Stewart J. Calculus
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PREFACE
In addition, I would like to thank George Bergman, David Cusick, Stuart Goldenberg,
Larry Peterson, Dan Silver, Norton Starr, Alan Weinstein, and Gail Wolkowicz for their
suggestions; Dan Clegg for his research in libraries and on the Internet; Al Shenk and Den-
nis Zill for permission to use exercises from their calculus texts; John Ringland for his
refinements of the multivariable Maple art; COMAP for permission to use project mate-
rial; George Bergman, David Bleecker, Dan Clegg, Victor Kaftal, Anthony Lam, Jamie
Lawson, Ira Rosenholtz, Paul Sally, Lowell Smylie, and Larry Wallen for ideas for exer-
cises; Dan Drucker for the roller derby project; Thomas Banchoff, Tom Farmer, Fred Gass,
John Ramsay, Larry Riddle, and Philip Straffin for ideas for projects; Dan Anderson, Dan
Clegg, Jeff Cole, Dan Drucker, and Barbara Frank for solving the new exercises and sug-
gesting ways to improve them; Marv Riedesel and Mary Johnson for accuracy in proof-
reading; and Jeff Cole and Dan Clegg for their careful preparation and proofreading of the
answer manuscript.
In addition, I thank those who have contributed to past editions: Ed Barbeau, Fred
Brauer, Andy Bulman-Fleming, Bob Burton, Tom DiCiccio, Garret Etgen, Chris Fisher,
Arnold Good, Gene Hecht, Harvey Keynes, Kevin Kreider, E. L. Koh, Zdislav Kovarik,
Emile LeBlanc, David Leep, Gerald Leibowitz, Lothar Redlin, Carl Riehm, Peter Rosen-
thal, Doug Shaw, and Saleem Watson.
I also thank Kathi Townes, Stephanie Kuhns, and Brian Betsill of TECHarts for their
production services and the following Brooks/Cole staff: Cheryll Linthicum, editorial pro-
duction project manager; Mark Santee, Melissa Wong, and Bryan Vann, marketing team;
Stacy Green, assistant editor, and Elizabeth Rodio, editorial assistant; Sam Subity, technol-
ogy project manager; Rob Hugel, creative director, and Vernon Boes, art director; and
Becky Cross, print buyer. They have all done an outstanding job.
I have been very fortunate to have worked with some of the best mathematics editors in
the business over the past two decades: Ron Munro, Harry Campbell, Craig Barth, Jeremy
Hayhurst, Gary Ostedt, and now Bob Pirtle. Bob continues in that tradition of editors who,
while offering sound advice and ample assistance, trust my instincts and allow me to write
the books that I want to write.
JAMES STEWART
Openmirrors.comOpenmirrors.com
Multimedia Manager Instructor’s Resource CD-ROM
ISBN 0-495-01222-X
Contains all art from the text in both jpeg and PowerPoint
formats, key equations and tables from the text, complete
pre-built PowerPoint lectures, and an electronic version of
the Instructor’s Guide.
Tools for Enriching™ Calculus
by James Stewart, Harvey Keynes, Dan Clegg,
and developer Hu Hohn
TEC provides a laboratory environment in which students
can explore selected topics. TEC also includes homework
hints for representative exercises. Available online at
www.stewartcalculus.com
.
Instructor’s Guide
by Douglas Shaw and James Stewart
ISBN 0-495-01214-9
Each section of the main text is discussed from several view-
points and contains suggested time to allot, points to stress, text
discussion topics, core materials for lecture, workshop /discus-
sion suggestions, group work exercises in a form suitable for
handout, and suggested homework problems. An electronic
version is available on the Multimedia Manager Instructor’s
Resource CD-ROM.
Instructor’s Guide for AP
®
Calculus
by Douglas Shaw and Robert Gerver, contributing author
ISBN 0-495-01223-8
Taking the perspective of optimizing preparation for the AP
exam, each section of the main text is discussed from several
viewpoints and contains suggested time to allot, points to
stress, daily quizzes, core materials for lecture, workshop/
discussion suggestions, group work exercises in a form suitable
for handout, tips for the AP exam, and suggested homework
problems.
Complete Solutions Manual
Single Variable
by Daniel Anderson, Jeffery A. Cole, and Daniel Drucker
ISBN 0-495-01232-7
Multivariable
by Dan Clegg and Barbara Frank
ISBN 0-495-01229-7
Includes worked-out solutions to all exercises in the text.
Printed Test Bank
by William Steven Harmon
ISBN 0-495-01221-1
Contains multiple-choice and short-answer test items that key
directly to the text.
ExamView
ISBN 0-495-38239-6
Create, deliver, and customize tests and study guides (both print
and online) in minutes with this easy-to-use assessment and
tutorial software on CD. Includes complete questions from the
Printed Test Bank.
JoinIn on TurningPoint
ISBN 0-495-11874-5
Enhance how your students interact with you, your lecture, and
each other. Thomson Brooks/Cole is now pleased to offer you
book-specific content for Response Systems tailored to Stewart’s
Calculus, allowing you to transform your classroom and assess
your students’ progress with instant in-class quizzes and polls.
Contact your local Thomson representative to learn more about
JoinIn on TurningPoint and our exclusive infrared and radio-
frequency hardware solutions.
Text-Specific DVDs
ISBN 0-495-01218-1
Text-specific DVD set, available at no charge to adopters. Each
disk features a 10- to 20-minute problem-solving lesson for
each section of the chapter. Covers both single- and multi-
variable calculus.
Solution Builder
www.thomsonedu.com/solutionbuilder
The online Solution Builder lets instructors easily build and save
personal solution sets either for printing or posting on password-
protected class websites. Contact your local sales representative
for more information on obtaining an account for this instructor-
only resource.
Stewart Specialty Website
www.stewartcalculus.com
Contents: Algebra Review
N
Additional Topics
N
Drill
exercises
N
Challenge Problems
N
Web Links
N
History of
Mathematics
N
Tools for Enriching Calculus (TEC)
Enhanced WebAssign
ISBN 0-495-10963-0
Instant feedback, grading precision, and ease of use are just
three reasons why WebAssign is the most widely used home-
work system in higher education. WebAssign’s homework deliv-
ery system lets instructors deliver, collect, grade and record
assignments via the web. And now, this proven system has been
enhanced to include end-of-section problems from Stewart’s
Calculus—incorporating exercises, examples, video skill-
builders and quizzes to promote active learning and provide
the immediate, relevant feedback students want.
ANCILLARIES FOR
INSTRUCTORS AND STUDENTS
TEC
ANCILLARIES
FOR INSTRUCTORS
||||
Electronic items
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Printed items (Table continues on page xxii.)
xxi
Openmirrors.comOpenmirrors.com
The Brooks/Cole Mathematics Resource Center Website
www.thomsonedu.com/math
When you adopt a Thomson Brooks/Cole mathematics text,
you and your students will have access to a variety of teaching
and learning resources. This website features everything from
book-specific resources to newsgroups. It’s a great way to
make teaching and learning an interactive and intriguing
experience..
Maple CD-ROM
ISBN 0-495-01237-8 (Maple 10)
ISBN 0-495-39052-6 (Maple 11)
Maple provides an advanced, high performance mathematical
computation engine with fully integrated numerics & symbolics,
all accessible from a WYSIWYG technical document environ-
ment. Available for bundling with your Stewart Calculus text at
a special discount.
Tools for Enriching™ Calculus
by James Stewart, Harvey Keynes, Dan Clegg,
and developer Hu Hohn
TEC provides a laboratory environment in which students
can explore selected topics. TEC also includes homework
hints for representative exercises. Available online at
www.stewartcalculus.com
.
Interactive Video SkillBuilder CD-ROM
ISBN 0-495-01217-3
Think of it as portable office hours! The Interactive Video
Skillbuilder CD-ROM contains more than eight hours of video
instruction. The problems worked during each video lesson are
shown next to the viewing screen so that students can try work-
ing them before watching the solution. To help students evalu-
ate their progress, each section contains a ten-question web
quiz (the results of which can be emailed to the instructor)
and each chapter contains a chapter test, with answers to
each problem.
Study Guide
Single Variable
by Richard St. Andre
ISBN 0-495-01233-5
Multivariable
by Richard St. Andre
ISBN 0-495-01227-0
Contains a short list of key concepts, a short list of skills
to master, a brief introduction to the ideas of the section,
an elaboration of the concepts and skills, including extra
worked-out examples, and links in the margin to earlier and
later material in the text and Study Guide.
Student Solutions Manual
Single Variable
by Daniel Anderson, Jeffery A. Cole, and Daniel Drucker
ISBN 0-495-01234-3
Multivariable
by Dan Clegg and Barbara Frank
ISBN 0-495-01228-9
Provides completely worked-out solutions to all odd-numbered
exercises within the text, giving students a way to check their
answers and ensure that they took the correct steps to arrive at
an answer.
CalcLabs with Maple
Single Variable
by Philip B. Yasskin, Maurice Rahe, David Barrow,
Art Belmonte, Albert Boggess, Jeffery Morgan, Kirby Smith,
and Michael Stecher
ISBN 0-495-01235-1
Multivariable
by Philip Yasskin, Maurice Rahe, and Art Belmonte
ISBN 0-495-01231-9
CalcLabs with Mathematica
Single Variable by Selwyn Hollis
ISBN 0-495-38245-0
Multivariable by Selwyn Hollis
ISBN 0-495-11890-7
Each of these comprehensive lab manuals will help students
learn to effectively use the technology tools available to them.
Each lab contains clearly explained exercises and a variety of
labs and projects to accompany the text.
A Companion to Calculus
by Dennis Ebersole, Doris Schattschneider, Alicia Sevilla,
and Kay Somers
ISBN 0-495-01124-X
Written to improve algebra and problem-solving skills of stu-
dents taking a calculus course, every chapter in this companion
is keyed to a calculus topic, providing conceptual background
and specific algebra techniques needed to understand and solve
calculus problems related to that topic. It is designed for calcu-
lus courses that integrate the review of precalculus concepts or
for individual use.
Linear Algebra for Calculus
by Konrad J. Heuvers, William P. Francis, John H. Kuisti,
Deborah F. Lockhart, Daniel S. Moak, and Gene M. Ortner
ISBN 0-534-25248-6
This comprehensive book, designed to supplement the calculus
course, provides an introduction to and review of the basic
ideas of linear algebra.
TEC
STUDENT
RESOURCES
||||
Electronic items
||||
Printed items
xxii
Openmirrors.com
Reading a calculus textbook is different from reading a news-
paper or a novel, or even a physics book. Don’t be discouraged
if you have to read a passage more than once in order to under-
stand it. You should have pencil and paper and calculator at
hand to sketch a diagram or make a calculation.
Some students start by trying their homework problems and
read the text only if they get stuck on an exercise. I suggest that
a far better plan is to read and understand a section of the text
before attempting the exercises. In particular, you should look
at the definitions to see the exact meanings of the terms. And
before you read each example, I suggest that you cover up the
solution and try solving the problem yourself. You’ll get a lot
more from looking at the solution if you do so.
Part of the aim of this course is to train you to think logically.
Learn to write the solutions of the exercises in a connected,
step-by-step fashion with explanatory sentences—not just a
string of disconnected equations or formulas.
The answers to the odd-numbered exercises appear at the
back of the book, in Appendix I. Some exercises ask for a ver-
bal explanation or interpretation or description. In such cases
there is no single correct way of expressing the answer, so don’t
worry that you haven’t found the definitive answer. In addition,
there are often several different forms in which to express a
numerical or algebraic answer, so if your answer differs from
mine, don’t immediately assume you’re wrong. For example,
if the answer given in the back of the book is and you
obtain , then you’re right and rationalizing the
denominator will show that the answers are equivalent.
The icon
;
indicates an exercise that definitely requires
the use of either a graphing calculator or a computer with
graphing software. (Section 1.4 discusses the use of these
graphing devices and some of the pitfalls that you may
encounter.) But that doesn’t mean that graphing devices can’t
be used to check your work on the other exercises as well. The
symbol is reserved for problems in which the full resources
CAS
1兾
(
1 ⫹
s
2
)
s
2
⫺ 1
of a computer algebra system (like Derive, Maple, Mathe-
matica, or the TI-89/92) are required.
You will also encounter the symbol
|
, which warns you
against committing an error. I have placed this symbol in the
margin in situations where I have observed that a large propor-
tion of my students tend to make the same mistake.
Tools for Enriching Calculus, which is a companion to this
text, is referred to by means of the symbol and can be
accessed from www.stewartcalculus.com. It directs you to mod-
ules in which you can explore aspects of calculus for which the
computer is particularly useful. TEC also provides Homework
Hints for representative exercises that are indicated by printing
the exercise number in red: These homework hints ask you
questions that allow you to make progress toward a solution
without actually giving you the answer. You need to pursue
each hint in an active manner with pencil and paper to work
out the details. If a particular hint doesn’t enable you to solve
the problem, you can click to reveal the next hint.
An optional CD-ROM that your instructor may have asked
you to purchase is the Interactive Video Skillbuilder, which con-
tains videos of instructors explaining two or three of the exam-
ples in every section of the text. Also on the CD is a video in
which I offer advice on how to succeed in your calculus course.
I recommend that you keep this book for reference purposes
after you finish the course. Because you will likely forget some
of the specific details of calculus, the book will serve as a
useful reminder when you need to use calculus in subsequent
courses. And, because this book contains more material than
can be covered in any one course, it can also serve as a valuable
resource for a working scientist or engineer.
Calculus is an exciting subject, justly considered to be one
of the greatest achievements of the human intellect. I hope you
will discover that it is not only useful but also intrinsically
beautiful.
JAMES STEWART
15.
TEC
TO THE STUDENT
xxiii
Openmirrors.com
DIAGNOSTIC TESTS
Success in calculus depends to a large extent on knowledge of the mathematics that
precedes calculus: algebra, analytic geometry, functions, and trigonometry. The fol-
lowing tests are intended to diagnose weaknesses that you might have in these areas.
After taking each test you can check your answers against the given answers and, if
necessary, refresh your skills by referring to the review materials that are provided.
1. Evaluate each expression without using a calculator.
(a) (b) (c)
(d) (e) (f)
2. Simplify each expression. Write your answer without negative exponents.
(a)
(b)
(c)
3. Expand and simplfy.
(a) (b)
(c) (d)
(e)
4. Factor each expression.
(a) (b)
(c) (d)
(e) (f)
5. Simplify the rational expression.
(a) (b)
(c) (d)
y
x
x
y
1
y
1
x
x
2
x
2
4
x 1
x 2
2x
2
x 1
x
2
9
ⴢ
x 3
2x 1
x
2
3x 2
x
2
x 2
x
3
y 4xy3x
3兾2
9x
1兾2
6x
1兾2
x
4
27xx
3
3x
2
4x 12
2x
2
5x 124x
2
25
共x 2兲
3
共2x 3兲
2
(
s
a
s
b
)(
s
a
s
b
)
共x 3兲共4x 5兲3共x 6兲 4共2x 5兲
冉
3x
3兾2
y
3
x
2
y
1兾2
冊
2
共3a
3
b
3
兲共4ab
2
兲
2
s
200
s
32
16
3兾4
冉
2
3
冊
2
5
23
5
21
3
4
3
4
共3兲
4
DIAGNOSTIC TEST: ALGEBRA
A
xxiv
DIAGNOSTIC TESTS
||||
xxv
6. Rationalize the expression and simplify.
(a) (b)
7. Rewrite by completing the square.
(a) (b)
8. Solve the equation. (Find only the real solutions.)
(a) (b)
(c) (d)
(e) (f)
(g)
9. Solve each inequality. Write your answer using interval notation.
(a) (b)
(c) (d)
(e)
10. State whether each equation is true or false.
(a) (b)
(c) (d)
(e) (f)
1兾x
a兾x b兾x
苷
1
a b
1
x y
苷
1
x
1
y
1 TC
C
苷 1 T
s
a
2
b
2
苷 a b
s
ab
苷
s
a
s
b
共p q兲
2
苷 p
2
q
2
2x 3
x 1
1
ⱍ
x 4
ⱍ
3x共x 1兲共x 2兲 0
x
2
2x 84
5 3x 17
2x共4 x兲
1兾2
3
s
4 x
苷 0
3
ⱍ
x 4
ⱍ
苷 10x
4
3x
2
2 苷 0
2x
2
4x 1 苷 0x
2
x 12 苷 0
2x
x 1
苷
2x 1
x
x 5 苷 14
1
2
x
2x
2
12x 11x
2
x 1
s
4 h
2
h
s
10
s
5
2
6. (a) (b)
7. (a) (b)
8. (a) (b) (c)
(d) (e) (f)
(g)
9. (a) (b)
(c) (d)
(e)
10. (a) False (b) True (c) False
(d) False (e) False (f) True
共1, 4兴
共1, 7兲共2, 0兲 傼 共1, 兲
共2, 4兲关4, 3兲
12
5
2
3
,
22
3
1,
s
2
1
1
2
s
2
3, 416
2共x 3兲
2
7
(
x
1
2
)
2
3
4
1
s
4 h
2
5
s
2
2
s
10
1. (a) (b) (c)
(d) (e) (f)
2. (a) (b) (c)
3. (a) (b)
(c) (d)
(e)
4. (a) (b)
(c) (d)
(e) (f)
5. (a) (b)
(c) (d) 共x y兲
1
x 2
x 1
x 3
x 2
x 2
xy共x 2兲共x 2兲3x
1兾2
共x 1兲共x 2兲
x共x 3兲共x
2
3x 9兲共x 3兲共x 2兲共x 2兲
共2x 3兲共x 4兲共2x 5兲共2x 5兲
x
3
6x
2
12x 8
4x
2
12x 9a b
4x
2
7x 1511x 2
x
9y
7
48a
5
b
7
6
s
2
1
8
9
4
25
1
81
8181
If you have had difficulty with these problems, you may wish to consult
the Review of Algebra on the website www.stewartcalculus.com.
ANSWERS TO DIAGNOSTIC TEST A: ALGEBRA
xxvi
||||
DIAGNOSTIC TESTS
1. Find an equation for the line that passes through the point and
(a) has slope
(b) is parallel to the -axis
(c) is parallel to the -axis
(d) is parallel to the line
2. Find an equation for the circle that has center and passes through the point .
3. Find the center and radius of the circle with equation .
4. Let and be points in the plane.
(a) Find the slope of the line that contains and .
(b) Find an equation of the line that passes through and . What are the intercepts?
(c) Find the midpoint of the segment .
(d) Find the length of the segment .
(e) Find an equation of the perpendicular bisector of .
(f) Find an equation of the circle for which is a diameter.
5. Sketch the region in the -plane defined by the equation or inequalities.
(a) (b)
(c) (d)
(e) (f) 9x
2
16y
2
苷 144x
2
y
2
4
y x
2
1
y
1
1
2
x
ⱍ
x
ⱍ
4 and
ⱍ
y
ⱍ
21 y 3
xy
AB
AB
AB
AB
BA
BA
B共5, 12兲A共7, 4兲
x
2
y
2
6x 10y 9 苷 0
共3, 2兲共1, 4兲
2x 4y 苷 3
y
x
3
共2, 5兲
DIAGNOSTIC TEST:
ANALYTIC GEOMETRY
B
5.
y
x12
0
y
x
0
y
x
0
4
3
_1
2
y
x
0
y
x
04_4
y
x
0
2
1
(a) (b) (c)
(d) (e) (f)
_1
3
2
_2
y=≈-1
≈+¥=4
y=1-
x
1
2
1. (a) (b)
(c) (d)
2.
3.
Center , radius 5
4. (a)
(b) ; -intercept , -intercept
(c)
(d)
(e)
(f) 共x 1兲
2
共y 4兲
2
苷 100
3x 4y 苷 13
20
共1, 4兲
16
3
y4x4x 3y 16 苷 0
4
3
共3, 5兲
共x 1兲
2
共y 4兲
2
苷 52
y 苷
1
2
x 6
x 苷 2
y 苷 5y 苷 3x 1
ANSWERS TO DIAGNOSTIC TEST B: ANALYTIC GEOMETRY
If you have had difficulty with these problems, you may wish to consult
the Review of Analytic Geometry on the website www.stewartcalculus.com.
DIAGNOSTIC TESTS
||||
xxvii
1.
The graph of a function is given at the left.
(a) State the value of .
(b) Estimate the value of .
(c) For what values of is ?
(d) Estimate the values of such that .
(e) State the domain and range of .
2.
If , evaluate the difference quotient and simplify your answer.
3.
Find the domain of the function.
(a) (b) (c)
4.
How are graphs of the functions obtained from the graph of ?
(a) (b) (c)
5.
Without using a calculator, make a rough sketch of the graph.
(a) (b) (c)
(d) (e) (f)
(g) (h)
6. Let
(a) Evaluate and . (b) Sketch the graph of .
7.
If and , find each of the following functions.
(a) (b) (c) t ⴰ t ⴰ tt ⴰ ff ⴰ t
t共x兲 苷 2x 3f 共x兲 苷 x
2
2x 1
ff 共1兲f 共2兲
f 共x兲 苷
再
1 x
2
2x 1
if x 0
if x 0
y 苷 1 x
1
y 苷 2
x
y 苷 2
s
x
y 苷
s
x
y 苷 4 x
2
y 苷 共x 2兲
3
3y 苷 共x 1兲
3
y 苷 x
3
y 苷 f 共x 3兲 2y 苷 2 f 共x兲 1y 苷 f 共x兲
f
h共x兲 苷
s
4 x
s
x
2
1t共x兲 苷
s
3
x
x
2
1
f 共x兲 苷
2x 1
x
2
x 2
f 共2 h兲 f 共2兲
h
f 共x兲 苷 x
3
f
f 共x兲 苷 0x
f 共x兲 苷 2x
f 共2兲
f 共1兲
f
DIAGNOSTIC TEST:
FUNCTIONS
C
6.
(a)
7.
(a)
(b) (b)
(c) 共t ⴰ t ⴰ t兲共x兲 苷 8x 21
共t ⴰ f 兲共x兲 苷 2x
2
4x 5
y
x
0
_1
1
共 f ⴰ t兲共x兲 苷 4x
2
8x 23, 3
y
(h)
x
0
1
1
(g)
y
x
0
1
_1
(f)
y
x
0
1
(e)
y
x
0
1
y
(d)
x
0
4
2
1.
(a) (b) 2.8
(c) (d)
(e)
2.
3.
(a)
(b)
(c)
4.
(a) Reflect about the -axis
(b) Stretch vertically by a factor of 2, then shift 1 unit downward
(c) Shift 3 units to the right and 2 units upward
5.
(c)
y
x
0
(2,3)
y
x
0
y
(a) (b)
1
1
x
0
1
_1
x
共, 1兴 傼 关1, 4兴
共, 兲
共, 2兲 傼 共2, 1兲 傼 共1, 兲
12 6h h
2
关3, 3兴, 关2, 3兴
2.5, 0.33, 1
2
y
0x
1
1
FIGURE FOR PROBLEM 1
ANSWERS TO DIAGNOSTIC TEST C: FUNCTIONS
If you have had difficulty with these problems, you should look at Sections 1.1–1.3 of this book.
Openmirrors.com
1.
Convert from degrees to radians.
(a) (b)
2.
Convert from radians to degrees.
(a) (b)
3.
Find the length of an arc of a circle with radius 12 cm if the arc subtends a central angle of .
4.
Find the exact values.
(a) (b) (c)
5.
Express the lengths and in the figure in terms of .
6.
If and , where and lie between and , evaluate .
7.
Prove the identities.
(a)
(b)
8.
Find all values of such that and .
9.
Sketch the graph of the function without using a calculator.y 苷 1 sin 2x
0 x 2
sin 2x 苷 sin xx
2 tan x
1 tan
2
x
苷 sin 2x
tan
sin
cos
苷 sec
sin共x y兲
20yxsec y 苷
5
4
sin x 苷
1
3
ba
sec共5
兾3兲sin共7
兾6兲tan共
兾3兲
30
25
兾6
18300
6.
8.
9.
_π π
x0
2
y
0,
兾3,
, 5
兾3, 2
1
15
(
4 6
s
2
)
1.
(a) (b)
2.
(a) (b)
3.
4.
(a) (b) (c)
5.
(a) (b) 24 cos
24 sin
2
1
2
s
3
2
cm
360兾
⬇ 114.6150
兾105
兾3
xxviii
||||
DIAGNOSTIC TESTS
DIAGNOSTIC TEST:
TRIGONOMETRY
D
a
¨
b
24
FIGURE FOR PROBLEM 5
ANSWERS TO DIAGNOSTIC TEST D: TRIGONOMETRY
If you have had difficulty with these problems, you should look at Appendix D of this book.
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SINGLE VARIABLE
CALCULUS
EARLY TRANSCENDENTALS