.
Finally, we solve the last equation above for y as a function of t, which gives
√
3 3 2
y(t) t + 3C.
2
Of course, the term 3C on the right-hand side represents 3 times an unknown constant. It
is, therefore, still an√unknown constant, which we will rewrite as C. We thus conclude that
the funtion y(t) 3 3 2
2 t + C is a solution to the original differential equation for any value
of C.
Notice that because this solution depends on the arbitrary constant C, we have found an
infinite family of solutions. This makes sense because we expect to find a unique solution
that corresponds to any given initial value.
dy t
For example, if we want to solve the initial value problem 2 , y(0) 2, we know that
dt y
√
the solution has the form y(t) 3 32 t 2 + C for some constant C. We therefore must find the
appropriate value for C that gives the initial value y(0) 2. Hence,
√
3 3 2 √3
2 y(0) 0 + C C,
2
which shows that C 23 8. The solution to the initial value problem is then
√
3 3 2
y(t) t + 8.
2
¹This is why we required that the left-hand side be written as a product in which dy/dt is one of the terms.
409
Chapter 7 Differential Equations
dy
The strategy of Example 7.4.1 may be applied to any differential equation of the form dt
1(y) · h(t), and any differential equation of this form is said to be separable. We work to solve
a separable differential equation by writing
1 dy
h(t),
1(y) dt
and then integrating both sides with respect to t. After integrating, we try to solve alge-
braically for y in order to write y as a function of t.
Example 7.4.2 Solve the differential equation
dy
3y.
dt
Solution. Following the same strategy as in Example 7.4.1, we have
1 dy
3.
y dt
Integrating both sides with respect to t,
∫ ∫
1 dy
dt 3 dt,
y dt
and thus ∫ ∫
1
dy 3 dt.
y
Antidifferentiating and including the integration constant, we find that
ln | y| 3t + C.
Finally, we need to solve for y. Here, one point deserves careful attention. By the definition
of the natural logarithm function, it follows that
| y| e 3t+C e 3t e C .
Since C is an unknown constant, e C is as well, though we do know that it is positive (because
e x is positive for any x). When we remove the absolute value in order to solve for y, however,
this constant may be either positive or negative. To account for a possible + or −, we denote
this updated constant by C to obtain
y(t) Ce 3t .
There is one more technical point to make. Notice that y 0 is an equilibrium solution to
this differential equation. In solving the equation above, we begin by dividing both sides
by y, which is not allowed if y 0. To be perfectly careful, therefore, we should consider
the equilibrium solutions separately. In this case, notice that the final form of our solution
captures the equilibrium solution by allowing C 0.
410
7.4 Separable differential equations
Activity 7.4.2. Suppose that the population of a town is growing continuously at an
annual rate of 3% per year.
a. Let P(t) be the population of the town in year t. Write a differential equation
that describes the annual growth rate.
b. Find the solutions of this differential equation.
c. If you know that the town’s population in year 0 is 10,000, find the population
P(t).
d. How long does it take for the population to double? This time is called the
doubling time.
e. Working more generally, find the doubling time if the annual growth rate is k
times the population.
Activity 7.4.3. Suppose that a cup of coffee is initially at a temperature of 105◦ F and
is placed in a 75◦ F room. Newton’s law of cooling says that
dT
−k(T − 75),
dt
where k is a constant of proportionality.
a. Suppose you measure that the coffee is cooling at one degree per minute at the
time the coffee is brought into the room. Use the differential equation to deter-
mine the value of the constant k.
b. Find all the solutions of this differential equation.
c. What happens to all the solutions as t → ∞? Explain how this agrees with your
intuition.
d. What is the temperature of the cup of coffee after 20 minutes?
e. How long does it take for the coffee to cool to 80◦ ?
Activity 7.4.4. Solve each of the following differential equations or initial value prob-
lems.
dy
a. dt − (2 − t)y 2 − t
1 dy 2 −2y
b. t dt et
c. y ′ 2y + 2, y(0) 2
d. y ′ 2y 2 , y(−1) 2
dy −2t y
e. dt t 2 +1
, y(0) 4
411
Chapter 7 Differential Equations
7.4.2 Summary
• A separable differential equation is one that may be rewritten with all occurrences of
the dependent variable multiplying the derivative and all occurrences of the indepen-
dent variable on the other side of the equation.
• We may find the solutions to certain separable differential equations by separating
variables, integrating with respect to t, and ultimately solving the resulting algebraic
equation for y.
• This technique allows us to solve many important differential equations that arise in
the world around us. For instance, questions of growth and decay and Newton’s Law
of Cooling give rise to separable differential equations. Later, we will learn in Sec-
tion 7.6 that the important logistic differential equation is also separable.
7.4.3 Exercises
dy
1. Initial value problem for dy/dx x 8 y. Find the equation of the solution to x6 y
dx
through the point (x, y) (1, 3).
2. Initial value problem for dy/dt 0.9(y − 300). Find the solution to the differential
equation
dy
0.6(y − 150)
dt
if y 25 when t 0.
3. Initial value problem for dy/dt y 2 (8 + t). Find the solution to the differential equa-
tion
dy
y 2 (6 + t),
dt
y 8 when t 1.
4. Initial value problem for du/dt e 6u+10t . Solve the separable differential equation for
u
du
e 5u+9t .
dt
Use the following initial condition: u(0) 16.
5. Initial value problem for dy/dx 170yx 16 . Find an equation of the curve that satisfies
dy
70yx 6
dx
and whose y-intercept is 3.
6. The mass of a radioactive sample decays at a rate that is proportional to its mass.
a. Express this fact as a differential equation for the mass M(t) using k for the con-
stant of proportionality.
b. If the initial mass is M0 , find an expression for the mass M(t).
412
7.4 Separable differential equations
c. The half-life of the sample is the amount of time required for half of the mass to
decay. Knowing that the half-life of Carbon-14 is 5730 years, find the value of k
for a sample of Carbon-14.
d. How long does it take for a sample of Carbon-14 to be reduced to one-quarter its
original mass?
e. Carbon-14 naturally occurs in our environment; any living organism takes in
Carbon-14 when it eats and breathes. Upon dying, however, the organism no
longer takes in Carbon-14. Suppose that you find remnants of a pre-historic
firepit. By analyzing the charred wood in the pit, you determine that the amount
of Carbon-14 is only 30% of the amount in living trees. Estimate the age of the
firepit.²
7. Consider the initial value problem
dy t
− , y(0) 8
dt y
a. Find the solution of the initial value problem and sketch its graph.
b. For what values of t is the solution defined?
c. What is the value of y at the last time that the solution is defined?
d. By looking at the differential equation, explain why we should not expect to find
solutions with the value of y you noted in (c).
8. Suppose that a cylindrical water tank with a hole in the bottom is filled with water.
The water, of course, will leak out and the height of the water will decrease. Let h(t)
denote the height of the water. A physical principle called Torricelli’s Law implies that
the height decreases at a rate proportional to the square root of the height.
a. Express this fact using k as the constant of proportionality.
b. Suppose you have two tanks, one with k −1 and another with k −10. What
physical differences would you expect to find?
c. Suppose you have a tank for which the height decreases at 20 inches per minute
when the water is filled to a depth of 100 inches. Find the value of k.
d. Solve the initial value problem for the tank in part (c), and graph the solution you
determine.
e. How long does it take for the water to run out of the tank?
f. Is the solution that you found valid for all time t? If so, explain how you know
this. If not, explain why not.
9. The Gompertz equation is a model that is used to describe the growth of certain popula-
²This approach is the basic idea behind radiocarbon dating.
413
Chapter 7 Differential Equations
tions. Suppose that P(t) is the population of some organism and that
( )
dP P
−P ln −P(ln P − ln 3).
dt 3
a. Sketch a slope field for P(t) over the range 0 ≤ P ≤ 6.
b. Identify any equilibrium solutions and determine whether they are stable or un-
stable.
c. Find the population P(t) assuming that P(0) 1 and sketch its graph. What
happens to P(t) after a very long time?
d. Find the population P(t) assuming that P(0) 6 and sketch its graph. What
happens to P(t) after a very long time?
e. Verify that the long-term behavior of your solutions agrees with what you pre-
dicted by looking at the slope field.
414
7.5 Modeling with differential equations
7.5 Modeling with differential equations
Motivating Questions
• How can we use differential equations to describe phenomena in the world around
us?
• How can we use differential equations to better understand these phenomena?
We have seen several ways that differential equations arise in the natural world, from the
growth of a population to the temperature of a cup of coffee. In this section, we look more
closely at how differential equations give us a natural way to describe various phenoma. As
we’ll see, the key is to understand the different factors that cause a quantity to change.
Preview Activity 7.5.1. Any time that the rate of change of a quantity is related to
the amount of a quantity, a differential equation naturally arises. In the following
two problems, we see two such scenarios; for each, we want to develop a differential
equation whose solution is the quantity of interest.
a. Suppose you have a bank account in which money grows at an annual rate of
3%.
i. If you have $10,000 in the account, at what rate is your money growing?
ii. Suppose that you are also withdrawing money from the account at $1,000
per year. What is the rate of change in the amount of money in the account?
What are the units on this rate of change?
b. Suppose that a water tank holds 100 gallons and that a salty solution, which
contains 20 grams of salt in every gallon, enters the tank at 2 gallons per minute.
i. How much salt enters the tank each minute?
ii. Suppose that initially there are 300 grams of salt in the tank. How much
salt is in each gallon at this point in time?
iii. Finally, suppose that evenly mixed solution is pumped out of the tank at the
rate of 2 gallons per minute. How much salt leaves the tank each minute?
iv. What is the total rate of change in the amount of salt in the tank?
7.5.1 Developing a differential equation
Preview Activity 7.5.1 demonstrates the kind of thinking we will be doing in this section. In
each of the two examples we considered, there is a quantity, such as the amount of money in
the bank account or the amount of salt in the tank, that is changing due to several factors. The
governing differential equation states that the total rate of change is the difference between
the rate of increase and the rate of decrease.
415
Chapter 7 Differential Equations
Example 7.5.1 In the Great Lakes region, rivers flowing into the lakes carry a great deal of
pollution in the form of small pieces of plastic averaging 1 millimeter in diameter. In order
to understand how the amount of plastic in Lake Michigan is changing, construct a model
for how this type pollution has built up in the lake.
Solution. First, some basic facts about Lake Michigan.
• The volume of the lake is 5 · 1012 cubic meters.
• Water flows into the lake at a rate of 5 · 1010 cubic meters per year. It flows out of the
lake at the same rate.
• Each cubic meter flowing into the lake contains roughly 3 · 10−8 cubic meters of plastic
pollution.
Let’s denote the amount of pollution in the lake by P(t), where P is measured in cubic meters
of plastic and t in years. Our goal is to describe the rate of change of this function; so we
want to develop a differential equation describing P(t).
First, we will measure how P(t) increases due to pollution flowing into the lake. We know
that 5 · 1010 cubic meters of water enters the lake every year and each cubic meter of water
contains 3 · 10−8 cubic meters of pollution. Therefore, pollution enters the lake at the rate of
( 3
)( 3 )
10 m water −8 m plastic
5 · 10 3 · 10 3
1.5 · 103 cubic m of plastic per year.
year m water
Second, we will measure how P(t) decreases due to pollution flowing out of the lake. If the
total amount of pollution is P cubic meters and the volume of Lake Michigan is 5 · 1012 cubic
meters, then the concentration of plastic pollution in Lake Michigan is
P
cubic m of plastic per cubic m of water.
5 · 1012
Since 5 · 1010 cubic meters of water flow out each year¹, then the plastic pollution leaves the
lake at the rate of
( )( )
P m 3 plastic 10 m
3water P
5 · 10 cubic m of plastic per year.
5 · 1012 m 3 water year 100
The total rate of change of P is thus the difference between the rate at which pollution enters
the lake and the rate at which pollution leaves the lake; that is,
dP P
1.5 · 103 −
dt 100
1
(1.5 · 105 − P).
100
We have now found a differential equation that describes the rate at which the amount of
pollution is changing. To understand the behavior of P(t), we apply some of the techniques
we have recently developed.
416
7.5 Modeling with differential equations
Because the differential equation is autonomous, we can sketch dP/dt as a function of P and
then construct a slope field, as shown in Figure 7.5.2 and Figure 7.5.3.
dP/dt P
1.5 · 105 P
t
Figure 7.5.2: Plot of dPdt vs. P for
Figure 7.5.3: Plot of the slope field for
dt 100 (1.5 · 10 − P).
dP 1 5
dP
dt 1
100 (1.5 · 10 5 − P).
These plots both show that P 1.5 · 105 is a stable equilibrium. Therefore, we should expect
that the amount of pollution in Lake Michigan will stabilize near 1.5 · 105 cubic meters of
pollution.
Next, assuming that there is initially no pollution in the lake, we will solve the initial value
problem
dP 1
(1.5 · 105 − P), P(0) 0.
dt 100
Separating variables, we find that
1 dP 1
.
1.5 · 105 − P dt 100
Integrating with respect to t, we have
∫ ∫
1 dP 1
dt dt,
1.5 · 105 − P dt 100
and thus changing variables on the left and antidifferentiating on both sides, we find that
∫ ∫
dP 1
dt
1.5 · 105 − P 100
1
− ln |1.5 · 105 − P| t+C
100
Finally, multiplying both sides by −1 and using the definition of the logarithm, we find that
1.5 · 105 − P Ce −t/100 . (7.5.1)
417
Chapter 7 Differential Equations
This is a good time to determine the constant C. Since P 0 when t 0, we have
1.5 · 105 − 0 Ce 0 C,
so C 1.5 · 105 .
Using this value of C in Equation (7.5.1) and solving for P, we arrive at the solution
P(t) 1.5 · 105 (1 − e −t/100 ).
Superimposing the graph of P on the slope field we saw in Figure 7.5.3, we see, as shown in
Figure 7.5.4
We see that, as expected, the amount of plastic pollution stabilizes around 1.5 · 105 cubic
meters.
P
t
Figure 7.5.4: The solution P(t) and the slope field for the differential equation
dt 100 (1.5 · 10 − P).
dP 1 5
There are many important lessons to learn from Example 7.5.1. Foremost is how we can
develop a differential equation by thinking about the “total rate = rate in - rate out” model.
In addition, we note how we can bring together all of our available understanding (plotting
dP
dt vs. P, creating a slope field, solving the differential equation) to see how the differential
equation describes the behavior of a changing quantity.
We can also explore what happens when certain aspects of the problem change. For in-
stance, let’s suppose we are at a time when the plastic pollution entering Lake Michigan has
stabilized at 1.5 · 105 cubic meters, and that new legislation is passed to prevent this type of
pollution entering the lake. So, there is no longer any inflow of plastic pollution to the lake.
How does the amount of plastic pollution in Lake Michigan now change? For example, how
long does it take for the amount of plastic pollution in the lake to halve?
Resetting t 0 at this time, we now have the initial value problem
dP 1
− P, P(0) 1.5 · 105 .
dt 100
¹and we assume that each cubic meter of water that flows out carries with it the plastic pollution it contains
418
7.5 Modeling with differential equations
It is a straightforward and familiar exercise to find that the solution to this equation is P(t)
1.5 · 105 e −t/100 . The time that it takes for half of the pollution to flow out of the lake is given
by T where P(T) 0.75 · 105 . Thus, we must solve the equation
0.75 · 105 1.5 · 105 e −T/100 ,
or
1
e −T/100 .
2
It follows that ( )
1
T −100 ln ≈ 69.3years.
2
In the upcoming activities, we explore some other natural settings in which differential equa-
tions model changing quantities.
Activity 7.5.2. Suppose you have a bank account that grows by 5% every year. Let
A(t) be the amount of money in the account in year t.
a. What is the rate of change of A with respect to t?
b. Suppose that you are also withdrawing $10,000 per year. Write a differential
equation that expresses the total rate of change of A.
c. Sketch a slope field for this differential equation, find any equilibrium solutions,
and identify them as either stable or unstable. Write a sentence or two that
describes the significance of the stability of the equilibrium solution.
d. Suppose that you initially deposit $100,000 into the account. How long does it
take for you to deplete the account?
e. What is the smallest amount of money you would need to have in the account
to guarantee that you never deplete the money in the account?
f. If your initial deposit is $300,000, how much could you withdraw every year
without depleting the account?
Activity 7.5.3. A dose of morphine is absorbed from the bloodstream of a patient at
a rate proportional to the amount in the bloodstream.
a. Write a differential equation for M(t), the amount of morphine in the patient’s
bloodstream, using k as the constant proportionality.
b. Assuming that the initial dose of morphine is M0 , solve the initial value problem
to find M(t). Use the fact that the half-life for the absorption of morphine is two
hours to find the constant k.
c. Suppose that a patient is given morphine intravenously at the rate of 3 mil-
ligrams per hour. Write a differential equation that combines the intravenous
administration of morphine with the body’s natural absorption.
419
Chapter 7 Differential Equations
d. Find any equilibrium solutions and determine their stability.
e. Assuming that there is initially no morphine in the patient’s bloodstream, solve
the initial value problem to determine M(t). What happens to M(t) after a very
long time?
f. To what rate should a doctor reduce the intravenous rate so that there is even-
tually 7 milligrams of morphine in the patient’s bloodstream?
7.5.2 Summary
• Differential equations arise in a situation when we understand how various factors
cause a quantity to change.
• We may use the tools we have developed so far—slope fields, Euler’s methods, and
our method for solving separable equations—to understand a quantity described by a
differential equation.
7.5.3 Exercises
1. Mixing problem. A tank contains 1060 L of pure water. A solution that contains 0.09
kg of sugar per liter enters the tank at the rate 7 L/min. The solution is mixed and
drains from the tank at the same rate.
(a) How much sugar is in the tank at the beginning?
(b) With S representing the amount of sugar (in kg) at time t (in minutes) write a dif-
ferential equation which models this situation.S′ f (t, S) .
(c) Find the amount of sugar (in kg) after t minutes.
(d) Find the amount of the sugar after 30 minutes.
2. Mixing problem. A tank contains 50 kg of salt and 2000 L of water. A solution of a
concentration 0.0125 kg of salt per liter enters a tank at the rate 5 L/min. The solution
is mixed and drains from the tank at the same rate.
(a) What is the concentration of our solution in the tank initially?
(b) Find the amount of salt in the tank after 4 hours.
(c) Find the concentration of salt in the solution in the tank as time approaches infinity.
3. Population growth problem. A bacteria culture starts with 320 bacteria and grows at
a rate proportional to its size. After 3 hours there will be 960 bacteria.
(a) Express the population after t hours as a function of t.
(b) What will be the population after 6 hours?
(c) How long will it take for the population to reach 1970 ?
420
7.5 Modeling with differential equations
4. Radioactive decay problem. An unknown radioactive element decays into non-ra-
dioactive substances. In 420 days the radioactivity of a sample decreases by 30 percent.
(a) What is the half-life of the element?
(b) How long will it take for a sample of 100 mg to decay to 98 mg?
5. Investment problem. A young person with no initial capital invests k dollars per year
in a retirement account at an annual rate of return 0.06. Assume that investments are
made continuously and that the return is compounded continuously.
Determine a formula for the sum S(t) -- (this will involve the parameter k). What value
of k will provide 2513000 dollars in 50 years?
6. Congratulations, you just won the lottery! In one option presented to you, you will be
paid one million dollars a year for the next 25 years. You can deposit this money in an
account that will earn 5% each year.
a. Set up a differential equation that describes the rate of change in the amount of
money in the account. Two factors cause the amount to grow—first, you are de-
positing one millon dollars per year and second, you are earning 5% interest.
b. If there is no amount of money in the account when you open it, how much money
will you have in the account after 25 years?
c. The second option presented to you is to take a lump sum of 10 million dollars,
which you will deposit into a similar account. How much money will you have
in that account after 25 years?
d. Do you prefer the first or second option? Explain your thinking.
e. At what time does the amount of money in the account under the first option
overtake the amount of money in the account under the second option?
7. When a skydiver jumps from a plane, gravity causes her downward velocity to increase
at the rate of 1 ≈ 9.8 meters per second squared. At the same time, wind resistance
causes her velocity to decrease at a rate proportional to the velocity.
a. Using k to represent the constant of proportionality, write a differential equation
that describes the rate of change of the skydiver’s velocity.
b. Find any equilibrium solutions and decide whether they are stable or unstable.
Your result should depend on k.
c. Suppose that the initial velocity is zero. Find the velocity v(t).
d. A typical terminal velocity for a skydiver falling face down is 54 meters per sec-
ond. What is the value of k for this skydiver?
e. How long does it take to reach 50% of the terminal velocity?
8. During the first few years of life, the rate at which a baby gains weight is proportional
to the reciprocal of its weight.
a. Express this fact as a differential equation.
b. Suppose that a baby weighs 8 pounds at birth and 9 pounds one month later.
421
Chapter 7 Differential Equations
How much will he weigh at one year?
c. Do you think this is a realistic model for a long time?
9. Suppose that you have a water tank that holds 100 gallons of water. A briny solution,
which contains 20 grams of salt per gallon, enters the tank at the rate of 3 gallons per
minute.
At the same time, the solution is well mixed, and water is pumped out of the tank at
the rate of 3 gallons per minute.
a. Since 3 gallons enters the tank every minute and 3 gallons leaves every minute,
what can you conclude about the volume of water in the tank.
b. How many grams of salt enters the tank every minute?
c. Suppose that S(t) denotes the number of grams of salt in the tank in minute t.
How many grams are there in each gallon in minute t?
d. Since water leaves the tank at 3 gallons per minute, how many grams of salt leave
the tank each minute?
e. Write a differential equation that expresses the total rate of change of S.
f. Identify any equilibrium solutions and determine whether they are stable or un-
stable.
g. Suppose that there is initially no salt in the tank. Find the amount of salt S(t) in
minute t.
h. What happens to S(t) after a very long time? Explain how you could have pre-
dicted this only knowing how much salt there is in each gallon of the briny solu-
tion that enters the tank.
422
7.6 Population Growth and the Logistic Equation
7.6 Population Growth and the Logistic Equation
Motivating Questions
• How can we use differential equations to realistically model the growth of a popula-
tion?
• How can we assess the accuracy of our models?
The growth of the earth’s population is one of the pressing issues of our time. Will the
population continue to grow? Or will it perhaps level off at some point, and if so, when?
In this section, we look at two ways in which we may use differential equations to help us
address these questions.
Before we begin, let’s consider again two important differential equations that we have seen
in earlier work this chapter.
Preview Activity 7.6.1. Recall that one model for population growth states that a
population grows at a rate proportional to its size.
a. We begin with the differential equation
dP 1
P.
dt 2
Sketch a slope field below as well as a few typical solutions on the axes provided.
4
P
3
2
1
t
2 4
b. Find all equilibrium solutions of the equation dP
dt 1
2P and classify them as
stable or unstable.
c. If P(0) is positive, describe the long-term behavior of the solution to dP
dt 12 P.
d. Let’s now consider a modified differential equation given by
dP 1
P(3 − P).
dt 2
423
Chapter 7 Differential Equations
As before, sketch a slope field as well as a few typical solutions on the following
axes provided.
4
P
3
2
1
t
2 4
e. Find any equilibrium solutions and classify them as stable or unstable.
f. If P(0) is positive, describe the long-term behavior of the solution.
7.6.1 The earth’s population
We will now begin studying the earth’s population. To get started, in Table 7.6.1 are some
data for the earth’s population in recent years that we will use in our investigations.
Year 1998 1999 2000 2001 2002 2005 2006 2007 2008 2009 2010
Pop
5.932 6.008 6.084 6.159 6.234 6.456 6.531 6.606 6.681 6.756 6.831
(billions)
Table 7.6.1: Some recent population data for planet Earth.
Activity 7.6.2. Our first model will be based on the following assumption:
The rate of change of the population is proportional to the population.
On the face of it, this seems pretty reasonable. When there is a relatively small number
of people, there will be fewer births and deaths so the rate of change will be small.
When there is a larger number of people, there will be more births and deaths so we
expect a larger rate of change.
If P(t) is the population t years after the year 2000, we may express this assumption
as
dP
kP
dt
where k is a constant of proportionality.
424
7.6 Population Growth and the Logistic Equation
a. Use the data in the table to estimate the derivative P ′(0) using a central differ-
ence. Assume that t 0 corresponds to the year 2000.
b. What is the population P(0)?
c. Use your results from (a) and (b) to estimate the constant of proportionality k
in the differential equation.
d. Now that we know the value of k, we have the initial value problem
dP
kP, P(0) 6.084.
dt
Find the solution to this initial value problem.
e. What does your solution predict for the population in the year 2010? Is this
close to the actual population given in the table?
f. When does your solution predict that the population will reach 12 billion?
g. What does your solution predict for the population in the year 2500?
h. Do you think this is a reasonable model for the earth’s population? Why or why
not? Explain your thinking using a couple of complete sentences.
Our work in Activity 7.6.2 shows that that the exponential model is fairly accurate for years
relatively close to 2000. However, if we go too far into the future, the model predicts increas-
ingly large rates of change, which causes the population to grow arbitrarily large. This does
not make much sense since it is unrealistic to expect that the earth would be able to support
such a large population.
The constant k in the differential equation has an important interpretation. Let’s rewrite the
dt kP by solving for k, so that we have
differential equation dP
dP/dt
k .
P
We see that k is the ratio of the rate of change to the population; in other words, it is the
contribution to the rate of change from a single person. We call this the per capita growth rate.
In the exponential model we introduced in Activity 7.6.2, the per capita growth rate is con-
stant. This means that when the population is large, the per capita growth rate is the same
as when the population is small. It is natural to think that the per capita growth rate should
decrease when the population becomes large, since there will not be enough resources to
support so many people. We expect it would be a more realistic model to assume that the
per capita growth rate depends on the population P.
In the previous activity, we computed the per capita growth rate in a single year by comput-
dt and P (which we did for t 0). If we return to the data in Table 7.6.1
ing k, the quotient of dP
and compute the per capita growth rate over a range of years, we generate the data shown
in Figure 7.6.2, which shows how the per capita growth rate is a function of the population,
425
Chapter 7 Differential Equations
P.
0.015
per capita growth rate
0.014
0.013
0.012
0.011
0.010
6.0 6.2 6.4 6.6 6.8 7.0
Figure 7.6.2: A plot of per capita growth rate vs. population P.
From the data, we see that the per capita growth rate appears to decrease as the population
increases. In fact, the points seem to lie very close to a line, which is shown at two different
scales in Figure 7.6.3.
0.015 0.03
per capita growth rate per capita growth rate
0.014
0.02
0.013
0.01
0.012
P
0.011 2 4 6 8 10 12 14 16 18 20
0.010 -0.01
6.0 6.2 6.4 6.6 6.8 7.0
Figure 7.6.3: The line that approximates per capita growth as a function of population, P.
Looking at this line carefully, we can find its equation to be
dP/dt
0.025 − 0.002P.
P
If we multiply both sides by P, we arrive at the differential equation
dP
P(0.025 − 0.002P).
dt
Graphing the dependence of dP/dt on the population P, we see that this differential equa-
tion demonstrates a quadratic relationship between dP
dt and P, as shown in Figure 7.6.4.
426
7.6 Population Growth and the Logistic Equation
0.10
dP
dt
0.05
P
2 4 6 8 10 12 14 16 18 20
-0.05
-0.10
Figure 7.6.4: A plot of dP
dt vs. P for the differential equation dP
dt P(0.025 − 0.002P).
dt P(0.025 − 0.002P) is an example of the logistic equation, and is the second
The equation dP
model for population growth that we will consider. We expect that it will be more realistic,
because the per capita growth rate is a decreasing function of the population.
Indeed, the graph in Figure 7.6.4 shows that there are two equilibrium solutions, P 0,
which is unstable, and P 12.5, which is a stable equilibrium. The graph shows that any
solution with P(0) > 0 will eventually stabilize around 12.5. Thus, our model predicts the
world’s population will eventually stabilize around 12.5 billion.
A prediction for the long-term behavior of the population is a valuable conclusion to draw
from our differential equation. We would, however, also like to answer some quantitative
questions. For instance, how long will it take to reach a population of 10 billion? To answer
this question, we need to find an explicit solution of the equation.
7.6.2 Solving the logistic differential equation
Since we would like to apply the logistic model in more general situations, we state the
logistic equation in its more general form,
dP
kP(N − P). (7.6.1)
dt
The equilibrium solutions here are P 0 and 1 − N P
0, which shows that P N. The
equilibrium at P N is called the carrying capacity of the population for it represents the
stable population that can be sustained by the environment.
We now solve the logistic equation (7.6.1). The equation is separable, so we separate the
variables
1 dP
k,
P(N − P) dt
and integrate to find that ∫ ∫
1
dP k dt.
P(N − P)
427
Chapter 7 Differential Equations
To find the antiderivative on the left, we use the partial fraction decomposition
[ ]
1 1 1 1
+ .
P(N − P) N P N − P
Now we are ready to integrate, with
∫ [ ] ∫
1 1 1
+ dP k dt.
N P N−P
1
On the left, observe that N is constant, so we can remove a factor of N and antidifferentiate
to find that
1
(ln |P| − ln |N − P|) kt + C.
N
Multiplying both sides of this last equation by N and using a rule of logarithms, we next
find that
P
ln kN t + C.
N−P
From the definition of the logarithm, replacing e C with C, and letting C absorb the absolute
value signs, we now know that
P
Ce kN t .
N−P
At this point, all that remains is to determine C and solve algebraically for P.
P0
If the initial population is P(0) P0 , then it follows that C N−P0 , so
P P0
e kN t .
N−P N − P0
We will solve this equation for P by multiplying both sides by (N − P)(N − P0 ) to obtain
P(N − P0 ) P0 (N − P)e kN t
P0 N e kN t − P0 Pe kN t .
Solving for P0 N e kN t , expanding, and factoring, it follows that
P0 N e kN t P(N − P0 ) + P0 Pe kN t
P(N − P0 + P0 e kN t ).
Dividing to solve for P, we see that
P0 N e kN t
P .
N − P0 + P0 e kN t
428
7.6 Population Growth and the Logistic Equation
1 −kN t
Finally, we choose to multiply the numerator and denominator by P0 e to obtain
N
P(t) ( ) .
N−P0
P0 e −kN t + 1
While that was a lot of algebra, notice the result: we have found an explicit solution to the
logistic equation.
Solution to the Logistic Equation.
The solution to the initial value problem
dP
kP(N − P), P(0) P0 ,
dt
is
N
P(t) ( ) . (7.6.2)
N−P0
P0 e −kN t + 1
For the logistic equation describing the earth’s population that we worked with earlier in
this section, we have
k 0.002, N 12.5, and P0 6.084.
This gives the solution
12.5
P(t) ,
1.0546e −0.025t + 1
whose graph is shown in Figure 7.6.5.
15
P
12
9
6
3
t
40 80 120 160 200
Figure 7.6.5: The solution to the logistic equation modeling the earth’s population.
The graph shows the population leveling off at 12.5 billion, as we expected, and that the
population will be around 10 billion in the year 2050. These results, which we have found
using a relatively simple mathematical model, agree fairly well with predictions made using
a much more sophisticated model developed by the United Nations.
429
Chapter 7 Differential Equations
The logistic equation is good for modeling any situation in which limited growth is possible.
For instance, it could model the spread of a flu virus through a population contained on a
cruise ship, the rate at which a rumor spreads within a small town, or the behavior of an
animal population on an island. Through our work in this section, we have completely
solved the logistic equation, regardless of the values of the constants N, k, and P0 . Anytime
we encounter a logistic equation, we can apply the formula we found in Equation (7.6.2).
Activity 7.6.3.
Consider the logistic equation
dP
dt
dP
kP(N − P)
dt
dP
with the graph of dt vs. P shown in Fig- N P
ure 7.6.6.
N/2
dP
Figure 7.6.6: Plot of dt vs. P.
a. At what value of P is the rate of change greatest?
b. Consider the model for the earth’s population that we created. At what value
of P is the rate of change greatest? How does that compare to the population in
recent years?
c. According to the model we developed, what will the population be in the year
2100?
d. According to the model we developed, when will the population reach 9 billion?
e. Now consider the general solution to the general logistic initial value problem
that we found, given by
N
P(t) ( ) .
N−P0
P0 e −kN t + 1
Verify algebraically that P(0) P0 and that limt→∞ P(t) N.
7.6.3 Summary
• If we assume that the rate of growth of a population is proportional to the population,
we are led to a model in which the population grows without bound and at a rate that
grows without bound.
430
7.6 Population Growth and the Logistic Equation
• By assuming that the per capita growth rate decreases as the population grows, we
are led to the logistic model of population growth, which predicts that the population
will eventually stabilize at the carrying capacity.
7.6.4 Exercises
1. Analyzing a logistic equation. The slope field for a population P modeled by dP/dt
2P − 4P 2 is shown in the figure below.
(a) On a print-out of the slope field, sketch three non-zero solution curves showing
different types of behavior for the population P. Give an initial condition P(0) that will
produce each.
(b) Is there a stable value of the population? If so, give the value; if not, enter none:
(c) Considering the shape of solutions for the population, give any intervals for which
the following are true. Note that your answers may reflect the fact that P is a population.
P is increasing when P is in
P is decreasing when P is in
Think about what these conditions mean for the population, and be sure that you are
able to explain that.
In the long-run, what is the most likely outcome for the population?
P→
Are there any inflection points in the solutions for the population?
Be sure you can explain what the meaning of the inflection points is for the population.
(d) Sketch a graph of dP/dt against P. Use your graph to answer the following ques-
tions.
When is dP/dt positive?
When is dP/dt negative?
When is dP/dt zero?
When is dP/dt at a maximum?
Be sure that you can see how the shape of your graph of dP/dt explains the shape of
solution curves to the differential equation.
431
Chapter 7 Differential Equations
2. Analyzing a logistic model. The table below gives the percentage, P, of households
with a VCR, as a function of year.
Year 1978 1979 1980 1981 1982 1983 1984
P 0.3 0.5 1.1 1.8 3.1 5.5 10.6
Year 1985 1986 1987 1988 1989 1990 1991
P 20.8 36.0 48.7 58.0 64.6 71.9 71.9
(a) A logistic model is a good one to use for these data. Explain why this might be the
case: logically, how large would the growth in VCR ownership be when they are first
introduced? How large can the ownership ever be?
We can also investigate this by estimating the growth rate of P for the given data. Do
this at the beginning, middle, and near the end of the data:
P ′(1980) ≈
P ′(1985) ≈
P ′(1990) ≈
Be sure you can explain why this suggests that a logistic model is appropriate.
(b) Use the data to estimate the year when the point of inflection of P occurs.
What percent of households had VCRs then? P
What limiting value L does this point of inflection predict (note that if the logistic model
is reasonable, this prediction should agree with the data for 1990 and 1991)?
(c) The best logistic equation (solution to the logistic differential equation) for these
data turns out to be the following.
75
P .
1 + 316.75e −0.699t
What limiting value does this predict?
3. Finding a logistic function for an infection model. The total number of people infected
with a virus often grows like a logistic curve. Suppose that 25 people originally have
the virus, and that in the early stages of the virus (with time, t, measured in weeks),
the number of people infected is increasing exponentially with k 1.7. It is estimated
that, in the long run, approximately 6500 people become infected.
(a) Use this information to find a logistic function to model this situation.
(b) Sketch a graph of your answer to part (a). Use your graph to estimate the length of
time until the rate at which people are becoming infected starts to decrease. What is
the vertical coordinate at this point?
4. Analyzing a population growth model. Any population, P, for which we can ignore
immigration, satisfies
dP
Birth rate − Death rate.
dt
For organisms which need a partner for reproduction but rely on a chance encounter
432
7.6 Population Growth and the Logistic Equation
for meeting a mate, the birth rate is proportional to the square of the population. Thus,
the population of such a type of organism satisfies a differential equation of the form
dP
aP 2 − bP with a, b > 0.
dt
This problem investigates the solutions to such an equation.
(a) Sketch a graph of dP/dt against P. Note when dP/dt is positive and negative.
dP/dt < 0 when P is in
dP/dt > 0 when P is in
(Your answers may involve a and b. Give your answers as an interval or list of intervals.
(b) Use this graph to sketch the shape of solution curves with various initial values: use
your answers in part (a), and where dP/dt is increasing and decreasing to decide what
the shape of the curves has to be. Based on your solution curves, why is P b/a called
the threshold population?
If P(0) > b/a, what happens to P in the long run?
If P(0) b/a, what happens to P in the long run?
If P(0) < b/a, what happens to P in the long run?
5. The logistic equation may be used to model how a rumor spreads through a group of
people. Suppose that p(t) is the fraction of people that have heard the rumor on day t.
The equation
dp
0.2p(1 − p)
dt
describes how p changes. Suppose initially that one-tenth of the people have heard the
rumor; that is, p(0) 0.1.
a. What happens to p(t) after a very long time?
b. Determine a formula for the function p(t).
c. At what time is p changing most rapidly?
d. How long does it take before 80% of the people have heard the rumor?
6. Suppose that b(t) measures the number of bacteria living in a colony in a Petri dish,
where b is measured in thousands and t is measured in days. One day, you measure
that there are 6,000 bacteria and the per capita growth rate is 3. A few days later, you
measure that there are 9,000 bacteria and the per capita growth rate is 2.
db/dt
a. Assume that the per capita growth rate b is a linear function of b. Use the
measurements to find this function and write a logistic equation to describe db
dt .
b. What is the carrying capacity for the bacteria?
c. At what population is the number of bacteria increasing most rapidly?
d. If there are initially 1,000 bacteria, how long will it take to reach 80% of the car-
rying capacity?
433
Chapter 7 Differential Equations
7. Suppose that the population of a species of fish is controlled by the logistic equation
dP
0.1P(10 − P),
dt
where P is measured in thousands of fish and t is measured in years.
a. What is the carrying capacity of this population?
b. Suppose that a long time has passed and that the fish population is stable at the
carrying capacity. At this time, humans begin harvesting 20% of the fish every
year. Modify the differential equation by adding a term to incorporate the har-
vesting of fish.
c. What is the new carrying capacity?
d. What will the fish population be one year after the harvesting begins?
e. How long will it take for the population to be within 10% of the carrying capacity?
434
CHAPTER 8
Sequences and Series
8.1 Sequences
Motivating Questions
• What is a sequence?
• What does it mean for a sequence to converge?
• What does it mean for a sequence to diverge?
We encounter sequences every day. Your
monthly utility payments, the annual inter-
est you earn on investments, the amount
you spend on groceries each week; all are
examples of sequences. Other sequences
with which you may be familiar include the
Fibonacci sequence 1, 1, 2, 3, 5, 8, . . ., where
each term is the sum of the two pre-
ceding terms, and the triangular numbers
1, 3, 6, 10, 15, 21, 28, 36, 45, 55, . . ., the num-
ber of vertices in the triangles shown in Fig-
ure 8.1.1.
Figure 8.1.1: Triangular numbers
Sequences of integers are of such interest to mathematicians and others that they have a
journal¹ devoted to them and an on-line encyclopedia² that catalogs a huge number of integer
sequences and their connections. Sequences are also used in digital recordings and images.
Our studies in calculus have dealt with continuous functions. Sequences model discrete in-
stead of continuous information. We will study ways to represent and work with discrete
¹The Journal of Integer Sequences at http://www.cs.uwaterloo.ca/journals/JIS/
²The On-Line Encyclopedia of Integer Sequences at http://oeis.org/
Chapter 8 Sequences and Series
information in this chapter as we investigate sequences and series, and ultimately see key
connections between the discrete and continuous.
Preview Activity 8.1.1. Suppose you receive $5000 through an inheritance. You de-
cide to invest this money into a fund that pays 8% annually, compounded monthly.
That means that each month your investment earns 0.08 12 · P additional dollars, where
P is your principal balance at the start of the month. So in the first month your invest-
ment earns ( )
0.08
5000
12
or $33.33. If you reinvest this money, you will then have $5033.33 in your account at
the end of the first month. From this point on, assume that you reinvest all of the
interest you earn.
a. How much interest will you earn in the second month? How much money will
you have in your account at the end of the second month?
b. Complete Table 8.1.2 to determine the interest earned and total amount of money
in this investment each month for one year.
Month Interest Total amount
earned of money
in the account
0 $0.00 $5000.00
1 $33.33 $5033.33
2
3
4
5
6
7
8
9
10
11
12
Table 8.1.2: Interest
c. As we will see later, the amount of money Pn in the account after month n is
given by
( )n
0.08
Pn 5000 1 + .
12
Use this formula to check your calculations in Table 8.1.2. Then find the amount
of money in the account after 5 years.
436
8.1 Sequences
d. How many years will it be before the account has doubled in value to $10000?
8.1.1 Sequences
As Preview Activity 8.1.1 illustrates, many discrete phenomena can be represented as lists
of numbers (like the amount of money in an account over a period of months). We call any
such list a sequence. A sequence is nothing more than list of terms in some order. We often
list the entries of the sequence with subscripts,
s1 , s2 , . . . , s n . . . ,
where the subscript denotes the position of the entry in the sequence.
Definition 8.1.3 A sequence is a list of terms s 1 , s 2 , s 3 , . . . in a specified order.
We can think of a sequence as a function f whose domain is the set of positive integers
where f (n) s n for each positive integer n. This alternative view will be be useful in many
situations.
We often denote the sequence
s1 , s2 , s3 , . . .
by {s n }. The value s n (alternatively s(n)) is called the nth term in the sequence. If the terms
are all 0 after some fixed value of n, we say the sequence is finite. Otherwise the sequence is
infinite. With infinite sequences, we are often interested in their end behavior and the idea
of convergent sequences.
Activity 8.1.2.
a. Let s n be the nth term in the sequence 1, 2, 3, . . .. Find a formula for s n and use
appropriate technological tools to draw a graph of entries in this sequence by
plotting points of the form (n, s n ) for some values of n. Most graphing calcula-
tors can plot sequences; directions follow for the TI-84.
• In the MODE menu, highlight SEQ in the FUNC line and press ENTER.
• In the Y= menu, you will now see lines to enter sequences. Enter a value
for nMin (where the sequence starts), a function for u(n) (the nth term in
the sequence), and the value of u(nMin).
• Set your window coordinates (this involves choosing limits for n as well as
the window coordinates XMin, XMax, YMin, and YMax.
• The GRAPH key will draw a plot of your sequence.
Using your knowledge of limits of continuous functions as x → ∞, decide if
this sequence {s n } has a limit as n → ∞. Explain your reasoning.
b. Let s n be the nth term in the sequence 1, 12 , 13 , . . .. Find a formula for s n . Draw
a graph of some points in this sequence. Using your knowledge of limits of
continuous functions as x → ∞, decide if this sequence {s n } has a limit as n →
∞. Explain your reasoning.
437
Chapter 8 Sequences and Series
c. Let s n be the nth term in the sequence 2, 32 , 43 , 54 , . . .. Find a formula for s n . Us-
ing your knowledge of limits of continuous functions as x → ∞, decide if this
sequence {s n } has a limit as n → ∞. Explain your reasoning.
Next we formalize the ideas from Activity 8.1.2.
Activity 8.1.3.
a. Recall our earlier work with limits involving infinity in Section 2.8. State clearly
what it means for a continuous function f to have a limit L as x → ∞.
b. Given that an infinite sequence of real numbers is a function from the integers
to the real numbers, apply the idea from part (a) to explain what you think it
means for a sequence {s n } to have a limit as n → ∞.
{ 1+n }
c. Based on your response to the part (b), decide if the sequence 2+n has a limit
as n → ∞. If so, what is the limit? If not, why not?
In Activities 8.1.2 and 8.1.3 we investigated a sequence {s n } that has a limit as n goes to
infinity. More formally, we make the following definition.
Definition 8.1.4 A sequence {s n } converges or is a convergent sequence provided that there
is a number L such that we can make s n as close to L as we want by taking n sufficiently large.
In this situation, we call L the limit of the convergent sequence and write
lim s n L.
n→∞
If the sequence {s n } does not converge, we say that the sequence {s n } diverges.
The idea of sequence having a limit as n → ∞ is the same as the idea of a continuous
function having a limit as x → ∞. The only difference is that sequences are discrete instead
of continuous.
Activity 8.1.4. Use graphical and/or algebraic methods to determine whether each
of the following sequences converges or diverges.
{ 1+2n }
a. 3n−2
{ 5+3n
}
b. 10+2n
{ 10n }
c. n! (where ! is the factorial symbol and n! n(n − 1)(n − 2) · · · (2)(1) for any
positive integer n (as convention we define 0! to be 1)).
8.1.2 Summary
• A sequence is a list of objects in a specified order. We will typically work with se-
quences of real numbers. We can think of a sequence as a function from the positive
438
8.1 Sequences
integers to the set of real numbers.
• A sequence {s n } of real numbers converges to a number L if we can make every value
of s k for k ≥ n as close as we want to L by choosing n sufficiently large.
• A sequence diverges if it does not converge.
8.1.3 Exercises
1. Limits of five sequences. Match the formulas with the descriptions of the behavior of
the sequence as n → ∞.
(a) s n 1 + cos(n)/n A. converges to zero through positive and negative
numbers
(b) s n n(n + 1) − 1
B. does not converge, but doesn’t go to ±∞
(c) s n (n + 1)/n
C. diverges to ∞
(d) s n (sin(n)/n)
D. converges to one from above and below
(e) s n n sin(n)/(n + 1)
E. converges to one from above
2. Formula for a sequence, given first terms. Find a formula for s n , n ≥ 1 for the sequence
0, 3, 8, 15, 24 . . .
3. Divergent or convergent sequences. For each of the sequences below, enter either di-
verges if the sequence diverges, or the limit of the sequence if the sequence converges
as n → ∞.
4n+8
A. n :
B. 4n :
4n+8
C. n2
:
sin n
D. 4n :
4. Terms of a sequence from sampling a signal. In electrical engineering, a continuous
function like f (t) sin t, where t is in seconds, is referred to as an analog signal. To
digitize the signal, we sample f (t) every ∆t seconds to form the sequence s n f (n∆t).
For example, sampling f every 1/10 second produces the sequence sin(1/10), sin(2/10),
sin(3/10),...
Suppose that the analog signal is given by
sin(1t)
f (t) .
t
Give the first 6 terms of a sampling of the signal every ∆t 1.5 seconds.
5. Finding limits of convergent sequences can be a challenge. However, there is a useful
tool we can adapt from our study of limits of continuous functions at infinity to use to
439
Chapter 8 Sequences and Series
find limits of sequences. We illustrate in this exercise with the example of the sequence
ln(n)
.
n
a. Calculate the first 10 terms of this sequence. Based on these calculations, do you
think the sequence converges or diverges? Why?
b. For this sequence, there is a corresponding continuous function f defined by
ln(x)
f (x) .
x
Draw the graph of f (x) on the interval [0, 10] and then plot the entries of the
sequence {on the
} graph. What conclusion do you think we can draw about the
ln(n)
sequence n if limx→∞ f (x) L? Explain.
c. Note that f (x) has the indeterminate form ∞∞ as x goes to infinity. What idea from
differential calculus can we use to calculate limx→∞ f (x)? Use this method to find
ln(n)
limx→∞ f (x). What, then, is limn→∞ n ?
6. We return to the example begun in Preview Activity 8.1.1 to see how to derive the for-
mula for the amount of money in an account at a given time. We do this in a general
setting. Suppose you invest P dollars (called the principal) in an account paying r% in-
r
terest compounded monthly. In the first month you will receive 12 (here r is in decimal
0.08
form; e.g., if we have 8% interest, we write 12 ) of the principal P in interest, so you
earn ( r )
P
12
dollars in interest. Assume that you reinvest all interest. Then at the end of the first
month your account will contain the original principal P plus the interest, or a total of
( r ) ( r )
P1 P + P P 1+
12 12
dollars.
a. Given that your principal is now P1 dollars, how much interest will you earn in
the second month? If P2 is the total amount of money in your account at the end
of the second month, explain why
( r ) ( r )2
P2 P1 1 + P 1+ .
12 12
b. Find a formula for P3 , the total amount of money in the account at the end of the
third month in terms of the original investment P.
c. There is a pattern to these calculations. Let Pn the total amount of money in the
account at the end of the third month in terms of the original investment P. Find
a formula for Pn .
440
8.1 Sequences
7. Sequences have many applications in mathematics and the sciences. In a recent paper³
the authors write
The incretin hormone glucagon-like peptide-1 (GLP-1) is capable of amelio-
rating glucose-dependent insulin secretion in subjects with diabetes. How-
ever, its very short half-life (1.5-5 min) in plasma represents a major limita-
tion for its use in the clinical setting.
The half-life of GLP-1 is the time it takes for half of the hormone to decay in its medium.
For this exercise, assume the half-life of GLP-1 is 5 minutes. So if A is the amount of
GLP-1 in plasma at some time t, then only A2 of the hormone will be present after t + 5
minutes. Suppose A0 100 grams of the hormone are initially present in plasma.
a. Let A1 be the amount of GLP-1 present after 5 minutes. Find the value of A1 .
b. Let A2 be the amount of GLP-1 present after 10 minutes. Find the value of A2 .
c. Let A3 be the amount of GLP-1 present after 15 minutes. Find the value of A3 .
d. Let A4 be the amount of GLP-1 present after 20 minutes. Find the value of A4 .
e. Let A n be the amount of GLP-1 present after 5n minutes. Find a formula for A n .
f. Does the sequence {A n } converge or diverge? If the sequence converges, find its
limit and explain why this value makes sense in the context of this problem.
g. Determine the number of minutes it takes until the amount of GLP-1 in plasma
is 1 gram.
8. Continuous data is the basis for analog information, like music stored on old cassette
tapes or vinyl records. A digital signal like on a CD or MP3 file is obtained by sam-
pling an analog signal at some regular time interval and storing that information. For
example, the sampling rate of a compact disk is 44,100 samples per second. So a digital
recording is only an approximation of the actual analog information. Digital infor-
mation can be manipulated in many useful ways that allow for, among other things,
noisy signals to be cleaned up and large collections of information to be compressed
and stored in much smaller space. While we won’t investigate these techniques in this
chapter, this exercise is intended to give an idea of the importance of discrete (digital)
techniques.
Let f be the continuous function defined by f (x) sin(4x) on the interval [0, 10]. A
graph of f is shown in Figure 8.1.5.
³Hui H, Farilla L, Merkel P, Perfetti R. The short half-life of glucagon-like peptide-1 in plasma does not reflect
its long-lasting beneficial effects, Eur J Endocrinol 2002 Jun;146(6):863-9.
441
Chapter 8 Sequences and Series
1.0
0.5
x
2 4 6 8 10
-0.5
-1.0
Figure 8.1.5: The graph of f (x) sin(4x) on the interval [0, 10]
We approximate f by sampling, that is by partitioning the interval [0, 10] into uniform
subintervals and recording the values of f at the endpoints.
a. Ineffective sampling can lead to several problems in reproducing the original sig-
nal. As an example, partition the interval [0, 10] into 8 equal length subintervals
and create a list of points (the sample) using the endpoints of each subinterval.
Plot your sample on graph of f in Figure Figure 8.1.5. What can you say about
the period of your sample as compared to the period of the original function?
b. The sampling rate is the number of samples of a signal taken per second. As
the part (a) illustrates, sampling at too small a rate can cause serious problems
with reproducing the original signal (this problem of inefficient sampling lead-
ing to an inaccurate approximation is called aliasing). There is an elegant theorem
called the Nyquist-Shannon Sampling Theorem that says that human perception
is limited, which allows that replacement of a continuous signal with a digital
one without any perceived loss of information. This theorem also provides the
lowest rate at which a signal can be sampled (called the Nyquist rate) without
such a loss of information. The theorem states that we should sample at double
the maximum desired frequency so that every cycle of the original signal will be
sampled at at least two points. Recall that the frequency of a sinusoidal function
is the reciprocal of the period. Identify the frequency of the function f and de-
termine the number of partitions of the interval [0, 10] that give us the Nyquist
rate.
c. Humans cannot typically pick up signals above 20 kHz. Explain why, then, that
information on a compact disk is sampled at 44,100 Hz.
442
8.2 Geometric Series
8.2 Geometric Series
Motivating Questions
• What is a geometric series?
• What is a partial sum of a geometric series? What is a simplified form of the nth
partial sum of a geometric series?
• Under what conditions does a geometric series converge? What is the sum of a con-
vergent geometric series?
Many important sequences are generated by addition. In Preview Activity 8.2.1, we see an
example of a sequence that is connected to a sum.
Preview Activity 8.2.1. Warfarin is an anticoagulant that prevents blood clotting; of-
ten it is prescribed to stroke victims in order to help ensure blood flow. The level of
warfarin has to reach a certain concentration in the blood in order to be effective.
Suppose warfarin is taken by a particular patient in a 5 mg dose each day. The drug is
absorbed by the body and some is excreted from the system between doses. Assume
that at the end of a 24 hour period, 8% of the drug remains in the body. Let Q(n) be
the amount (in mg) of warfarin in the body before the (n + 1)st dose of the drug is
administered.
a. Explain why Q(1) 5 × 0.08 mg.
b. Explain why Q(2) (5 + Q(1)) × 0.08 mg. Then show that
Q(2) (5 × 0.08) (1 + 0.08) mg.
c. Explain why Q(3) (5 + Q(2)) × 0.08 mg. Then show that
( )
Q(3) (5 × 0.08) 1 + 0.08 + 0.082 mg.
d. Explain why Q(4) (5 + Q(3)) × 0.08 mg. Then show that
( )
Q(4) (5 × 0.08) 1 + 0.08 + 0.082 + 0.083 mg.
e. There is a pattern that you should see emerging. Use this pattern to find a for-
mula for Q(n), where n is an arbitrary positive integer.
f. Complete Table 8.2.1 with values of Q(n) for the provided n-values (reporting
Q(n) to 10 decimal places). What appears to be happening to the sequence Q(n)
as n increases?
n 1 2 3 4 5 6 7 8 9 10
Q(n) 0.40
Table 8.2.1: Values of Q(n) for selected values of n
443
Chapter 8 Sequences and Series
8.2.1 Geometric Series
In Preview Activity 8.2.1 we encountered the sum
( )
(5 × 0.08) 1 + 0.08 + 0.082 + 0.083 + · · · + 0.08n−1
for the long-term level of Warfarin in the patient’s system. This sum has the form
a + ar + ar 2 + · · · + ar n−1 (8.2.1)
where a 5 × 0.08 and r 0.08. Such a sum is called a finite geometric series with ratio r.
Activity 8.2.2. Let a and r be real numbers (with r , 1) and let
S n a + ar + ar 2 + · · · + ar n−1 .
In this activity we will find a shortcut formula for S n that does not involve a sum of n
terms.
a. Multiply S n by r. What does the resulting sum look like?
b. Subtract rS n from S n and explain why
S n − rS n a − ar n . (8.2.2)
c. Solve equation (8.2.2) for S n to find a simple formula for S n that does not involve
adding n terms.
The sum of the terms of a sequence is called a series. We summarize the result of Activity 8.2.2
in the following way.
A finite geometric series S n is a sum of the form
S n a + ar + ar 2 + · · · + ar n−1 , (8.2.3)
where a and r are real numbers such that r , 1. The finite geometric series S n can be
written more simply as
a(1 − r n )
S n a + ar + ar 2 + · · · + ar n−1 . (8.2.4)
1−r
We now apply Equation (8.2.4) to the example involving Warfarin from Preview Activ-
ity 8.2.1. Recall that
( )
Q(n) (5 × 0.08) 1 + 0.08 + 0.082 + 0.083 + · · · + 0.08n−1 mg,
so Q(n) is a geometric series with a 5 × 0.08 0.4 and r 0.08. Thus,
( )
1 − 0.08n 1
Q(n) 0.4 (1 − 0.08n ) .
1 − 0.08 2.3
444
8.2 Geometric Series
Notice that as n goes to infinity, the value of 0.08n goes to 0. So,
1 1
lim Q(n) lim (1 − 0.08n ) ≈ 0.435.
n→∞ n→∞ 2.3 2.3
1
Therefore, the long-term level of Warfarin in the blood under these conditions is 2.3 , which
is approximately 0.435 mg.
To determine the long-term effect of Warfarin, we considered a finite geometric series of n
terms, and then considered what happened as n was allowed to grow without bound. In
this sense, we were actually interested in an infinite geometric series (the result of letting n
go to infinity in the finite sum).
Definition 8.2.2 An infinite geometric series is an infinite sum of the form
∑
∞
a + ar + ar 2 + · · · ar n . (8.2.5)
n0
The value of r in the geometric series (8.2.5) is called the common ratio of the series because
the ratio of the (n + 1)st term, ar n , to the nth term, ar n−1 , is always r:
ar n
r.
ar n−1
Geometric series are common in mathematics and arise naturally in many different situa-
tions. As a familiar example, suppose we want to write the number with repeating decimal
expansion
N 0.121212
as a rational number. Observe that
N 0.12 + 0.0012 + 0.000012 + · · ·
( ) ( )( ) ( )( )2
12 12 1 12 1
+ + + ···.
100 100 100 100 100
This is an infinite geometric series with a 12
100 and r 1
100 .
By using the formula for the value of a finite geometric sum, we can also develop a formula
for the value of an infinite geometric series. We explore this idea in the following activity.
Activity 8.2.3. Let r , 1 and a be real numbers and let
S a + ar + ar 2 + · · · ar n−1 + · · ·
be an infinite geometric series. For each positive integer n, let
S n a + ar + ar 2 + · · · + ar n−1 .
Recall that
1 − rn
Sn a .
1−r
445
Chapter 8 Sequences and Series
a. What should we allow n to approach in order to have S n approach S?
b. What is the value of limn→∞ r n for |r | > 1? for |r| < 1? Explain.
c. If |r | < 1, use the formula for S n and your observations in (a) and (b) to explain
why S is finite and find a resulting formula for S.
We can now find the value of the geometric series
( ) ( )( ) ( )( )2
12 12 1 12 1
N + + + ···.
100 100 100 100 100
Using a 12
100 and r 1
100 , we see that
( ) ( )
12 1 12 100 4
N .
100 1 − 100
1 100 99 33
The sum of a finite number of terms of an infinite geometric series is often called a partial
sum of the series. Thus,
∑
n−1
S n a + ar + ar 2 + · · · + ar n−1 ar k .
k0
∑∞
is called the nth partial sum of the series k0 ar k . We summarize our recent work with
geometric series as follows.
• An infinite geometric series is an infinite sum of the form
∑
∞
a + ar + ar 2 + · · · ar n , (8.2.6)
n0
where a and r are real numbers such that r , 0.
• The nth partial sum S n of an infinite geometric series is
S n a + ar + ar 2 + · · · + ar n−1 .
n
• If |r | < 1, then using the fact that S n a 1−r
1−r , it follows that the sum S of the
infinite geometric series (8.2.6) is
1 − rn a
S lim S n lim a
n→∞ n→∞ 1−r 1−r
446
8.2 Geometric Series
Activity 8.2.4. The formulas we have derived for an infinite geometric series and its
partial sum have assumed we begin indexing the sums at n 0. If instead we have
a sum that does not begin at n 0, we can factor out common terms and use the
established formulas. This process is illustrated in the examples in this activity.
a. Consider the sum
∑
∞ ( )k ( ) ( )2 ( )3
1 1 1 1
(2) (2) + (2) + (2) + ···.
3 3 3 3
k1
(1)
Remove the common factor of (2) 3 from each term and hence find the sum of
the series.
b. Next let a and r be real numbers with −1 < r < 1. Consider the sum
∑
∞
ar k ar 3 + ar 4 + ar 5 + · · · .
k3
Remove the common factor of ar 3 from each term and find the sum of the series.
c. Finally, we consider the most general case. Let a and r be real numbers with
−1 < r < 1, let n be a positive integer, and consider the sum
∑
∞
ar k ar n + ar n+1 + ar n+2 + · · · .
kn
Remove the common factor of ar n from each term to find the sum of the series.
8.2.2 Summary
• An infinite geometric series is an infinite sum of the form
∑
∞
ar k
k0
where a and r are real numbers and r , 0.
∑∞
• The nth partial sum of the geometric series k0 ar k is
∑
n−1
Sn ar k .
k0
A formula for the nth partial sum of a geometric series is
1 − rn
Sn a .
1−r
∑∞
If |r | < 1, the infinite geometric series k0 ar k has the finite sum a
1−r .
447
Chapter 8 Sequences and Series
8.2.3 Exercises
1. Seventh term of a geometric sequence. Find the 4th term of the geometric sequence
−1, −3.5, −12.25, ...
2. A geometric series. Find the sum of the series
2 2 2
2+ + + ... + n−1 + ....
7 49 7
3. A series that is not geometric. Determine the sum of the following series.
∞ ( n
∑ )
3 + 8n
12n
n1
4. Two sums of geometric sequences. Find the sum of each of the geometric series given
below. For the value of the sum, enter an expression that gives the exact value, rather
than entering an approximation.
A. −15 + 5 − 5
3 + 5
9 − 5
27 + 5
81 −···
17 ( ) n
∑ 1
B. 2
n4
5. There is an old question that is often used to introduce the power of geometric growth.
Here is one version. Suppose you are hired for a one month (30 days, working every
day) job and are given two options to be paid.
Option 1. You can be paid $500 per day or
Option 2. You can be paid 1 cent the first day, 2 cents the second day, 4 cents the third
day, 8 cents the fourth day, and so on, doubling the amount you are paid each
day.
a. How much will you be paid for the job in total under Option 1?
b. Complete Table 8.2.3 to determine the pay you will receive under Option 2 for the
first 10 days.
c. Find a formula for the amount paid on day n, as well as for the total amount paid
by day n. Use this formula to determine which option (1 or 2) you should take.
448
8.2 Geometric Series
Day Pay on this day Total amount paid to date
1 $0.01 $0.01
2 $0.02 $0.03
3
4
5
6
7
8
9
10
Table 8.2.3: Option 2 payments
6. Suppose you drop a golf ball onto a hard surface from a height h. The collision with the
ground causes the ball to lose energy and so it will not bounce back to its original height.
The ball will then fall again to the ground, bounce back up, and continue. Assume that
at each bounce the ball rises back to a height 34 of the height from which it dropped.
Let h n be the height of the ball on the nth bounce, with h0 h. In this exercise we will
determine the distance traveled by the ball and the time it takes to travel that distance.
a. Determine a formula for h1 in terms of h.
b. Determine a formula for h2 in terms of h.
c. Determine a formula for h3 in terms of h.
d. Determine a formula for h n in terms of h.
e. Write an infinite series that represents the total distance traveled by the ball. Then
determine the sum of this series.
f. Next, let’s determine the total amount of time the ball is in the air.
i) When the ball is dropped from a height H, if we assume the only force acting
on it is the acceleration due to gravity, then the height of the ball at time t is
given by
1
H − 1t 2 .
2
Use this formula to determine the time it takes for the ball to hit the ground
after being dropped from height H.
ii) Use your work in the preceding item, along with that in (a)-(e) above to de-
termine the total amount of time the ball is in the air.
7. Suppose you play a game with a friend that involves rolling a standard six-sided die.
Before a player can participate in the game, he or she must roll a six with the die. As-
sume that you roll first and that you and your friend take alternate rolls. In this exercise
we will determine the probability that you roll the first six.
a. Explain why the probability of rolling a six on any single roll (including your first
turn) is 16 .
449
Chapter 8 Sequences and Series
b. If you don’t roll a six on your first turn, then in order for you to roll the first six on
your second turn, both you and your friend had to fail to roll a six on your first
turns, and then you had to succeed in rolling a six on your second turn. Explain
why the probability of this event is
( )( )( ) ( )2 ( )
5 5 1 5 1
.
6 6 6 6 6
c. Now suppose you fail to roll the first six on your second turn. Explain why the
probability is
( ) ( ) ( ) ( ) ( ) ( )4 ( )
5 5 5 5 1 5 1
6 6 6 6 6 6 6
that you to roll the first six on your third turn.
d. The probability of you rolling the first six is the probability that you roll the first
six on your first turn plus the probability that you roll the first six on your second
turn plus the probability that your roll the first six on your third turn, and so on.
Explain why this probability is
( )2 ( ) ( )4 ( )
1 5 1 5 1
+ + + ···.
6 6 6 6 6
Find the sum of this series and determine the probability that you roll the first
six.
8. The goal of a federal government stimulus package is to positively affect the economy.
Economists and politicians quote numbers like “k million jobs and a net stimulus to
the economy of n billion of dollars.” Where do they get these numbers? Let’s consider
one aspect of a stimulus package: tax cuts. Economists understand that tax cuts or
rebates can result in long-term spending that is many times the amount of the rebate.
For example, assume that for a typical person, 75% of her entire income is spent (that
is, put back into the economy). Further, assume the government provides a tax cut or
rebate that totals P dollars for each person.
a. The tax cut of P dollars is income for its recipient. How much of this tax cut will
be spent?
b. In this simple model, we will say that the spent portion of the tax cut/rebate from
part (a) then becomes income for another person who, in turn, spends 75% of this
income. After this ``second round” of spent income, how many total dollars have
been added to the economy as a result of the original tax cut/rebate?
c. This second round of spending becomes income for another group who spend
75% of this income, and so on. In economics this is called the multiplier effect. Ex-
plain why an original tax cut/rebate of P dollars will result in multiplied spend-
ing of
0.75P(1 + 0.75 + 0.752 + · · · ).
dollars.
d. Based on these assumptions, how much stimulus will a 200 billion dollar tax cut/
450
8.2 Geometric Series
rebate to consumers add to the economy, assuming consumer spending remains
consistent forever.
9. Like stimulus packages, home mortgages and foreclosures also impact the economy. A
problem for many borrowers is the adjustable rate mortgage, in which the interest rate
can change (and usually increases) over the duration of the loan, causing the monthly
payments to increase beyond the ability of the borrower to pay. Most financial analysts
recommend fixed rate loans, ones for which the monthly payments remain constant
throughout the term of the loan. In this exercise we will analyze fixed rate loans.
When most people buy a large ticket item like car or a house, they have to take out a loan
to make the purchase. The loan is paid back in monthly installments until the entire
amount of the loan, plus interest, is paid. With a loan, we borrow money, say P dollars
(called the principal), and pay off the loan at an interest rate of r%. To pay back the
loan we make regular monthly payments, some of which goes to pay off the principal
and some of which is charged as interest. In most cases, the interest is computed based
on the amount of principal that remains at the beginning of the month. We assume a
fixed rate loan, that is one in which we make a constant monthly payment M on our
loan, beginning in the original month of the loan.
Suppose you want to buy a house. You have a certain amount of money saved to make
a down payment, and you will borrow the rest to pay for the house. Of course, for the
privilege of loaning you the money, the bank will charge you interest on this loan, so
the amount you pay back to the bank is more than the amount you borrow. In fact, the
amount you ultimately pay depends on three things: the amount you borrow (called
the principal), the interest rate, and the length of time you have to pay off the loan plus
interest (called the duration of the loan). For this example, we assume that the interest
rate is fixed at r%.
To pay off the loan, each month you make a payment of the same amount (called install-
ments). Suppose we borrow P dollars (our principal) and pay off the loan at an interest
rate of r% with regular monthly installment payments of M dollars. So in month 1 of
the loan, before we make any payments, our principal is P dollars. Our goal in this
exercise is to find a formula that relates these three parameters to the time duration of
the loan.
We are charged interest every month at an annual rate of r%, so each month we pay
r
12 % interest on the principal that remains. Given that the original principal is P dollars,
( )
we will pay 0.0r12 P dollars in interest on our first payment. Since (we )paid M dollars
in total for our first payment, the remainder of the payment (M − 12 r
P) goes to pay
down the principal. So the principal remaining after the first payment (let’s call it P1 )
is the original principal minus what we paid on the principal, or
( ( r ) ) ( r )
P1 P − M − P 1+ P − M.
12 12
As long as P1 is positive, we still have to keep making payments to pay off the loan.
a. Recall that the amount of interest we pay each time depends on the principal
that remains. How much interest, in terms of P1 and r, do we pay in the second
installment?
451
Chapter 8 Sequences and Series
b. How much of our second monthly installment goes to pay off the principal? What
is the principal P2 , or the balance of the loan, that we still have to pay off after
making the second installment of the loan? Write your response in the form P2
( )P1 − ( )M, where you fill in the parentheses.
( )
r 2
[ ( )]
c. Show that P2 1 + 12 P− 1+ 1+ r
12 M.
d. Let P3 be the amount of principal that remains after the third installment. Show
that ( [ ]
r )3 ( r ) ( r )2
P3 1 + P− 1+ 1+ + 1+ M.
12 12 12
e. If we continue in the manner described in the problems above, then the remaining
principal of our loan after n installments is
[ ]
( r )n
n−1 (
∑ r )k
Pn 1 + P− 1+ M. (8.2.7)
12 12
k0
This is a rather complicated formula and one that is difficult to use. However,
we can simplify the sum if we recognize part of it as a partial sum of a geometric
series. Find a formula for the sum
n−1 (
∑ r )k
1+ . (8.2.8)
12
k0
and then a general formula for Pn that does not involve a sum.
f. It is usually more convenient to write our formula for Pn in terms of years rather
than months. Show that P(t), the principal remaining after t years, can be written
as ( )
12M ( r ) 12t 12M
P(t) P − 1+ + . (8.2.9)
r 12 r
g. Now that we have analyzed the general loan situation, we apply formula (8.2.9)
to an actual loan. Suppose we charge $1,000 on a credit card for holiday expenses.
If our credit card charges 20% interest and we pay only the minimum payment of
$25 each month, how long will it take us to pay off the $1,000 charge? How much
in total will we have paid on this $1,000 charge? How much total interest will we
pay on this loan?
h. Now we consider larger loans, e.g., automobile loans or mortgages, in which we
borrow a specified amount of money over a specified period of time. In this situa-
tion, we need to determine the amount of the monthly payment we need to make
to pay off the loan in the specified amount of time. In this situation, we need to
find the monthly payment M that will take our outstanding principal to 0 in the
specified amount of time. To do so, we want to know the value of M that makes
P(t) 0 in formula (8.2.9). If we set P(t) 0 and solve for M, it follows that
( )
r 12t
rP 1 + 12
M (( ) ).
r 12t
12 1+ 12 − 1
452
8.2 Geometric Series
i) Suppose we want to borrow $15,000 to buy a car. We take out a 5 year loan
at 6.25%. What will our monthly payments be? How much in total will we
have paid for this $15,000 car? How much total interest will we pay on this
loan?
ii) Suppose you charge your books for winter semester on your credit card. The
total charge comes to $525. If your credit card has an interest rate of 18% and
you pay $20 per month on the card, how long will it take before you pay off
this debt? How much total interest will you pay?
iii) Say you need to borrow $100,000 to buy a house. You have several options
on the loan:
• 30 years at 6.5%
• 25 years at 7.5%
• 15 years at 8.25%.
a. What are the monthly payments for each loan?
b. Which mortgage is ultimately the best deal (assuming you can afford
the monthly payments)? In other words, for which loan do you pay the
least amount of total interest?
453
Chapter 8 Sequences and Series
8.3 Series of Real Numbers
Motivating Questions
• What is an infinite series?
• What is the nth partial sum of an infinite series?
• How do we add up an infinite number of numbers? In other words, what does it
mean for an infinite series of real numbers to converge?
• What does it mean for an infinite series of real numbers to diverge?
In Section 8.2, we encountered infinite geometric series. For example, by writing
12 12 1 12 1
N 0.1212121212 · · · + · + · +···
100 100 100 100 1002
as a geometric series, we found a way to write N as a single fraction: N 33
4
. In this section,
we explore other types of infinite sums. In Preview Activity 8.3.1 we see how one such sum
is related to the famous number e.
Preview Activity 8.3.1. Have you ever wondered how your calculator can produce a
numeric approximation for complicated numbers like e, π or ln(2)? After all, the only
operations a calculator can really perform are addition, subtraction, multiplication,
and division, the operations that make up polynomials. This activity provides the first
steps in understanding how this process works. Throughout the activity, let f (x) e x .
a. Find the tangent line to f at x 0 and use this linearization to approximate
e. That is, find a formula L(x) for the tangent line, and compute L(1), since
L(1) ≈ f (1) e.
b. The linearization of e x does not provide a good approximation to e since 1 is
not very close to 0. To obtain a better approximation, we alter our approach
a bit. Instead of using a straight line to approximate e, we put an appropriate
bend in our estimating function to make it better fit the graph of e x for x close
to 0. With the linearization, we had both f (x) and f ′(x) share the same value as
the linearization at x 0. We will now use a quadratic approximation P2 (x) to
f (x) e x centered at x 0 which has the property that P2 (0) f (0), P2′ (0)
f ′(0), and P2′′(0) f ′′(0).
i) Let P2 (x) 1 + x + x2 . Show that P2 (0) f (0), P2′ (0) f ′(0), and P2′′(0)
2
f ′′(0). Then, use P2 (x) to approximate e by observing that P2 (1) ≈ f (1).
ii) We can continue approximating e with polynomials of larger degree whose
higher derivatives agree with those of f at 0. This turns out to make the
polynomials fit the graph of f better for more values of x around 0. For
example, let P3 (x) 1 + x + x2 + x6 . Show that P3 (0) f (0), P3′ (0) f ′(0),
2 3
′′ ′′ ′′′ ′′′
P3 (0) f (0), and P3 (0) f (0). Use P3 (x) to approximate e in a way
similar to how you did so with P2 (x) above.
454
8.3 Series of Real Numbers
8.3.1 Infinite Series
Preview Activity 8.3.1 shows that an approximation to e using a linear polynomial is 2, an
approximation to e using a quadratic polynomial is 2.5, and an approximation using a cubic
polynomial is 2.6667. If we continue this process we can obtain approximations from quartic
(degree 4) and quintic (degree 5) polynomials, giving us the following approximations to e:
linear 1+1 2
quadratic 1+1+ 1
2 2.5
cubic 1+1+ 1
2 + 1
6 2.6
quartic 1+1+ 1
2 + 1
6 + 1
24 2.7083
quintic 1+1+ 1
2 + 1
6 + 1
24 + 1
120 2.716
We see an interesting pattern here. The number e is being approximated by the sum
1 1 1 1 1
1+1+ + + + +···+ (8.3.1)
2 6 24 120 n!
for increasing values of n.
We can use summation notation as a shorthand¹ for writing the sum in Equation (8.3.1) to
get
1 1 1 1 1 ∑n
1
e ≈1+1+ + + + +···+ . (8.3.2)
2 6 24 120 n! k!
k0
We can calculate this sum using as large an n as we want, and the larger n is the more
accurate the approximation (8.3.2) is. This suggests that we can write the number e as the
infinite sum
∑∞
1
e . (8.3.3)
k!
k0
This sum is an example of } infinite series. Note that the series (8.3.3) is the sum of the terms
{ 1an
of the infinite sequence n! . In general, we use the following notation and terminology.
Definition 8.3.1 An infinite series of real numbers is the sum of the entries in an infinite
sequence of real numbers. In other words, an infinite series is sum of the form
∑
∞
a1 + a2 + · · · + a n + · · · ak ,
k1
where a1 , a2 , . . ., are real numbers.
We use summation notation to identify a series. If the series adds the entries of a sequence
{a n }n≥1 , we will write the series as ∑
ak
k≥1
or
∑
∞
ak .
k1
¹Note that 0! appears in Equation (8.3.2). By definition, 0! 1.
455
Chapter 8 Sequences and Series
Each of these notations is simply shorthand for the infinite sum a1 + a 2 + · · · + a n + · · ·.
Is it even possible to sum an infinite list of numbers? That doing so is possible in some situa-
tions shouldn’t come as a surprise. We have already examined the special case of geometric
series in the previous section. Moreover, the definite integral is defineed as the limit of a
sequence of finite sums, which provides insight into how we’ll think of infinite sums gener-
ally. As we investigate other infinite sums, there are two key questions we seek to answer:
(1) is the sum finite? and (2) if yes, what is its value?
Activity 8.3.2. Consider the series
∑
∞
1
.
k2
k1
While it is physically impossible to add an infinite collection of numbers, we can, of
course, add any finite collection of them. In what follows, we investigate how under-
standing how to find the nth partial sum (that is, the sum of the first n terms) enables
us to make sense of the infinite sum.
a. Sum the first two numbers in this series. That is, find a numeric value for
∑
2
1
k2
k1
b. Next, add the first three numbers in the series.
c. Continue adding terms in this series to complete the list below. Carry each sum
to at least 8 decimal places.
∑
1
1 ∑
4
1 ∑
7
1 ∑
10
1
1
k2 k2 k2 k2
k1 k1 k1 k1
∑2
1 ∑5
1 ∑8
1
k2 k2 k2
k1 k1 k1
∑
3
1 ∑
6
1 ∑
9
1
k2 k2 k2
k1 k1 k1
∑
d. The sums in the table in part c form a sequence whose nth term is S n nk1 k12 .
Based on your calculations in the table, do you think the sequence {S n } con-
verges or diverges?
∑ Explain. How do you think this sequence {S n } is related to
the series ∞ 1
k1 k 2 ?
The example in Activity 8.3.2 illustrates how we define the sum of an infinite series. We
construct a new sequence of numbers (called the sequence of partial sums) where the nth
term in the sequence consists of the sum of the first n terms of the series. If this sequence
converges, the corresponding infinite series is said to converge, and we say that we can find
456
8.3 Series of Real Numbers
the sum of the series. More formally, we have the following definition.
∑∞ ∑n
Definition 8.3.2 The nth partial sum of the series k1 a k is the finite sum S n k1 ak .
In other words, the nth partial sum S n of a series is the sum of the first n terms in the series,
Sn a1 + a2 + · · · + a n .
We investigate the behavior of the series by examining the sequence
S1 , S2 , . . . , S n , . . .
of its partial sums. If the sequence of partial sums converges to some finite number, we say
that the corresponding series converges. Otherwise, we say the series diverges. From our work
in Activity 8.3.2, the series
∑∞
1
k2
k1
appears to converge to some number near 1.54977. We formalize the concept of convergence
and divergence of an infinite series in the following definition.
Definition 8.3.3 The infinite series
∑
∞
ak
k1
converges (or is convergent) if the sequence {S n } of partial sums converges, where
∑
n
Sn ak .
k1
∑∞
If limn→∞ S n S, then we call S the sum of the series k1 a k . That is,
∑
∞
a k lim S n S.
n→∞
k1
If the sequence of partial sums does not converge, then the series
∑
∞
ak
k1
diverges (or is divergent).
The early terms in a series do not influence whether or not the series converges or diverges.
Rather, the convergence or divergence of a series
∑
∞
ak
k1
457
Chapter 8 Sequences and Series
is determined by what happens to the terms a k for very large values of k. To see why, suppose
that m is some constant larger than 1. Then
∑
∞ ∑
∞
a k (a 1 + a2 + · · · + a m ) + ak .
k1 km+1
∞ ∑
1 + a 2 + · · · + a m is a finite number, the series
Since a∑ k1 a k will converge if and only if the
∞
series km+1 a k converges. Because the starting index of the series doesn’t affect whether
the series converges or diverges, we will often just write
∑
ak
when we are interested in questions of convergence/divergence as opposed to the exact sum
of a series.
In Section 8.2 we encountered ∑the special family of infinite geometric series. Recall that a
geometric series has the form ∞ k0 ar , where a and r are real numbers (and r , 1). We
k
found that the nth partial sum S n of a geometric series is given by the convenient formula
1 − rn
Sn ,
1−r
and thus a geometric series converges if |r| < 1. Geometric series diverge for all other values
of r.
It is generally a difficult question to determine if a given nongeometric series converges or
diverges. There are several tests we can use that we will consider in the following sections.
8.3.2 The Divergence Test
The first question we ask about any infinite series is usually “Does the series converge or
diverge?” There is a straightforward way to check that certain series diverge, and we explore
this test in the next activity.
∑
Activity 8.3.3. If the series a k converges, then an important result necessarily follows
regarding the sequence {a n }. This activity explores this result.
∑∞
Assume that the series k1 a k converges and has sum equal to L.
∑∞
a. What is the nth partial sum S n of the series k1 ak ?
∑∞
b. What is the (n − 1)st partial sum S n−1 of the series k1 ak ?
c. What is the difference
∑ between the nth partial sum and the (n − 1)st partial sum
of the series ∞ a
k1 k ?
∑
d. Since we are assuming that ∞k1 a k L, what does that tell us about limn→∞ S n ?
Why? What does that tell us about limn→∞ S n−1 ? Why?
458
8.3 Series of Real Numbers
e. Combine the results of the previous two parts of this activity to determine
lim a n lim (S n − S n−1 ).
n→∞ n→∞
The result of Activity 8.3.3 is the following important conditional statement:
∑∞
If the series k1 a k converges, then the sequence {a k } of kth terms converges to
0.
It is logically
∑ equivalent to say that if the sequence {a k } does not converge to 0, then the
series ∞ a
k1 k cannot converge. This statement is called the Divergence Test.
The Divergence Test.
∑
If limk→∞ a k , 0, then the series a k diverges.
Activity 8.3.4. Determine if the Divergence Test applies to the following series. If the
test does not apply, explain why. If the test does apply, what does it tell us about the
series?∑ ∑ ∑1
a. k
k+1 b. (−1)k c. k
Note well: be very careful with the Divergence Test. This test only tells us what happens to
a series if the terms of the corresponding sequence do not converge to 0. If the sequence of
the terms of the series does converge to 0, the Divergence Test does not apply: indeed, as we
will soon see, a series whose terms go to zero may either converge or diverge.
8.3.3 The Integral Test
∑
The Divergence Test settles the questions of divergence or convergence of ∑ series a k in
which limk→∞ a k , 0. Determining the convergence or divergence of series a k in which
limk→∞ a k 0 turns out to be more complicated. Often, we have to investigate the sequence
of partial sums or apply some other technique.
Next, we consider the harmonic series ²
∑
∞
1
.
k
k1
The first 9 partial sums of this series are shown following.
∑
1
1 ∑
4
1 ∑
7
1
1 2.083333333 2.592857143
k k k
k1 k1 k1
²This series is called harmonic because each term in the series after the first is the harmonic mean of the term
2ab
before it and the term after it. The harmonic mean of two numbers a and b is a+b . See “What’s Harmonic about
the Harmonic Series”, by David E. Kullman (in the College Mathematics Journal, Vol. 32, No. 3 (May, 2001), 201-203)
for an interesting discussion of the harmonic mean.
459
Chapter 8 Sequences and Series
∑
2
1 ∑
5
1 ∑
8
1
1.5 2.283333333 2.717857143
k k k
k1 k1 k1
∑
3
1 ∑
6
1 ∑
9
1
1.833333333 2.450000000 2.828968254
k k k
k1 k1 k1
∑
This information alone doesn’t seem to be enough to tell us if the series ∞ 1
k1 k converges or
diverges. The partial sums could eventually level off to some fixed number or continue to
∑
grow without bound. Even if we look at larger partial sums, such as 1000 n1 k ≈ 7.485470861,
1
the result isn’t particularly convincing one way or another. The Integral Test is one way to
determine whether or not the harmonic series converges, and we explore this test further in
the next activity.
∑
Activity 8.3.5. Consider the harmonic series ∞ 1
k1 k . Recall that the harmonic series
will converge provided
∑ that its sequence of partial sums converges. The nth partial
sum S n of the series ∞ 1
k1 k is
∑
n
1
Sn
k
k1
1 1 1
1+ + +···+
2 3( ) n( ) ( )
1 1 1
1(1) + (1) + (1) + · · · + (1) .
2 3 n
Through this last expression for S n , we can visualize this partial sum as a sum of areas
of rectangles with heights m1 and bases of length 1, as shown in Figure 8.3.4, which
uses the 9th partial sum.
1.00
ak
0.75
0.50
0.25
k
0.00
0 1 2 3 4 5 6 7 8 9 10
Figure 8.3.4: A picture of the 9th partial sum of the harmonic series as a sum of
areas of rectangles.
The graph of the continuous function f defined by f (x) 1
x is overlaid on this plot.
a. Explain how this picture represents a particular Riemann sum.
460
8.3 Series of Real Numbers
b. What is the definite integral that corresponds to the Riemann sum you consid-
ered in (a)?
c. Which is larger, the definite integral in (b), or the corresponding partial sum S9
of the series? Why?
d. If instead of considering the 9th partial sum, we consider∑ the 1nth partial sum,
and we let n go to infinity, we can then compare the series ∞
k1 k to the improper
∫∞
1
integral 1 x dx. Which of these quantities is larger? Why?
∫∞
e. Does the improper integral 1 x1 dx converge or diverge? What does that result,
∑
together with your work in (d), tell us about the series ∞ 1
k1 k ?
The ideas from Activity 8.3.5 hold more generally. Suppose that f is a continuous decreasing
∑
function and that a k f (k) for each value of k. Consider the corresponding series ∞k1 a k .
The partial sum
∑
n
Sn ak
k1
can always be viewed as a left hand Riemann sum of f (x), using rectangles with bases of
width 1 and heights given by the values a k . A representative picture is shown at left in
Figure 8.3.5. Since f is a decreasing function, we have that
∫ n
Sn > f (x) dx.
1
Taking the limit as n goes to infinity shows that
∑
∞ ∫ ∞
ak > f (x) dx.
k1 1
∫∞ ∑∞
Therefore, if the improper integral 1
f (x) dx diverges, so does the series k1 ak .
1.00 1.00
ak ak
0.75 0.75
0.50 0.50
0.25 0.25
k k
0.00 0.00
0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10
Figure 8.3.5: Comparing an improper integral to a series
461
Chapter 8 Sequences and Series
What’s more, if we look at the right hand Riemann sums of f on [1, n] as shown at right in
Figure 8.3.5, we see that
∫ ∞ ∑
∞
f (x) dx > ak .
1 k2
∫∞ ∑ ∑
So if 1 f (x) dx converges, then so does ∞k2 a k , which also means that the series
∞
k1 a k
also converges. Our preceding discussion has demonstrated the truth of the Integral Test.
The Integral Test.
Let f be a real valued function and assume f is decreasing and positive for all x larger
than some number c. Let a k f (k) for each positive integer k.
∫∞ ∑∞
a. If the improper integral c
f (x) dx converges, then the series k1 a k converges.
∫∞ ∑∞
b. If the improper integral c
f (x) dx diverges, then the series k1 a k diverges.
The Integral Test compares a given infinite series to a natural, corresponding improper in-
tegral and says that the infinite series and corresponding improper integral both have the
same convergence status. In the next activity, we apply the Integral Test to determine the
convergence or divergence of a class of important series.
1∑
Activity 8.3.6. The series k p are special series called p-series. We have already
seen that the p-series with p 1 (the harmonic series) diverges. We investigate the
behavior of other p-series in this activity.
∫∞ ∑∞
1 1
a. Evaluate the improper integral 1 x2
dx. Does the series k1 k 2 converge or
diverge? Explain.
∫∞
b. Evaluate the improper integral 1 x1p dx where p > 1. For which values of p
∑
can we conclude that the series ∞ 1
k1 k p converges?
∫∞
c. Evaluate the improper integral 1 x1p dx where p < 1. What does this tell us
∑
about the corresponding p-series ∞ 1
k1 k p ?
d. Summarize your work in this activity by completing the following statement.
∑∞ 1
The p-series k1 k p converges if and only if .
8.3.4 The Limit Comparison Test
The Integral Test allows us to determine the convergence of an entire family of series: the
p-series. However, we have seen that it is often difficult to integrate functions, so the Integral
Test is not one that we can use all of the time. In fact, even for a relatively simple series such
∑ 2 +1
as k 4k+2k+2 , the Integral Test is not an option. In what follows we develop a test that applies
to series of rational functions by comparing their behavior to the behavior of p-series.
462
8.3 Series of Real Numbers
k+1 ∑
Activity 8.3.7. Consider the series k 3 +2
. Since the convergence or divergence of
a series only depends on the behavior of the series for large values of k, we might
examine the terms of this series more closely as k gets large.
a. By computing the value of kk+1 3 +2 for k 100 and k 1000, explain why the terms
k+1 k
k 3 +2
are essentially k3
when k is large.
b. Let’s formalize our observations in (a) a bit more. Let a k k+1
k 3 +2
and b k k
k3
.
Calculate
ak
lim .
k→∞ b k
What does the value of the limit tell you about a k and b k for large values of k?
Compare your response from part (a).
∑
c. Does the series kk3 converge or diverge? Why? What do you think that tells us
∑
about the convergence or divergence of the series kk+1
3 +2 ? Explain.
Activity 8.3.7 illustrates how we can compare one series with∑positive ∑ terms to another
whose convergence status we know. Suppose we have two ∑ series a k and b k with positive
terms and we know the convergence status of the series a k . Recall that the convergence or
divergence of a series depends only on the terms of the series for large ∑ k, so if we
∑values of
know that a k and b k are proportional for large k, then the two series a k and b k should
behave the same way. In other words, if there is a positive finite constant c such that
bk
lim c,
k→∞ ak
then b k ≈ ca k for large values of k. So
∑ ∑ ∑
bk ≈ ca k c ak .
Since multiplying by a nonzero constant
∑ does
∑ not affect the convergence or divergence of
a series, it follows that the series a k and b k either both converge or both diverge. The
formal statement of this fact is called the Limit Comparison Test.
The Limit Comparison Test.
∑ ∑
Let a k and b k be series with positive terms. If
bk
lim c
k→∞ ak
∑ ∑
for some positive (finite) constant c, then a k and b k either both converge or both
diverge.
∑ p(k)
The Limit Comparison Test shows that if we have a series q(k)
of rational functions where
∑ p(k)
p(k) is a polynomial of degree m and q(k) a polynomial of degree l, then the series q(k)
will
∑ km
behave like the series k l . So this test allows us to determine the convergence or divergence
of series whose terms are rational functions.
463
Chapter 8 Sequences and Series
Activity 8.3.8. Use the Limit Comparison Test to determine the convergence or diver-
gence of the series
∑ 3k 2 + 1
.
5k 4 + 2k + 2
∑ 1
by comparing it to the series k2
.
8.3.5 The Ratio Test
The Limit Comparison Test works well if we can find a series with known behavior to com-
pare. But such series are not always easy to find. In this section we will examine a test that
allows us to examine the behavior of a series by comparing it to a geometric series, without
knowing in advance which geometric series we need.
Activity 8.3.9. Consider the series defined by
∑
∞
2k
. (8.3.4)
k1
3k − k
This series is not a geometric series, but this activity will illustrate
∑ how we might
compare this series to a geometric one. Recall that a series a k is geometric if the
k
ratio aak+1
k
is always the same. For the series in (8.3.4), note that a k 3k2−k .
∑ k
a. To see if 3k2−k is comparable to a geometric series, we analyze the ratios of
successive terms in the series. Complete Table 8.3.6, listing your calculations to
at least 8 decimal places.
k 5 10 20 21 22 23 24 25
a k+1
ak
∑ 2k
Table 8.3.6: Ratios of successive terms in the series 3k −k
a k+1
b. Based on your calculations in Table 8.3.6, what can we say about the ratio ak if
k is large?
∑ 2k
c. Do you agree or disagree with the statement: “the series 3k −k
is approximately
∑ 2k
geometric when k is large”? If not, why not? If so, do you think the series 3k −k
converges or diverges? Explain.
We
∑ can generalize the argument in Activity 8.3.9 in the following way. Consider the series
a k . If
a k+1
≈r
ak
464
8.3 Series of Real Numbers
∑
∑ k, kthen a k+1 ≈ ra k for large k and the series a k is approximately the
for large values of
geometric series ar for large k.∑Since the geometric series with ratio r converges only for
−1 < r < 1, we see that the series a k will converge if
a k+1
lim r
k→∞ ak
for a value of r such that |r| < 1. This result is known as the Ratio Test.
The Ratio Test.
∑
Let a k be an infinite series. Suppose
|a k+1 |
lim r.
k→∞ |a k |
∑
a. If 0 ≤ r < 1, then the series a k converges.
∑
b. If 1 < r, then the series a k diverges.
c. If r 1, then the test is inconclusive.
Note well: The Ratio Test looks at the limit of the ratio of consecutive terms of a given series;
in so doing, the test is asking, “is this series approximately geometric?” If so, the test uses
the limit of the ratio of consecutive terms to determine if the given series converges.
We have now encountered several tests for determining convergence or divergence of series.
• The Divergence Test can be used to show that a series diverges, but never to prove that
a series converges.
• We used the Integral Test to determine the convergence status of an entire class of
series, the p-series.
• The Limit Comparison Test works well for series that involve rational functions and
which can therefore by compared to p-series.
• Finally, the Ratio Test allows us to compare our series to a geometric series; it is partic-
ularly useful for series that involve nth powers and factorials.
• Two other tests, the Direct Comparison Test and the Root Test, are discussed in the
exercises.
One of the challenges of determining whether a series converges or diverges is finding which
test answers that question.
Activity 8.3.10. Determine whether each of the following series converges or diverges.
Explicitly
∑ k state which test you use. ∑ 10k
a. 2k
c. k!
∑ k 3 +2 ∑ k 3 −2k 2 +1
b. k 2 +1
d. k 6 +4
465
Chapter 8 Sequences and Series
8.3.6 Summary
• An infinite series is a sum of the elements in an infinite sequence. In other words, an
infinite series is a sum of the form
∑
∞
a1 + a2 + · · · + a n + · · · ak
k1
where a k is a real number for each positive integer k.
∑∞
• The nth partial sum S n of the series k1 a k is the sum of the first n terms of the series.
That is,
∑
n
Sn a1 + a2 + · · · + a n ak .
k1
∑∞
• The sequence {S n } of partial sums of a series k1 a k tells us about the convergence or
divergence of the series. In particular
∑
◦ The series ∞ k1 a k converges if the sequence {S n } of partial sums converges. In
this case we say that the series is the limit of the sequence of partial sums and
write
∑
∞
a k lim S n .
n→∞
k1
∑∞
◦ The series k1 a k diverges if the sequence {S n } of partial sums diverges.
8.3.7 Exercises
1. Convergence of a sequence and its series. Given:
90
An
9n
Determine:
∑
∞
(a) whether (A n ) is convergent.
n1
(b) whether {A n } is convergent.
If convergent, find the limit of convergence.
∑
∞
2
2. Two partial sums. Consider the series . Let s n be the n-th partial sum; that is,
n+7
n1
∑
n
2
sn .
i+7
i1
Find s4 and s8
466
8.3 Series of Real Numbers
3. Convergence of a series and its sequence. Let
3n
an .
6n + 13
Decide whether the given sequence or series is convergent or divergent. If convergent,
find the limit (for a sequence) or the sum (for a series).
∑
∞
3n
(a) The series .
6n + 13
n1
{ }
3n
(b) The sequence .
6n + 13
4. Convergence of an integral and a related series. Compute the value of the following
improper integral. ∫ ∞
3 dx
1 x2 + 1
∑
∞
4
Does the series converge or diverge?
n1
n2 + 1
{ bn }
5. In this exercise we investigate the sequence n! for any constant b.
∑ 10k
a. Use the Ratio Test to determine if the series k! converges or diverges.
∑ bk
b. Now apply the Ratio Test to determine if the series k! converges for any constant
b.
{ n }
c. Use your result from (b) to decide whether the sequence bn! converges or di-
{ n}
verges. If the sequence bn! converges, to what does it converge? Explain your
reasoning.
6. There is a test∑for convergence similar to the Ratio Test called the Root Test. Suppose we
have a series a k of positive terms so that a n → 0 as n → ∞.
a. Assume √
n
an → r
as n goes to infinity. Explain why this tells us that a n ≈ r n for large values of n.
∑
b. Using the result of part (a), explain why a k looks like a geometric
∑ series when
n is big. What is the ratio of the geometric series to which a k is comparable?
c. Use
∑ what we know about geometric series to determine that values of r so that
√
a k converges if n a n → r as n → ∞.
7. The associative and distributive
∑ laws ∑
of addition allow us to add finite sums in any
order we want. That is, if nk0 a k and nk0 b k are finite sums of real numbers, then
∑
n ∑
n ∑
n
ak + bk (a k + b k ).
k0 k0 k0
467
Chapter 8 Sequences and Series
However, we do need to be careful extending rules like this to infinite series.
a. Let a n 1 + 1
2n and b n −1 for each nonnegative integer n.
∑∞ ∑∞
i. Explain why the series k0 a k and k0 b k both diverge.
∑∞
ii. Explain why the series k0 (a k + b k ) converges.
iii. Explain why
∑
∞ ∑
∞ ∑
∞
ak + bk , (a k + b k ).
k0 k0 k0
∑∞
This
∑∞ shows that it is possible ∑ to have to two divergent series k0 a k and
∞
k0 b k but yet have the series k0 (a k + b k ) converge.
b. While part (a) shows that we cannot add series term by term in general, we can
under reasonable conditions. The problem in part ∑ (a) is that∑we tried to add di-
vergent series.
∑ In this exercise we will show that if a k and b k are convergent
series, then (a k + b k ) is a convergent series and
∑ ∑ ∑
(a k + b k ) ak + bk .
∑∞ ∑∞
i. Let A n and B n be the nth partial sums of the series k1 a k and k1 bk ,
respectively. Explain why
∑
n
A n + Bn (a k + b k ).
k1
ii. Use the previous result and properties of limits to show that
∑
∞ ∑
∞ ∑
∞
(a k + b k ) ak + bk .
k1 k1 k1
(Note that the starting point of the sum is irrelevant in this problem, so it
doesn’t matter where we begin the sum.)
∑∞ 2k +3k
c. Use the prior result to find the sum of the series k0 5k
.
8. In the Limit Comparison Test we compared the behavior of a series to one whose be-
havior we know. In that test we use the limit of the ratio of corresponding terms of
the series to determine if the comparison is valid. In this exercise we see how we can
compare two series directly, term by term, without using a limit of sequence. First we
consider an example.
a. Consider the series ∑ 1 ∑ 1
and .
k2 k2 + k
∑ ∑
We know that the series k12 is a p-series with p 2 > 1 and so k12 converges.
∑
In this part of the exercise we will see how to use information about k12 to de-
∑ 1
termine information about k 2 +k . Let a k k12 and b k k 21+k .
468
8.3 Series of Real Numbers
∑ 1 ∑ 1
i) Let S n be the nth partial sum of k2
and Tn the nth partial sum of k 2 +k
.
Which is larger, S1 or T1 ? Why?
ii) Recall that
S2 S1 + a2 and T2 T1 + b 2 .
Which is larger, a 2 or b 2 ? Based on that answer, which is larger, S2 or T2 ?
iii) Recall that
S3 S2 + a3 and T3 T2 + b 3 .
Which is larger, a 3 or b 3 ? Based on that answer, which is larger, S3 or T3 ?
iv) Which is larger, a n or b n ? Explain. Based on that answer, which is larger, S n
or Tn ?
v) Based on your response to the previous part
∑ 1of this∑exercise, what relation-
1
ship do you expect there to be between and ? Do you expect
∑ 1 k 2 k 2 +k
k 2 +k
to converge or diverge? Why?
b. The example in the previous part of this exercise illustrates a more general result.
Explain why the Direct Comparison Test, stated here, works.
The Direct Comparison Test.
If
0 ≤ bk ≤ ak
for every k, then we must have
∑ ∑
0≤ bk ≤ ak
∑ ∑
a. If a k converges, then b k converges.
∑ ∑
b. If b k diverges, then a k diverges.
Note 8.3.7 This comparison test applies only to series with nonnegative terms.
i) Use the Direct
∑Comparison Test to determine the convergence or divergence
1
of the series k−1 . Hint: Compare to the harmonic series.
ii) Use the Direct
∑Comparison Test to determine the convergence or divergence
of the series k 3k+1 .
469
Chapter 8 Sequences and Series
8.4 Alternating Series
Motivating Questions
• What is an alternating series?
• Under what conditions does an alternating series converge? Why?
• How well does the nth partial sum of a convergent alternating series approximate
the actual sum of the series? Why?
So far, we’ve considered series with exclusively nonnegative terms. Next, we consider series
that have some negative terms. For instance, the geometric series
( )n
4 8 2
2− + −···+2 − + ···,
3 9 3
has a 2 and r − 32 , so that every other term alternates in sign. This series converges to
a 2 6
S ( 2) .
1−r 1 − −3 5
In Preview Activity 8.4.1 and our following discussion, we investigate the behavior of similar
series where consecutive terms have opposite signs.
Preview Activity 8.4.1. Preview Activity 8.3.1 showed how we can approximate the
number e with linear, quadratic, and other polynomial approximations. We use a sim-
ilar approach in this activity to obtain linear and quadratic approximations to ln(2).
Along the way, we encounter a type of series that is different than most of the ones
we have seen so far. Throughout this activity, let f (x) ln(1 + x).
a. Find the tangent line to f at x 0 and use this linearization to approximate
ln(2). That is, find L(x), the tangent line approximation to f (x), and use the fact
that L(1) ≈ f (1) to estimate ln(2).
b. The linearization of ln(1 + x) does not provide a very good approximation to
ln(2) since 1 is not that close to 0. To obtain a better approximation, we alter our
approach; instead of using a straight line to approximate ln(2), we use a qua-
dratic function to account for the concavity of ln(1 + x) for x close to 0. With
the linearization, both the function’s value and slope agree with the lineariza-
tion’s value and slope at x 0. We will now make a quadratic approximation
P2 (x) to f (x) ln(1 + x) centered at x 0 with the property that P2 (0) f (0),
P2′ (0) f ′(0), and P2′′(0) f ′′(0).
i. Let P2 (x) x − x2 . Show that P2 (0) f (0), P2′ (0) f ′(0), and P2′′(0) f ′′(0).
2
Use P2 (x) to approximate ln(2) by using the fact that P2 (1) ≈ f (1).
ii. We can continue approximating ln(2) with polynomials of larger degree
whose derivatives agree with those of f at 0. This makes the polynomials
470
8.4 Alternating Series
fit the graph of f better for more values of x around 0. For example, let
P3 (x) x − x2 + x3 . Show that P3 (0) f (0), P3′ (0) f ′(0), P3′′(0) f ′′(0),
2 3
and P3′′′(0) f ′′′(0). Taking a similar approach to preceding questions, use
P3 (x) to approximate ln(2).
iii. If we used a degree 4 or degree 5 polynomial to approximate ln(1+x), what
approximations of ln(2) do you think would result? Use the preceding
questions to conjecture a pattern that holds, and state the degree 4 and
degree 5 approximation.
8.4.1 The Alternating Series Test
Preview Activity 8.4.1 gives us several approximations to ln(2). The linear approximation is
1, and the quadratic approximation is 1 − 12 21 . If we continue this process, cubic, quartic
(degree 4), quintic (degree 5), and higher degree polynomials give us the approximations to
ln(2) in Table 8.4.1.
linear 1 1
quadratic 1− 1
2 0.5
cubic 1− 1
2 + 1
3 0.83
quartic 1− 1
2 + 1
3 − 1
4 0.583
quintic 1− 1
2 + 1
3 − 1
4 + 1
5 0.783
Table 8.4.1
The pattern here shows that ln(2) can be approximated by the partial sums of the infinite
series
∑
∞
1
(−1)k+1 (8.4.1)
k
k1
where the alternating signs are indicated by the factor (−1)k+1 . We call such a series an
alternating series.
Using computational technology, we find that the sum of the first 100 terms in this series
is 0.6881721793. As a comparison, ln(2) ≈ 0.6931471806. This shows that even though the
series (8.4.1) converges to ln(2), it must do so quite slowly, since the sum of the first 100 terms
isn’t particularly close to ln(2). We will investigate the issue of how quickly an alternating
series converges later in this section.
Definition 8.4.2 An alternating series is a series of the form
∑
∞
(−1)k a k ,
k0
where a k > 0 for each k.
471
Chapter 8 Sequences and Series
We have some flexibility in how we write an alternating series; for example, the series
∑
∞
(−1)k+1 a k ,
k1
whose index starts at k 1, is also alternating. As we will soon see, there are several very
nice results that hold for alternating series, while alternating series can also demonstrate
some unusual behaivior.
It is important to remember that most of the series tests we have seen in previous sections
apply only to series with nonnegative terms. Alternating series require a different test.
Activity 8.4.2. Remember that, by definition, a series converges if and only if its cor-
responding sequence of partial sums converges.
a. Calculate the first few partial sums (to 10 decimal places) of the alternating series
∑
∞
1
(−1)k+1 .
k
k1
∑n
Label each partial sum with the notation S n k1 (−1)
k+1 1
k for an appropriate
choice of n.
b. Plot the sequence of partial sums from part (a). What do you notice about this
sequence?
Activity 8.4.2 illustrates the general behavior of any convergent alternating series. We see
that the partial sums of the alternating harmonic series oscillate around a fixed number that
turns out to be the sum of the series.
∑
Recall that if limk→∞ a k , 0, then the series a k diverges by the Divergence Test. From this
point forward, we will thus only consider alternating series
∑
∞
(−1)k+1 a k
k1
in which the sequence a k consists of positive numbers that decrease to 0. The nth partial
sum S n is
∑
n
Sn (−1)k+1 a k .
k1
Notice that
• S2 a1 − a2 , and since a1 > a 2 we have 0 < S2 < S1 .
• S3 S2 + a3 and so S2 < S3 . But a3 < a 2 , so S3 < S1 . Thus, 0 < S2 < S3 < S1 .
• S4 S3 − a4 and so S4 < S3 . But a4 < a 3 , so S2 < S4 . Thus, 0 < S2 < S4 < S3 < S1 .
• S5 S4 + a5 and so S4 < S5 . But a 5 < a4 , so S5 < S3 . Thus, 0 < S2 < S4 < S5 < S3 < S1 .
472
8.4 Alternating Series
This pattern continues as illustrated in Fig-
an
ure 8.4.3 (with n odd) so that each partial sum
lies between the previous two partial sums.
S2 S4 S6 . . . Sn−1 Sn S5 S3 S1
Figure 8.4.3: Partial sums of an
alternating series
Note further that the absolute value of the difference between the (n − 1)st partial sum S n−1
and the nth partial sum S n is
|S n − S n−1 | a n .
Because the sequence {a n } converges to 0, the distance between successive partial sums be-
comes as close to zero as we’d like, and thus the sequence of partial sums converges (even
though we don’t know the exact value to which it converges).
The preceding discussion has demonstrated the truth of the Alternating Series Test.
The Alternating Series Test.
∑
Given an alternating series (−1)k a k , if the sequence {a k } of positive terms decreases
to 0 as k → ∞, then the alternating series converges.
Note that if the limit of the sequence {a k } is not 0, then the alternating series diverges.
Activity 8.4.3. Which series converge and which diverge? Justify your answers.
∑∞
(−1)k ∑∞
(−1)k+1 2k ∑∞
(−1)k
a. b. c.
k2 + 2 k+5 ln(k)
k1 k1 k2
8.4.2 Estimating Alternating Sums
If the series converges, the argument for the Alternating Series Test also provides us with a
method to determine how close the nth partial sum S n is to the actual sum of the series. To
see how this works, let S be the sum of a convergent alternating series, so
∑
∞
S (−1)k a k .
k1
Recall that the sequence of partial sums oscillates around the sum S so that
|S − S n | < |S n+1 − S n | a n+1 .
Therefore, the value of the term a n+1 provides an error estimate for how well the partial
sum S n approximates the actual sum S. We summarize this fact in the statement of the
Alternating Series Estimation Theorem.
473
Chapter 8 Sequences and Series
Alternating Series Estimation Theorem.
∑∞ ∑
n
k1 (−1)
If the alternating series k+1 a converges and has sum S, and S (−1)k+1 a k
k n
k1
is the nth partial sum of the alternating series, then
∑
∞
(−1)k+1 a k − S n ≤ a n+1 .
k1
Example 8.4.4 Determine how well the 100th partial sum S100 of
∑
∞
(−1)k+1
k
k1
approximates the sum of the series.
∑∞ (−1)k+1
Solution. If we let S be the sum of the series k1 k , then we know that
|S100 − S| < a101 .
Now
1
a 101
≈ 0.0099,
101
so the 100th partial sum is within 0.0099 of the sum of the series. We have discussed the fact
(and will later verify) that
∑∞
(−1)k+1
S ln(2),
k
k1
and so S ≈ 0.693147 while
∑
100
(−1)k+1
S100 ≈ 0.6881721793.
k
k1
We see that the actual difference between S and S100 is approximately 0.0049750013, which
is indeed less than 0.0099.
Activity 8.4.4. Determine the number of terms it takes to approximate the sum of the
convergent alternating series
∑∞
(−1)k+1
k4
k1
to within 0.0001.
474
8.4 Alternating Series
8.4.3 Absolute and Conditional Convergence
A series such as
1 1 1 1 1 1 1 1 1
1− − + + + − − − − +··· (8.4.2)
4 9 16 25 36 49 64 81 100
whose terms are neither all nonnegative nor alternating is different from any series that we
have considered so far. The behavior of such a series can be rather complicated, but there
is an important connection between a series with some negative terms and series with all
positive terms.
Activity 8.4.5.
a. Explain why the series
1 1 1 1 1 1 1 1 1
1− − + + + − − − − +···
4 9 16 25 36 49 64 81 100
must have a sum that is less than the series
∑
∞
1
.
k2
k1
b. Explain why the series
1 1 1 1 1 1 1 1 1
1− − + + + − − − − +···
4 9 16 25 36 49 64 81 100
must have a sum that is greater than the series
∑
∞
1
− .
k2
k1
c. Given that the terms in the series
1 1 1 1 1 1 1 1 1
1− − + + + − − − − +···
4 9 16 25 36 49 64 81 100
converge to 0, what do you think the previous two results tell us about the con-
vergence status of this series?
∑ ∑
∑ if a series a k has some negative∑terms but |a k |
As the example in Activity 8.4.5 suggests,
converges, then
∑ the original series, a k , must also converge. That is, if |a k | converges,
then so must a k .
As we just observed, this is the case for the series (8.4.2), because
∑ the corresponding series
of the absolute values of its terms is the convergent p-series k12 . But there are series, such
∑
as the alternating harmonic series (−1)k+1 1k , that converge while the corresponding series
∑1
of absolute values, k , diverges. We distinguish between these behaviors by introducing
the following language.
475
Chapter 8 Sequences and Series
∑
Definition 8.4.5 Consider a series ak .
∑ ∑
a. The series a k converges absolutely (or is absolutely convergent) provided that |a k | con-
verges.
∑ ∑
b. The series a∑
k converges conditionally (or is conditionally convergent) provided that |a k |
diverges and a k converges.
In this terminology, the series (8.4.2) converges absolutely while the alternating harmonic
series is conditionally convergent.
Activity 8.4.6.
∑ ln(k)
a. Consider the series (−1)k k .
i. Does this series converge? Explain.
ii. Does this series converge absolutely? Explain what test you use to deter-
mine your answer.
∑ ln(k)
b. Consider the series (−1)k k2
.
i. Does this series converge? Explain.
√
ii. Does this series converge absolutely? Hint: Use the fact that ln(k) < k for
large values of k and then compare to an appropriate p-series.
Conditionally convergent series
∑ turn out to be very interesting. If the sequence ∑ {a n } de-
creases to 0, but the series a k diverges, the conditionally convergent series (−1)k a k is
right on the borderline of being a divergent series. As a result, any conditionally conver-
gent series converges very slowly. Furthermore, some very strange things can happen with
conditionally convergent series, as illustrated in some of the exercises.
8.4.4 Summary of Tests for Convergence of Series
We have discussed several tests for convergence/divergence of series in our sections and in
exercises. We close this section of the text with a summary of all the tests we have encoun-
tered, followed by an activity that challenges you to decide which convergence test to apply
to several different series.
∑
Geometric Series The geometric series ar k with ratio r converges for −1 < r < 1 and
diverges for |r| ≥ 1.
∑
∞
The sum of the convergent geometric series ar k is a
1−r .
k0
∑
Divergence Test If the sequence a n does not converge to 0, then the series a k diverges.
This is the first test to apply because the conclusion is simple. However, if limn→∞ a n
0, no conclusion can be drawn.
476
8.4 Alternating Series
Integral Test Let f be a positive, decreasing function on an interval [c, ∞) and let a k f (k)
for each positive integer k ≥ c.
∫∞ ∑
• If c
f (t) dt converges, then a k converges.
∫∞ ∑
• If c
f (t) dt diverges, then a k diverges.
Use this test when f (x) is easy to integrate.
Direct Comp. Test (see Ex 4 in Section 8.3)
Let 0 ≤ a k ≤ b k for each positive integer k.
∑ ∑
• If b k converges, then a k converges.
∑ ∑
• If a k diverges, then b k diverges.
Use this test when you have a series with known behavior that you can compare to —
this test can be difficult to apply.
Limit Comp. Test Let a n and b n be sequences of positive terms. If
ak
lim L
k→∞ bk
∑ ∑
for some positive finite number L, then the two series a k and b k either both con-
verge or both diverge.
Easier to apply in general than the comparison test, but you must have a series with
known behavior to compare. Useful to apply to series of rational functions.
Ratio Test Let a k , 0 for each k and suppose
|a k+1 |
lim r.
k→∞ |a k |
∑
• If r < 1, then the series a k converges absolutely.
∑
• If r > 1, then the series a k diverges.
• If r 1, then test is inconclusive.
This test is useful when a series involves factorials and powers.
Root Test (see Exercise 2 in Section 8.3)
Let a k ≥ 0 for each k and suppose
√k
lim a k r.
k→∞
∑
• If r < 1, then the series a k converges.
∑
• If r > 1, then the series a k diverges.
• If r 1, then test is inconclusive.
In general, the Ratio Test can usually be used in place of the Root Test. However, the
Root Test can be quick to use when a k involves kth powers.
477
Chapter 8 Sequences and Series
Alt. Series Test If a n is a positive, decreasing sequence so that lim a n 0, then the alter-
∑ n→∞
nating series (−1)k+1 a k converges.
This test applies only to alternating series — we assume that the terms a n are all posi-
tive and that the sequence {a n } is decreasing.
∑
n
Alt. Series Est. Let S n (−1)k+1 a k be the nth partial sum of the alternating series
k1
∑
∞
(−1)k+1 a k . Assume a n > 0 for each positive integer n, the sequence a n decreases to
k1
0 and lim S n S. Then it follows that |S − S n | < a n+1 .
n→∞
This bound can be used to determine the accuracy of the partial sum S n as an approx-
imation of the sum of a convergent alternating series.
Activity 8.4.7. For (a)-(j), use appropriate tests to determine the convergence or diver-
gence of the following series. Throughout, if a series is a convergent geometric series,
find its sum.
∑
∞
2
a. √
k3 k−2
∑
∞
k
b.
1 + 2k
k1
∑
∞
2k 2 + 1
c.
k3 + k + 1
k0
∑
∞
100k
d.
k!
k0
∑
∞
2k
e.
k1
5k
∑
∞
k3 − 1
f.
k5 + 1
k1
∑
∞
3k−1
g.
k2
7k
∑
∞
1
h.
k2
kk
∑
∞
(−1)k+1
i. √
k1 k+1
478
8.4 Alternating Series
∑
∞
1
j.
k ln(k)
k2
k. Determine a value of n so that the nth partial sum S n of the alternating series
∑
∞
(−1)n
approximates the sum to within 0.001.
ln(n)
n2
8.4.5 Summary
• An alternating series is a series whose terms alternate in sign. It has the form
∑
(−1)k a k
where a k is a positive real number for each k.
• The sequence of partial sums of a convergent alternating series oscillates around the
sum of the series if the sequence of nth terms converges
∑ to 0. That is why the Alternat-
ing Series Test shows that the alternating series ∞ k1 (−1) a k converges whenever the
k
sequence {a n } of nth terms decreases to 0.
• The difference between the n ∑ − 1st partial sum S n−1 and the nth partial sum S n of a
convergent alternating series ∞ k1 (−1) a k is |S n − S n−1 | a n . Since the partial sums
k
oscillate around the sum S of the series, it follows that
|S − S n | < a n .
∑∞
k1 (−1)
So the nth partial sum of a convergent alternating series ka approximates the
k
actual sum of the series to within a n .
8.4.6 Exercises
1. Testing convergence for an alternating series. (a) Carefully determine the convergence
∑
∞
(−1)n
of the series 3n .
n1
∑
∞
(−1)n
(b) Carefully determine the convergence of the series 3n .
n1
2. Estimating the sum of an alternating series. For the following alternating series,
∑
∞
(0.5)3 (0.5)5 (0.5)7
a n 0.5 − + − + ...
3! 5! 7!
n1
how many terms do you have to compute in order for your approximation (your partial
sum) to be within 0.0000001 from the convergent value of that series?
479
Chapter 8 Sequences and Series
3. Estimating the sum of a different alternating series. For the following alternating
series,
∑
∞
(0.4)2 (0.4)4 (0.4)6 (0.4)8
an 1 − + − + − ...
2! 4! 6! 8!
n1
how many terms do you have to go for your approximation (your partial sum) to be
within 0.0000001 from the convergent value of that series?
4. Estimating the sum of one more alternating series. For the following alternating se-
ries,
∑
∞
1 1 1
an 1 − + − + ...
10 100 1000
n1
how many terms do you have to go for your approximation (your partial sum) to be
within 1e-08 from the convergent value of that series?
5. Conditionally convergent series converge very slowly. As an example, consider the
famous formula¹
π 1 1 1 ∑∞
1
1− + − +··· (−1)k . (8.4.3)
4 3 5 7 2k + 1
k0
In theory, the partial sums of this series could be used to approximate π.
a. Show that the series in (8.4.3) converges conditionally.
b. Let S n be the nth partial sum of the series in (8.4.3). Calculate the error in approx-
imating π4 with S100 and explain why this is not a very good approximation.
c. Determine the number of terms it would take in the series (8.4.3) to approximate π4
to 10 decimal places. (The fact that it takes such a large number of terms to obtain
even a modest degree of accuracy is why we say that conditionally convergent
series converge very slowly.)
∑
6. We have shown that if (−1)k+1 a k is a convergent alternating series, then the sum S
of the series lies between any two consecutive partial sums S n . This suggests that the
average Sn +S
2
n+1
is a better approximation to S than is S n .
S n +S n+1
a. Show that 2 S n + 12 (−1)n+2 a n+1 .
b. Use this revised approximation in (a) with n 20 to approximate ln(2) given that
∑
∞
1
ln(2) (−1)k+1 .
k
k1
Compare this to the approximation using just S20 . For your convenience, S20
155685007
232792560 .
¹We will derive this formula in upcoming work.
480
8.4 Alternating Series
7. In this exercise, we examine one of the conditions of the Alternating Series Test. Con-
sider the alternating series
1 1 1 1 1 1
1−1+ − + − + − + ···,
2 4 3 9 4 16
{1} { }
where the terms are selected alternately from the sequences n and − n12 .
a. Explain why the nth term of the given series converges to 0 as n goes to infinity.
b. Rewrite the given series by grouping terms in the following manner:
( ) ( ) ( )
1 1 1 1 1 1
(1 − 1) + − + − + − + ···.
2 4 3 9 4 16
Use this regrouping to determine if the series converges or diverges.
c. Explain why the condition that the sequence {a n } decreases to a limit of 0 is in-
cluded in the Alternating Series Test.
8. Conditionally convergent series exhibit interesting and unexpected behavior. In this
∑ (−1)k+1
exercise we examine the conditionally convergent alternating harmonic series ∞k1 k
and discover that addition is not commutative for conditionally convergent series. We
will also encounter Riemann’s Theorem concerning rearrangements of conditionally
convergent series. Before we begin, we remind ourselves that
∑
∞
(−1)k+1
ln(2),
k
k1
a fact which will be verified in a later section.
a. First we make a quick analysis of the positive and negative terms of the alternating
harmonic series.
∑∞ 1
i. Show that the series k1 2k diverges.
∑∞ 1
ii. Show that the series k1 2k+1 diverges.
iii. Based on the results
∑ of1 the previous
∑∞ parts of this exercise, what can we say
about the sums ∞ 1
kC 2k and kC 2k+1 for any positive integer C? Be specific
in your explanation.
b. Recall addition of real numbers is commutative; that is
a+bb+a
for any real numbers a and b. This property is valid for any sum of finitely
many terms, but does this property extend when we add infinitely many terms
together?
The answer is no, and something even more odd happens. Riemann’s Theorem
(after the nineteenth-century mathematician Georg Friedrich Bernhard Riemann)
states that a conditionally convergent series can be rearranged to converge to any
prescribed sum. More specifically, this means that if we choose any real number
481
Chapter 8 Sequences and Series
∑ (−1)k+1
S, we can rearrange the terms of the alternating harmonic series ∞k1 k so
that the sum is S. To understand how Riemann’s Theorem works, let’s assume
for the moment that the number S we want our rearrangement to converge to is
positive. Our job is to find a way to order the sum of terms of the alternating
harmonic series to converge to S.
i. Explain how we know that, regardless of the value of S, we can find a partial
sum P1
∑
n1
1 1 1 1
P1 1+ + +···+
2k + 1 3 5 2n1 + 1
k1
of the positive terms of the alternating harmonic series that equals or exceeds
S. Let
S1 P1 .
ii. Explain how we know that, regardless of the value of S1 , we can find a partial
sum N1
∑
m1
1 1 1 1 1
N1 − − − − −···−
2k 2 4 6 2m 1
k1
so that
S2 S1 + N1 ≤ S.
iii. Explain how we know that, regardless of the value of S2 , we can find a partial
sum P2
∑
n2
1 1 1 1
P2 + +···+
2k + 1 2(n1 + 1) + 1 2(n1 + 2) + 1 2n2 + 1
kn 1 +1
of the remaining positive terms of the alternating harmonic series so that
S3 S2 + P2 ≥ S.
iv. Explain how we know that, regardless of the value of S3 , we can find a partial
sum
∑m2
1 1 1 1
N2 − − − −···−
2k 2(m 1 + 1) 2(m 1 + 2) 2m 2
km1 +1
of the remaining negative terms of the alternating harmonic series so that
S4 S3 + N2 ≤ S.
v. Explain why we can continue this process indefinitely and find a sequence
{S n } whose terms are partial sums of a rearrangement of the terms in the
alternating harmonic series so that limn→∞ S n S.
482
8.5 Taylor Polynomials and Taylor Series
8.5 Taylor Polynomials and Taylor Series
Motivating Questions
• What is a Taylor polynomial? For what purposes are Taylor polynomials used?
• What is a Taylor series?
• How do we determine the accuracy when we use a Taylor polynomial to approximate
a function?
So far, each infinite series we have discussed has been a series of real numbers, such as
1 1 1 ∑ 1 ∞
1+ + +···+ k +··· . (8.5.1)
2 4 2 k0
2k
In the remainder of this chapter, we will include series that involve a variable. For instance,
if in the geometric series in Equation (8.5.1) we replace the ratio r 12 with the variable x,
we have the infinite (still geometric) series
∑
∞
1 + x + x2 + · · · + x k + · · · xk. (8.5.2)
k0
Here we see something very interesting: because a geometric series converges whenever its
ratio r satisfies |r | < 1, and the sum of a convergent geometric series is 1−r
a
, we can say that
for |x| < 1,
1
1 + x + x2 + · · · + x k + · · · . (8.5.3)
1−x
1
Equation (8.5.3) states that the non-polynomial function 1−x on the right is equal to the in-
finite polynomial expresssion on the left. Because the terms on the left get very small as k
gets large, we can truncate the series and say, for example, that
1
1 + x + x2 + x3 ≈
1−x
for small values of x. This shows one way that a polynomial function can be used to approx-
imate a non-polynomial function; such approximations are one of the main themes in this
section and the next.
In Preview Activity 8.5.1, we begin our exploration of approximating functions with poly-
nomials.
Preview Activity 8.5.1. Preview Activity 8.3.1 showed how we can approximate the
number e using linear, quadratic, and other polynomial functions; we then used sim-
ilar ideas in Preview Activity 8.4.1 to approximate ln(2). In this activity, we review
and extend the process to find the “best” quadratic approximation to the exponential
483
Chapter 8 Sequences and Series
function e x around the origin. Let f (x) e x throughout this activity.
a. Find a formula for P1 (x), the linearization of f (x) at x 0. (We label this lin-
earization P1 because it is a first degree polynomial approximation.) Recall that
P1 (x) is a good approximation to f (x) for values of x close to 0. Plot f and P1
near x 0 to illustrate this fact.
b. Since f (x) e x is not linear, the linear approximation eventually is not a very
good one. To obtain better approximations, we want to develop a different ap-
proximation that “bends” to make it more closely fit the graph of f near x 0.
To do so, we add a quadratic term to P1 (x). In other words, we let
P2 (x) P1 (x) + c2 x 2
for some real number c2 . We need to determine the value of c 2 that makes the
graph of P2 (x) best fit the graph of f (x) near x 0.
Remember that P1 (x) was a good linear approximation to f (x) near 0; this is
because P1 (0) f (0) and P1′ (0) f ′(0). It is therefore reasonable to seek a value
of c 2 so that
P2 (0) f (0), P2′ (0) f ′(0), and P2′′(0) f ′′(0).
Remember, we are letting P2 (x) P1 (x) + c 2 x 2 .
i. Calculate P2 (0) to show that P2 (0) f (0).
ii. Calculate P2′ (0) to show that P2′ (0) f ′(0).
iii. Calculate P2′′(x). Then find a value for c 2 so that P2′′(0) f ′′(0).
iv. Explain why the condition P2′′(0) f ′′(0) will put an appropriate “bend”
in the graph of P2 to make P2 fit the graph of f around x 0.
8.5.1 Taylor Polynomials
Preview Activity 8.5.1 illustrates the first steps in the process of approximating functions
with polynomials. Using this process we can approximate trigonometric, exponential, loga-
rithmic, and other nonpolynomial functions as closely as we like (for certain values of x) with
polynomials. This is extraordinarily useful in that it allows us to calculate values of these
functions to whatever precision we like using only the operations of addition, subtraction,
multiplication, and division, which can be easily programmed in a computer.
We next extend the approach in Preview Activity 8.5.1 to arbitrary functions at arbitrary
points. Let f be a function that has as many derivatives as we need at a point x a. Recall
that P1 (x) is the tangent line to f at (a, f (a)) and is given by the formula
P1 (x) f (a) + f ′(a)(x − a).
P1 (x) is the linear approximation to f near a that has the same slope and function value as
f at the point x a.
484
8.5 Taylor Polynomials and Taylor Series
We next want to find a quadratic approximation
P2 (x) P1 (x) + c2 (x − a)2
so that P2 (x) more closely models f (x) near x a. Consider the following calculations of
the values and derivatives of P2 (x):
P2 (x) P1 (x) + c 2 (x − a)2 P2 (a) P1 (a) f (a)
P2′ (x) P1′ (x) + 2c2 (x − a) P2′ (a) P1′ (a) f ′(a)
P2′′(x) 2c 2 P2′′(a) 2c2 .
To make P2 (x) fit f (x) better than P1 (x), we want P2 (x) and f (x) to have the same concavity
at x a, in addition to having the same slope and function value. That is, we want to have
P2′′(a) f ′′(a).
This implies that
2c 2 f ′′(a)
and thus
f ′′(a)
c2 .
2
Therefore, the quadratic approximation P2 (x) to f centered at x a is
f ′′(a)
P2 (x) f (a) + f ′(a)(x − a) + (x − a)2 .
2!
This approach extends naturally to polynomials of higher degree. We define polynomials
P3 (x) P2 (x) + c 3 (x − a)3 ,
P4 (x) P3 (x) + c 4 (x − a)4 ,
P5 (x) P4 (x) + c 5 (x − a)5 ,
and in general
Pn (x) Pn−1 (x) + c n (x − a)n .
The defining property of these polynomials is that for each n, Pn (x) and all its first n deriv-
atives must agree with those of f at x a. In other words we require that
(k)
Pn (a) f (k) (a)
for all k from 0 to n.
To see the conditions under which this happens, suppose
Pn (x) c 0 + c 1 (x − a) + c 2 (x − a)2 + · · · + c n (x − a)n .
485
Chapter 8 Sequences and Series
Then
(0)
Pn (a) c 0
(1)
Pn (a) c 1
(2)
Pn (a) 2c 2
(3)
Pn (a) (2)(3)c 3
(4)
Pn (a) (2)(3)(4)c 4
(5)
Pn (a) (2)(3)(4)(5)c5
and, in general,
(k)
Pn (a) (2)(3)(4) · · · (k − 1)(k)c k k!c k .
(k)
So having Pn (a) f (k) (a) means that k!c k f (k) (a) and therefore
f (k) (a)
ck
k!
for each value of k. Using this expression for c k , we have found the formula for the poly-
nomial approximation of f that we seek. Such a polynomial is called a Taylor polynomial.
Taylor Polynomials.
The nth order Taylor polynomial of f centered at x a is given by
f ′′(a) f (n) (a)
Pn (x) f (a) + f ′(a)(x − a) + (x − a)2 + · · · + (x − a)n
2! n!
∑
n
f (k) (a)
(x − a)k .
k!
k0
This degree n polynomial approximates f (x) near x a and has the property that
(k)
Pn (a) f (k) (a) for k 0, 1, . . . , n.
Example 8.5.1 Determine the third order Taylor polynomial for f (x) e x , as well as the
general nth order Taylor polynomial for f centered at x 0.
Solution. We know that f ′(x) e x and so f ′′(x) e x and f ′′′(x) e x . Thus,
f (0) f ′(0) f ′′(0) f ′′′(0) 1.
So the third order Taylor polynomial of f (x) e x centered at x 0 is
f ′′(0) f ′′′(0)
P3 (x) f (0) + f ′(0)(x − 0) + (x − 0)2 + (x − 0)3
2! 3!
x2 x3
1+x+ + .
2 6
486
8.5 Taylor Polynomials and Taylor Series
In general, for the exponential function f we have f (k) (x) e x for every positive integer k.
Thus, the kth term in the nth order Taylor polynomial for f (x) centered at x 0 is
f (k) (0) 1
(x − 0)k x k .
k! k!
Therefore, the nth order Taylor polynomial for f (x) e x centered at x 0 is
x2 1 ∑ xk
n
Pn (x) 1 + x + + · · · + xn .
2! n! k!
k0
Activity 8.5.2. We have just seen that the nth order Taylor polynomial centered at
a 0 for the exponential function e x is
∑
n
xk
.
k!
k0
In this activity, we determine small order Taylor polynomials for several other familiar
functions, and look for general patterns.
a. Let f (x) 1
1−x .
i. Calculate the first four derivatives of f (x) at x 0. Then find the fourth
order Taylor polynomial P4 (x) for 1−x
1
centered at 0.
ii. Based on your results from part (i), determine a general formula for f (k) (0).
b. Let f (x) cos(x).
i. Calculate the first four derivatives of f (x) at x 0. Then find the fourth
order Taylor polynomial P4 (x) for cos(x) centered at 0.
ii. Based on your results from part (i), find a general formula for f (k) (0). (Think
about how k being even or odd affects the value of the kth derivative.)
c. Let f (x) sin(x).
i. Calculate the first four derivatives of f (x) at x 0. Then find the fourth
order Taylor polynomial P4 (x) for sin(x) centered at 0.
ii. Based on your results from part (i), find a general formula for f (k) (0). (Think
about how k being even or odd affects the value of the kth derivative.)
It is possible that an nth order Taylor polynomial is not a polynomial of degree n; that is,
the order of the approximation can be different from the degree of the polynomial. For
example, in Activity 8.5.3 we found that the second order Taylor polynomial P2 (x) centered
at 0 for sin(x) is P2 (x) x. In this case, the second order Taylor polynomial is a degree 1
polynomial.
487
Chapter 8 Sequences and Series
8.5.2 Taylor Series
In Activity 8.5.2 we saw that the fourth order Taylor polynomial P4 (x) for sin(x) centered at
0 is
x3
P4 (x) x − .
3!
The pattern we found for the derivatives f (k) (0) describe the higher-order Taylor polynomi-
als, e.g.,
x 3 x (5)
P5 (x) x − + ,
3! 5!
x 3 x (5) x (7)
P7 (x) x − + − ,
3! 5! 7!
x 3 x (5) x (7) x (9)
P9 (x) x − + − + ,
3! 5! 7! 9!
and so on. It is instructive to consider the graphical behavior of these functions; Figure 8.5.2
shows the graphs of a few of the Taylor polynomials centered at 0 for the sine function.
3 3
y y
2 2
1 1
x x
-4 -2 2 4 -4 -2 2 4
-1 -1
-2 -2
-3 -3
3 3
y y
2 2
1 1
x x
-4 -2 2 4 -4 -2 2 4
-1 -1
-2 -2
-3 -3
Figure 8.5.2: The order 1, 5, 7, and 9 Taylor polynomials centered at x 0 for f (x) sin(x).
488
8.5 Taylor Polynomials and Taylor Series
Notice that P1 (x) is close to the sine function only for values of x that are close to 0, but as
we increase the degree of the Taylor polynomial the Taylor polynomials provide a better fit
to the graph of the sine function over larger intervals. This illustrates the general behavior
of Taylor polynomials: for any sufficiently well-behaved function, the sequence {Pn (x)} of
Taylor polynomials converges to the function f on larger and larger intervals (though those
intervals may not necessarily increase without bound). If the Taylor polynomials ultimately
converge to f on its entire domain, we write
∑
∞
f (k) (a)
f (x) (x − a)k .
k!
k0
Definition 8.5.3 Let f be a function all of whose derivatives exist at x a. The Taylor series
for f centered at x a is the series T f (x) defined by
∑
∞
f (k) (a)
T f (x) (x − a)k .
k!
k0
In the special case where a 0 in Definition 8.5.3, the Taylor series is also called the Maclau-
rin series for f . From Example 8.5.1 we know the nth order Taylor polynomial centered at
0 for the exponential function e x ; thus, the Maclaurin series for e x is
∑
∞
xk
.
k!
k0
Activity 8.5.3. In Activity 8.5.2 we determined small order Taylor polynomials for a
few familiar functions, and also found general patterns in the derivatives evaluated
at 0. Use that information to write the Taylor series centered at 0 for the following
functions.
a. f (x) 1
1−x
b. f (x) cos(x) (You will need to carefully consider how to indicate that many of
the coefficients are 0. Think about a general way to represent an even integer.)
c. f (x) sin(x) (You will need to carefully consider how to indicate that many of
the coefficients are 0. Think about a general way to represent an odd integer.)
d. f (x) 1
1+x
Activity 8.5.4.
a. Plot the graphs of several of the Taylor polynomials centered at 0 (of order at
least 5) for e x and convince yourself that these Taylor polynomials converge to
e x for every value of x.
b. Draw the graphs of several of the Taylor polynomials centered at 0 (of order at
least 6) for cos(x) and convince yourself that these Taylor polynomials converge
to cos(x) for every value of x. Write the Taylor series centered at 0 for cos(x).
489
Chapter 8 Sequences and Series
1
c. Draw the graphs of several of the Taylor polynomials centered at 0 for 1−x . Based
on your graphs, for what values of x do these Taylor polynomials appear to
1
converge to 1−x ? How is this situation different from what we observe with e x
1
and cos(x)? In addition, write the Taylor series centered at 0 for 1−x .
1
The Maclaurin series for e x , sin(x), cos(x), and 1−x will be used frequently, so we should be
certain to know and recognize them well.
8.5.3 The Interval of Convergence of a Taylor Series
In the previous section (in Figure 8.5.2 and Activity 8.5.4) we observed that the Taylor poly-
nomials centered at 0 for e x , cos(x), and sin(x) converged to these functions for all values of
1 1
x in their domain, but that the Taylor polynomials centered at 0 for 1−x converge to 1−x on
the interval (−1, 1) and diverge for all other values of x. So the Taylor series for a function
f (x) does not need to converge for all values of x in the domain of f .
Our observations suggest two natural questions: can we determine the values of x for which
a given Taylor series converges? And does the Taylor series for a function f actually converge
to f (x)?
Example 8.5.4 Graphical evidence suggests that the Taylor series centered at 0 for e x con-
verges for all values of x. To verify this, use the Ratio Test to determine all values of x for
which the Taylor series
∑∞
xk
(8.5.4)
k!
k0
converges absolutely.
Solution. Recall that the Ratio Test applies only to series of nonnegative terms. In this
example, the variable x may have negative values. But we are interested in absolute conver-
gence, so we apply the Ratio Test to the series
∑
∞
xk ∑
∞
|x| k
.
k! k!
k0 k0
Now, observe that
|x| k+1
a k+1 (k+1)!
lim lim
k→∞ a k k→∞ |x| k
k
|x| k+1 k!
lim
k→∞ |x| k (k + 1)!
|x|
lim
k→∞ k + 1
0
for any value of x. So the Taylor series (8.5.4) converges absolutely for every value of x, and
thus converges for every value of x.
490
8.5 Taylor Polynomials and Taylor Series
One question still remains: while the Taylor series for e x converges for all x, what we have
done does not tell us that this Taylor series actually converges to e x for each x. We’ll return
to this question when we consider the error in a Taylor approximation near the end of this
section.
We can apply the main idea from Example 8.5.4 in general. To determine the values of x for
which a Taylor series
∑
∞
c k (x − a)k ,
k0
centered at x a will converge, we apply the Ratio Test with a k |c k (x − a)k |. The series
converges if limk→∞ aak+1
k
< 1.
Observe that
a k+1 |c k+1 |
|x − a| ,
ak |c k |
so when we apply the Ratio Test, we get
a k+1 |c k+1 |
lim lim |x − a| .
k→∞ ak k→∞ |c k |
Note suppose that
|c k+1 |
lim L,
k→∞ |c k |
so that
a k+1
lim |x − a| · L.
k→∞ ak
There are three possibilities for L: L can be 0, it can be a finite positive value, or it can be
infinite. Based on this value of L, we can determine for which values of x the original Taylor
series converges.
• If L 0, then the Taylor series converges on (−∞, ∞).
• If L is infinite, then the Taylor series converges only at x a.
• If L is finite and nonzero, then the Taylor series converges absolutely for all x that
satisfy
|x − a| · L < 1
or for all x such that
1
|x − a| < ,
L
which is the interval ( )
1 1
a− ,a+ .
L L
Because the Ratio Test is inconclusive when the |x − a| · L 1, the endpoints a ± L1 have
to be checked separately.
491
Chapter 8 Sequences and Series
It is important to notice that the set of x values at which a Taylor series converges is always
an interval centered at x a. For this reason, the set on which a Taylor series converges is
called the interval of convergence. Half the length of the interval of convergence is called the
radius of convergence. If the interval of convergence of a Taylor series is infinite, then we say
that the radius of convergence is infinite.
Activity 8.5.5.
a. Use the Ratio Test to explicitly determine the interval of convergence of the Tay-
lor series for f (x) 1−x
1
centered at x 0.
b. Use the Ratio Test to explicitly determine the interval of convergence of the Tay-
lor series for f (x) cos(x) centered at x 0.
c. Use the Ratio Test to explicitly determine the interval of convergence of the Tay-
lor series for f (x) sin(x) centered at x 0.
The Ratio Test allows us to determine the set of x values for which a Taylor series converges
absolutely. However, just because a Taylor series for a function f converges, we cannot be
certain that the Taylor series actually converges to f (x). To show why and where a Taylor
series does in fact converge to the function f , we next consider the error that is present in
Taylor polynomials.
8.5.4 Error Approximations for Taylor Polynomials
We now know how to find Taylor polynomials for functions such as sin(x), and how to de-
termine the interval of convergence of the corresponding Taylor series. We next develop an
error bound that will tell us how well an nth order Taylor polynomial Pn (x) approximates its
generating function f (x). This error bound will also allow us to determine whether a Taylor
series on its interval of convergence actually equals the function f from which the Taylor
series is derived. Finally, we will be able to use the error bound to determine the order of
the Taylor polynomial Pn (x) that we will ensure that Pn (x) approximates f (x) to the desired
degree of accuracy.
For this argument, we assume throughout that we center our approximations at 0 (but a
similar argument holds for approximations centered at a). We define the exact error, E n (x),
that results from approximating f (x) with Pn (x) by
E n (x) f (x) − Pn (x).
We are particularly interested in |E n (x)|, the distance between Pn and f . Because
(k)
Pn (0) f (k) (0)
for 0 ≤ k ≤ n, we know that
(k)
E n (0) 0
for 0 ≤ k ≤ n. Furthermore, since Pn (x) is a polynomial of degree less than or equal to n,
we know that
(n+1)
Pn (x) 0.
492
8.5 Taylor Polynomials and Taylor Series
(n+1) (n+1)
Thus, since E n (x) f (n+1) (x) − Pn (x), it follows that
(n+1)
En (x) f (n+1) (x)
for all x.
Suppose that we want to approximate f (x) at a number c close to 0 using Pn (c). If we assume
| f (n+1) (t)| is bounded by some number M on [0, c], so that
f (n+1) (t) ≤ M
for all 0 ≤ t ≤ c, then we can say that
(n+1)
En (t) f (n+1) (t) ≤ M
for all t between 0 and c. Equivalently,
(n+1)
− M ≤ En (t) ≤ M (8.5.5)
on [0, c]. Next, we integrate the three terms in Inequality (8.5.5) from t 0 to t x, and
thus find that ∫ x ∫ x ∫ x
(n+1)
−M dt ≤ E n (t) dt ≤ M dt
0 0 0
(n)
for every value of x in [0, c]. Since E n (0) 0, the First FTC tells us that
(n)
−Mx ≤ E n (x) ≤ Mx
for every x in [0, c].
Integrating this last inequality, we obtain
∫ x ∫ x ∫ x
(n)
−Mt dt ≤ E n (t) dt ≤ Mt dt
0 0 0
and thus
x2 (n−1) x2
−M ≤ E n (x) ≤ M
2 2
for all x in [0, c].
Integrating n times, we arrive at
x n+1 x n+1
−M ≤ E n (x) ≤ M
(n + 1)! (n + 1)!
for all x in [0, c]. This enables us to conclude that
|x| n+1
|E n (x)| ≤ M
(n + 1)!
for all x in [0, c], and we have found a bound on the approximation’s error, E n .
Our work above was based on the approximation centered at a 0; the argument may be
generalized to hold for any value of a, which results in the following theorem.
493
Chapter 8 Sequences and Series
The Lagrange Error Bound for Pn (x).
Let f be a continuous function with n + 1 continuous derivatives. Suppose that M is
a positive real number such that f (n+1) (x) ≤ M on the interval [a, c]. If Pn (x) is the
nth order Taylor polynomial for f (x) centered at x a, then
|c − a| n+1
Pn (c) − f (c) ≤ M .
(n + 1)!
We can use this error bound to tell us important information about Taylor polynomials and
Taylor series, as we see in the following examples and activities.
Example 8.5.5 Determine how well the 10th order Taylor polynomial P10 (x) for sin(x), cen-
tered at 0, approximates sin(2).
Solution. To answer this question we use f (x) sin(x), c 2, a 0, and n 10 in the
Lagrange error bound formula. We also need to find an appropriate value for M. Note that
the derivatives of f (x) sin(x) are all equal to ± sin(x) or ± cos(x). Thus,
f (n+1) (x) ≤ 1
for any n and x. Therefore, we can choose M to be 1. Then
|2 − 0| 11 211
P10 (2) − f (2) ≤ (1) ≈ 0.00005130671797.
(11)! (11)!
So P10 (2) approximates sin(2) to within at most 0.00005130671797. A computer algebra sys-
tem tells us that
P10 (2) ≈ 0.9093474427 and sin(2) ≈ 0.9092974268
with an actual difference of about 0.0000500159.
Activity 8.5.6. Let Pn (x) be the nth order Taylor polynomial for sin(x) centered at
x 0. Determine how large we need to choose n so that Pn (2) approximates sin(2) to
20 decimal places.
Example 8.5.6 Show that the Taylor series for sin(x) actually converges to sin(x) for all x.
Solution. Recall from the previous example that since f (x) sin(x), we know
f (n+1) (x) ≤ 1
for any n and x. This allows us to choose M 1 in the Lagrange error bound formula. Thus,
|x| n+1
|Pn (x) − sin(x)| ≤ (8.5.6)
(n + 1)!
for every x.
494
8.5 Taylor Polynomials and Taylor Series
∑
We showed in earlier work that the Taylor series ∞ x k
k0 k! converges for every value of x.
Because the terms of any convergent series must approach zero, it follows that
x n+1
lim 0
n→∞ (n + 1)!
for every value of x. Thus, taking the limit as n → ∞ in the inequality (8.5.6), it follows that
lim |Pn (x) − sin(x)| 0.
n→∞
As a result, we can now write
∑
∞
(−1)n x 2n+1
sin(x)
(2n + 1)!
n0
for every real number x.
Activity 8.5.7.
a. Show that the Taylor series centered at 0 for cos(x) converges to cos(x) for every
real number x.
b. Next we consider the Taylor series for e x .
i. Show that the Taylor series centered at 0 for e x converges to e x for every
nonnegative value of x.
ii. Show that the Taylor series centered at 0 for e x converges to e x for every
negative value of x.
iii. Explain why the Taylor series centered at 0 for e x converges to e x for every
real number x. Recall that we earlier showed that the Taylor series centered
at 0 for e x converges for all x, and we have now completed the argument
that the Taylor series for e x actually converges to e x for all x.
c. Let Pn (x) be the nth order Taylor polynomial for e x centered at 0. Find a value
of n so that Pn (5) approximates e 5 correct to 8 decimal places.
8.5.5 Summary
• We can use Taylor polynomials to approximate functions. This allows us to approxi-
mate values of functions using only addition, subtraction, multiplication, and division
of real numbers. The nth order Taylor polynomial centered at x a of a function f is
f ′′(a) f (n) (a)
Pn (x) f (a) + f ′(a)(x − a) + (x − a)2 + · · · + (x − a)n
2! n!
∑
n
f (k) (a)
(x − a)k .
k!
k0
495
Chapter 8 Sequences and Series
• The Taylor series centered at x a for a function f is
∑
∞
f (k) (a)
(x − a)k .
k!
k0
The nth order Taylor polynomial centered at a for f is the nth partial sum of its Taylor
series centered at a. So the nth order Taylor polynomial for a function f is an approx-
imation to f on the interval where the Taylor series converges; for the values of x for
which the Taylor series converges to f we write
∑
∞
f (k) (a)
f (x) (x − a)k .
k!
k0
• The Lagrange Error Bound shows us how to determine the accuracy in using a Taylor
polynomial to approximate a function. More specifically, if Pn (x) is the nth order Tay-
lor polynomial for f centered at x a and if M is an upper bound for f (n+1) (x) on
the interval [a, c], then
|c − a| n+1
Pn (c) − f (c) ≤ M .
(n + 1)!
8.5.6 Exercises
1. Determining Taylor polynomials from a function formula. Find the Taylor polyno-
mials of degree n approximating sin(4x) for x near 0 for n 1, 3, 5.
2. Determining Taylor polynomials from given derivative values. Suppose 1 is a func-
tion which has continuous derivatives, and that 1(8) 4, 1 ′(8) 1, 1 ′′(8) 4, 1 ′′′(8) 5.
(a) What is the Taylor polynomial of degree 2 for 1 near 8?
(b) What is the Taylor polynomial of degree 3 for 1 near 8?
(c) Use the two polynomials that you found in parts (a) and (b) to approximate 1(8.1).
3. Finding the Taylor series for a given rational function. Find the first four terms of the
1
Taylor series for the function about the point a 1.
x
4. Finding the Taylor series for a given trigonometric function. Find the first four terms
of the Taylor series for the function sin(x) about the point a π/4.
5. Finding the Taylor series for a given logarithmic function. Find the first five terms of
the Taylor series for the function f (x) ln(x) about the point a 4.
6. In this exercise we investigation the Taylor series of polynomial functions.
a. Find the 3rd order Taylor polynomial centered at a 0 for f (x) x 3 −2x 2 +3x −1.
Does your answer surprise you? Explain.
b. Without doing any additional computation, find the 4th, 12th, and 100th order
Taylor polynomials (centered at a 0) for f (x) x 3 − 2x 2 + 3x − 1. Why should
you expect this?
496
8.5 Taylor Polynomials and Taylor Series
c. Now suppose f (x) is a degree m polynomial. Completely describe the nth order
Taylor polynomial (centered at a 0) for each n.
7. The examples we have considered in this section have all been for Taylor polynomials
and series centered at 0, but Taylor polynomials and series can be centered at any value
of a. We look at examples of such Taylor polynomials in this exercise.
a. Let f (x) sin(x). Find the Taylor polynomials up through order four of f cen-
tered at x π2 . Then find the Taylor series for f (x) centered at x π2 . Why should
you have expected the result?
b. Let f (x) ln(x). Find the Taylor polynomials up through order four of f centered
at x 1. Then find the Taylor series for f (x) centered at x 1.
c. Use your result from (b) to determine which Taylor polynomial will approximate
ln(2) to two decimal places. Explain in detail how you know you have the desired
accuracy.
8. We can use known Taylor series to obtain other Taylor series, and we explore that idea
in this exercise, as a preview of work in the following section.
a. Calculate the first four derivatives of sin(x 2 ) and hence find the fourth order Tay-
lor polynomial for sin(x 2 ) centered at a 0.
b. Part (a) demonstrates the brute force approach to computing Taylor polynomials
and series. Now we find an easier method that utilizes a known Taylor series.
Recall that the Taylor series centered at 0 for f (x) sin(x) is
∑
∞
x 2k+1
(−1)k . (8.5.7)
(2k + 1)!
k0
i. Substitute x 2 for x in the Taylor series (8.5.7). Write out the first several terms
and compare to your work in part (a). Explain why the substitution in this
problem should give the Taylor series for sin(x 2 ) centered at 0.
ii. What should we expect the interval of convergence of the series for sin(x 2 )
to be? Explain in detail.
9. Based on the examples we have seen, we might expect that the Taylor series for a func-
tion f always converges to the values f (x) on its interval
{ of convergence. We explore
e −1/x
2
if x , 0,
that idea in more detail in this exercise. Let f (x)
0 if x 0.
a. Show, using the definition of the derivative, that f ′(0) 0.
b. It can be shown that f (n) (0) 0 for all n ≥ 2. Assuming that this is true, find the
Taylor series for f centered at 0.
c. What is the interval of convergence of the Taylor series centered at 0 for f ? Ex-
plain. For which values of x the interval of convergence of the Taylor series does
the Taylor series converge to f (x)?
497
Chapter 8 Sequences and Series
8.6 Power Series
Motivating Questions
• What is a power series?
• What are some important uses of power series?
• What is the connection between power series and Taylor series?
We have noted in our work with Taylor polynomials and Taylor series that polynomial func-
tions are the simplest possible functions in mathematics, in part because they require only
addition and multiplication to evaluate. From the point of view of calculus, polynomials are
especially nice: we can easily differentiate or integrate any polynomial. In light of our work
in Section 8.5, we now know that several important non-polynomials have polynomial-like
expansions. For example, for any real number x,
x2 x3 xn
ex 1 + x + + +···+ + ···.
2! 3! n!
There are two settings where other series like the one for e x arise: we may be given an
expression such as
1 + 2x + 3x 2 + 4x 3 + · · ·
and we seek the values of x for which the expression makes sense. Or we may be trying to
find an unknown function f that has expression
f (x) a 0 + a1 x + a 2 x 2 + · · · + a k x k + · · · ,
and we try to determine the values of the constants a 0 , a1 , . . .. The latter situation is explored
in Preview Activity 8.6.1.
Preview Activity 8.6.1. In Chapter 7, we learned some of the many important ap-
plications of differential equations, and learned some approaches to solve or analyze
them. Here, we consider an important approach that will allow us to solve a wider
variety of differential equations.
Let’s consider the familiar differential equation from exponential population growth
given by
y ′ k y, (8.6.1)
where k is the constant of proportionality. While we can solve this differential equa-
tion using methods we have already learned, we take a different approach now that
can be applied to a much larger set of differential equations. For the rest of this ac-
tivity, let’s assume that k 1. We will use our knowledge of Taylor series to find a
solution to the differential equation (8.6.1).
498
8.6 Power Series
To do so, we assume that we have a solution y f (x) and that f (x) has a Taylor series
that can be written in the form
∑
∞
y f (x) ak x k ,
k0
where the coefficients a k are undetermined. Our task is to find the coefficients.
a. Assume that we can differentiate a power series term by term. By taking the de-
rivative of f (x) with respect to x and substituting the result into the differential
equation (8.6.1), show that the equation
∑
∞ ∑
∞
ka k x k−1
ak x k
k1 k0
∑∞
must be satisfied in order for f (x) k0 a k x k to be a solution of the DE.
b. Two series are equal if and only if they have the same coefficients on like power
terms. Use this fact to find a relationship between a1 and a0 .
c. Now write a2 in terms of a1 . Then write a2 in terms of a0 .
d. Write a3 in terms of a 2 . Then write a3 in terms of a0 .
e. Write a4 in terms of a 3 . Then write a4 in terms of a0 .
f. Observe that there is a pattern in (b)-(e). Find a general formula for a k in terms
of a0 .
g. Write the series expansion for y using only the unknown coefficient a0 . From
this, determine what familiar functions satisfy the differential equation (8.6.1).
(Hint: Compare to a familiar Taylor series.)
8.6.1 Power Series
As Preview Activity 8.6.1 shows, it can be useful to treat an unknown function as if it has a
Taylor series, and then determine the coefficients from other information. In other words, we
define a function as an infinite series of powers of x and then determine the coefficients based
on something besides a formula for the function. This method allows us to approximate
solutions to many different types of differential equations, even if we cannot solve them
explicitly. This is different from our work with Taylor series since we are not using an original
function f to generate the coefficients of the series.
Definition 8.6.1 A power series centered at x a is a function of the form
∑
∞
c k (x − a)k (8.6.2)
k0
499
Chapter 8 Sequences and Series
where {c k } is a sequence of real numbers and x is an independent variable.
A power series defines a function f whose domain is the set of x values for which the power
series converges. We therefore write
∑
∞
f (x) c k (x − a)k .
k0
It turns out that¹, on its interval of convergence, every power series is in fact the Taylor series
of the function it defines, so all of the techniques we developed in the previous section can
be applied to power series as well.
Example 8.6.2 Consider the power series defined by
∑
∞
xk
f (x) .
k0
2k
( )
What are f (1) and f 32 ? Find a general formula for f (x) and determine the values for which
this power series converges.
Solution. If we evaluate f at x 1 we obtain the series
∑
∞
1
k0
2k
which is a geometric series with ratio 12 . So we can sum this series and find that
1
f (1) 2.
1− 1
2
Similarly,
∞ ( )k
∑ 3 1
f (3/2) 4.
k0
4 1− 3
4
In general, f (x) is a geometric series with ratio x2 , so
∞ ( )k
∑ x 1 2
f (x)
2 1− x
2 2−x
k0
provided that −1 < x2 < 1 (which ensures that the ratio is less than 1 in absolute value).
Thus, the power series that defines f (x) 2−x
2
converges for −2 < x < 2.
As we did for Taylor series, we define the interval of convergence of a power series (8.6.2) to
be the set of values of x for which the series converges. And as we did with Taylor series,
we typically use the Ratio Test to find the values of x for which the power series converges
absolutely, and then check the endpoints separately if the radius of convergence is finite.
¹See Exercise 8.6.4.4 in this section.
500
8.6 Power Series
∑∞ xk
Example 8.6.3 Let f (x) k1 k 2 . Determine the interval of convergence of this power
series.
Solution. First we will plot some of the partial sums of this power series to get an idea of
the interval of convergence. Let
∑
n
xk
S n (x)
k2
k1
for each n ≥ 1. Figure 8.6.4 shows plots of S10 (x) (in red), S25 (x) (in blue), and S50 (x) (in
green).
10
y
5
x
-2 -1 1 2
-5
-10
∑∞ xk
Figure 8.6.4: Graphs of some partial sums of the power series k1 k 2 .
The behavior of S50 in particular suggests that f (x) appears to be converging to a particular
curve on the interval (−1, 1), while growing without bound outside of that interval. Thus,
the interval of convergence might be −1 < x < 1. To verify our conjecture, we apply the
Ratio Test. Now,
xk
ak 2 ,
k
so
|x| k+1
|a k+1 | (k+1)2
lim lim
k→∞ |a k | k→∞ |x| k
k2
( )2
k
lim |x|
k→∞ k+1
( )2
k
|x| lim
k→∞ k + 1
|x|.
Therefore, the Ratio Test tells us that f (x) converges absolutely when |x| < 1 and diverges
when |x| > 1. Because the Ratio Test is inconclusive when |x| 1, we need to check x 1
and x −1 individually.
501
Chapter 8 Sequences and Series
When x 1, observe that
∑
∞
1
f (1) .
k2
k1
This is a p-series with p > 1, which we know converges. When x −1, we have
∑
∞
(−1)k
f (−1) .
k2
k1
{ }
This is an alternating series, and since the sequence n12 decreases to 0, the power series
converges by the Alternating Series Test. Thus, the interval of convergence of this power
series is −1 ≤ x ≤ 1.
Activity 8.6.2. Determine the interval of convergence of each power series.
∑ (x−1)k ∑
a. ∞ k1 3k
d. ∞ xk
k1 (2k)!
∑∞
b. k1 kx k
∑∞ k 2 (x+1)k ∑∞
c. k1 4k
e. k1 k!x k
8.6.2 Manipulating Power Series
We know power series expansions for important functions such as sin(x) and e x . Often, we
can use a known power series expansion to find a power series for a different, but related,
function. The next activity demonstrates one way to do this.
Activity 8.6.3. Our goal in this activity is to find a power series expansion for f (x)
1
1+x 2
centered at x 0.
While we could use the methods of Section 8.5 and differentiate f (x) 1+x1
2 several
times to look for patterns and find the Taylor series for f (x), we seek an alternate
approach because of how complicated the derivatives of f (x) quickly become.
a. What is the Taylor series expansion for 1(x) 1
1−x ? What is the interval of
convergence of this series?
b. How is 1(−x 2 ) related to f (x)? Explain, and hence substitute −x 2 for x in the
power series expansion for 1(x). Given the relationship between 1(−x 2 ) and
f (x), how is the resulting series related to f (x)?
c. For which values of x will this power series expansion for f (x) be valid? Why?
In a previous section we found several important Maclaurin series and their intervals of
convergence. Here, we list these key functions and their corresponding expansions.
502
8.6 Power Series
∑
∞
(−1)k x 2k+1
sin(x) for − ∞ < x < ∞
(2k + 1)!
k0
∑∞
(−1)k x 2k
cos(x) for − ∞ < x < ∞
(2k)!
k0
∑∞
xk
ex for − ∞ < x < ∞
k!
k0
1 ∑∞
xk for − 1 < x < 1
1−x
k0
As we saw in Activity 8.6.3, we can use these known series to find other power series expan-
3
sions for related functions such as sin(x 2 ), e 5x , and cos(x 5 ).
Activity 8.6.4. Let f be the function given by the power series expansion
∑
∞
x 2k
f (x) (−1)k .
(2k)!
k0
a. Assume that we can differentiate a power series term by term, just like we can
differentiate a (finite) polynomial. Use the fact that
x2 x4 x6 x 2k
f (x) 1 − + − + · · · + (−1)k +···
2! 4! 6! (2k)!
to find a power series expansion for f ′(x).
b. Observe that f (x) and f ′(x) have familiar Taylor series. What familiar functions
are these? What known relationship does our work demonstrate?
c. What is the series expansion for f ′′(x)? What familiar function is f ′′(x)?
Our work in Activity 8.6.3 holds more generally. The corresponding theorem, which we will
not prove, states that we can differentiate a power series for a function f term by term and
obtain the series expansion for f ′, and similarly we can integrate
∫ a series expansion for a
function f term by term and obtain the series expansion for f (x) dx. For both, the radius
of convergence of the resulting series is the same as the original, though it is possible that the
convergence status of the various series may differ at the endpoints. The formal statement
of the Power Series Differentiation and Integration Theorem follows.
503
Chapter 8 Sequences and Series
Power Series Differentiation and Integration Theorem.
Suppose f (x) has a power series expansion
∑
∞
f (x) ck x k
k0
and that the series
∑ converges absolutely to f (x) on the interval −r < x < r. Then, the
power series ∞ k1 kc k x
k−1 obtained by differentiating the power series for f (x) term
by term converges absolutely to f ′(x) on the interval −r < x < r. That is,
∑
∞
f ′(x) kc k x k−1 , for |x| < r.
k1
∑
Similarly, the power series ∞
k+1
k0 c k k+1 obtained by integrating the power series for f (x) term
x
by term converges absolutely on the interval −r < x < r, and
∫ ∑
∞
x k+1
f (x) dx ck + C, for |x| < r.
k+1
k0
This theorem validates the steps we took in Activity 8.6.4. It tells us that we can differentiate
and integrate term by term on the interior of the interval of convergence, but it does not tell
us what happens at the endpoints of this interval. We always need to check what happens at
the endpoints separately. More importantly, we can use use the approach of differentiating
or integrating a series term by term to find new series.
Example 8.6.5 Find a series expansion centered at x 0 for arctan(x), as well as its interval
of convergence.
Solution. While we could differentiate arctan(x) repeatedly and look for patterns in the
derivative values at x 0 in an attempt to find the Maclaurin series for arctan(x) from
the definition, it turns out to be far easier to use a known series in an insightful way. In
Activity 8.6.3, we found that
1 ∑∞
(−1)k x 2k
1 + x2
k0
for −1 < x < 1. Recall that
d 1
[arctan(x)] ,
dx 1 + x2
and therefore ∫
1
dx arctan(x) + C.
1 + x2
1
It follows that we can integrate the series for 1+x 2 term by term to obtain the power series
expansion for arctan(x). Doing so, we find that
∫ (∑
∞
)
arctan(x) (−1)k x 2k dx
k0
504
8.6 Power Series
∞ (∫
∑ )
(−1)k x 2k dx
k0
( )
∑∞
x 2k+1
(−1) k
+ C.
2k + 1
k0
The Power Series Differentiation and Integration Theorem tells us that this equality is valid
for at least −1 < x < 1.
To find the value of the constant C, we can use the fact that arctan(0) 0. So
( )
∑
∞
02k+1
0 arctan(0) (−1) k
+ C C,
2k + 1
k0
and we must have C 0. Therefore,
∑
∞
x 2k+1
arctan(x) (−1)k (8.6.3)
2k + 1
k0
for at least −1 < x < 1.
It is a straightforward exercise to check that the power series
∑
∞
x 2k+1
(−1)k
2k + 1
k0
converges both when x −1 and when x 1; in each case, we have an alternating series with
1
terms 2k+1 that decrease to 0, and thus the interval of convergence for the series expansion
for arctan(x) in Equation (8.6.3) is −1 ≤ x ≤ 1.
Activity 8.6.5. Find a power series expansion for ln(1 + x) centered at x 0 and
determine its interval of convergence.
8.6.3 Summary
• A power series is a series of the form
∑
∞
ak x k .
k0
• We can often assume a solution to a given problem can be written as a power series,
then use the information in the problem to determine the coefficients in the power se-
ries. This method allows us to approximate solutions to certain problems using partial
sums of the power series; that is, we can find approximate solutions that are polyno-
mials.
505
Chapter 8 Sequences and Series
• The connection between power series and Taylor series is that they are essentially the
same thing: on its interval of convergence a power series is the Taylor series of its sum.
8.6.4 Exercises
1. Finding coefficients in a power series expansion of a rational function. Represent the
5 ∑ ∞
function as a power series f (x) c n x n . Find c 0 , c 1 , c2 , c3 , c 4 and the radius
(1 − 6x)
n0
of convergence R.
2. Finding coefficients in a power series expansion of a function involving arctan(x).
∑
∞
The function f (x) 5x arctan(3x) is represented as a power series f (x) cn x n .
n0
Find the coefficients c0 , c1 , c 2 , c 3 , c 4 and the radius of convergence R.
3. We can use power series to approximate definite integrals to which known techniques
of integration do not apply. We will illustrate this in this exercise with the definite
∫1
integral 0
sin(x 2 ) ds.
a. Use the Taylor series for sin(x) to find the Taylor series for sin(x 2 ). What is the
interval of convergence for the Taylor series for sin(x 2 )? Explain.
b. Integrate the∫ Taylor series for sin(x 2 ) term by term to obtain a power series ex-
pansion for sin(x 2 ) dx.
∫1
c. Use the result from part (b) to explain how to evaluate sin(x 2 ) dx. Determine
∫ 10
the number of terms you will need to approximate 0
sin(x 2 ) dx to 3 decimal
places.
4. There is an important connection between power series and Taylor series. Suppose f
is defined by a power series centered at 0 so that
∑
∞
f (x) ak x k .
k0
a. Determine the first 4 derivatives of f evaluated at 0 in terms of the coefficients a k .
b. Show that f (n) (0) n!a n for each positive integer n.
c. Explain how the result of (b) tells us the following:
On its interval of convergence, a power series is the Taylor series of its
sum.
5. In this exercise we will begin with a strange power series and then find its sum. The
Fibonacci sequence { f n } is a famous sequence whose first few terms are
f0 0, f1 1, f2 1, f3 2, f4 3, f5 5, f6 8, f7 13, · · · ,
where each term in the sequence after the first two is the sum of the preceding two
506
8.6 Power Series
terms. That is, f0 0, f1 1 and for n ≥ 2 we have
f n f n−1 + f n−2 .
Now consider the power series
∑
∞
F(x) fk x k .
k0
We will determine the sum of this power series in this exercise.
a. Explain why each of the following is true.
∑∞
i. xF(x) k1 f k−1 x
k
∑∞
ii. x 2 F(x) k2 f k−2 x
k
b. Show that
F(x) − xF(x) − x 2 F(x) x.
c. Now use the equation
F(x) − xF(x) − x 2 F(x) x
to find a simple form for F(x) that doesn’t involve a sum.
d. Use a computer algebra system or some other method to calculate the first 8 de-
x
rivatives of 1−x−x 2 evaluated at 0. Why shouldn’t the results surprise you?
6. Airy’s equation²
y ′′ − x y 0, (8.6.4)
can be used to model an undamped vibrating spring with spring constant x (note that
y is an unknown function of x). So the solution to this differential equation will tell
us the behavior of a spring-mass system as the spring ages (like an automobile shock
absorber). Assume that a solution y f (x) has a Taylor series that can be written in
the form
∑
∞
y ak x k ,
k0
where the coefficients are undetermined. Our job is to find the coefficients.
(a) Differentiate the series for y term by term to find the series for y ′. Then repeat to
find the series for y ′′.
(b) Substitute your results from part (a) into the Airy equation and show that we can
write Equation (8.6.4) in the form
∑
∞ ∑
∞
(k − 1)ka k x k−2 − a k x k+1 0. (8.6.5)
k2 k0
(c) At this point, it would be convenient if we could combine the series on the left in
(8.6.5), but one written with terms of the form x k−2 and the other with terms in
507
Chapter 8 Sequences and Series
the form x k+1 . Explain why
∑
∞ ∑
∞
(k − 1)ka k x k−2 (k + 1)(k + 2)a k+2 x k . (8.6.6)
k2 k0
(d) Now show that
∑
∞ ∑
∞
a k x k+1 a k−1 x k . (8.6.7)
k0 k1
(e) We can now substitute (8.6.6) and (8.6.7) into (8.6.5) to obtain
∑
∞ ∑
∞
(n + 1)(n + 2)a n+2 x n − a n−1 x n 0. (8.6.8)
n0 n1
Combine the like powers of x in the two series to show that our solution must
satisfy
∑
∞
2a2 + [(k + 1)(k + 2)a k+2 − a k−1 ] x k 0. (8.6.9)
k1
(f) Use equation (8.6.9) to show the following:
i. a 3k+2 0 for every positive integer k,
ii. a 3k 1
a
(2)(3)(5)(6)···(3k−1)(3k) 0
for k ≥ 1,
iii. a 3k+1 1
a
(3)(4)(6)(7)···(3k)(3k+1) 1
for k ≥ 1.
(g) Use the previous part to conclude that the general solution to the Airy equation
(8.6.4) is
( )
∑
∞
x 3k
y a0 1 +
(2)(3)(5)(6) · · · (3k − 1)(3k)
k1
( )
∑∞
x 3k+1
+ a1 x + .
(3)(4)(6)(7) · · · (3k)(3k + 1)
k1
Any values for a0 and a1 then determine a specific solution that we can approxi-
mate as closely as we like using this series solution.
²The general differential equations of the form y ′′ ± k 2 x y 0 is called Airy’s equation. These equations arise in
many problems, such as the study of diffraction of light, diffraction of radio waves around an object, aerodynamics,
and the buckling of a uniform column under its own weight.
508
APPENDIX A
A Short Table of Integrals
∫ (u)
a. du
a 2 +u 2
1
a arctan a +C
∫ √
b. √ du ln |u + u 2 ± a 2 | + C
u 2 ±a 2
∫ √ √ 2 √
c. u 2 ± a 2 du u2 u 2 ± a 2 ± a2 ln |u + u 2 ± a 2 | + C
∫ 2 √ 2 √
d. √u du u u 2 ± a 2 ∓ a ln |u + u 2 ± a 2 | + C
2 2 u ±a2 2
∫ √
a+ u 2 +a 2
e. √ du − 1a ln u +C
u u 2 +a 2
∫ (u)
f. √ du 1
a arcsec a +C
u u 2 −a 2
∫ (u)
g. √ du arcsin a +C
a 2 −u 2
∫ √ √ (u)
a2
h. a 2 − u 2 du u
2 a2 − u2 + 2 arcsin a +C
∫ 2 √ (u)
a2
i. √ u du − u2 a 2 − u 2 + 2 arcsin a +C
a 2 −u 2
∫ √
a+ a 2 −u 2
j. √ du − 1a ln u +C
u a 2 −u 2
∫ √
a 2 −u 2
k. √du − a2 u
+C
u 2 a 2 −u 2
Appendix A A Short Table of Integrals
510
APPENDIX B
Answers to Activities
This appendix contains answers to all activities in the text. Answers for preview activities
are not included.
1 · Understanding the Derivative
1.1 · How do we measure velocity?
Activity 1.1.2 Answer.
a. AV[0.4,0.8] 12.8 ft/sec; AV[0.7,0.8] 8 ft/sec; the other average velocities are, respec-
tively, 6.56, 6.416, 0, 4.8, 6.24, 6.384, all in ft/sec.
b. m 12.8 is the average velocity of the ball between t 0.4 and t 0.8.
feet
s
64
B
A
56
48
sec
0.4 0.8 1.2
c. Like a straight line with slope about 6.4.
d. About 6.4 feet per second.
Activity 1.1.3 Answer.
a. AV[1.5,2] −24 ft/sec, which is negative.
b. The instantaneous velocity at t 1.5 is approximately −16 ft/sec; at t 2, the instan-
taneous velocity is about −32 ft/sec, and −16 > −32.
Appendix B Answers to Activities
c. When the ball is rising, its instantaneous velocity is positive, while when the ball is
falling, its instantaneous velocity is negative.
d. Zero.
Activity 1.1.4 Answer. AV[2,2+h] −32 − 16h
1.2 · The notion of limit
Activity 1.2.2 Answer.
a. 2.
b. 12.
1
c. 2.
Activity 1.2.3 Answer.
a. 6 + h.
b. 6.2 meters/min.
c. 6 meters per minute.
Activity 1.2.4 Answer.
a. AV[0.5,1] 1−1
1−0.5 0, AV[1.5,2.5] 3−1
2.5−1.5 2, and AV[0,5] 5−0
5−0 1.
b. Take shorter and shorter time intervals and draw the lines whose slopes represent
average velocity. If those lines’ slopes are approaching a single number, that number
represents the instantaneous velocity.
c. The instantaneous velocity at t 2 is greater than the average velocity on [1.5, 2.5].
1.3 · The derivative of a function at a point
Activity 1.3.2 Answer.
a. f is linear.
b. The average rate of change on [1, 4], [3, 7], and [5, 5 + h] is −2.
c. f ′(1) −2.
√
d. f ′(2) −2, f ′(π) −2, and f ′(− 2) −2, since the slope of a linear function is the
same at every point.
Activity 1.3.3 Answer.
a. The vertex is ( 12 , 36).
s(2)−s(1)
b. 2−1 −32 feet per second.
c. s ′(1) −16.
d.
512
y secant line
32
tangent line
16
t
1 2
e. s ′(a) is positive whenever 0 ≤ a < 12 ; s ′(a) to be negative whenever 1
2 < a < 2; s ′( 21 ) 0.
Activity 1.3.4 Answer.
a.
y (in thousands)
80
P
60
40
20
t (decades)
1 2 3 4
b. AV[2,4] ≈ 9171 people per decade is expected to be the average rate of change of the
city’s population over the two decades from 2030 to 2050.
c.
P(2 + h) − P(2)
P ′(2) lim ≈ 7458.5
h→0 h
which is measured in people per decade.
d. See the graph provided in (a) above. The magenta line has slope equal to the average
rate of change of P on [2, 4], while the green line is the tangent line at (2, P(2)) with
slope P ′(2).
e. It appears that the tangent line’s slope at the point (a, P(a)) will increase as a increases.
1.4 · The derivative function
513
Appendix B Answers to Activities
Activity 1.4.2 Answer.
f g
x
x
f′ g′
x x
p q
x
x
p′ q′
x x
514
r s
x x
r′ s′
x
x
w z
x x
w′ z′
x x
Activity 1.4.3 Answer.
a. f ′(x) 0.
515
Appendix B Answers to Activities
b. 1 ′(t) 1.
c. p ′(z) 2z.
d. q ′(s) 3s 2 .
−1
e. F′(t) t2
.
f. G′(y) 1
√
2 y
.
1.5 · Interpreting, estimating, and using the derivative
Activity 1.5.2 Answer.
a. F′(30) ≈ 3.85 degrees per minute.
b. F′(60) ≈ 1.56 degrees per minute.
c. F′(75) > F′(90).
d. The value F(64) 330.28 is the temperature of the potato in degrees Fahrenheit at
time 64, while F′(64) 1.341 measures the instantaneous rate of change of the potato’s
temperature with respect to time at the instant t 64, and its units are degrees per
minute. Because at time t 64 the potato’s temperature is increasing at 1.341 degrees
per minute, we expect that at t 65, the temperature will be about 1.341 degrees
greater than at t 64, or in other words F(65) ≈ 330.28 + 1.341 331.621. Similarly,
Û
at t 66, two minutes have elapsed from t 64, so we expect an increase of 21.341
degrees: F(66) ≈ 330.28 + 2 · 1.341 332.962.
e. Throughout the time interval [0, 90], the temperature F of the potato is increasing. But
as time goes on, the rate at which the temperature is rising appears to be decreasing.
That is, while the values of F continue to get larger as time progresses, the values of
F′ are getting smaller (while still remaining positive). We thus might say that “the
temperature of the potato is increasing, but at a decreasing rate.”
Activity 1.5.3 Answer.
a. It costs $800 to make 2000 feet of rope.
b. “dollars per foot.”
c. C(2100) ≈ 835,.
d. Either C′(2000) C′(3000) or C′(2000) > C′(3000).
e. Impossible. The total cost function C(r) can never decrease.
Activity 1.5.4 Answer.
a. f ′(90) 0.0006 liters per kilometer per kilometer per hour.
b. At 80 kilometers per hour, the car is using fuel at a rate of 0.015 liters per kilometer.
c. When the car is traveling at 90 kilometers per hour, its rate of fuel consumption per
kilometer is increasing at a rate of 0.0006 liters per kilometer per kilometer per hour.
516
1.6 · The second derivative
Activity 1.6.2 Answer.
a. Increasing: 0 < t < 2, 3 < t < 5, 7 < t < 9, and 10 < t < 12. Decreasing: never.
b. Velocity is increasing on 0 < t < 1, 3 < t < 4, 7 < t < 8, and 10 < t < 11; y v(t) is
decreasing on 1 < t < 2, 4 < t < 5, 8 < t < 9, and 11 < t < 12. Velocity is constant on
2 < t < 3, 5 < t < 7, and 9 < t < 10.
c. a(t) v ′(t) and a(t) s ′′(t).
d. s ′′(t) is positive since s ′(t) is increasing.
e. • increasing.
• decreasing.
• constant.
• increasing.
• decreasing.
• constant.
• concave up.
• concave down.
• linear.
Activity 1.6.3 Answer.
a. Degrees Fahrenheit per minute.
b. F′′(30) ≈ −0.119.
c. At the moment t 30, the temperature of the potato is 251 degrees; its temperature
is rising at a rate of 3.85 degrees per minute; and the rate at which the temperature is
rising is falling at a rate of -0.119 degrees per minute per minute.
d. Increasing at a decreasing rate.
Activity 1.6.4 Answer.
517
Appendix B Answers to Activities
f f
x x
f′ f′
x x
f ′′ f ′′
x x
1.7 · Limits, Continuity, and Differentiability
Activity 1.7.2 Answer.
a. f (−2) 1; f (−1) is not defined; f (0) 73 ; f (1) 2; f (2) 2.
b.
lim f (x) 2 and lim + f (x) 1
x→−2− x→−2
5 5
lim − f (x) and lim + f (x)
x→−1 3 x→−1 3
7 7
lim f (x) and lim+ f (x)
x→0− 3 x→0 3
lim− f (x) 3 and lim+ f (x) 3
x→1 x→1
lim f (x) 2 and lim+ f (x) 2
x→2− x→2
518
c. limx→−2 f (x) does not exist. The values of the limits as x → a for a −1, 0, 1, 2 are
3 , 3 , 3, 2.
5 7
d. a −2, a −1, and a 1.
e.
4
y
3
2
1
x
-3 -2 -1 1 2 3
-1
-2
Activity 1.7.3 Answer.
a. a −2; a +2.
b. a 3.
c. a −1; a 3.
d. a −2; a 2; a 3; a −1.
e. “If f is continuous at x a, then f has a limit at x a.”
Activity 1.7.4 Answer.
a. 1 is piecewise linear.
b.
1(0 + h) − 1(0)
1 ′(0) lim
h→0 h
|0 + h| − |0|
lim
h→0 h
|h|
lim
h→0 h
|h| |h|
c. limh→0+ h 1, but limh→0− h −1.
d. a −3, −2, −1, 1, 2, 3.
e. True.
1.8 · The Tangent Line Approximation
519
Appendix B Answers to Activities
Activity 1.8.2 Answer.
a. L(−1) −2; L′(−1) 3.
b. 1(−1) −2; 1 ′(−1) 3.
c. Less.
d. 1(−1.03) ≈ L(−1.03) −2.09.
e. Concave up.
f. The illustration below shows a possible graph of y 1(x) near x −1, along with the
tangent line y L(x) through (−1, 1(−1)).
y = g(x)
(−1, −2)
y = L(x)
Activity 1.8.3 Answer.
a. L(x) −1 + 2(x − 2).
b. f (2.07) ≈ L(2.07) −0.86.
c. See the image in part e.
d. Neither.
e. See the image below, which shows, at left, a possible graph of y f (x) near x 2,
along with the tangent line y L(x) through (2, f (2)).
y y y
y= f ′ (x)
2 y = f (x) 2 2
y = f ′′ (x)
x x x
2 2 2
520
f. Too large.
2 · Computing Derivatives
2.1 · Elementary derivative rules
Activity 2.1.2 Answer.
a. f ′(t) 0.
b. 1 ′(z) 7z ln(7).
c. h ′(w) 34 w −1/4 .
dp
d. dx 0.
√ √
e. r ′(t) ( 2)t ln( 2).
−1 −q −2 .
dq [q ]
d
f.
g. dm
dt −3t −4 − t34 .
Activity 2.1.3 Answer.
a. f ′(x) 53 x 2/3 − 4x 3 + 2x ln(2).
b. 1 ′(x) 14e x + 3 · 5x 4 − 1.
c. h ′(z) 12 z −1/2 − 4z −5 + 5z ln(5).
√
dt 53 · 7t − πe .
d. dr 6 t
e. ds
dy 4y 3 .
f. q ′(x) 2x − 2x −2 .
g. p ′(a) 12a 3 − 6a 2 + 14a − 1.
Activity 2.1.4 Answer.
a. h ′(4) 3
16 .
b. (i.)P ′(4) 2(1.37)4 ln(1.37) ≈ 2.218 million cells per day; (ii.) the population is growing
at an increasing rate.
c. y − 25 −33(a + 1).
d. The slope is a number, while the equation is, well, an equation.
2.2 · The sine and cosine functions
Activity 2.2.2 Answer.
a. Figure B.0.1.
b. 1, 0, −1, 0, 1, 0, −1, 0, 1.
c. f ′(0) f ′(−2π) f ′(2π) 1.
521
Appendix B Answers to Activities
d. Figure B.0.1.
dx [sin(x)] cos(x).
d
e.
1 1
2π
−2π −π
-1
π −2π −π
-1
π 2π
Figure B.0.1: At left, the graph of y f (x) sin(x). At right, the graph of y f ′(x).
Activity 2.2.3 Answer.
a. Figure B.0.2.
b. 0, −1, 0, 1, 0, −1, 0, 1, 0.
c. 1 ′( π2 ) 1 ′(− 3π
2 ) −1.
d. Figure B.0.2.
dx [cos(x)] − sin(x).
d
e.
1 1
−2π −π
-1
π 2π −2π −π
-1
π 2π
Figure B.0.2: At left, the graph of y 1(x) cos(x). At right, the graph of y 1 ′(x)
Activity 2.2.4 Answer.
a. dh
dt −3 sin(t) − 4 cos(t).
√
b. f ′( π6 ) 2 + 3
4 .
π2
c. y − 4 (π − 2)(x − π2 ).
d. p ′(z) 4z 3 + 4z ln(4) − 4 sin(z).
e. P ′(2) 8 cos(2) ≈ −3.329 hundred animals per decade.
2.3 · The product and quotient rules
Activity 2.3.2 Answer.
a. m ′(w) 3w 17 · 4w ln(4) + 4w · 51w 16 .
b. h ′(t) (sin(t) + cos(t)) · 4t 3 + t 4 · (cos(t) − sin(t)).
c. f ′(1) e(cos(1) + sin(1)) ≈ 3.756.
522
d. L(x) − 12 (x + 1).
Activity 2.3.3 Answer.
(z 4 +1)3z ln(3)−3z (4z 3 )
a. r ′(z) (z 4 +1)2
.
(cos(t)+t 2 ) cos(t)−sin(t)(− sin(t)+2t)
b. v ′(t) (cos(t)+t 2 )2
.
c. R′(0) 29 .
−100 −400
d. I ′(0.5) e50 ′
0.5 ≈ 30.327, I (2) e2
≈ −13.534, and I ′(5) e4
≈ −2.695, each in
candles per millisecond.
Activity 2.3.4 Answer.
a. f ′(r) (5r 3 + sin(r))[4r ln(4) + 2 sin(r)] + (4r − 2 cos(r))[15r 2 + cos(r)].
t 6 ·6t [− sin(t)]−cos(t)[t 6 ·6t ln(6)+6t ·6t 5 ]
b. p ′(t) (t 6 ·6t )2
.
(z 2 +1)1−z(2z)
c. 1 ′(z) 3[z 7 e z + 7z 6 e z ] − 2[z 2 cos(z) + 2z sin(z)] + (z 2 +1)2
.
−2 sin(1)−4 cos(1)
d. s ′(1) e1
≈ −1.414 feet per second.
e. p ′(3) 30 and q ′(3) 13
8 .
2.4 · Derivatives of other trigonometric functions
Activity 2.4.2 Answer.
π
a. All real numbers x such that x , 2 + kπ, where k ±1, ±2, . . ..
b. h ′(x)
sin(x)
cos2 (x)
.
c. h ′(x) sec(x) tan(x).
π
d. h and h ′ have the same domain: all real numbers x such that x , 2 + kπ, where
k ±1, ±2, . . ..
Activity 2.4.3 Answer.
a. All real numbers x such that x , kπ, where k 0, ±1, ±2, . . ..
b. h ′(x) − sin2 (x) .
cos(x)
c. h ′(x) − csc(x) cot(x).
d. p and p ′ have the same domain: all real numbers x such that x , kπ, where k
0, ±1, ±2, . . ..
Activity 2.4.4 Answer.
√
a. m f ′( π3 ) 10 3 + 43 .
√ √
b. p ′( π4 ) π2
16 2+ 2π
2 + π
2 − 1.
(t 2 +1) sec2 (t)−2t tan(t)
c. h ′(t) (t 2 +1)2
+ 2e t sin(t) − 2e t cos(t).
523
Appendix B Answers to Activities
d. 1 ′(r)
r sec(r) tan(r)+sec(r)−rln(5) sec(r)
5r .
e. s ′(2)
15 cos(2)−15 sin(2)
e2
≈ −2.69 inches per second.
2.5 · The chain rule
Activity 2.5.2 Answer.
a. h ′(x) −4x 3 sin(x 4 ).
2 (x)
b. h ′(x) √
sec
.
2 tan(x)
c. h ′(x) 2sin(x) ln(2) cos(x).
d. h ′(x) −5 cot4 (x) csc2 (x).
e. h ′(x) 9(sec(x) + e x )8 (sec(x) tan(x) + e x ).
Activity 2.5.3 Answer.
4(6r 5 +2e r )
a. p ′(r) √ .
2 r 6 +2e r
b. m ′(v) −3v 2 sin(v 2 ) sin(v 3 ) + 2v cos(v 3 ) cos(v 2 ).
(e 4y +1)[−10 sin(10y)]−cos(10y)[4e 4y ]
c. h ′(y) (e 4y +1)2
.
d. s ′(z) 2z
2 sec(z) ln(2)[z 2 sec(z) tan(z) + sec(z) · 2z].
e. c ′(x) cos(e )[e x2 x2 · 2x].
Activity 2.5.4 Answer.
a. y − 2 14 (x − 0).
b. v(1) s ′(1) − 38 inches per second; the particle is moving left at the instant t 1.
c. P ′(1000) 30e −0.0323 (−0.0000323) ≈ −0.000938 inches of mercury per foot.
d. C′(2) −10; D ′(−1) −20.
2.6 · Derivatives of Inverse Functions
Activity 2.6.2 Answer.
a. h ′(x) x + 2x ln(x).
(e t +1) 1t −ln(t)·e t
b. p ′(t) (e t +1)2
.
c. s ′(y) 1
cos(y)+2
· (− sin(y)).
d. z ′(x) sec2 (ln(x)) · x1 .
e. m ′(z) 1
ln(z)
· 1z .
Activity 2.6.3 Answer.
a. tan(r(x)) x.
524
b. r ′(x) cos2 (r(x)).
c. r ′(x) cos2 (arctan(x)).
d. With θ arctan(x),
√
1 + x2
x
θ
1
e. cos(arctan(x)) √ 1 .
1+x 2
f. r ′(x) 1
1+x 2
.
Activity 2.6.4 Answer.
[ ] [ ]
a. f ′(x) x 3 · 1
1+x 2
+ arctan(x) · 3x 2 + e x · 1
x + ln(x) · e x .
b. p ′(t) 2t arcsin(t) ln(2)[t · √1 + arcsin(t) · 1].
1−t 2
[ ]
c. h ′(z) 27(arcsin(5z) + arctan(4 − z))26 √ 1 ·5+ 1
1+(4−z)2
· (−1) .
1−(5z)2
d. s ′(y) − y12 .
e. m ′(v) 1
sin2 (v)+1
· [2 sin(v) cos(v)].
[ ]
(1+w 2 ) w1 −ln(w)·2w
f. 1 ′(w) (
1
ln(w)
)2 · (1+w 2 )2
1+
1+w 2
2.7 · Derivatives of Functions Given Implicitly
Activity 2.7.2 Answer.
a. The graph of the curve fails the vertical line test.
dy
b. dx 1
5y 4 −15y 2 +4
.
c. y − 16 x + 1.
d. (1.418697, 0.543912), (−1.418697, −0.543912), (−3.63143, 1.64443), and
(3.63143, −1.64443).
Activity 2.7.3 Answer.
a. Horizontal at x ≈ 0.42265, thus (0.42265, −1.05782); (0.42265, 0.229478);
(0.42265, 0.770522); (0.42265, 2.05782). There are four more points where x ≈ 1.57735.
525
Appendix B Answers to Activities
√
1± 5
b. When y 12 , 2 , so one point is (2.21028, 12 ).
c. y − 1 12 (x − 1).
Activity 2.7.4 Answer.
dy
dx (−3y − 6x) 6y − 3x 2 and the tangent line has equation y − 3 1(x + 3).
a. 2
dy 3x 2 +1
b. dx cos(y)+1
. and the tangent line has equation y 12 x.
dy −x ye −x y +e −x y
c. dx x 2 e −x y +2
. and the tangent line is y − 1 0.110794(x − 0.571433).
2.8 · Using Derivatives to Evaluate Limits
Activity 2.8.2 Answer.
ln(1+x)
a. limx→0 x 1.
cos(x)
b. limx→π x − π1 .
2 ln(x)
c. limx→1 1−e x−1
−2.
sin(x)−x
d. limx→0 cos(2x)−1
0.
Activity 2.8.3 Answer.
f (x)
a. limx→2 1(x)
18 .
p(x)
b. limx→2 q(x)
1.
r(x)
c. limx→2 s(x)
< 0.
Activity 2.8.4 Answer.
a. limx→∞ x
ln(x)
∞.
e x +x
b. limx→∞ 2e x +x 2
12 .
ln(x)
c. limx→0+ 1 0.
x
tan(x)
d. limx→ π2 − x− π2 −∞.
e. limx→∞ xe −x 0.
3 · Using Derivatives
3.1 · Using derivatives to identify extreme values
Activity 3.1.2 Answer.
a. x −4 or x 1.
b. 1 has a local maximum at x −4 and neither a max nor min at x 1.
526
c. 1 does not have a global minimum; it is unclear (at this point in our work) if 1 increases
without bound, so we can’t say for certain whether or not 1 has a global maximum.
d. limx→∞ 1 ′(x) ∞.
e. A possible graph of 1 is the following.
g
-6 -4 -2 2
Activity 3.1.3 Answer.
a. x −1 is an inflection point of 1.
b. 1 is concave up for x < −1, concave down for −1 < x < 2, and concave down for x > 2.
c. 1 has a local minimum at x −1.67857351.
d. 1 is a degree 5 polynomial.
Activity 3.1.4 Answer.
a. In the graph below, h(x) x 2 +cos(3x) is given in dark blue, while h(x) x 2 +cos(1.6x)
is shown in light blue.
12
8
4
-2 2
527
Appendix B Answers to Activities
b. If 2
> 1, then the equation cos(kx) k22 has no solution. Hence, whenever k 2 < 2,
k2 √
or k < 2 ≈ 1.414, it follows that the equation cos(kx) k22 has no solutions x, which
means that h ′′(x) is never zero (indeed, for these k-values, h ′′(x) is always positive
√
so that h is always concave up). On the other hand, if k ≥ 2, then k22 ≤ 1, which
guarantees that cos(kx) k22 has infinitely many solutions, due to the periodicity of
the cosine function. At each such point, h ′′(x) 2 − k 2 cos(kx) changes sign, and
√
therefore h has infinitely many inflection points whenever k ≥ 2.
c. To see why h can only have a finite number of critical numbers regardless of the value
of k, consider the equation
0 h ′(x) 2x − k sin(kx),
which implies that 2x k sin(kx). Since −1 ≤ sin(kx) ≤ 1, we know that −k ≤
k sin(kx) ≤ k. Once |x| is sufficiently large, we are guaranteed that |2x| > k, which
means that for large x, 2x and k sin(kx) cannot intersect. Moreover, for relatively small
values of x, the functions 2x and k sin(kx) can only intersect finitely many times since
k sin(kx) oscillates a finite number of times. This is why h can only have a finite number
of critical numbers, regardless of the value of k.
3.2 · Using derivatives to describe families of functions
Activity 3.2.2 Answer.
√a
a. p has two critical numbers (x ± 3) whenever a > 0 and no critical numbers when
a < 0.
b. When a < 0, p is always increasing and has no relative extreme values. When a > 0, p
√ √
has a relative maximum at x − 3a and a relative minimum at x + 3a .
c. p is CCD for x < 0 and p is CCU for x > 0, making x 0 an inflection point.
d.
p p
− a3
p pa
3
528
Activity 3.2.3 Answer.
a. h is an always increasing function.
b. h is always concave down.
c. limx→∞ a(1 − e −bx ) a, and limx→∞ a(1 − e −bx ) −∞.
d. If b is large and x is close to zero, h ′(x) is relatively large near x 0, and the curve’s
slope will quickly approach zero as x increases. If b is small, the graph is less steep
near x 0 and its slope goes to zero less quickly as x increases.
e.
a
h(x) = a(1 − e−bx )
Activity 3.2.4 Answer.
a. L is an always increasing function.
(1)
b. L is concave up for all t < − 1k ln c and concave up for all other values of t.
c. limt→∞ A
1+ce −kt
A, and
A
lim 0.
t→∞ 1 + ce −kt
(1)
d. The inflection point on the graph of L is (− 1k ln c , A2 ).
e.
529
Appendix B Answers to Activities
y=A
y = L(x)
A (− 1k ln( 1c ), A2 )
(0, 1+c )
y=0
3.3 · Global Optimization
Activity 3.3.2 Answer.
√
a. x ± 2 ≈ ±1.414.
b.
g g g
2 2 2
-2 3 -2 2 -2 1
√
c. On [−2, 3], 1 has a global maximum at x 3 and a global minimum at x 2.
√ √
d. On [−2, 2], 1 has a global maximum at x − 2 and a global minimum at x 2.
√
e. On [−2, 3], 1 has a global maximum at x − 2 and a global minimum at x 1.
Activity 3.3.3 Answer.
a. Absolute maximum: e −1 ; absolute minimum: 0.
√
b. Absolute maximum: 2; absolute minimum: −1.
c. Absolute maximum: 9.8; absolute minimum: 8.
d. Absolute minimum at x 2; no absolute maximum.
Activity 3.3.4 Answer.
a.
530
10 − 2x
x
x
15 − 2x
b. V(x) x(10 − 2x)(15 − 2x) 4x 3 − 50x 2 + 150x.
c. 1 ≤ x ≤ 3.
√
25±5 7
d. x 6 ≈ 6.371459426, 1.961873908.
e. • V(1.961873908) 132.0382370
• V(1) 104
• V(3) 108
f. Absolute maximum: 132.0382370; absolute minimum: 104.
3.4 · Applied Optimization
Activity 3.4.2 Answer.
a. Let the can have radius r and height h.
b. V πr 2 h; S 2πr 2 + 2πrh; C 2πr 2 · 0.027 + 2πrh · 0.015.
c. C(r) 0.054πr 2 + 0.48 1r , r > 0.
√
d. r 3 0.48
0.108π ≈ 1.12259; h ≈ 4.041337; minimum cost C(1.12259) ≈ 0.64137.
Activity 3.4.3 Answer. The absolute minimum time the hiker can achieve is 0.99302 hours,
which is attained by hiking about 2.2 km from P to Q and then turning into the woods for
the remainder of the trip.
√
Activity 3.4.4 Answer. Maximum area: A( √5 ) 500 9 3 ≈ 96.225. Maximum perimeter:
√ 3
82−1
P(1) 52. At x 3 the absolute maximum of combined perimeter and area occurs.
Activity 3.4.5 Answer. A(1.19606) ≈ 2.2018 is the absolute maximum cross-sectional area,
which leads to the absolute maximum volueme.
3.5 · Related Rates
531
Appendix B Answers to Activities
Activity 3.5.2 Answer.
a.
6
r
8
h
b. r 34 h.
c. V 16 πh .
3 3
d. dV
dt 16 πh dt .
9 2 dh
e. dh
dt h3 64
81π ≈ 0.2515 feet per minute.
f. Most rapidly when h 3.
Activity 3.5.3 Answer.
a.
rocket
z
h
θ
camera
4000
b. dh
dt 4000 sec2 (θ) dθ
dt .
dt z dt .
c. h dh dz
d. dz
dt h3000 360 feet/sec; dθ
dt h3000 12
125 radians per second.
e. greater.
Activity 3.5.4 Answer.
a.
532
lamp
15
6
x s
b. 3s 2x.
dt 2 dt .
c. 3 ds dx
d. ds
dt x8 2 feet per second.
e. at a constant rate.
dy
f. Let y represent the location of the tip of the shadow; dt 5 feet/sec.
Activity 3.5.5 Answer. Let x denote the position of the ball at time t and z the distance
from the ball to first base, as pictured below.
z
90
x
dz
dt x45 100
√ ≈ 44.7214 feet/sec.
5
Let r be the runner’s position at time t and let s be the distance between the runner and the
ball, as pictured.
533
Appendix B Answers to Activities
s
x
r
ds
dt x45 430
√ ≈ 104.2903 feet/sec.
17
4 · The Definite Integral
4.1 · Determining distance traveled from velocity
Activity 4.1.2 Answer.
a.
mph
3
y = v(t)
2
1
hrs
1 2
A v(0.0) · 0.5 + v(0.5) · 0.5 + v(1.0) · 0.5 + v(1.5) · 0.5
1.500 · 0.5 + 1.9375 · 0.5 + 2.000 · 0.5 + 2.0625 · 0.5
3.75
Thus, D ≈ 3.75 miles.
534
b. Using 8 rectangles of width 0.25, D ≈ 3.875.
c. s(t) 18 t 4 − 12 t 3 + 34 t 2 + 32 t.
d. s(2) − s(0) 18 24 − 12 23 + 34 22 + 32 2 4.
Activity 4.1.3 Answer.
a. On (0, 1), s is increasing because velocity is positive.
b. s(t) 32t − 16t 2 .
c. s(1) − s( 12 ) 4.
d. A 4 feet is the total distance the ball traveled vertically on [ 12 , 1].
e. s(1) − s(0) 16 is the vertical distance the ball traveled on the interval [0, 1]. Equiva-
lently, the area under the velocity curve on [0, 1] is A 16 feet.
f. s(2) − s(0) 0, so the ball has zero change in position on the interval [0, 2].
Activity 4.1.4 Answer.
a. Total distance traveled is 2; change in position is 0.
b. 0 < t < 1 and 4 < t < 8.
c. s(8) − s(0) 5 m, while the distance traveled on [0, 8] is D 13, and thus these two
quantities are different.
d. See the figure below.
m/sec
4 8
y = v(t) y = s(t)
2 3 3 4
1 1 1 sec
1 2 1 4 6 8 2 4 6 8
2
-2 -4
-4 -8
4.2 · Riemann Sums
Activity 4.2.2 Answer.
a. 65
b. 32
535
Appendix B Answers to Activities
c.
∑
7
3 + 7 + 11 + 15 + · · · + 27 4k − 1.
k1
d.
∑
8
4 + 8 + 16 + 32 + · · · + 256 2i .
i2
e.
∑
6
1 63
.
i1
2i 64
Activity 4.2.3 Answer.
a.
y = v(t)
4
2
2 5
b. L4 311
48 ≈ 6.47917, R 4 335
48 ≈ 6.97917, and M4 637
96 ≈ 6.63542.
c.
L 4 + M4 646 637
, M4 .
2 96 96
d. L n is an under-estimate; R n is an over-estimate.
Activity 4.2.4 Answer.
25 −1.44
a. M5 − 36
536
y = v(t)
1
1 2 3 4 5
-1
b. The change in position is approximately −1.44 feet.
c. D ≈ 2.336.
d. − 43 is the object’s total change in position on [1, 5].
4.3 · The Definite Integral
Activity 4.3.2 Answer.
∫1
a. 0
3x dx 32 .
∫4
b. −1
(2 − 2x) dx −5.
∫1 √
π
c. −1
1 − x 2 dx 2.
∫4
d. −3
1(x) dx 3π
4 − 32 .
Activity 4.3.3 Answer.
∫2
a. 5
f (x) dx −2.
∫5
b. 0
1(x) dx 3.
∫5
c. 0
( f (x) + 1(x)) dx 2.
∫5
d. 2
(3x 2 − 4x 3 ) dx −492.
∫0
e. 5
(2x 3 − 71(x)) dx − 583
2 .
Activity 4.3.4 Answer.
√
a. y v(t) 4 − (t − 2)2 is the top half of the circle (t − 2)2 + y 2 4, which has radius
2 and is centered at (2, 0).
∫4
b. 0
v(t) dt 2π.
c. The object moved 2π meters in 4 minutes.
π
d. vAVG [0, 4] 2, meters per minute,.
537
Appendix B Answers to Activities
e.
4
y = v(t)
2
2 4
π
The height of the rectangle is the average value of v, vAVG [0, 4] 2 ≈ 1.57.
f. D 2π.
4.4 · The Fundamental Theorem of Calculus
Activity 4.4.2 Answer.
∫4
a. −1
(2 − 2x) dx −5.
∫ π
b. 0
2
sin(x) dx 1.
∫1
c. 0
e x dx e − 1.
∫1
d. −1
x 5 dx 0.
∫2
e. 0
(3x 3 − 2x 2 − e x ) dx 23
3 − e 2.
Activity 4.4.3 Answer.
538
given function, f (x) antiderivative, F(x)
k, (k , 0) kx
x n , n , −1 1
n+1 x
n+1
x, x > 0
1
ln(x)
sin(x) − cos(x)
cos(x) sin(x)
sec(x) tan(x) sec(x)
csc(x) cot(x) − csc(x)
sec2 (x) tan(x)
csc2 (x) − cot(x)
ex ex
a x (a > 1) 1
ln(a)
ax
1
1+x 2
arctan(x)
√ 1 arcsin(x)
1−x 2
∫1( )
a. 0
x 3 − x − e x + 2 dx 11
4 − e.
∫ π/3 √ π2
b. 0
(2 sin(t) − 4 cos(t) + sec2 (t) − π) dt 1 − 3 − 3 .
∫1 √
c. 0
( x − x 2 ) dx 13 .
Activity 4.4.4 Answer.
400
a. The person burned exactly 3 calories in the first 10 minutes of the workout.
∫ 40 ∫ 40
b. C(40) − C(0) 0
C′(t) dt 0
c(t) dt is the total calories burned on [0, 40].
c. The exact average rate at which the person burned calories on 0 ≤ t ≤ 40 is
∫ 40
1 1 1700 1700
c AVG[0,40] c(t) dt · ≈ 14.17 cal/min.
40 − 0 0 40 3 120
d. One time at which the instantaneous √ rate at which calories are burned equals the av-
erage rate on [0, 40] is t 53 (6 − 6) ≈ 5.918.
5 · Evaluating Integrals
5.1 · Constructing Accurate Graphs of Antiderivatives
Activity 5.1.2 Answer.
a. F is increasing on (0, 2) and (5, 7); F is decreasing on (2, 5).
b. F is concave up on (0, 1), (4, 6); concave down on (1, 3), (6, 7); neither on (3, 4).
c. A relative maximum at x 2; a relative minimum at x 5.
d. F(1) − 21 ; F(2) π4 − 12 ; F(3) π4 − 1; F(4) π
4 − 2; F(5) π
4 − 25 ; F(6) π
2 − 25 ;
F(7) 3π
4 − 2 ; F(8) 4 − 2 ; and F(−1) −1.
5 3π 5
539
Appendix B Answers to Activities
e. Use the function values found in (d) and the earlier information regarding the shape
of F.
f. G(x) F(x) + 1.
4
Activity 5.1.3 Answer.
a.
3
2
1
-5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
-1
-2
4
-3
b. H(x) − cos(x) + 2. 3
-4
c.
2
-5
1
-6
-4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 1
-1
Activity 5.1.4 Answer.
-2
a. A is increasing on (0, 1.5), (4, 6); A is decreasing on (1.5, 4).
b. A is concave up on (0, 1) and (3, 5);-3 A is concave down on (1, 3) and (5, 6).
c. At x 1.5, A has a relative maximum; A has a relative minimum at x 4.
-4
d. A(0) − 12 ; A(1) 0; A(2) 0; A(3) −2; A(4) −3.5, A(5) −2, A(6) −0.5.
e. Use your work in (a)-(d) appropriately.
-5
f. B(x) A(x) + 12 .
-6
5.2 · The Second Fundamental Theorem of Calculus
540
Activity 5.2.2 Answer.
a. A′(x) f (x).
b. A(1) − π4 .
c. A is increasing wherever f is positive; A is CCU wherever f is increasing. A(2) 0,
A(3) −0.5, A(4) −1.5, A(5) −2, A(6) −2 + π4 , and A(7) −2 + π2 .
π
d. F and A differ by the constant 4 − 12 .
e. B and C have the same shape as A and F, and differ from A by a constant. Observe
that B(3) 0 and C(1) 0.
Activity 5.2.3 Answer.
a. See the plot at below left.
b. F′ f .
c. F is increasing for all x > 0; F is decreasing for x < 0
d. F is CCU on −1 < x < 1 and CCD for x < −1 and x > 1.
e. F(5) ≈ 2.35973; F(10) ≈ 2.35973.
f. See the graph at below right.
F
0.5 1.25
f
-5 5 -5 5
-0.5 -1.25
Activity 5.2.4 Answer.
[∫ x ]
2 2
a. d
dx 4
e t dt e x .
∫x [ ]
t4 x4
b. d
−2 dt 1+t 4
dt 1+x 4
− 16
17 .
[∫ 1 ]
c. d
dx x
cos(t 3 ) dt − cos(x 3 ).
541
Appendix B Answers to Activities
∫x [ ]
d. d
3 dt
ln(1 + t 2 ) dt ln(1 + x 2 ) − ln(10).
[∫ x 3 ]
e. d
dx 4
sin(t 2 ) dt sin(x 6 ) · 3x 2 .
5.3 · Integration by Substitution
Activity 5.3.2 Answer.
∫
a. sin(8 − 3x) dx − 13 (− cos(8 − 3x)) + C.
∫
b. sec2 (4x) dx 1
4 tan(4x) + C.
∫
c. 1
11x−9 dx 1
11 ln |11x − 9| + C.
∫
d. csc(2x + 1) cot(2x + 1) dx − 12 cot(2x + 1) + C.
∫
e. √ 1 dx 1
4 arcsin(4x) + C
1−16x 2
∫
f. 5−x dx − ln(5)
1
5−x + C.
Activity 5.3.3 Answer.
∫
x2
a. 5x 3 +1
dx 1
15 ln(5x 3 + 1) + C.
∫
b. e x sin(e x ) dx − cos(e x ) + C.
∫ √
cos( x) √
c. √
x
dx 2 sin( x) + C.
Activity 5.3.4 Answer.
∫ x2
a. x
x1 1+4x 2
dx 18 (ln(17) − ln(5)).
∫1
b. 0
e −x (2e −x + 3)9 dx − 20
1
(2e −1 + 3)10 + 20 (2e
1 0 + 3)10 .
∫ 4/πcos( x1 )
√
2
c. 2/π x2
dx 1 − 2 .
5.4 · Integration by Parts
Activity 5.4.2 Answer.
∫
a. te −t dt −te −t − e −t .
∫ 4
b. 4x sin(3x)dx − x cos(3x) + 94 sin(3x) + c.
3
∫
c. z sec2 (z)dz z tan(z) + ln | cos(z)| + c.
∫
d. x ln(x)dx 21 x 2 ln(x) − 41 x 2 + c.
Activity 5.4.3 Answer.
∫ ( )
a. arctan(x)dx x arctan(x) − 12 ln |1 + x 2 | + c.
∫
b. ln(z)dz z ln(z) − z + c.
542
∫ ( ( ) ( ))
c. t 3 sin(t 2 )dt 1
2 −t 2 cos t 2 + sin t 2 .
∫ ( )
3 3 3
d. s 5 e s ds 1
3 s3 e s − e s + c.
∫ ( ) ( ) ( )
e. e 2t cos e t dt e t sin e t + cos e t + c.
Activity 5.4.4 Answer.
∫
a. x 2 sin(x)dx −x 2 cos(x) + 2x sin(x) + 2 cos(x) + c.
∫
b. t 3 ln(t)dt 14 t 4 ln(t) − 1 4
16 t + c.
∫
c. e z sin(z)dz − 12 e z cos(z) + 12 e z sin(z) + c.
∫
d. s 2 e 3s ds 13 s 2 e 3s − 92 se 3s + 2 3s
27 e + c.
∫
e. t arctan(t)dt 12 t 2 arctan(t) − 21 t − 12 arctan(t) + c.
5.5 · Other Options for Finding Algebraic Antiderivatives
Activity 5.5.2 Answer.
∫
a. 1
x 2 −2x−3
dx 1
4 ln |x − 3| − 14 ln |x + 1| + C.
∫
x 2 +1
b. x 3 −x 2
dx − ln |x| + x −1 + 2 ln |x − 1| + C.
∫
c. x−2
x 4 +x 2
dx ln |x| + 2x −1 − 12 ln |1 + x 2 | + 2 arctan(x) + C.
Activity 5.5.3 Answer.
∫ √
a. x 2 + 4 dx 12 arctan( x2 ) + C.
∫ √
b. √ x dx x 2 + 4 + C.
2 x +4
∫ √
c. √ 2
dx 2
5 ln |5x + 16 + 25x 2 | + C.
16+25x 2
∫ √
49−36x 2
d. √ 1 dx − 49x + C.
x 2 49−36x 2
5.6 · Numerical Integration
Activity 5.6.2 Answer.
∫2 1 1
a. 1 2 dx .
x 2
b. The table below gives values of the trapezoid rule and corresponding errors for differ-
ent n-values.
n Tn ET,n
4 0.50899 0.00899
8 0.50227 0.00227
16 0.50057 0.00057
c. The table below gives values of the midpoint rule and corresponding errors for differ-
ent n-values.
543
Appendix B Answers to Activities
n Mn E M,n
4 0.49555 −0.00445
8 0.49887 −0.00113
16 0.49972 −0.00028
d. The trapezoid rule overestimates; the midpoint rule underestimates.
1
e. f (x) is concave up on [1, 2].
x2
Activity 5.6.3 Answer.
a. Plot the data.
∫ 1.8
b. 0
v(t)dt.
c.
L3 165.6 ft R 3 105.6 ft T3 135.6 ft .
R 3 and T3 are underestimates.
d. M3 143.4 ft ; overestimate.
e. S6 140.8 ft .
∫ 1.8
f. Simpson’s rule gives the best approximation of the distance traveled, 0
v(t)dt ≈
140.8 ft .
Activity 5.6.4 Answer.
a. For L1 and T1 :
f 1 h
L1 2 L1 2 L1 2
R1 1 R1 1 R1 1
Table B.0.3: Left and Trapezoid rules.
The values of L1 and R 1 are the same for all three.
b. For the M1 ,
f 1 h
M1 7
4 M1 15
8 M1 31
16
Table B.0.4: Midpoint Rule.
c. For T1 and S2 ,
544
f 1 h
T1 3
2 T1 3
2 T1 3
2
S2 5
3 ≈ 1.6667 S2 7
4 S2 43
24 ≈ 1.79167
Table B.0.5: Trapezoid and Simpson’s Rule.
d.
∫ 1 ∫ 1 ∫ 1
5 7 9
f (x)dx 1(x)dx h(x)dx
0 3 0 4 0 5
e. Left endpoint rule results are overestimates; right endpoint rules are underestimates;
midpoint rules are overestimates; trapezoid rules are underestimates. Simpson’s rule
∫1
is exact for both f and 1, while a slight overestimate of 0
h(x)dx.
6 · Using Definite Integrals
6.1 · Using Definite Integrals to Find Area and Length
Activity 6.1.2 Answer.
∫ 16 √
a. A 0 ( x − 14 x) dx 32
3 .
∫ √20/3 √5
160
b. A √
− 20/3
((12 − x 2 ) − (x 2 − 8)) dx 3
3
≈ 68.853.
∫ π √
c. A 0
4
(cos(x) − sin(x)) dx 2 − 1.
d. The left-hand region has area
∫ √
0 ( ( )) 13 + 5 5
A1 √ x2 − x3 − x dx ≈ 1.007514.
1− 5
2
24
The right-hand region has area
∫
√
1+ 5 √
2 (( ) ) 13 − 5 5
A2 x 3 − x − x 2 dx ≈ 0.075819.
0 24
Activity 6.1.3 Answer.
∫ √
y 2 √
a. A √ (6 − 2y 2 − y 2 ) dy 8 2 ≈ 11.314.
y− 2
∫ y1
b. A y−1
(2 − 2y 2 − (1 − y 2 )) dy 43 .
∫ √
y4−2 3
( √
) √
2−y
c. A y0 2 − y dy 11
3 − 2 3 ≈ 0.2026
∫3
d. A 0
(y − (y 2 − 2y)) dy 92 .
545
Appendix B Answers to Activities
Activity 6.1.4 Answer.
a. L ≈ 2.95789.
∫2 √
b. L −2
4
4−x 2
dx 2π.
∫1√
c. L 0
1 + e 6x (9x 2 + 6x + 1) dx ≈ 20.1773.
∫b√
d. We will usually have to estimate the value of a
1 + f ′(x)2 dx using computational
technology.
e. Approximately (14.9165, f (14.9165)) (14.9165, 23.2502).
6.2 · Using Definite Integrals to Find Volume
Activity 6.2.2 Answer.
∫4 √ ∫4
a. V 0 π( x)2 dx 0 πx dx 8π.
∫4 √ ∫4
b. V 0
π(4 − ( x)2 ) dx 0 π(4 − x) dx 8π.
∫1
c. V 0
π(x − x 6 ) dx 14 π.
5
∫ √3 √
136 3
d. V √
− 3
π((x 2 + 4)2 − (2x 2 + 1)2 ) dx 5 π.
∫2
e. V 0
π y 4 dy 5 π.
32
Activity 6.2.3 Answer.
∫2
a. V 0
π y 4 ∆ dy.
∫2
b. V 0
π(16 − y 4 ) dy.
√
c. V int0 2 π(4x 2 − x 6 ) dx.
∫ 2√2
d. V 0
π(y 2/3 − y 2 /4) dy.
∫3
e. V 0
π((y + 1)2 − (y − 1)4 ) dy.
Activity 6.2.4 Answer.
a. √
∫ 2 √
4
V π((2x + 2)2 − (x 3 + 2)2 ) dx (21 + 8 2)π ≈ 19.336.
0 21
b. √ )
∫ √
2 (
32 2
V π((4 − x ) − (4 − 2x) ) dx 8 −
3 2 2
π ≈ 18.3626.
0 21
c. √
∫ 2 2 √
1 2
V π((y 1/3 + 1)2 − ( y + 1)2 ) dy (15 + 8 2)π ≈ 11.022.
0 2 15
546
d. √
∫ 2 2 √
1 2 2
V π((5 − y) − (5 − y 1/3 )2 ) dy (75 − 8 2)π ≈ 26.677.
0 2 15
6.3 · Density, Mass, and Center of Mass
Activity 6.3.2 Answer.
a. M 0.1 − 0.1e −2 ≈ 0.0181269 grams.
∫5
b. i. V 0
π(4 − 45 x)2 dx 80π
3 ≈ 83.7758m .
3
ii. M 64000π
3 ≈ 67020.6433kg.
∫5
iii. M 0
(400+ 1+x
200
2 )·π(4− 5 x) dx 128π( 3 +24 arctan(5)−5 ln(26))
4 2 265
≈ 42224.8024kg
c. b ≈ 3.0652.
Activity 6.3.3 Answer.
x1 +x2
a. x 2 3.
x1 +x2 +x3 +x 4
b. x 4 3.
x1 +x2 +x3 +x 4
c. x 4 2.75.
2x 1 +3x2 +1x 3 +1x 4
d. x 7 16
7 .
2x 1 +3x2 +1x 3 +1x 4
e. x 7 17
7 .
2x 1 +3x2 +1x 3 +2x 4
f. x 7 22
7 .
g. Answers will vary.
h. If we have an existing arrangement and balancing point, moving one of the locations to
the left will move the balancing point to the left; similarly, moving one of the locations
to the right will move the balancing point to the right. If instead we add weight to an
existing location, if that location is left of the balancing point, the balancing point will
move left; the behavior is similar if on the right.
Activity 6.3.4 Answer.
∫ 20
a. M 0
4 + 0.1x dx 100 g.
b. Greater than 10.
∫ 20
x(4+0.1x)) dx
c. x 0
∫ 20 32
3 .
0
4+0.1x dx
d. 5 g/cm.
e. Slightly to the right of the center of mass for ρ(x).
∫ 20
x4e 0.020732x dx
f. x ∫0 20 ≈ 10.6891,
0
4e 0.020732x dx
6.4 · Physics Applications: Work, Force, and Pressure
547
Appendix B Answers to Activities
Activity 6.4.2 Answer.
∫ 200
a. W 0
0.3(200 − h) dh 6000 foot-pounds.
∫ 100
b. W 0
(40 − 0.1h) dh 3500foot-pounds.
c. BAVG[0,100] ≈ 25.9798 pounds.
d. For the given spring,
i. k 15.
∫1
ii. W 0
15x dx 15
2 foot-pounds.
∫ 1.5
iii. W 1
15x dx 9.375 foot-pounds.
Activity 6.4.3 Answer.
a. ∫ 3
W 9.81 · 4000π · x dx 308 190 newton-meters.
2
b. ∫ 8
W 62.4π(100 − x 2 )(x + 5) dx ≈ 673593 foot-pounds.
3
c. ∫ 3
25
W 62.4(50 − x)x dx 5720 foot-pounds.
1 2
Activity 6.4.4 Answer.
∫ x50
a. F x0
(6240x)dx 7 800 000 pounds .
∫ x30 √
b. F x10
124.8(x − 10) 900 − x 2 dx 800 244 pounds .
∫ x4
c. F x1
62.4(x − 1)(5 − 1.25x)dx 351 pounds .
6.5 · Improper Integrals
Activity 6.5.2 Answer.
∫ 10 ∫ 1000 ∫ 100000
a. i. 1
1
x dx ln(10) 1
1
x dx ln(1000) 1
1
x dx ln(100000)
∫b
ii. 1
1 x
dx ln(b).
∫b
iii. limb→∞ 1
1 x
dx limb→∞ ln(b) ∞
∫ 10 ∫ 1000 ∫ 100000
b. i. 1
1
x 3/2
dx 2− √2
1
1
x 3/2
dx 2− √2
1
1
x 3/2
dx 2− √ 2
10 1000 100000
∫b
ii. 1
1 x 3/2
dx 2− √2 .
b
∫b ( )
iii. limb→∞ 1
1
x 3/2
dx limb→∞ 2 − √2 2
b
c. Both graphs have a vertical asymptote at x 0 and for both graphs, the x-axis is a
548
horizontal asymptote. However, the graph of y 1
x 3/2
will ’’approach the x-axis faster’’
than the graph of y x1 .
d. The area bounded by the graph of y x1 , the x-axis, and the vertical line x 1 is infinite
or unbounded. However, The area bounded by the graph of y x 3/2 1
, the x-axis, and
the vertical line x 1 is equal to 2.
Activity 6.5.3 Answer.
∫∞
a. 1
1
x2
dx 1
∫∞
b. 0
e −x/4 dx 4
∫∞
c. 2
9
(x+5)2/3
dx ∞
∫∞
d. 4
3
(x+2)5/4
dx 12
61/4
∫∞
e. 0
xe −x/4 dx 16
∫∞
f. If 0 < p < 1, 1
1
xp dx diverges, while if p > 1, the integral converges.
Activity 6.5.4 Answer.
∫1
a. 1
0 x 1/3
dx 3
2
∫2
b. 0
e −x dx 1 − e −2
∫4
c. 0
√ 1 dx 4
4−x
∫2
1
d. −2 x 2
dx diverges.
∫ π/2
e. 0
tan(x)dx ∞
∫1
π
f. 0
√ 1 dx 2
1−x 2
7 · Differential Equations
7.1 · An Introduction to Differential Equations
Activity 7.1.2 Answer.
a. Let P be the population t the time in years; dP
dt 0.0125P.
b. Let m be the mass t the time in days; dm
dt −0.056m.
c. Let B be the balance t be time in years; dB
dt 0.04B − 1000.
d. Let t be time in minutes H the temperature of the hot chocolate; dH
dt −0.1(H − 70).
e. Let t be time in minutes and H the temperature of the soda;
dH
0.1(70 − H) −0.1(H − 70).
dt
549
Appendix B Answers to Activities
Activity 7.1.3 Answer.
a. For the skydiver:
dv dv dv
≈ 1.5 ≈ 1.2 ≈ 0.9
dt (v0.5) dt (v1) dt (v1.5)
dv dv
≈ 0.6 ≈ 0.3
dt (v2) dt (v2.5)
b. For the meteorite:
dv dv
≈ −0.3 ≈ −0.6
dt (v3.5) dt (v4)
dv dv
≈ −0.9 ≈ −1.2
dt (v4.5) dt (v5)
A graph of the points from parts (a) and (b) is shown in the following diagram:
c. dv
dt −0.6v + 1.8.
d. The rate of change of velocity with respect to time is a linear function of velocity.
e. 0 < v < 3.
f. 3 < v < 5.
g. v 3.
Activity 7.1.4 Answer.
a. v(t) 1.5t − 0.25t 2 is not a solution to the given DE.
b. v(t) 3 + 2e −0.5t is a solution to the given DE.
c. v(t) 3 is a solution to the given DE.
d. v(t) 3 + Ce −0.5t is a solution to the given DE for any choice of C.
550
7.2 · Qualitative behavior of solutions to DEs
Activity 7.2.2 Answer.
a. When y < 4, y is an increasing function of t. When y > 4, y is a decreasing function
of t.
b.
c.
d. ( )
dy 1
2 − e −t/2 −e −t/2
dt 2
551
Appendix B Answers to Activities
and
1 1( )
− (y − 4) − 4 + 2e −t/2 −e −t/2
2 2
In addition, y(0) 4 + 2e 0 6.
e. A constant function.
Activity 7.2.3 Answer.
a. When y < 0 and when y > 4, y is a decreasing function of t. When 0 < y < 4, y is a
increasing function of t.
b. y 0 and y 4.
c.
d.
e. y 4 is stable; y 0 is unstable.
f. Tend to 4.
g. Figure 7.2.12 is for an ustable equilibrium; Figure 7.2.13 is for a stable equilibrium.
552
7.3 · Euler’s method
Activity 7.3.2 Answer.
a.
ti yi dy/dt ∆y
0 0 −1 −0.2
0.2 −0.2 −0.6 −0.12
0.4 −0.32 −0.2 −0.04
0.6 −0.36 0.2 0.04
0.8 −0.32 0.6 0.12
1.0 −0.2 1 0.2
b. y t 2 − t, with errors e1 0.04, e2 0.08, e3 0.12, e4 0.16, e5 0.2.
dy
c. If we first think about how y1 is generated for the initial value problem dt f (t)
2t − 1, y(0) 0, we see that y1 y0 + ∆t · f (t0 ). Since y0 0, we have y1 ∆t · f (t0 ).
From there, we know that y2 is given by y2 y1 + ∆t f (t1 ). Substituting our earlier
result for y1 , we see that y2 ∆t · f (t0 ) + ∆t f (t1 ). Continuing this process up to y5 , we
get
y5 ∆t · f (t0 ) + ∆t f (t1 ) + ∆t f (t2 ) + ∆t f (t3 ) + ∆t f (t4 )
This is precisely the left Riemann sum with five subintervals for the definite integral
∫1
0
(2t − 1) dt.
d. Solutions to this differential equation all differ by only a constant.
Activity 7.3.3 Answer.
a.
553
Appendix B Answers to Activities
b. y 0 or y 6; y 0 is unstable, y 6 is stable.
c. The solution will tend to y 6.
d.
ti yi dy/dt ∆y
0.0 1.0000 5.0000 1.0000
0.2 2.0000 8.0000 1.6000
0.4 3.6000 8.6400 1.7280
0.6 5.3280 3.5804 0.7161
0.8 6.0441 −0.2664 −0.0533
1.0 5.9908 0.0551 0.0110
e. The value of y i 6 for every value of i.
7.4 · Separable differential equations
Activity 7.4.2 Answer.
a. dP
dt 0.03P
b. P Ce 0.03t .
554
c. P 10000e 0.03t .
ln(2)
d. The doubling time is t 0.03 ≈ 23.105 years.
e. The doubling time is t 1
k ln(2).
Activity 7.4.3 Answer.
a. k 1
30
b. T 75 + Ce −t/30
c. The temperature of the coffee tends to 75 degrees.
d. T(20) 75 + 30e −2/3 ≈ 90.4◦ F.
(1)
e. t −30 ln 6 ≈ 53.75 minutes.
Activity 7.4.4 Answer.
( 2
)
2t− t2
a. y −1 + Ce .
( )
2
b. y 1
2 ln e t + C .
c. y −1 + 3e 2t .
d. y − 2t+
1
3 − 4t+3 .
2
2
e. y 4
t 2 +1
.
7.5 · Modeling with differential equations
Activity 7.5.2 Answer.
a. dA
dt 0.05A.
b. dA
dt 0.05A − 10000.
c.
The only equilibrium solution is A 200000.
d. t 20 ln(2) ≈ 13.86years.
555
Appendix B Answers to Activities
e. At least $200000.
f. Up to $15000 every year.
Activity 7.5.3 Answer.
a. dM
dt −kM, where k is a positive constant.
(1)
b. k − 12 ln 2 ≈ 0.34657.
c. dM
dt 3 − kM, where k is a positive constant.
d. The equilibrium solution mM 3
k is stable.
( −kt
)
e. M 3
k 1−e .
f. About 2.426 milligrams per hour.
7.6 · Population Growth and the Logistic Equation
Activity 7.6.2 Answer.
a. P ′(0) ≈ 0.0755.
b. P(0) 6.084.
c. k ≈ 0.012041.
d. P(t) 6.084e 0.012041t .
e. P(10) ≈ 6.8878.
( )
f. t 1
0.012041 ln 12
6.084 ≈ 56.41, or in the year 2056.
g. P(500) ≈ 3012.3 billion.
Activity 7.6.3 Answer.
a. When P N
2.
b. When the population is 6.125 billion.
c. P 12.5
1.0546e −0.025t +1
; P(100) 11.504 billion.
( )
( 12.5
9 −1)
d. t 1
−0.025 ln 1.0546 ≈ 39.9049 (so in about year 2040).
e. limt→∞ P(t) N.
8 · Sequences and Series
8.1 · Sequences
Activity 8.1.2 Answer.
a. s n n. A plot of the first 50 points in the sequence is shown here.
556
20
sn
18
16
14
12
10
8
6
4
2
n
2 4 6 8 10 12 14 16 18 20
This sequence does not have a limit as n goes to infinity.
b. s n is s n n1 . A plot of the first 50 points in the sequence is shown here.
1.00
sn
0.75
0.50
0.25
n
2 4 6 8 10 12 14 16 18 20
This sequence has a limit of 0 as n goes to infinity.
c. s n n+1
n . A plot of the first 50 points in the sequence is shown here.
2.0
sn
1.5
1.0
0.5
n
2 4 6 8 10 12 14 16 18 20
This sequence has a limit of 1 as n goes to infinity.
Activity 8.1.4 Answer.
{ 1+2n }
a. The sequence 3n−2 converges to 23 .
{ 5+3n
}
b. The sequence 10+2n diverges to infinity.
10n
c. n! → 0 as n → ∞.
8.2 · Geometric Series
Activity 8.2.2 Answer.
a. rS n ar + ar 2 + ar 3 + · · · + ar n .
b. S n − rS n a − ar n .
557
Appendix B Answers to Activities
n
c. S n a 1−r
1−r .
Activity 8.2.3 Answer.
a. Observe that
S lim S n .
n→∞
b. If r > 1, then limn→∞ r n ∞. If 0 < r < 1, then limn→∞ r n 0.
c. Since
1 − rn
S lim S n lim a
n→∞ n→∞ 1−r
and
lim r n 0
n→∞
for 0 < r < 1, we conclude that
1 − rn a
S lim a
n→∞ 1−r 1−r
when 0 < r < 1.
Activity 8.2.4 Answer.
(1) [ (1) ( 1 )2 ]
a. (2) 3 1+ 3 + 3 +··· .
ar 3
b. ar 3 + ar 4 + ar 5 + · · · 1−r .
ar n
c. r n + ar n+1 + ar n+2 + · · · 1−r .
8.3 · Series of Real Numbers
Activity 8.3.2 Answer.
a. See the table in part c.
b. See the table in part c.
∑
1
1 ∑
6
1
c. 1 1.491388889
k2 k2
k1 k1
∑
2
1 ∑
7
1
1.25 1.511797052
k2 k2
k1 k1
∑
3
1 ∑
8
1
1.361111111 1.527422052
k2 k2
k1 k1
∑
4
1 ∑
9
1
1.423611111 1.539767731
k2 k2
k1 k1
∑
5
1 ∑
10
1
1.463611111 1.549767731
k2 k2
k1 k1
558
d. It appears {S n } converges to something a bit larger than 1.5.
Activity 8.3.3 Answer.
∑n
a. S n k1 ak .
∑n−1
b. S n−1 k1 ak .
∑n ∑n−1
c. k1 ak − k1 ak an .
d. limn→∞ S n L and limn→∞ S n−1 L.
e. We have limn→∞ a n limn→∞ (S n − S n−1 ) 0.
Activity 8.3.4 Answer.
∑ k
a. k+1 diverges.
∑
b. (−1)k diverges.
c. The Divergence Test does not apply.
Activity 8.3.5 Answer.
∑∞
a. The nth partial sum of the series 1
k1 k is the left hand Riemann sum of f (x) on the
interval [1, n].
∑n ∫n
b. 1
k1 k > 1
1
x dx.
∑∞ ∫∞
c. 1
k1 k > 1
1
x dx.
∫∞ ∫t ∑∞
d. 1
f (x) dx limt→∞ 1
1 x
dx∞ so the series 1
k1 k diverges.
Activity 8.3.6 Answer.
∑∞ 1
a. k1 k 2 converges.
∑∞
b. 1
k1 k p converges when p > 1.
∑∞
c. 1
k1 k p diverges when p < 1.
∑∞
d. The p-series 1
k1 k p converges if and only if p > 1.
Activity 8.3.7 Answer.
k+1 k
a. k 3 +2
looks like k3
when k is large.
ak
b. limk→∞ bk 1 so a k ≈ b k for large values of k.
∑ k+1
c. k 3 +2
converges.
∑ k 2 +1
Activity 8.3.8 Answer. k 4 +2k+2
converges.
Activity 8.3.9 Answer.
a.
559
Appendix B Answers to Activities
a k+1
k
ak
5 0.6583679115
10 0.6665951585
20 0.6666666642
21 0.6666666658
22 0.6666666664
23 0.6666666666
24 0.6666666666
25 0.6666666667
a k+1
b. ak ≈ 2
3 when k is large.
∑ 2k
c. 3k −k
converges.
Activity 8.3.10 Answer.
∑ n
a. 2n converges.
∑ k 3 +2
b. k 2 +1
diverges.
∑ 10k
c. k! converges.
∑ k 3 −2k 2 +1
d. k 6 +4
converges.
8.4 · Alternating Series
Activity 8.4.2 Answer.
a.
∑1 ∑6
(−1)k+1 1k 1 (−1)k+1 1k 0.6166666667
∑2k1 ∑7k1
(−1)k+1 1k 0.5 (−1)k+1 1k 0.7595238095
∑3k1 ∑8k1
(−1)k+1 1k 0.8333333333 (−1)k+1 1k 0.6345238095
∑4k1 ∑9k1
(−1)k+1 1k 0.5833333333 k1 (−1)
k+1 1 0.7456349206
∑5k1 ∑10 k
k1 (−1) k1 (−1)
k+1 1 0.7833333333 k+1 1 0.6456349206
k k
b. There appears to be a limit for the sequence of partial sums.
Activity 8.4.3 Answer.
∑∞ (−1)k
a. k1 k 2 +2 converges.
∑∞ 2(−1)k+1 k
b. k1 k+5 diverges.
∑∞ (−1)k
c. k2 ln(k) converges.
Activity 8.4.4 Answer. S10 ≈ 0.9469925924; 720 π
7 4 ≈ 0.9470328299.
Activity 8.4.5 Answer.
∑∞
a. 1 − 1
4 − 1
9 + 1
16 + 1
25 + 1
36 − 1
49 − 1
64 − 1
81 − 1
100 +··· < 1
k1 k 2 .
∑∞
b. 1 − 1
4 − 1
9 + 1
16 + 1
25 + 1
36 − 1
49 − 1
64 − 1
81 − 1
100 +··· > k1 − k 2 .
1
560
∑∞ ∑∞
k1 − k 2
1 1
c. We expect this series to converge to some finite number between and k1 k 2 .
Activity 8.4.6 Answer.
∑ ln(k)
a. i. (−1)k k converges.
∑ ln(k)
ii. (−1) k
k converges conditionally.
∑ ln(k)
i. (−1)kk2
converges.
∑ ln(k)
ii. (−1)k k 2 converges absolutely.
Activity 8.4.7 Answer.
∑∞
a. √2 diverges.
k3 k−2
∑∞ k
b. k1 1+2k diverges.
∑∞ 2k 2 +1
c. k0 k 3 +k+1 diverges.
∑∞ 100k
d. k0 k! converges.
∑∞ 2k 2
e. k1 5k is a geometric series with ratio 5 and sum 23 .
∑∞ k 3 −1
f. k1 k 5 +1 converges.
∑∞ 3k−1
g. k2 7k
is a convergent geometric series with a 3
49 and r 3
7 and sum 3
28 .
∑∞ 1
h. k2 k k converges.
∑∞ (−1)k+1
i. √ converges.
k1 k+1
∑∞ 1
j. k2 k ln(k) diverges.
∑∞ (−1)k
k. k2 ln(k) converges very slowly.
8.5 · Taylor Polynomials and Taylor Series
Activity 8.5.2 Answer.
a. i. f k (0) 0 if k is odd, and f 2k (0) (−1)k .
ii. Pn (x) 1 − x2 x4 x6 n/2 x n x2 x4 x6
2 + 4! − 6! + · · · + (−1) n! if n is even and Pn (x) 1 − 2 + 4! − 6! +
(n−1)/2 x ( n−1)
···+ (−1) (n−1)!
if n is odd.
b. i. f k (0) 0 if k is even and f 2k+1 (0) (−1)k .
x3 x5 x7 n x3 x5
ii. Pn (x) x − 3! + 5! − 7! + · · · + (−1)(n−1)/2 xn! if n is odd and Pn (x) x − 3! + 5! −
x7 n/2+1 x n−1 if n is even.
7! + · · · + (−1) (n−1)!
c. i. f (k) (0) k!.
561
Appendix B Answers to Activities
ii.
∑
n
Pn (x) xk.
k0
Activity 8.5.3 Answer.
a. P(x) 1 + x + x 2 + x 3 + · · · + x n + · · ·
b. P(x) 1 − 1 2
2! x + 1 4
4! x − · · · + (−1)n (2n)!
1
x 2n + · · ·.
c. P(x) x − 1 3
3! x + 1 5
5! x − · · · + (−1)n (2n+1)!
1
x 2n+1 + · · ·.
d. Pn (x) 1 + x + x 2 + x 3 + · · · + x n
Activity 8.5.4 Answer.
a. It appears that as we increase the order of the Taylor polynomials, they fit the graph
of f better and better over larger intervals.
b. It appears that as we increase the order of the Taylor polynomials, they fit the graph
of f better and better over larger intervals.
c. The Taylor polynomials converge to 1
1−x only on the interval (−1, 1).
Activity 8.5.5 Answer.
a. (−∞, ∞).
b. (−∞, ∞).
c. The interval (−1, 1).
Activity 8.5.6 Answer. n ≥ 27.
Activity 8.5.7 Answer.
a. Compare Example 8.5.6.
b. i. Use the fact that that | f (n) (x)| ≤ e c on the interval [0, c] for any fixed positive
value of c.
ii. Repeat the argument in (a) but replace e c with 1, and everything else holds in the
same way.
iii. Combine the results of (a) and (b)
c. n 28.
8.6 · Power Series
Activity 8.6.2 Answer.
a. [0, 2).
b. (−1, 1).
c. (−5, 3).
d. (−∞, ∞).
e. {0}.
562
Activity 8.6.3 Answer.
∑∞
a. 1
1−x k0 x k for −1 < x < 1.
∑∞
b. f (x) 1(−x 2 ) k0 (−1)
k x 2k .
c. −1 < x < 1.
Activity 8.6.4 Answer.
x3 x5 k−1
a. f ′(x) −x + 3! − 5! + · · · + (−1)k (k−1)!
x
+ · · ·.
b. d
dx cos(x) − sin(x).
x2 x4
c. f ′′(x) −1 + 2! − 4! + · · ·.
∑∞ k x k+1
Activity 8.6.5 Answer. ln(1 + x) k0 (−1) k+1 . for −1 < x < 1.
563
Appendix B Answers to Activities
564
APPENDIX C
Answers to Selected Exercises
This appendix contains answers to all non-WeBWorK exercises in the text. For WeBWorK
exercises, please use the html version of the text for access to answers and solutions.
1 · Understanding the Derivative
1.1 · How do we measure velocity?
1.1.4.6. Answer.
a. s(15) − s(0) ≈ −98.75.
b.
s(15) − s(0)
AV[0,15] ≈ −6.58
15 − 0
s(2) − s(0)
AV[0,2] ≈ −47.63
2−0
s(6) − s(1)
AV[1,6] ≈ −13.25
6−1
s(10) − s(8)
AV[8,10] ≈ −7.35
10 − 8
c. Most negative average velocity on [0, 4]; most positive average velocity on [4, 8].
d. 21.31+22.25
2 21.78 feet per second.
e. The average velocities are negative; the instantaneous velocity was positive. Down-
ward motion corresponds to negative average velocity; upward motion to positive av-
erage velocity.
1.1.4.7. Answer.
a. Sketch a plot where the diver’s height at time t is on the vertical axis. For instance,
h(2.45) 0.
b. AV[2.45,7] ≈ −3.5−0 −3.5
7−2.45 4.55 −0.7692 m/sec. The average velocity is not the same on
every time interval within [2.45, 7].
c. When the diver is going upward, her velocity is positive. When she is going down-
ward, her velocity is negative. At the peak of her dive and when her feet touch the
Appendix C Answers to Selected Exercises
bottom of the pool.
d. It looks like when the position function is steep, the velocity function’s value is farther
away from zero, and that whenever the height/position function is rising/increasing,
the velocity function has a positive value. Similarly, whenever the position function is
decreasing, the velocity is negative.
1.1.4.8. Answer.
a. 15957 people.
b. In an average year the population grew by about 798 people/year.
c. The slope of a secant line through the points (a, f (a)) and (b, f (b)).
d. AV[0,20] ≈ 798 people per year.
e.
AV[5,10] ≈ 734.50
AV[5,9] ≈ 733.06
AV[5,8] ≈ 731.62
AV[5,7] ≈ 730.19
AV[5,6] ≈ 728.7535
1.2 · The notion of limit
1.2.4.5. Answer.
a. All real numbers except x ±2.
b.
x f (x)
2.1 −8.41
2.01 −8.0401
2.001 −8.004001
1.999 −7.996001
1.99 −7.9601
1.9 −7.61
limx→2 f (x) −8.
16−x 4
c. limx→2 x 2 −4
−8.
d. False.
e. False.
f.
566
1.2.4.6. Answer.
a. All real numbers except x −3.
b.
x 1(x)
−2.9 −1
−2.99 −1
−2.999 −1
−3.001 1
−3.01 −1
−3.1 −1
The limit does not exist.
c. If x > −3,
|x + 3| x+3
− − −1;
x+3 x+3
if x < −3, it follows that
|x + 3| −(x + 3)
− − +1.
x+3 x+3
Hence the limit does not exist.
d. False.
e. False.
f.
567
Appendix C Answers to Selected Exercises
1.2.4.7. Answer.
a.
b.
568
1.2.4.8. Answer.
a.
100 cos(0.75(1 + h)) · e −0.2(1+h) − 100 cos(0.75) · e −0.2
AV[1,1+h]
h
b.
lim AV1,1+h ≈ −53.837.
h→0
c. The instantaneous velocity of the bungee jumper at the moment t 1 is approximately
−53.837 ft/sec.
1.3 · The derivative of a function at a point
1.3.3.6. Answer.
a.
y
f
4
x
-4 4
-4
b. AV[−3,−1] ≈ 1.15; AV[0,2] ≈ −0.4.
c. f ′(−3) ≈ 3; f ′(0) ≈ − 12 .
1.3.3.7. Answer.
a. For instance, you could let f (−3) 3 and have f pass through the points (−3, 3),
(−1, −2), (0, −3), (1, −2), and (3, −1) and draw the desired tangent lines accordingly.
b. For instance, you could draw a function 1 that passes through the points (−2, 3), (−1, 2),
(1, 0), (2, 0), and (3, 3) in such a way that the tangent line at (−1, 2) is horizontal and
the tangent line at (2, 0) has slope 1.
1.3.3.8. Answer.
a. AV[0,7] 0.1175
7 ≈ 0.01679 billion people per year; P ′(7) ≈ 0.1762 billion people per
year; P ′(7) > AV[0,7] .
b. AV[19,29] ≈ 0.02234 billion people/year.
569
Appendix C Answers to Selected Exercises
c. We will say that today’s date is July 1, 2015, which means that t 22.5;
115(1.014)22.5+h − 115(1.014)22.5
P ′(22.5) lim ;
h→0 h
P ′(22.5) ≈ 0.02186 billions of people per year.
d. y − 1.57236 0.02186(t − 22.5).
1.3.3.9. Answer.
a. All three approaches show that f ′(2) 1.
b. All three approaches show that f ′(1) −1.
c. All three approaches show that f ′(1) 12 .
d. All three approaches show that f ′(1) does not exist.
e. The first two approaches show that f ′( π2 ) 0.
1.4 · The derivative function
1.4.3.6. Answer.
a. See the figure below.
b. See the figure below.
3 3 y = f ′ (x)
y = f (x)
-3 3 -3 3
-3 -3
c. One example of a formula for f is f (x) 12 x 2 − 1.
1.4.3.7. Answer.
a. 1 ′(x) 2x − 1.
b.
570
c. p ′(x) 10x − 4.
d. The constants 3 and 12 don’t seem to affect the results at all. The coefficient −4 on the
linear term in p(x) appears to make the ``−4’’ appear in p ′(x) 10x − 4. The leading
coefficient 5 in (x) 5x 2 − 4x + 12 leads to the coefficient of ``10’’ in p ′(x) 10x − 4.
1.4.3.8. Answer.
a. 1 is linear.
b. On −3.5 < x < −2, −2 < x < 0 and 2 < x < 3.5.
c. At x −2, 0, 2; 1 must have sharp corners at these points.
d.
2 2 y = g′ (x)
y = g(x)
-2 2 -2 2
-2 -2
571
Appendix C Answers to Selected Exercises
1.4.3.9. Answer.
f
f
x x
f′ f′
x x
f f
x x
f′ f′
x
x
1.5 · Interpreting, estimating, and using the derivative
1.5.4.5. Answer.
a. F′(10) ≈ −3.33592.
b. The coffee’s temperature is decreasing at about 3.33592 degrees per minute.
c. F′(20).
572
d. We expect F′ to get closer and closer to 0 as time goes on.
1.5.4.6. Answer.
a. If a patient takes a dose of 50 ml of a drug, the patient will experience a body temper-
ature change of 0.75 degrees F.
b. ``degrees Fahrenheit per milliliter.’’
c. For a patient taking a 50 ml dose, adding one more ml to the dose leads us to expect
a temperature change that is about 0.02 degrees less than the temperature change in-
duced by a 50 ml dose.
1.5.4.7. Answer.
a. t 0.
b. v ′(1) −32.
c. ``feet per second per second’’; v ′(1) −32 tells us that the ball’s velocity is decreasing
at a rate of 32 feet per second per second.
d. The acceleration of the ball.
1.5.4.8. Answer.
a. AV[40000,55000] ≈ −0.153 dollars per mile.
b. h ′(55000) ≈ −0.147 dollars per mile. During 550001st mile, we expect the car’s value
to drop by 0.147 dollars.
c. h ′(30000) < h ′(80000).
d. The graph of h might have the general shape of the graph of y e −x for positive values
of x: always positive, always decreasing, and bending upwards while tending to 0 as
x increases.
1.6 · The second derivative
1.6.5.6. Answer.
a. f is increasing and concave down near x 2.
b. Greater.
c. Less.
d.
573
Appendix C Answers to Selected Exercises
1.6.5.7. Answer.
a. 1 ′(2) ≈ 1.4.
b. At most one.
c. 9.
d. 1 ′′(2) ≈ 5.5.
1.6.5.8. Answer.
a. h ′(4.5) ≈ 14.3; h ′(5) ≈ 21.2; h ′(5.5) ≈ 23.9; rising most rapidly at t 5.5.
b. h ′(5) ≈ 9.6.
c. Acceleration of the bungee jumper in feet per second per second.
d. 0 < t < 2, 6 < t < 10.
1.6.5.9. Answer.
a.
574
b.
c.
d.
575
Appendix C Answers to Selected Exercises
1.7 · Limits, Continuity, and Differentiability
1.7.5.5. Answer.
a. a 0.
b. a 0, 3.
c. a −2, 0, 1, 2, 3.
d.
p
3 3
p′
-3 3 -3 3
-3 -3
1.7.5.6. Answer.
a. f (x) |x − 2|.
b. Impossible.
c. Let f be the function defined to be f (x) 1 for every value of x , −2, and such that
f (−2) 4.
576
d.
3
p
2
1
-2 -1 1 2
-1
1.7.5.7. Answer.
a. h must be piecewise linear with slope of 1 or −1, depending on the interval.
b. h ′(x) is not defined for x −2, 0, 2.
c. It is possible that h is not continuous at x −2, 0, 2.
d. Two of the many possible graphs for h are shown in the following figure.
3 3
2
y = h(x) y = h′ (x)
1
-3 3 -3 -2 -1 1 2 3
-1
-2
-3 -3
1.7.5.8. Answer.
a. At x 0.
1(0 + h) − 1(0)
1 ′(0) lim
h→0 h
577
Appendix C Answers to Selected Exercises
√ √
|h| − |0|
lim
h→0 h
√
|h|
lim
h→0 h
b.
h 0.1 0.01 0.001 0.0001 −0.1 −0.01 −0.001 −0.0001
√
|h|/h 3.162 10 31.62 100 −3.162 −10 −31.62 −100
1 ′(0) does not exist.
c.
3 3
y = g(x) y = g′ (x)
2
1
-3 3 -3 -2 -1 1 2 3
-1
-2
-3 -3
1.8 · The Tangent Line Approximation
1.8.4.5. Answer.
a. p(3) −1 and p ′(3) −2.
b. p(2.79) ≈ −0.58.
c. Too large.
d.
578
2 3 4
= p(x) (3, −1)
y = L(x)
1.8.4.6. Answer.
a. F′(60) ≈ 1.56 degrees per minute.
b. L(t) ≈ 1.56(t − 60) + 324.5.
c. F(63) ≈ L(63) ≈ 329.18 degrees F.
d. Overestimate.
1.8.4.7. Answer.
a. s(9.34) ≈ L(9.34) 3.592.
b. underestimate.
c. The object is slowing down as it moves toward toward its starting position at t 4.
1.8.4.8. Answer.
a. x 1.
b. On −0.37 < x < 1.37; f is concave up.
c. f (1.88) ≈ −3.0022, and this estimate is larger than the true value of f (1.88).
2 · Computing Derivatives
2.1 · Elementary derivative rules
2.1.5.10. Answer.
a. h(2) 27; h ′(2) −19/2.
b. L(x) 27 − 2 (x
19
− 2).
c. p is increasing at x 2.
d. p(2.03) ≈ −11.44.
579
Appendix C Answers to Selected Exercises
2.1.5.11. Answer.
a. p is not differentiable at x −1 and x 1; q is not differentiable at x −1 and x 1.
b. r is not differentiable at x −1 and x 1.
c. r ′(−2) 4; r ′(0) 12 .
d. y 4.
2.1.5.12. Answer.
a. w ′(t) 3t t (ln(t) + 1) + 2 √ 1 2 .
1−t
b. L(t) ( √3 − 3 )
2π
+ ( √3 (ln( 12 ) + 1) + √4 )(t − 12 ).
2 2 3
c. v is decreasing at t 12 .
2.1.5.13. Answer.
a.
a x+h − a x
f ′(x) lim
h→0 h
a · ah − ax
x
lim
h→0 h
a (a − 1)
x h
lim ,
h→0 h
b. Since a x does not depend at all on h, we may treat a x as constant in the noted limit and
thus write the value a x in front of the limit being taken.
c. When a 2, L ≈ 0.6931; when a 3, L ≈ 1.0986.
d. a ≈ 2.71828 (for which L ≈ 1.000)
dx [2 ] 2x · ln(2) and dx [3 ] 3x · ln(3)
d x d x
e.
f.
d x
[e ] e x .
dx
2.2 · The sine and cosine functions
2.2.3.1. Answer.
a. V ′(2) 24 · 1.072 · ln(1.07) + 6 cos(2) ≈ −0.63778 thousands of dollars per year.
b. V ′′(2) 24 · 1.072 · ln(1.07)2 − 6 sin(2) ≈ −5.33 thousands of dollars per year per year.
At this moment, V ′ is decreasing and we expect the derivative’s value to decrease by
about 5.33 thousand dollars per year over the course of the next year.
c. See the figure below. Adding the term 6 sin(t) to A to create the function V adds volatil-
ity to the value of the portfolio.
580
2.2.3.2. Answer.
(π) (√ )
a. f ′ 4 −5 2
2
.
b. L(x) 3 + 2(x − π).
c. Decreasing.
d. The tangent line to f lies above the curve at this point.
2.2.3.3. Answer.
a. Hint: in the numerator of the difference quotient, combine the first and last terms and
remove a factor of sin(x).
b. Hint: divide each part of the numerator by h and consider the sum of two separate
limits.
( ) ( )
cos(h)−1 sin(h)
c. limh→0 h 0 and limh→0 h 1.
d. f ′(x) sin(x) · 0 + cos(x) · 1.
e. Hint: cos(α + β) is cos(α + β) cos(α) cos(β) − sin(α) sin(β).
2.3 · The product and quotient rules
2.3.5.10. Answer.
a. h(2) −15; h ′(2) 23/2.
b. L(x) −15 + 23/2(x − 2).
c. Increasing.
d. r(2.06) ≈ −0.5796.
2.3.5.11. Answer.
a. w ′(t) t t (ln t + 1) · (arccos t) + t t · √ −1 .
1−t 2
581
Appendix C Answers to Selected Exercises
b. L(t) ≈ 0.740 − 0.589(t − 0.5).
c. Increasing.
2.3.5.12. Answer.
a. r ′(−2) 5 and r ′(0) 1.
b. At x −1 and x 1.
c. L(x) 2.
d. z ′(0) − 14 and z ′(2) −1.
e. At x −1, x 1, x −1.5, and x 1.
2.3.5.13. Answer.
a. C(t) A(t)Y(t) bushels in year t.
b. 1190000 bushels of corn.
c. C′(t) A(t)Y ′(t) + A′(t)Y(t).
d. C′(0) 158000 bushels per year.
e. C(1) ≈ 1348000bushels.
2.3.5.14. Answer.
a. 1(80) 20 kilometers per liter, and 1 ′(80) −0.16. kilometers per liter per kilometer
per hour.
b. h(80) 4 liters per hour and h ′(80) 0.082 liters per hour per kilometer per hour.
c. Think carefully about units and how each of the three pairs of values expresses funda-
mentally the same facts.
2.4 · Derivatives of other trigonometric functions
2.4.3.6. Answer.
−2 sin(2)−2 cos(2) ln(1.2)
a. h ′(2) 1.22
≈ −1.1575 feet per second.
cos(2)(−2+2ln 2 (1.2))+4 ln(1.2) sin(2))
b. h ′′(2) 1.22
≈ 1.0193 feet per second per second.
c. The object is falling and slowing down.
2.4.3.7. Answer.
a. f ′(x) sin(x) · (− csc2 (x)) + cot(x) · cos(x).
b. False.
− sin2 (x) π
c. f ′(x) sin(x)
− sin(x) for x , 2 + kπ for some integer value of k.
2.4.3.8. Answer.
(z 2 sec(z)+1)(z sec2 (z)+tan(z))−z tan(z)(z 2 sec(z) tan(z)+2z sec(z))
a. p ′(z) 2 + 3e z
2 (z sec(z)+1)
b. y − 4 3(x − 0).
582
c. Increasing.
2.5 · The chain rule
2.5.5.8. Answer.
(π)
a. h ′ 4 3
√ .
2 2
b. r ′(0.25) cos(0.253 ) · 3(0.25)2 ≈ 0.1875 > h ′(0.25) 3 sin2 (0.25) · cos(0.25) ≈ 0.1779; r
is changing more rapidly.
c. h ′(x) is periodic; r ′(x) is not.
2.5.5.9. Answer.
a. p ′(x) e u(x) · u ′(x).
b. q ′(x) u ′(e x ) · e x .
c. r ′(x) − csc2 (u(x) · u ′(x).
d. s ′(x) u ′(cot(x)) · (− csc2 (x)).
e. a ′(x) u ′(x 4 ) · 4x 3 .
f. b ′(x) 4(u(x))3 · u ′(x).
2.5.5.10. Answer.
a. C′(0) 0 and C′(3) − 12 .
b. Consider C′(1). By the chain rule, we’d expect that C′(1) p ′(q(1)) · q ′(1), but we know
that q ′(1) does not exist since q has a corner point at x 1. This means that C′(1) does
not exist either.
c. Since Y(x) q(q(x)), the chain rule implies that Y ′(x) q ′(q(x)) · q ′(x), and thus
Y ′(−2) q ′(q(−2)) · q ′(−2) q ′(−1) · q ′(−2). But q ′(−1) does not exist, so Y ′(−2) also
fails to exist. Using Z(x) q(p(x)) and the chain rule, we have Z′(x) q ′(p(x)) · p ′(x).
Therefore Z′(0) q ′(p(0)) · p ′(0) q ′(−0.5) · p ′(0) 0 · 0.5 0.
2.5.5.11. Answer.
a. dV
dh h1 7π cubic feet per foot.
b. h ′(2) π cos(2π) π feet per hour.
c. dV
dt t2 7π 2 cubic feet per hour.
d. In (a) we are determining the instantaneous rate at which the volume changes as we in-
crease the height of the water in the tank, while in (c) we are finding the instantaneous
rate at which volume changes as we increase time.
2.6 · Derivatives of Inverse Functions
2.6.6.9. Answer.
( )
a. f ′(x) 1
2 arctan(x)+3 arcsin(x)+5
· 2
1+x 2
+ √ 3 .
1−x 2
583
Appendix C Answers to Selected Exercises
( )
b. r ′(z) 1
· 1
· √ 1 .
1+(ln(arcsin(z)))2 arcsin(z) 1−z 2
[ ( )] [ ( )]
c. q ′(t) arctan2 (3t) · 4 arcsin3 (7t) √ 7 + arcsin4 (7t) · 2 arctan(3t) 3
1+(3t)2
.
1−(7t)2
( )
(arcsin(v)+v 2 )· 1
−arctan(v)· √ 1
+2v
1+v 2
d. 1 ′(v) 1
arctan(v) · (arcsin(v)+v 2 )2
1−v 2
arcsin(v)+v 2
2.6.6.10. Answer.
a. f ′(1) ≈ 2.
b.
y = f −1 (x)
y = f (x)
c. ( f −1 )′(−1) ≈ 1/2.
2.6.6.11. Answer.
a. f passes the horizontal line test.
√3
b. f −1 (x) 1(x) 4x − 16.
c. f ′(x) 34 x 2 ; f ′(2) 3. 1 ′(x) 13 (4x − 16)−2/3 · 4; 1 ′(6) 13 . These two derivative values
are reciprocals.
2.6.6.12. Answer.
a. h passes the horizontal line test.
b. The equation y x + sin(x) can’t be solved for x in terms of y.
c. (h −1 )′( π2 + 1) 1.
2.7 · Derivatives of Functions Given Implicitly
2.7.3.6. Answer. Horizontal tangent lines: (0, −1), (0, −0.618), (0, 1.618), (1, −1), (1, −0.618),
(1, 1.618), (0.5, −1.0493), (0.5, 0.2104), (0.5, 1.6139). Vertical tangent lines: (−0.1756, −0.379),
(0.2912, −0.379), (0.7088, −0.379), (1.1756, −0.379), (−0.8437, 1.235), and (1.8437, 1.235).
584
π
( π
)
2.7.3.7. Answer. y 2 − x− 2 .
2.7.3.8. Answer.
ln(y)
a. x ln(a)
.
1 dy
b. 1 1
ln(a)
· y dx .
dx [a ] a x ln(a).
d x
c.
2.8 · Using Derivatives to Evaluate Limits
2.8.4.5. Answer. limx→3 h(x) −2.
585
Appendix C Answers to Selected Exercises
g
1 2 3 4 5
f
3(a−b)
2.8.4.6. Answer. Horizontal asymptote: y 35 ; vertical asymptote: x c; hole: (a, 5(a−c)
).
R is not continuous at x a and x c.
2.8.4.7. Answer.
a. ln(x 2x ) 2x · ln(x).
b. x 1
1 .
x
c. limx→0+ h(x) 0.
d. limx→0+ 1(x) limx→0+ x 2x 1.
2.8.4.8. Answer.
ln(x)
a. Show that limx→∞ √
x
0.
ln(x)
b. Show that limx→∞ √n
x
0.
p(x)
c. Consider limx→∞ e x By repeated application of LHR, the numerator will eventually
be simply a constant (after n applications of LHR), and thus with e x still in the denom-
586
inator, the overall limit will be 0.
ln(x)
d. Show that limx→∞ xn 0
e. For example, f (x) 3x 2 + 1 and 1(x) −0.5x 2 + 5x − 2.
3 · Using Derivatives
3.1 · Using derivatives to identify extreme values
3.1.4.4. Answer.
a. f ′ is positive for −1 < xlt1 and for x > 1; f ′ is negative for all x < −1. f has a local
minimum at x −1.
b. A possible graph of y f ′′(x) is shown at right in the figure.
c. f ′′(x) is negative for −0.35 < x < 1; f ′′(x) is positive everywhere else; f has points of
inflection at x ≈ −0.35 and x 1.
d. A possible graph of y f (x) is shown at left in the figure.
f′ f ′′
2
f
x x
x
1 1 1
(0, −0.25)
3.1.4.5. Answer.
a. Neither.
b. 1 ′′(2) 0; 1 ′′ is negative for 1 < x < 2 and positive for 2 < x < 3.
c. 1 has a point of inflection at x 2.
3.1.4.6. Answer.
a. h can have no, one, or two real zeros.
587
Appendix C Answers to Selected Exercises
b. One root is negative and the other positive.
c. h will look like a line with slope 3.
d. h is concave up everywhere; h is almost linear for large values of |x|.
3.1.4.7. Answer.
a. p ′′(x) is negative for −1 < x < 2 and positive for all other values of x; p has points of
inflection at x −1 and x 2.
b. Local maximum.
c. Neither.
3.2 · Using derivatives to describe families of functions
3.2.3.3. Answer.
a.
b. x 0 and x 2a
3 .
588
c. x 3a ; p ′′(x) changes sign from negative to positive at x 3a .
d. As we increase the value of a, both the location of the critical number and the inflection
point move to the right along with a.
3.2.3.4. Answer.
e −x e −x
a. x c is a vertical asymptote because limx→c + x−c ∞ and limx→c − x−c −∞.
e −x e −x
b. limx→∞ x−c 0; limx→−∞ x−c −∞.
c. The only critical number for q is x c − 1.
d. When x < c −1, q ′(x) > 0; when x > c −1, q ′(x) < 0; q has a local maximum at x c −1.
e.
3.2.3.5. Answer.
a. x m.
b. E is increasing for x < m and decreasing for x > m, with a local maximum at x m.
c. x m ± s.
d. limx→∞ E(x) limx→−∞ E(x) 0.
e.
589
Appendix C Answers to Selected Exercises
3.3 · Global Optimization
3.3.4.1. Answer.
a. Not enough information is given.
b. Global minimum at x b.
c. Global minimum at x a; global maximum at x b.
d. Not enough information is provided.
3.3.4.2. Answer.
( ) 3
a. Absolute maximum p(0) p(a) 0; absolute minimum p √a − 2a√ .
3 3 3
(1) (2)
b. Absolute max r b ≈ 0.368 ba ; absolute min r b ≈ 0.270 ba .
c. Absolute minimum 1(b) a(1 − e −b ); absolute maximum 1(3b) a(1 − e −3b ).
2 2
(π) ( 5π )
d. Absolute max s 2k 1; absolute min s 6k 12 .
3.3.4.3. Answer.
a. Global maximum at x a; global minimum at x b.
b. Global maximum at x c; global minimum at either x a or x b.
c. Global minimum at x a and x b; global maximum somewhere in (a, b).
d. Global minimum at x c; global maximum value at x a.
3.3.4.4. Answer.
12 ) 8; absolute min s( 12 ) 2.
a. Absolute max s( 5π 11π
√
3 3
12 ) 8; absolute min s(0) 5 −
b. Absolute max s( 5π 2 ≈ 2.402.
12 ) 8; absolute min s( 12 ) 2. (There are other points at which the
c. Absolute max s( 5π 11π
function achieves these values on the given interval.)
590
12 ) 8; absolute min s( 6 ) ≈ 2.402.
d. Absolute max s( 5π 5π
3.4 · Applied Optimization
(√ ) (√ ) (√ ) 3
3.4.3.6. Answer. The absolute maximum volume is V 5
3 15
12
5
3 − 14 5
3 ≈ 1.07583
cubic feet.
3.4.3.7. Answer. The maximum possible area that each of the four pens can enclose is
390625 square feet.
3.4.3.8. Answer. 172.047 feet of cable.
3.4.3.9. Answer. The minimum cost is $1165.70.
3.5 · Related Rates
3.5.3.4. Answer. The boat is approaching the dock at a rate of 13
6 ≈ 2.167 feet per second.
3.5.3.5. Answer. The depth of the water is increasing at
dh
1.28
dt h5
feet per minute. The depth of the water is increasing at a decreasing rate.
3.5.3.6. Answer. dθ
dt x30 −0.24 radians per second.
3.5.3.7. Answer. dh
dt V1000 10
(√ )2 ≈ 0.0328 feet per minute.
3 3000
π π
4 · The Definite Integral
4.1 · Determining distance traveled from velocity
4.1.5.7. Answer.
a. At time t 1, 12 miles north of the lake.
b. s(2) − s(0) 1 mile north of the lake.
c. 40 miles.
d.
591
Appendix C Answers to Selected Exercises
mph miles
10 y = v(t) 10 (1, s(1))
6 6 y = s(t)
2 (4, s(4))
hrs 2 (0, s(0)) hrs
-2 1 2 3 4 5 -2 1 2 3 4 5
-6 -6 (3, s(3))
-10 -10
4.1.5.8. Answer.
a. t 500
32 125
8 15.625 is when the rocket reaches its maximum height.
b. A 3906.25, the vertical distance traveled on [0, 15.625].
c. s(t) 500t − 16t 2 .
d. s(15.625) − s(0) 3906.25 is the change of the rocket’s position on [0, 15.625].
e. s(5) − s(1) 1616; the rocket rose 1616 feet on [1, 5].
4.1.5.9. Answer.
a. 1
2 + 41 π ≈ 1.285.
b. s(5) − s(2) −2 is the change in position of the object on [2, 5].
c. On the time interval [5, 7].
d. s is increasing on the intervals (0, 2) and (5, 7); the position function has a relative
maximum at t 2.
4.1.5.10. Answer.
a.
592
b. Think about the product of the units involved: ``units of pollution per day’’ times
``days’’. Connect this to the area of a thin vertical rectangle whose height is given by
the curve.
c. An underestimate is 336 units of pollution.
4.2 · Riemann Sums
4.2.5.4. Answer.
a. M4 43.5.
b. A 87
2 .
c. The rectangles with heights that come from the midpoint have the same area as the
trapezoids that are formed by the function values at the two endpoints of each subin-
terval.
M n will give the exact area for any value of n. Neither L n nor R n will be exact for any
593
Appendix C Answers to Selected Exercises
n.
d. For any linear function 1 of the form 1(x) mx + b such that 1(x) ≥ 0 on the interval
of interest.
4.2.5.5. Answer.
a. f (x) x 2 + 1 on the interval [1, 3].
b. If S is a left Riemann sum, f (x) x 2 + 1 on the interval [1.4, 3.4]. If S is a middle
Riemann sum, f (x) x 2 + 1 on the interval [1.2, 3.2].
c. The area under f (x) x 2 + 1 on [1, 3].
∑10 ( )
d. R 10 i1 (1 + 0.2i)2 + 1 · 0.2.
4.2.5.6. Answer.
a.
b. M3 99.6 feet.
c. L6 114,
R 6 84,
2 (L 6 + R 6 ) 99.
1
and
d. 114 feet.
4.2.5.7. Answer.
a. M4 ≈ 6.4.
594
b. The total tonnage of pollution escaping the scrubbing process in the time interval [0, 4]
weeks.
c. L 5 ≈ 5.19620599.
d. 6.4 tons.
4.3 · The Definite Integral
4.3.5.7. Answer.
∫4
a. The total change in position is P 0
v(t)dt.
b. P −2.625 feet.
c. D 3.375 feet.
d. AV −0.65625 feet per second.
e. s(t) −t 2 + t.
595
Appendix C Answers to Selected Exercises
4.3.5.8. Answer.
∫1 ∫3 ∫4 ∫4
a. The total change in position, P, is P 0
v(t) dt + 1
v(t) dt + 3
v(t) dt 0
v(t) dt.
∫4
b. P 0
v(t) dt ≈ 2.665.
∫1 ∫3 ∫4
c. The total distance traveled, D, is D 0
v(t) dt − 1
v(t) dt + 3
v(t) dt.
d. D ≈ 8.00016.
e.
vAVG[0,4] ≈ 0.66625
feet per second.
4.3.5.9. Answer.
∫1
a. 0
[ f (x) + 1(x)] dx 1 − π4 .
∫4
b. 1
[2 f (x) − 31(x)] dx − 15
2 − 3π.
c. h AVG[0,4] 5
8 + 3π
16 .
d. c − 38 + 3π
16 .
4.3.5.10. Answer.
∫1
a. A1 −1
(3 − x 2 ) dx.
∫1
b. A2 −1
2x 2 dx.
596
∫1 ∫1
c. The exact area between the two curves is −1
(3 − x 2 ) dx − −1
2x 2 dx.
d. Use the sum rule for definite integrals over the same interval.
e. Think about subtracting the area under q from the area under p.
4.4 · The Fundamental Theorem of Calculus
4.4.5.7. Answer.
a. 20 meters.
b. vAVG[12,24] 12.5 meters per minute.
c. The object’s maximum acceleration is 3 meters per minute per minute at the instant
t 2.
d. c 5.
4.4.5.8. Answer.
a. − 56 .
1
b. fAVG[0,5] .
2
c. 1(x) f (x) for 0 ≤ x < 5 and 1(x) − 54 (x − 5) on 5 ≤ x ≤ 7.
597
Appendix C Answers to Selected Exercises
4.4.5.9. Answer.
a.
h (feet) 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10,000
c (ft/min) 925 875 830 780 730 685 635 585 535 490 440
1 1 1 1 1 1 1 1 1 1 1
m (min/ft) 925 875 830 780 730 685 635 585 535 490 440
b. The antiderivative function tells the total number of minutes it takes for the plane to
climb to an altitude of h feet.
∫ 10000
c. M 0
m(h) dh.
d. It takes the plane aabout M5 ≈ 15.27 minutes.
4.4.5.10. Answer. Yes.
4.4.5.11. Answer.
a. G′(x) 1
−x · (−1) x1 .
b. Since for those values of x, G′(x) x1 .
c. If x < 0, then H(x) ln(|x|) ln(−x) G(x); if x > 0, then H(x) ln(|x|) ln(x)
F(x).
d. For x < 0, H ′(x) G′(x) x1 ; for x > 0, H ′(x) F′(x) 1
x for all x , 0.
5 · Evaluating Integrals
5.1 · Constructing Accurate Graphs of Antiderivatives
5.1.5.5. Answer.
a. s(1) 53 , s(3) −1, s(5) − 11
3 , s(6) − 2 .
5
b. s is increasing on 0 < t < 1 and 5 < t < 6; decreasing for 1 < t < 5.
c. s is concave down for t < 2; concave up for t > 2.
d.
598
3 3
v (1, 5/3)
s
1
A1
t t
1 6 2 4 6
A2 -1
(3, −1)
-3 -3
e. s(t) −2t + 16 (t − 3)3 + 5.
5.1.5.6. Answer.
a. C measures the total number of calories burned in the workout since t 0.
b. C(5) 12.5, C(10) 50, C(15) 125, C(20) 187.5, C(25) 237.5, C(30) 262.5.
c.
cal/min cal
15 300
c
10 200
C
5 100
min min
10 20 30 10 20 30
d. C(t) 12.5 + 7.5(t − 5) on this interval.
5.1.5.7. Answer.
a. B(−1) −1, B(0) 0, B(1) 12 , B(2) 0, B(3) −1, B(4) − 32 , B(5) −1, B(6) 0.
Also, A(x) 1 + B(x) and C(x) B(x) − 21 .
x −1 0 1 2 3 4 5 6
A(x) 0 1 1.5 1 0 −0.5 0 1
B(x) −1 0 0.5 0 −1 −1.5 −1 0
C(x) −1.5 −0.5 0 −0.5 −1.5 −2 −1.5 −0.5
b.
599
Appendix C Answers to Selected Exercises
c. A, B, and C are vertical translations of each other.
d. A′ f .
5.2 · The Second Fundamental Theorem of Calculus
5.2.5.4. Answer. F is increasing on x < −1, 0.5 < x < 4, and 5 < x < 6.5; decreasing on
−1 < x < 0.5 and 4 < x < 5; concave up on approximately −0.4 < x < 2 and 4.5 < x < 6;
concave down on approximately 2 < x < 4.5 and x > 6; F(2) 0; F(0.5) −6.06; F(−1)
−1.77; F(4) 6.69; F(5) 6.33; F(6.5) 8.12.
6 15
4 10
y = g(t)
A2
2 5 (5, 6.33)
A4 y = F(x)
A1 1 2 3 4 5 6 -1 1 2 3 4 5 6
-2 -5
A3
(0.5, −6.06)
-4 -10
5.2.5.5. Answer.
a. The total sand removed on this time interval is
∫ 6 [ ( )]
4πt
2 + 5 sin dt.
0 25
b. The total amount of sand on the beach at time x is given by
∫ x ∫ x [ ( ( ))]
15t 4πt
Y(x) [S(t) − R(t)] dt − 2 + 5 sin dt.
0 0 1 + 3t 25
600
c. Y ′(4) S(4) − R(4) ≈ −1.90875 cubic yards per hour.
d. Y has an absolute minimum on [0, 6] of Y(5.118) ≈ 2492.368.
5.2.5.6. Answer.
(a) m(h) 1
c(h)
.
h (feet) 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10,000
c (ft/min) 925 875 830 780 730 685 635 585 535 490 440
1 1 1 1 1 1 1 1 1 1 1
m (min/ft) 925 875 830 780 730 685 635 585 535 490 440
(b) The antiderivative function tells us the total number of minutes that it takes for the
plane to climb to an altitude of h feet.
(c) The number of minutes required for the airplane to ascend to 10,000 feet of altitude is
given by the definite integral
∫ 10000
M m(h) dh.
0
(d) The number of minutes required for the airplane to ascend to h feet of altitude is given
by the definite integral
∫ h
M(h) m(t) dt.
0
(e) Estimating the desired integral using 3 subintervals and midpoints,
∫ 6000
m(h) dh ≈ 7.77.
0
Using 5 subintervals and midpoints,
∫ 10000
m(h) dh ≈ 15.27.
0
5.3 · Integration by Substitution
5.3.5.7. Answer.
∫
a. tan(x) dx ln (| sec(x)|) + C.
∫
b. cot(x) dx − ln (| csc(x)|) + C.
∫ sec2 (x)+sec(x) tan(x)
c. sec(x)+tan(x)
dx ln (| sec(x) + tan(x)|) + C.
sec2 (x)+sec(x) tan(x)
d. sec(x)+tan(x)
sec(x).
∫
e. sec(x) dx ln (| sec(x) + tan(x)|) + C.
∫
f. csc(x) dx − ln (| csc(x) + cot(x)|) + C.
601
Appendix C Answers to Selected Exercises
5.3.5.8. Answer.
∫ √ ∫ √
a. x x − 1 dx (u + 1) u du.
∫ √
b. x x − 1 dx 25 (x − 1)5/2 + 23 (x − 1)3/2 + C.
∫ √
c. x 2 x − 1 dx 27 (x − 1)7/2 + 45 (x − 1)5/2 + 32 (x − 1)3/2 + C.
∫ √
x x 2 − 1 dx 13 (x 2 − 1)3/2 + C.
5.3.5.9. Answer.
a. We don’t have a function-derivative pair.
b. sin3 (x) sin(x)(1 − cos2 (x)).
c. u cos(x) and du − sin(x) dx.
∫
d. sin3 (x) dx 1
3 cos3 (x) − cos(x) + C.
∫
e. cos3 (x) dx sin(x) − 13 sin3 (x) + C.
5.3.5.10. Answer.
a. The model is reasonable because it appears to be periodic and the rate of consumption
seems to peak at the times of day where people are most active in their homes.
b. The total power consumed in 24 hours, measured in megawatt-hours.
∫ 24
c. 0
r(t) dt ≈ 95.7809 megawatts of power used in 24 hours.
d. rAVG[0,24] ≈ 3.99087 megawatts.
5.4 · Integration by Parts
5.4.7.5. Answer.
a. F′(x) xe −2x .
b. F(x) − 12 xe −2x − 14 e −2x + 1
4
c. Increasing.
5.4.7.6. Answer.
∫ ∫
a. e 2x cos(e x ) dx z cos(z) dz.
∫
b. e 2x cos(e x ) dx e x sin(e x ) + cos(e x ) + C.
∫
c. e 2x cos(e 2x ) dx 1
2 sin(e 2x ) + C
602
∫
d. • e 2x sin(e x ) dx sin(e x ) − z cos(e x ) + C.
∫
• e 3x sin(e 3x ) dx − 31 cos(e 3x ) + C.
∫ 2 2 2 2
• xe x cos(e x ) sin(e x ) dx 1
4 sin2 (e x ) + C.
5.4.7.7. Answer.
∫
a. u-substitution; x 2 cos(x 3 ) dx 1
3 sin(x 3 ) + C.
∫ ( )
b. Both are needed; x 5 cos(x 3 ) dx 1
3 x 3 sin(x 3 ) + cos(x 3 ) + C.
∫
x2 x2
c. Integration by parts; xln(x 2 ) dx 2 ln(x 2 ) − 2 + C.
d. Neither.
∫
e. u-substitution; x 3 sin(x 4 ) dx − 14 cos(x 4 ) + C.
∫
f. Both are needed; x 7 sin(x 4 ) dx − 14 x 4 cos(x 4 ) + 14 sin(x 4 ) + C.
5.5 · Other Options for Finding Algebraic Antiderivatives
5.5.5.6. Answer.
a. ∫
x3 + x + 1 1 1 3
dx − arctan(x) + ln |x + 1| + ln |x − 1| + C
x4 − 1 2 4 4
b.
∫
x5 + x2 + 3 x3 255 5
dx + 3x 2 + 25x + ln |x − 3| − 39 ln |x − 2| + ln |x − 1| + C.
x − 6x + 11x − 6
3 2 3 2 2
c. ∫
x2 − x − 1 5 5
dx ln |x − 3| − − + C.
(x − 3)3 x − 3 2(x − 3)2
5.5.5.7. Answer.
∫ √
5+ 9x 2 +52
a. √ 1 dx − 15 ln 3x + C.
x 9x 2 +25
∫ √ ( 2√ √ )
b. x 1 + x 4 dx 1
2
x
2 x 4 + 1 + 21 ln |x 2 + x 4 + 1| + C
∫ √ √ √
ex
c. e x 4 + e 2x dx e 2x + 4 + 2 ln |e x + e 2x + 4| + C
2
∫ √
3+ 9−cos2 (x)
d. √ tan(x) dx 3 ln
1
cos(x)
+ C.
9−cos2 (x)
5.5.5.8. Answer.
√
a. Try u 1 + x 2 or u x + 1 + x 2 .
√ √
b. Try u x + 1 + x 2 and dv 1
x dx.
c. No.
603
Appendix C Answers to Selected Exercises
√ √
x+ 1+x 2
d. It appears that the function x does not have an elementary antiderivative.
5.6 · Numerical Integration
5.6.6.5. Answer.
a. u-substitution fails since there’s not a composite function present; try showing that
each of the choices of u x and dv tan(x) dx, or u tan(x) and dv x dx, fail to
produce an integral that can be evaluated by parts.
b. • L4 0.25892
• R 4 0.64827
• M4 0.41550
L 4 +R4
• T4 2 0.45360
2M4 +T4
• S8 3 0.42820
c. L4 and M4 are underestimates; R 4 and M4 are overestimates.
5.6.6.6. Answer.
a. Decreasing.
b. Concave down.
∫6
c. 3
f (x) ≈ 7.03.
5.6.6.7. Answer.
∫ 60
a. 0
r(t) dt.
∫ 60
b. 0
r(t) dt > M3 204000.
∫ 60
c. 0
r(t) dt ≈ S6 619000
3 ≈ 206333.33.
d. 1
60 S6≈ 3438.89; 2000+2100+2400+3000+3900+5100+6500
7 25000
7 ≈ 3571.43. each estimates the
average rate at which water flows through the dam on [0, 60], and the first is more
accurate.
6 · Using Definite Integrals
6.1 · Using Definite Integrals to Find Area and Length
6.1.5.5. Answer.
√
∫ 3+ 3 √
a. A √
3− 3
2
−2y 2 + 6y − 3 dy 3.
2
∫ 3π/4 √
b. A π/4
sin(x) − cos(x) dx 2.
∫ y+1
c. A −1
5/2 2 − (y 2 − y − 2) dy 343
48 .
√
∫ m+ m 2 +4 ( ) ( √ )3 ( √ )3 ( √ )2
m+ m 2 +4 m− m 2 +4 m+ m 2 +4
d. A √2 mx − x 2 − 1 dx − 13 2 − 1
3 2 + m
2 2 −
m− m 2 +4
2
604
( √ )2 ( √ √ )
m− m 2 +4 m+ m 2 +4 m− m 2 +4
m
2 2 + 2 − 2 .
6.1.5.6. Answer. a 12 .
6.1.5.7. Answer.
a. r 34 .
b.
√
4 6
c. A1 A2 27 .
d. Yes.
6.2 · Using Definite Integrals to Find Volume
6.2.5.7. Answer.
√ ( ( ) )2
∫ 1.84257
x3
a. L 0
1 + −3 sin 4 · 34 x 2 dx ≈ 4.10521.
∫ 1.84527 ( 3)
b. A 0
3 cos x
4 dx ≈ 4.6623.
∫ 1.84527 ( 3)
c. V 0
π · 9 cos2 x
4 dx ≈ 40.31965.
∫3 ( ( y ) ) 2/3
d. V 0
π 4 arccos 3 dy ≈ 23.29194.
6.2.5.8. Answer.
a.
605
Appendix C Answers to Selected Exercises
∫ π
b. A 0
4
(cos(x) − sin(x)) dx.
∫ π
c. V 0
4
π(cos2 (x) − sin2 (x)) dx.
√
∫ 2 ∫1
d. V 0
2
π arcsin2 (y) dy + √
2 π arccos2 (y) dy
2
∫ π
e. V 0
4
π[(2 − sin(x))2 − (2 − cos(x))2 ] dx.
√
∫ 2 ∫1
f. V 0
2
π[(1 + arcsin(y))2 − 12 ] dy + √
2 π[(1 + arccos(y))2 − 12 ] dy
2
6.2.5.9. Answer.
∫ 1.5
a. A 0
1 + 12 (x − 2)2 − 12 x 2 dx 2.25.
∫ 1.5 [( )2 ( )2]
b. V 0
π 2 + 12 (x − 2)2 − 1 + 12 x 2 dx 32 π
315
∫ 1.125 (√ )2 ∫3 ( √ )2
c. V 0
π 2y dy + 1.125
π 2 − 2(y − 1) dy ≈ 7.06858347.
∫ 1.5 √ √
d. P 3 + 0
1 + (x − 2)2 + 1 + x 2 dx ≈ 7.387234642.
6.3 · Density, Mass, and Center of Mass
6.3.5.5. Answer.
a. a −10 ln(0.7) ≈ 3.567 cm.
b. Left of the midpoint.
c. x ≈ 50.3338
30 ≈ 1.687.
d. q −10 ln(0.85) ≈ 1.625 cm.
606
6.3.5.6. Answer.
a. M1 arctan(10) ≈ 1.47113; M2 10 − 10e −1 ≈ 6.32121.
b. x1 ≈ 1.56857; x2 ≈ 4.18023.
c. (i)
∫ 10 ∫ 10
M ρ(x) dx + p(x) dx ≈ 1.47113 + 6.32121 7.79234.
0 0
(ii)
∫ 10
x(ρ(x) + p(x))) dx 28.73167.
0
(iii) False.
6.3.5.7. Answer.
a.
∫ 30
b. V 0
π(2xe −1.25x + (30 − x)e −0.25(30−x) )2 dx ≈ 52.0666 cubic inches.
c. W ≈ 0.6 · 52.0666 31.23996 ounces.
d. At a given x-location, the amount of weight concentrated there is approximately the
weight density (0.6 ounces per cubic inch) times the volume of the slice, which is
Vtextslice ≈ π f (x)2 .
e. x ≈ 23.21415
6.4 · Physics Applications: Work, Force, and Pressure
6.4.5.6. Answer.
a.
607
Appendix C Answers to Selected Exercises
W ≈ 1646.79 foot-pounds.
∫h ( 3)
b. W 0
3744x cos x
4 dx.
c. F ≈ 462.637 pounds.
6.4.5.7. Answer.
a. W 5904(19π − 8) ≈ 305179.3 foot-pounds.
b.
F ≈ 1123.2
pounds.
6.5 · Improper Integrals
6.5.5.6. Answer.
a. Diverges.
b. Diverges.
c. Converges to 1.
∫∞
d. e
1
x(ln(x))p
dx diverges if p ≤ 1 and converges to 1
p−1 if p > 1.
e. Diverges.
f. Converges to −1.
6.5.5.7. Answer.
a. converges
b. diverges
c. diverges
7 · Differential Equations
7.1 · An Introduction to Differential Equations
608
7.1.5.4. Answer.
dt |T105 −2; when T 105, the coffee’s temperature is decreasing at an instanta-
dT
a.
neous rate of −2 degrees F per minute.
b. T decreasing at t 0.
c. T(1) ≈ 103 degrees F.
d.
5
4 dT
dt
3
2
1
T
30 60 90 120
-1
-2
-3
e. For T < 75, T increases. For T > 75, T decreases.
f. Room temperature is 75 degrees F.
g. Substitute T(t) 75 + 30e −t/15 in for T in the differential equation dT
dt − 15 T + 5 and
1
verify the equality holds; T(0) 75 + 30e 0 75 + 30 105; T(t) 75 + 30e −t/15 → 75
as t → ∞.
7.1.5.5. Answer.
a. 1 < P < 3.
b. P < 1 and 3 < P < 4.
c. P will not change at all.
d. The population will decrease toward P 0 with P always being positive.
e. The population will increase toward P 3 with P always being between 1 and 3.
f. The population will decrease toward P 3 with P always being above 3.
g. There’s a maximum threshold of P 3.
7.1.5.6. Answer.
a. i. y(t) t + 1 + 2e t is a solution to the DE.
ii. y(t) t + 1 is a solution to the DE.
iii. y(t) t + 2 is a not solution to the DE.
609
Appendix C Answers to Selected Exercises
b. k 9.
7.2 · Qualitative behavior of solutions to DEs
7.2.4.5. Answer.
a.
y
3
2
1
t
-3 -2 -1 1 2 3
-1
-2
-3
b. Sketch curves through appropriate points in the slope field above.
c. y(t) t − 1.
d. t and y are equal.
7.2.4.6. Answer.
a.
4
y
3
2
1
t
1 2 3 4
b. Any solution curve that starts with P(0) > 3 will decrease to P(t) 3 as t → ∞; any
curve that starts with 1 < P(0) < 3 will increase to P(t) 3; any curve that starts with
0 < P(0) < 1 will decrease to P(t) 0.
610
c. P 0, P 1, and P 3. P 1 is unstable; P 0 and P 3 are stable.
d. The population will stabilize either at the value P 3 or at P 0.
e. P(t) 1 is the threshold.
7.2.4.7. Answer.
a. A graph of f against P is given in blue in the figure below. The equilibrium solutions
are P 0 (unstable) and P 6 (stable).
dP/dt
9
6
3
P
3
b. dP
dt 1(P) P(6 − P) − 1 ; the equilibrium at P ≈ 0.172 is unstable; the equilibrium at
P ≈ 5.83 is stable.
√ √
6− 32 6− 32
c. If P < 2 , then the fish population will die out. If 2 <, then the fish population
√
6+ 32
will approach 2 thousand fish.
√ √
6+ 36−4h 6− 36−4h
d. dP
dt 1(P) P(6 − P) − h; equilibrium solutions P 2 , 2 .
e. 9000 fish; harvesting at that rate will maintain the number of fish we start with, pro-
vided it’s at least 3000.
7.2.4.8. Answer.
dy
a. dt 20y.
dy y
b. dt 20y − C 2+y
y y
c. For positive y near 0, M(y) 2+y ≈ 0; for large values of y, M(y) 2+y ≈ 1.
d. The only equilibrium solution is y 0, which is unstable.
e. The equilibrium solutions are y 0 (stable) and y 1 (unstable).
f. At least 41 cats.
7.3 · Euler’s method
611
Appendix C Answers to Selected Exercises
7.3.4.4. Answer.
a. Alice’s coffee: dT
dt |T100 −0.5(30) −15 degrees per minute; Bob’s coffee:
A dTB
dt |T100
−0.1(30) −3 degrees per minute.
b. Consider the insulation of the containers.
c. Alice’s coffee:
dTA
−0.5(TA − (70 + 10 sin t)),
dt
with the inital condition TA (0) 100.
t TA (t)
0.0 100
0.1 98.5
0.2 97.12492
0.3 95.86801
0.4 94.72237
.. ..
. .
49.6 65.56715
49.7 65.48008
49.8 65.43816
49.9 65.44183
50 65.49103
d.
t TA (t)
0.0 100
0.1 99.7
0.2 99.41298
0.3 99.13872
0.4 98.87689
.. ..
. .
49.6 69.39515
49.7 69.33946
49.8 69.29248
49.9 69.25467
50 69.22638
e. Compare the rate of initial decrease and amplitude of oscillation.
612
7.3.4.5. Answer.
a. K 1.054; y(1) 2.6991.
b. K 1.272; y(1) 2.7169.
c. K 0.122 and y(0.3) 0.0412.
7.3.4.6. Answer.
a. y(1) ≈ y5 2.7027.
b. y(1) ≈ y10 2.7141.
c. The square of ∆t.
7.4 · Separable differential equations
7.4.3.6. Answer.
a. dM
dt kM.
b. M(t) M0 e kt .
ln(2)
c. M(t) M0 e − 5730 t ≈ M0 e −0.000121t
5730 ln(4)
d. t ln(2)
≈ 11460 years.
5730 ln(0.3)
e. t − ln(2)
≈ 9952.8 years.
7.4.3.7. Answer.
√
a. y 64 − t 2 .
b. −8 ≤ t ≤ 8.
c. y(8) 0.
dy
d. dt − yt is not defined when y 0.
7.4.3.8. Answer.
√
a. dh
dt k h.
b. The tank with k −10 has water leaving the tank much more rapidly.
c. k −5.
d. h(t) (5 − 2.5t)2 .
e. 2.5 minutes.
f. No.
7.4.3.9. Answer.
(a)
613
Appendix C Answers to Selected Exercises
(b) P 3 is stable.
(c) P(t) 3e ln( 3 )e .
1 −t
−t
(d) P(t) 3e ln(2)e .
(e) Yes.
7.5 · Modeling with differential equations
7.5.3.6. Answer.
a. dA
dt 1 + 0.05A
b. A(25) 49.80686 million dollars.
c. A(25) 34.90343 million dollars.
d. The first.
e. t 20 ln(2) ≈ 13.86 years.
7.5.3.7. Answer.
a. dv
dt 9.8 − kv
b. v 9.8
k is a stable equilibrium.
9.8−9.8e −kt
c. v(t) k
d. k 9.8/54 ≈ 0.181481.
ln(0.5)
e. t −0.181481 ≈ 3.1894 seconds.
7.5.3.8. Answer.
a. dw
dt k
w.
√ √
b. w(t) 7t + 64; w(12) 148 ≈ 12.17 pounds.
c. The model is unrealistic.
614
7.5.3.9. Answer.
a. The inflow and outflow are at the same rate.
b. 60 grams per minute.
S(t) grams
c. 100 gallon
3S(t) grams
d. 100 minute .
e. dS
dt 60 − 3
100 S.
f. S 2000 is a stable equilibrium solution.
g. S(t) 2000 − 2000e − 100 t .
3
h. S(t) → 2000.
7.6 · Population Growth and the Logistic Equation
7.6.4.5. Answer.
a. p(t) → 1 as t → ∞ provided p(0) > 0.
b. p(t) 1
9e −0.2t +1
.
c. t −5 ln(1/9) ≈ 10.986 days.
d. t −5 ln(0.2/9) ≈ 19.033 days.
7.6.4.6. Answer.
a. db
dt 1
3000 b(15000 − b)
b. b 15000.
c. When b 7500.
d. t − 15 ln(1/70) ≈ 0.8497 days.
7.6.4.7. Answer.
a. 10000 fish.
b. dP
dt 0.1P(10 − P) − 0.2P.
c. 8000 fish.
d. P(1) ≈ 8.7899 thousand fish.
e. t −1.2 ln(5/11) ≈ 0.9461 years.
8 · Sequences and Series
8.1 · Sequences
8.1.3.5. Answer.
a. Unclear whether it converges or diverges.
615
Appendix C Answers to Selected Exercises
n 1 2 3 4 5
ln(n)
n 0 0.3466 0.3663 0.3466 0.3218
t 6 7 8 9 10
ln(n)
n 0.2987 0.2781 0.2599 0.2442 0.2303
ln(n)
b. If limx→∞ f (x) L, then limn→∞ n L as well.
c.
ln(n) ln(x)
lim lim 0.
n→∞ n x→∞ x
8.1.3.6. Answer.
(r) ( )
r 2
a. P1 12 in interest in the second month; at the end of the second month, P2 P 1 + 12 .
( )
r 3
b. P3 P 1 + 12 .
( )
r n
c. Pn P 1 + 12 is a pattern to these calculations.
8.1.3.7. Answer.
a. A1 12 (100).
( 1 )2
b. A2 2 · 100.
( 1 )3
c. A3 2 · 100.
( 1 )4
d. A4 2 · 100.
( 1 )n
e. A n 2 · 100.
f. A n → 0 as time goes on.
g. It takes about 6.6439 half-lives to elapse to get down to 1 gram remaining, or 5·6.6439
33.2193 minutes.
8.1.3.8. Answer.
a. The data points do not appear periodic at all.
b. At least 13 samples, so at least every 10/13 ≈ 0.76923 seconds.
c. 44100 Hz is slightly more than double 20 KHz.
8.2 · Geometric Series
616
8.2.3.5. Answer.
a. 30 · 500 1500 dollars.
b.
Day Pay on this day Total amount paid to date
1 $0.01 $0.01
2 $0.02 $0.03
3 $0.04 $0.07
4 $0.08 $0.15
5 $0.16 $0.31
6 $0.32 $0.63
7 $0.64 $1.27
8 $1.28 $2.55
9 $2.56 $5.11
10 $5.12 $10.23
( )
c. $0.01 230 − 1 $10, 737, 418.23.
8.2.3.6. Answer.
(3)
a. h1 4 h.
(3) ( 3 )2
b. h2 4 h1 4 h.
(3) ( 3 )3
c. h3 4 h2 4 h.
(3) ( 3 )n
d. h n 4 h n−1 4 h.
e. The distance traveled by the ball is 7h, which is finite.
8.2.3.7. Answer.
a. There are 6 equally possible outcomes when we roll one die.
b. The three rolls are independent so the probability of the overall outcome is the product
of the three probabilities.
c. See (b).
d. P 6
11 .
8.2.3.8. Answer.
a. 0.75P dollars spent.
b. 0.75P + 0.75(0.75P) 0.75P(1 + 0.75) dollars.
c. 0.75P + 0.752 P + 0.752 P + · · · 0.75P(1 + 0.75 + 0.752 + · · · ) dollars.
d. A stimulus of 200 billion dollars adds 600 billion dollars to the economy.
8.2.3.9. Answer.
(r)
a. 12 P1 dollars.
617
Appendix C Answers to Selected Exercises
( )
b. P2 1 + r
12 P1 − M.
( )
r 2
[ ( )]
c. P2 1 + 12 P− 1+ 1+ r
12 M.
( )
d. P3 1 + r
12 P2 − M.
( )3 [ ( ) ( ) ]
r 2
P3 1 + r
12 P− 1+ 1+ r
12 + 1+ 12 M.
( )
r n
( 12M ) ( ( )
r n
)
e. Pn P 1 + 12 − r 1+ 12 −1 .
( )
r 12t
( 12M ) ( ( )
r 12t
)
f. P(t) P 1 + 12 − r 1+ 12 −1
( )
12(25) ( )
0.2 12t 12(25)
g. A(t) 1000 − 0.2 1+ 12 + 0.2 .
t ≈ 5.5.
We pay $659 dollars in interest on our $1000 loan.
h. $291.74 each month to complete the loan in 5 years; we pay $2,504.40 in interest.
8.3 · Series of Real Numbers
8.3.7.5. Answer.
∑ 10k
a. k! converges.
∑ 10k
b. k! converges.
{ bn }
c. The sequence n! has to converge to 0.
8.3.7.6. Answer.
√
a. n a n ≈ r for large n.
a n+1
b. an ≈ r.
c. 0 < r < 1.
8.3.7.7. Answer.
a. i. a n 1 + 1
2n → 1 , 0 and b n −1 → −1 , 0.
ii. The series is geometric with r 21 .
iii. Since the two individual series diverge, neither sum is a finite number, so it doesn’t
make any sense to add them.
b. i. Note that A n + B n (a1 + b 1 ) + (a 2 + b2 ) + · · · + (a n + b n ).
∑n (∑n ∑n )
ii. Note that limn→∞ k1 (a k + b k ) limn→∞ k1 ak + k1 bk .
∑∞ 2k +3k
c. k0 5k
25
6 .
8.3.7.8. Answer.
a. i) S1 1 and T1 12 .
ii) S2 > T2 .
iii) S3 > T3 .
618
iv) S n > Tn .
∑ ∑ ∑
v) 1
k2
> 1
k 2 +k
; 1
k 2 +k
converges.
∑ ∑ ∑
b. If b k diverges, then b k is infinite, and anything larger must also be infinite; if ak
is convergent then anything smaller and positive must also be finite.
i) Note that 0 < 1
k < 1
k−1 .
ii) Note that 1
k3
> 1
k 3 +1
.
8.4 · Alternating Series
8.4.6.5. Answer.
∑∞ 1
a. k0 2k+1 diverges by comparison to the Harmonic series.
∑∞
b. |S100 − k0 (−1) 2k+1 |
k 1 <≈ 0.0049.
c. n > 4,999,999,999.5
8.4.6.6. Answer.
S n +S n+1 S n +S n +(−1)n+2 a n+1
a. 2 2 .
S20 +S21
b. S20 0.668771403 . . .; 2 161227687
232792560 0.692580926 . . ., accurate to within about
0.0006.
8.4.6.7. Answer.
{1} { }
a. n and − n12 converge to 0.
b. Notice that 1
k − 1
k2
k−1
k2
and compare to the Harmonic series.
c. It is possible for a series to alternate, have the terms go to zero, have the terms not
decrease to zero, and the series diverge.
8.5 · Taylor Polynomials and Taylor Series
8.5.6.6. Answer.
a. P3 (x) −1 + 3x − 4 2
2! x + 6 3
3! x , which is the same polynomial as f (x).
b. For n ≥ 3, Pn (x) f (x).
c. For n ≥ m, Pn (x) f (x).
8.5.6.7. Answer.
a.
π) (
P1 (x) 1 + 0 x − 1
2
( π ) 1 ( π )2 1 ( π )2
P2 (x) 1 + 0 x − − x− 1− x−
2 2! 2 2! 2
( π) 1 ( π )2 0 ( π )3
P3 (x) 1 + 0 x − − x− + x−
2 2! 2 3! 2
1 ( π )2
1− x− P2 (x)
2! 2
619
Appendix C Answers to Selected Exercises
( π) 1 ( π )2 0 ( π )3 1 ( π )4
P4 (x) 1 + 0 x − − x− + x− + x−
2 2! 2 3! 2 4! 2
1 ( π )2 1 ( π )4
1− x− + x−
2! 2 4! 2
1 ( )
π 2 1 ( π )4 1 ( π )6
P(x) 1 − x− + x− − x− +···
2! 2 4! 2 6! 2
b.
1 2 6
P4 (x) 0 + 1(x − 1) −
(x − 1)2 + (x − 1)3 − (x − 1)4 .
2! 3! 4!
1 1 1 1
P(x) 1(x − 1) − (x − 1)2 + (x − 1)3 − (x − 1)4 + (x − 1)5 − · · ·
2 3 4 5
c. P101 (1) ≈ 0.698073
8.5.6.8. Answer.
a. P4 (x) x 2 .
b. i. 1(x) P(x 2 ) x 2 − 1 6
3! x + 1 1
5! x 0 − · · ·.
ii. All real numbers.
8.6 · Power Series
8.6.4.3. Answer.
∑∞ k x 2(2k+1)
a. sin(x 2 ) k0 (−1) (2k+1)! , with interval of convergence (−∞, ∞).
∫ ∑∞ x 4k+3
b. sin(x 2 ) dx k0 (−1) (2k+1)!(4k+3)
k + C.
∫1 ∑∞
c. 0
sin(x 2 ) dx k0 (−1) (2k+1)!(4k+3) .
k 1
Use n 1 to generate the desired estimate.
8.6.4.4. Answer.
a.
b. Then
∑
∞
f ′(x) ka k x k−1
k1
∑
∞
f ′′(x) k(k − 1)a k x k−2
k2
∑
∞
f (3) (x) k(k − 1)(k − 2)a k x k−3
k3
.. ..
. .
∑
∞
f (n) (x) k(k − 1)(k − 2) · · · (k − n + 1)a k x k−n
kn
.. ..
. .
620
So
f (0) a 0
f ′(0) a 1
f ′′(0) 2!a 2
f (3) (0) 3!a 3
.. ..
. .
f (k) (0) k!a k
.. ..
. .
and
f (k) (0)
ak
k!
for each k ≥ 0. But these are just the coefficients of the Taylor series expansion of f ,
which leads us to the following observation.
8.6.4.6. Answer. The results from the various part of this exercise show that
( )
∑
∞
x 3k
y a0 1 +
(2)(3)(5)(6) · · · (3k − 1)(3k)
k1
( )
∑∞
x 3k+1
+ a1 x + .
(3)(4)(6)(7) · · · (3k)(3k + 1)
k1
621
Appendix C Answers to Selected Exercises
622
Index
u-substitution, 284 concave up, 60
concavity, 59
absolute convergence, 475 conditional convergence, 475
acceleration, 60 constant multiple rule, 92
alternating series, 471 continuous, 72
alternating series estimation theorem, continuous at x a, 72
473 converge
alternating series test, 473 sequence, 438
antiderivative, 206 convergence
general, 249, 263 absolute, 475
graph, 261 conditional, 475
antidifferentiation, 205 convergent sequence, 438
arc length, 330, 331 cosecant, 116
arcsine, 133 cotangent, 116
area, 326 critical number, 159
under velocity function, 203 critical point, 159
asymptote, 151 critical value, 159
horizontal, 151 cusp, 75
vertical, 151
autonomous, 382 decreasing, 54
average rate of change, 22 definite integral
average value, 238 definition, 230, 231
average value of a function, 237 density, 345
average velocity, 2, 5 derivative
arcsine, 134
backward difference, 47 constant function, 91
cosine, 101
carrying capacity, 427 cotangent, 117
center of mass (continuous mass definition, 23, 36
distribution), 350 exponential function, 91
center of mass (point-masses, 350 inverse, 136
central difference, 47 logarithm, 131
chain rule, 122 power function, 91
codomain, 130 sine, 101
composition, 120 tangent, 117
concave down, 60 difference quotient, 47
623
Index
differentiable, 24, 73 unbounded region of integration,
differential equation, 376 367
autonomous, 382 increasing, 54
first order, 382 indefinite integral, 283
solution, 379 evaluate, 283
disk method, 336 indeterminate form, 14
distance traveled, 205 infinite series, 455
diverge infinity, 150
sequence, 438 inflection point, 164
Divergence Test, 459 initial condition, 263
domain, 130 instantaneous rate of change, 23, 46
dominates, 156 instantaneous velocity, 3, 17
integral function, 264
equilibrium solution, 390 integral sign, 231
stable, 390 integral test, 459, 462
unstable, 390 integrand, 231
error, 313 integration by parts, 292
error function, 274, 306 interval of convergence, 490
Euler’s Method, 398
error, 402 left limit, 69
extreme value, 158 lemniscate, 140
extreme value theorem, 180 limit
definition, 13
Fibonacci sequence, 435 one-sided, 69
finite geometric series, 444 limit comparison test, 462
first derivative test, 159, 160 limits of integration, 231
foot-pound, 355 local linearization, 81
forward difference, 47 locally linear, 74
FTC, 247 logistic, 427
function, 130 logistic equation, 427
function-derivative pair, 284 solution, 429
Fundamental Theorem of Calculus
First, 270 Maclaurin series, 489
Second, 272 mass, 345
fundamental theorem of calculus, 245, maximum
247 absolute, 157
global, 157
geometric series, 445 local, 157
common ratio, 445 relative, 157
midpoint rule
harmonic series, 459 error, 313
Hooke’s Law, 357 minimum
absolute, 157
implicit function, 140 global, 157
improper integral, 366 local, 157
converges, 368 relative, 157
diverges, 368
unbounded integrand, 368 net signed area, 222
624
Newton’s Law of Cooling, 383 series
Newton-meter, 355 converges, 457
diverges, 457
one-to-one, 130 geometric, 445
onto, 130 sigma notation, 217
Simpson’s rule, 316
partial fractions, 302
slope field, 387, 388
partial sum, 446, 456
solid of revolution, 335
per capita growth rate, 425
stable, 390
position, 2
sum rule, 93
power series, 499
definition, 499
tangent, 115
product rule, 106
tangent line, 24
quotient rule, 108 equation, 81
Taylor polynomial, 486
ratio test, 464 error, 492
related rates, 193 Taylor polynomials, 484
Riemann sum, 219 Taylor series, 488, 489
left, 219 interval of convergence, 490
middle, 220 radius of convergence, 492
right, 219 total change theorem, 250, 251
right limit, 69 trapezoid rule, 312
error, 313
secant, 116 triangular numbers, 435
secant line, 24 trigonometry, 115
second derivative, 57 fundamental trigonometric identity,
second derivative test, 162 115
second fundamental theorem of calculus,
273 unstable, 390
separable, 407, 408
sequence, 437 washer method, 337
term, 437 weighted average, 316, 348
sequence of partial sums, 456 work, 355, 356
625
Index
626
Colophon
This book was authored in PreTeXt.