Multivariable calculus 4e james stewart

Multivariable Calculus
Concepts and Contexts | 4e

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Courtesy of Frank O. Gehry

Courtesy of Frank O. Gehry

The cover photograph shows the
DZ Bank in Berlin, designed and
built 1995–2001 by Frank Gehry
and Associates. The interior atrium
is dominated by a curvaceous fourstory stainless steel sculptural
shell that suggests a prehistoric
creature and houses a central conference space.
The highly complex structures
that Frank Gehry designs would be
impossible to build without the computer.
The CATIA software that his architects and engineers use to produce the
computer models is based on principles of
calculus—fitting curves by matching tangent
lines, making sure the curvature isn’t too

large, and controlling parametric surfaces.
“Consequently,” says Gehry, “we have a lot
of freedom. I can play with shapes.”
The process starts with Gehry’s initial
sketches, which are translated into a succession of physical models. (Hundreds of different
physical models were constructed during the design
of the building, first with basic wooden blocks and then
evolving into more sculptural forms.) Then an engineer
uses a digitizer to record the coordinates of a series of
points on a physical model. The digitized points are fed
into a computer and the CATIA software is used to link
these points with smooth curves. (It joins curves so that
their tangent lines coincide; you can use the same idea to
design the shapes of letters in the Laboratory Project on
page 208 of this book.) The architect has considerable freedom in creating these curves, guided by displays of the
curve, its derivative, and its curvature. Then the curves are

Courtesy of Frank O. Gehry

Calculus and the Architecture of Curves


Courtesy of Frank O. Gehry
thomasmayerarchive.com

Courtesy of Frank O. Gehry

connected to each other by a parametric surface,
and again the architect can do so in many possible
ways with the guidance of displays of the geometric

characteristics of the surface.
The CATIA model is then used to produce
another physical model, which, in turn, suggests
modifications and leads to additional computer
and physical models.

The CATIA program was developed in France
by Dassault Systèmes, originally for designing
airplanes, and was subsequently employed in
the automotive industry. Frank Gehry, because of
his complex sculptural shapes, is the first to use
it in architecture. It helps him answer his question, “How wiggly can you get and still make a
building?”

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Multivariable Calculus
Concepts and Contexts | 4e

James Stewart
McMaster University
and

University of Toronto

Australia . Brazil . Japan . Korea . Mexico . Singapore . Spain . United Kingdom . United States

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Multivariable Calculus: Concepts and Contexts,
Fourth Edition
James Stewart
Publisher: Richard Stratton
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For product information and technology assistance, contact us at
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For permission to use material from this text or product,
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K03T09

Trademarks
Derive is a registered trademark of Soft Warehouse, Inc.
Maple is a registered trademark of Waterloo Maple, Inc.
Mathematica is a registered trademark of Wolfram Research, Inc.
Tools for Enriching is a trademark used herein under license.


Contents
Preface

xi

To the Student

8

xx

Infinite Sequences and Series
8.1

Sequences

554

Laboratory Project
8.2
8.3
8.4
8.5
8.6
8.7

thomasmayerarchive.com

Writing Project

Logistic Sequences

■

■

An Elusive Limit

Applied Project

■

618

9.3
9.4

Radiation from the Stars

9.7

■

■

Laboratory Project

Review

634

662

663

Putting 3D in Perspective

Functions and Surfaces 673
Cylindrical and Spherical Coordinates
■

633

The Geometry of a Tetrahedron

Equations of Lines and Planes
Laboratory Project

9.6

627

631

Three-Dimensional Coordinate Systems
Vectors
639
The Dot Product 648
The Cross Product 654
Discovery Project

9.5

618

619

Vectors and the Geometry of Space

9.2

575

628

Focus on Problem Solving

9.1

564

How Newton Discovered the Binomial Series

Applications of Taylor Polynomials
Review

9

thomasmayerarchive.com

■

Series 565

The Integral and Comparison Tests; Estimating Sums
Other Convergence Tests
585
Power Series
592
Representations of Functions as Power Series 598
Taylor and Maclaurin Series 604
Laboratory Project

8.8

553

Families of Surfaces

672

682
687

688

Focus on Problem Solving

691

vii

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viii

CONTENTS

10 Vector Functions
10.1
10.2

Courtesy of Frank O. Gehry

10.3
10.4

Vector Functions and Space Curves 694
Derivatives and Integrals of Vector Functions 701
Arc Length and Curvature 707
Motion in Space: Velocity and Acceleration 716
Applied Project

10.5

693

■

Kepler’s Laws

Parametric Surfaces
Review
733

727

Focus on Problem Solving

11 Partial Derivatives
11.1
11.2
11.3
11.4
11.5
11.6
11.7

737

■

Courtesy of Frank O. Gehry

Discovery Project

Designing a Dumpster
■

Lagrange Multipliers

12.3
12.4
12.5
12.6
12.7

Rocket Science

Applied Project

■

820

Hydro-Turbine Optimization

821

822
827

829

Double Integrals over Rectangles 830
Iterated Integrals 838
Double Integrals over General Regions 844
Double Integrals in Polar Coordinates 853
Applications of Double Integrals 858
Surface Area 868
Triple Integrals 873

Discovery Project

12.8

812

813

Applied Project

12 Multiple Integrals

thomasmayerarchive.com

Quadratic Approximations and Critical Points

Focus on Problem Solving

12.2

811

■

Review

12.1

735

Functions of Several Variables 738
Limits and Continuity 749
Partial Derivatives 756
Tangent Planes and Linear Approximations 770
The Chain Rule 780
Directional Derivatives and the Gradient Vector 789
Maximum and Minimum Values 802
Applied Project

11.8

726

■

Volumes of Hyperspheres

883

Triple Integrals in Cylindrical and Spherical Coordinates
Applied Project

■

Discovery Project

Roller Derby
■

883

889

The Intersection of Three Cylinders

890

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CONTENTS

12.9

Change of Variables in Multiple Integrals
Review
899
Focus on Problem Solving

13 Vector Calculus
13.1
13.2
13.3
13.4
13.5
13.6

thomasmayerarchive.com

13.7

13.9

903

905

Vector Fields 906
Line Integrals 913
The Fundamental Theorem for Line Integrals
Green’s Theorem 934
Curl and Divergence 941
Surface Integrals 949
Stokes’ Theorem
960
Writing Project

13.8

891

■

Three Men and Two Theorems

The Divergence Theorem
Summary
973

Review
974
Focus on Problem Solving

Appendixes

925

966

967

977

A1

D

Precise Definitions of Limits

A2

E

A Few Proofs

H

Polar Coordinates

I

Complex Numbers

J

Answers to Odd-Numbered Exercises

A3
A6
A22
A31

Index A51

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ix


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Preface
When the first edition of this book appeared twelve years ago, a heated debate about calculus reform was taking place. Such issues as the use of technology, the relevance of rigor,
and the role of discovery versus that of drill were causing deep splits in mathematics

departments. Since then the rhetoric has calmed down somewhat as reformers and traditionalists have realized that they have a common goal: to enable students to understand and
appreciate calculus.
The first three editions were intended to be a synthesis of reform and traditional
approaches to calculus instruction. In this fourth edition I continue to follow that path by
emphasizing conceptual understanding through visual, verbal, numerical, and algebraic
approaches. I aim to convey to the student both the practical power of calculus and the
intrinsic beauty of the subject.

What’s New In the Fourth Edition?
The changes have resulted from talking with my colleagues and students at the University
of Toronto and from reading journals, as well as suggestions from users and reviewers.
Here are some of the many improvements that I’ve incorporated into this edition:
■

The majority of examples now have titles.

■

Some material has been rewritten for greater clarity or for better motivation. See,
for instance, the introduction to series on page 565.

■

New examples have been added and the solutions to some of the existing examples
have been amplified.

■

A number of pieces of art have been redrawn.

■

The data in examples and exercises have been updated to be more timely.

■

Sections 8.7 and 8.8 have been merged into a single section. I had previously
featured the binomial series in its own section to emphasize its importance. But
I learned that some instructors were omitting that section, so I decided to incorporate binomial series into 8.7.

■

More than 25% of the exercises in each chapter are new. Here are a few of my
favorites: 8.2.35, 9.1.42, 11.1.10–11, 11.6.37–38, 11.8.20–21, and 13.3.21–22.

■

There are also some good new problems in the Focus on Problem Solving sections.
See, for instance, Problem 13 on page 632, Problem 8 on page 692, Problem 9 on
page 736, and Problem 11 on page 904.

Features
Conceptual Exercises

The most important way to foster conceptual understanding is through the problems that
we assign. To that end I have devised various types of problems. Some exercise sets begin
with requests to explain the meanings of the basic concepts of the section. (See, for
instance, the first couple of exercises in Sections 8.2, 11.2, and 11.3. I often use them as
xi

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xii

PREFACE

a basis for classroom discussions.) Similarly, review sections begin with a Concept Check
and a True-False Quiz. Other exercises test conceptual understanding through graphs or
tables (see Exercises 8.7.2, 10.2.1–2, 10.3.33–37, 11.1.1–2, 11.1.9–18, 11.3.3–10, 11.6.1–2,
11.7.3–4, 12.1.5–10, 13.1.11–18, 13.2.15–16, and 13.3.1–2).
Graded Exercise Sets

Each exercise set is carefully graded, progressing from basic conceptual exercises and
skill-development problems to more challenging problems involving applications and
proofs.

Real-World Data

My assistants and I have spent a great deal of time looking in libraries, contacting companies and government agencies, and searching the Internet for interesting real-world data
to introduce, motivate, and illustrate the concepts of calculus. As a result, many of the
examples and exercises deal with functions defined by such numerical data or graphs. See,
for instance, Example 3 in Section 9.6 (wave heights).
Functions of two variables are illustrated by a table of values of the wind-chill index as
a function of air temperature and wind speed (Example 1 in Section 11.1). Partial derivatives are introduced in Section 11.3 by examining a column in a table of values of the heat
index (perceived air temperature) as a function of the actual temperature and the relative
humidity. This example is pursued further in connection with linear approximations
(Example 3 in Section 11.4). Directional derivatives are introduced in Section 11.6 by

using a temperature contour map to estimate the rate of change of temperature at Reno
in the direction of Las Vegas. Double integrals are used to estimate the average snowfall
in Colorado on December 20–21, 2006 (Example 4 in Section 12.1). Vector fields are
introduced in Section 13.1 by depictions of actual velocity vector fields showing San Francisco Bay wind patterns.

Projects

One way of involving students and making them active learners is to have them work (perhaps in groups) on extended projects that give a feeling of substantial accomplishment
when completed. Applied Projects involve applications that are designed to appeal to the
imagination of students. The project after Section 11.8 uses Lagrange multipliers to determine the masses of the three stages of a rocket so as to minimize the total mass while
enabling the rocket to reach a desired velocity. Discovery Projects explore aspects of
geometry: tetrahedra (after Section 9.4), hyperspheres (after Section 12.7), and intersections of three cylinders (after Section 12.8). The Laboratory Project on page 687 uses technology to discover how interesting the shapes of surfaces can be and how these shapes
evolve as the parameters change in a family. The Writing Project on page 966 explores the
historical and physical origins of Green’s Theorem and Stokes’ Theorem and the interactions of the three men involved. Many additional projects are provided in the Instructor’s Guide.

Technology

The availability of technology makes it not less important but more important to understand clearly the concepts that underlie the images on the screen. But, when properly used,
graphing calculators and computers are powerful tools for discovering and understanding
those concepts. I assume that the student has access to either a graphing calculator or a
computer algebra system. The icon ; indicates an exercise that definitely requires the use
of such technology, but that is not to say that a graphing device can’t be used on the other
exercises as well. The symbol CAS is reserved for problems in which the full resources of a
computer algebra system (like Derive, Maple, Mathematica, or the TI-89/92) are required.
But technology doesn’t make pencil and paper obsolete. Hand calculation and sketches are
often preferable to technology for illustrating and reinforcing some concepts. Both instructors and students need to develop the ability to decide where the hand or the machine is
appropriate.

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PREFACE

xiii

Tools for Enriching™ Calculus

TEC is a companion to the text and is intended to enrich and complement its contents. (It
is now accessible from the Internet at www.stewartcalculus.com.) Developed by Harvey
Keynes, Dan Clegg, Hubert Hohn, and myself, TEC uses a discovery and exploratory
approach. In sections of the book where technology is particularly appropriate, marginal
icons direct students to TEC modules that provide a laboratory environment in which they
can explore the topic in different ways and at different levels. Visuals are animations of figures in the text; Modules are more elaborate activities and include exercises. Instructors
can choose to become involved at several different levels, ranging from simply encouraging students to use the Visuals and Modules for independent exploration, to assigning specific exercises from those included with each Module, or to creating additional exercises,
labs, and projects that make use of the Visuals and Modules.
TEC also includes Homework Hints for representative exercises (usually odd-numbered) in every section of the text, indicated by printing the exercise number in red. These
hints are usually presented in the form of questions and try to imitate an effective teaching
assistant by functioning as a silent tutor. They are constructed so as not to reveal any more
of the actual solution than is minimally necessary to make further progress.

Enhanced WebAssign

Technology is having an impact on the way homework is assigned to students, particularly in large classes. The use of online homework is growing and its appeal depends on
ease of use, grading precision, and reliability. With the fourth edition we have been working with the calculus community and WebAssign to develop an online homework system.
Many of the exercises in each section are assignable as online homework, including free
response, multiple choice, and multi-part formats. The system also includes Active Examples,
in which students are guided in step-by-step tutorials through text examples, with links to
the textbook and to video solutions.

Website: www.stewartcalculus.com

This website includes the following.
■

Algebra Review

■

Lies My Calculator and Computer Told Me

■

History of Mathematics, with links to the better historical websites

■

Additional Topics (complete with exercise sets):
Trigonometric Integrals, Trigonometric Substitution, Strategy for Integration,
Strategy for Testing Series, Fourier Series, Formulas for the Remainder Term in
Taylor Series, Linear Differential Equations, Second-Order Linear Differential
Equations, Nonhomogeneous Linear Equations, Applications of Second-Order
Differential Equations, Using Series to Solve Differential Equations, Rotation
of Axes, and (for instructors only) Hyperbolic Functions

■

Links, for each chapter, to outside Web resources

■

Archived Problems (drill exercises that appeared in previous editions, together
with their solutions)

■

Challenge Problems (some from the Focus on Problem Solving sections of prior
editions)

Content
8

■

Infinite Sequences and Series

Tests for the convergence of series are considered briefly, with intuitive rather than formal justifications. Numerical estimates of sums of series are based on which test was
used to prove convergence. The emphasis is on Taylor series and polynomials and their
applications to physics. Error estimates include those from graphing devices.

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xiv

PREFACE

9 ■ Vectors and The Geometry of Space

The dot product and cross product of vectors are given geometric definitions, motivated by
work and torque, before the algebraic expressions are deduced. To facilitate the discussion
of surfaces, functions of two variables and their graphs are introduced here.
10 ■ Vector Functions

The calculus of vector functions is used to prove Kepler’s First Law of planetary motion,
with the proofs of the other laws left as a project. In keeping with the introduction of parametric curves in Chapter 1, parametric surfaces are introduced as soon as possible, namely,
in this chapter. I think an early familiarity with such surfaces is desirable, especially with
the capability of computers to produce their graphs. Then tangent planes and areas of parametric surfaces can be discussed in Sections 11.4 and 12.6.
11

■

Partial Derivatives

Functions of two or more variables are studied from verbal, numerical, visual, and algebraic points of view. In particular, I introduce partial derivatives by looking at a specific
column in a table of values of the heat index (perceived air temperature) as a function of
the actual temperature and the relative humidity. Directional derivatives are estimated from
contour maps of temperature, pressure, and snowfall.
12

■

Multiple Integrals

Contour maps and the Midpoint Rule are used to estimate the average snowfall and average
temperature in given regions. Double and triple integrals are used to compute probabilities, areas of parametric surfaces, volumes of hyperspheres, and the volume of intersection
of three cylinders.

13 ■ Vector Fields

Vector fields are introduced through pictures of velocity fields showing San Francisco Bay
wind patterns. The similarities among the Fundamental Theorem for line integrals, Green’s
Theorem, Stokes’ Theorem, and the Divergence Theorem are emphasized.

Ancillaries
Multivariable Calculus: Concepts and Contexts, Fourth Edition, is supported by a complete set of ancillaries developed under my direction. Each piece has been designed to
enhance student understanding and to facilitate creative instruction. The table on pages
xviii and xix lists ancillaries available for instructors and students.

Acknowledgments
I am grateful to the following reviewers for sharing their knowledge and judgment with
me. I have learned something from each of them.
Fourth Edition Reviewers

Jennifer Bailey, Colorado School of Mines
Lewis Blake, Duke University
James Cook, North Carolina State University
Costel Ionita, Dixie State College
Lawrence Levine, Stevens Institute of Technology
Scott Mortensen, Dixie State College

Drew Pasteur, North Carolina State University
Jeffrey Powell, Samford University
Barbara Tozzi, Brookdale Community College
Kathryn Turner, Utah State University
Cathy Zucco-Tevelof, Arcadia University

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PREFACE
Previous Edition Reviewers

Irfan Altas, Charles Sturt University
William Ardis, Collin County Community College
Barbara Bath, Colorado School of Mines
Neil Berger, University of Illinois at Chicago
Jean H. Bevis, Georgia State University
Martina Bode, Northwestern University
Jay Bourland, Colorado State University
Paul Wayne Britt, Louisiana State University
Judith Broadwin, Jericho High School (retired)
Charles Bu, Wellesley University
Meghan Anne Burke, Kennesaw State University
Robert Burton, Oregon State University
Roxanne M. Byrne, University of Colorado at Denver
Maria E. Calzada, Loyola University–New Orleans
Larry Cannon, Utah State University
Deborah Troutman Cantrell,
Chattanooga State Technical Community College
Bem Cayco, San Jose State University
John Chadam, University of Pittsburgh
Robert A. Chaffer, Central Michigan University
Dan Clegg, Palomar College
Camille P. Cochrane, Shelton State Community College
James Daly, University of Colorado

Richard Davis, Edmonds Community College
Susan Dean, DeAnza College
Richard DiDio, LaSalle University
Robert Dieffenbach, Miami University–Middletown
Fred Dodd, University of South Alabama
Helmut Doll, Bloomsburg University
William Dunham, Muhlenberg College
David A. Edwards, The University of Georgia
John Ellison, Grove City College
Joseph R. Fiedler, California State University–Bakersfield
Barbara R. Fink, DeAnza College
James P. Fink, Gettysburg College
Joe W. Fisher, University of Cincinnati
Robert Fontenot, Whitman College
Richard L. Ford, California State University Chico
Laurette Foster, Prairie View A & M University
Ronald C. Freiwald, Washington University in St. Louis
Frederick Gass, Miami University
Gregory Goodhart, Columbus State Community College
John Gosselin, University of Georgia

Daniel Grayson,
University of Illinois at Urbana–Champaign
Raymond Greenwell, Hofstra University
Gerrald Gustave Greivel, Colorado School of Mines
John R. Griggs, North Carolina State University
Barbara Bell Grover, Salt Lake Community College
Murli Gupta, The George Washington University
John William Hagood, Northern Arizona University
Kathy Hann, California State University at Hayward

Richard Hitt, University of South Alabama
Judy Holdener, United States Air Force Academy
Randall R. Holmes, Auburn University
Barry D. Hughes, University of Melbourne
Mike Hurley, Case Western Reserve University
Gary Steven Itzkowitz, Rowan University
Helmer Junghans, Montgomery College
Victor Kaftal, University of Cincinnati
Steve Kahn, Anne Arundel Community College
Mohammad A. Kazemi,
University of North Carolina, Charlotte
Harvey Keynes, University of Minnesota
Kandace Alyson Kling, Portland Community College
Ronald Knill, Tulane University
Stephen Kokoska, Bloomsburg University
Kevin Kreider, University of Akron
Doug Kuhlmann, Phillips Academy
David E. Kullman, Miami University
Carrie L. Kyser, Clackamas Community College
Prem K. Kythe, University of New Orleans
James Lang, Valencia Community College–East Campus
Carl Leinbach, Gettysburg College
William L. Lepowsky, Laney College
Kathryn Lesh, University of Toledo
Estela Llinas, University of Pittsburgh at Greensburg
Beth Turner Long,
Pellissippi State Technical Community College
Miroslav Lovri´c, McMaster University
Lou Ann Mahaney, Tarrant County Junior College–Northeast
John R. Martin, Tarrant County Junior College

Andre Mathurin, Bellarmine College Prep
R. J. McKellar, University of New Brunswick
Jim McKinney,
California State Polytechnic University–Pomona

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xv


xvi

PREFACE

Richard Eugene Mercer, Wright State University
David Minda, University of Cincinnati
Rennie Mirollo, Boston College
Laura J. Moore-Mueller, Green River Community College
Scott L. Mortensen, Dixie State College
Brian Mortimer, Carleton University
Bill Moss, Clemson University
Tejinder Singh Neelon,
California State University San Marcos
Phil Novinger, Florida State University
Richard Nowakowski, Dalhousie University
Stephen Ott, Lexington Community College
Grace Orzech, Queen’s University
Jeanette R. Palmiter, Portland State University

Bill Paschke, University of Kansas
David Patocka, Tulsa Community College–Southeast Campus
Paul Patten, North Georgia College
Leslie Peek, Mercer University
Mike Pepe, Seattle Central Community College
Dan Pritikin, Miami University
Fred Prydz, Shoreline Community College
Denise Taunton Reid, Valdosta State University
James Reynolds, Clarion University
Hernan Rivera, Texas Lutheran University
Richard Rochberg, Washington University
Gil Rodriguez, Los Medanos College
David C. Royster, University of North Carolina–Charlotte
Daniel Russow, Arizona Western College
Dusty Edward Sabo, Southern Oregon University

Daniel S. Sage, Louisiana State University
N. Paul Schembari, East Stroudsburg University
Dr. John Schmeelk, Virginia Commonwealth University
Bettina Schmidt, Auburn University at Montgomery
Bernd S.W. Schroeder, Louisiana Tech University
Jeffrey Scott Scroggs, North Carolina State University
James F. Selgrade, North Carolina State University
Brad Shelton, University of Oregon
Don Small,
United States Military Academy–West Point
Linda E. Sundbye, The Metropolitan State College of Denver
Richard B. Thompson,The University of Arizona
William K. Tomhave, Concordia College
Lorenzo Traldi, Lafayette College

Alan Tucker,
State University of New York at Stony Brook
Tom Tucker, Colgate University
George Van Zwalenberg, Calvin College
Dennis Watson, Clark College
Paul R. Wenston, The University of Georgia
Ruth Williams, University of California–San Diego
Clifton Wingard, Delta State University
Jianzhong Wang, Sam Houston State University
JingLing Wang, Lansing Community College
Michael B. Ward, Western Oregon University
Stanley Wayment, Southwest Texas State University
Barak Weiss, Ben Gurion University–Be’er Sheva, Israel
Teri E. Woodington, Colorado School of Mines
James Wright, Keuka College

In addition, I would like to thank Ari Brodsky, David Cusick, Alfonso Gracia-Saz,
Emile LeBlanc, Tanya Leise, Joe May, Romaric Pujol, Norton Starr, Lou Talman, and Gail
Wolkowicz for their advice and suggestions; Al Shenk and Dennis Zill for permission to
use exercises from their calculus texts; COMAP for permission to use project material;
Alfonso Gracia-Saz, B. Hovinen, Y. Kim, Anthony Lam, Romaric Pujol, Felix Recio, and
Paul Sally for ideas for exercises; Dan Drucker for the roller derby project; and Tom
Farmer, Fred Gass, John Ramsay, Larry Riddle, V. K. Srinivasan, and Philip Straffin for
ideas for projects. I’m grateful to Dan Clegg, Jeff Cole, and Tim Flaherty for preparing the
answer manuscript and suggesting ways to improve the exercises.
As well, I thank those who have contributed to past editions: Ed Barbeau, George
Bergman, David Bleecker, Fred Brauer, Andy Bulman-Fleming, Tom DiCiccio, Martin
Erickson, Garret Etgen, Chris Fisher, Stuart Goldenberg, Arnold Good, John Hagood,
Gene Hecht, Victor Kaftal, Harvey Keynes, E. L. Koh, Zdislav Kovarik, Kevin Kreider,
Jamie Lawson, David Leep, Gerald Leibowitz, Larry Peterson, Lothar Redlin, Peter Rosenthal, Carl Riehm, Ira Rosenholtz, Doug Shaw, Dan Silver, Lowell Smylie, Larry Wallen,

Saleem Watson, and Alan Weinstein.

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PREFACE

xvii

I also thank Stephanie Kuhns, Rebekah Million, Brian Betsill, and Kathi Townes of
TECH-arts for their production services; Marv Riedesel and Mary Johnson for their careful proofing of the pages; Thomas Mayer for the cover image; and the following Brooks/
Cole staff: Cheryll Linthicum, editorial production project manager; Jennifer Jones,
Angela Kim, and Mary Anne Payumo, marketing team; Peter Galuardi, media editor; Jay
Campbell, senior developmental editor; Jeannine Lawless, associate editor; Elizabeth
Neustaetter, editorial assistant; Bob Kauser, permissions editor; Becky Cross, print/media
buyer; Vernon Boes, art director; Rob Hugel, creative director; and Irene Morris, cover
designer. 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 three decades: Ron Munro, Harry Campbell, Craig Barth, Jeremy Hayhurst, Gary Ostedt, Bob Pirtle, and now Richard Stratton. Special thanks go to all of them.
JAMES STEWART

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Ancillaries for Instructors
PowerLecture CD-ROM with JoinIn and ExamView

ISBN 0-495-56049-9

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. Also contains JoinIn on TurningPoint
personal response system questions and ExamView algorithmic test generation. See below for complete descriptions.
TEC 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.

ExamView
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 full algorithmic generation of problems and complete questions from the Printed Test
Bank.
JoinIn on TurningPoint
Enhance how your students interact with you, your lecture, and
each other. Brooks/Cole, Cengage Learning 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 Cengage representative
to learn more about JoinIn on TurningPoint and our exclusive
infrared and radio-frequency hardware solutions.
Text-Specific DVDs
ISBN 0-495-56050-2

Instructor’s Guide
by Douglas Shaw and James Stewart
ISBN 0-495-56047-2

Each section of the main text is discussed from several viewpoints and contains suggested time to allot, points to stress,
text discussion topics, core materials for lecture, workshop/
discussion suggestions, group work exercises in a form suitable for handout, and suggested homework problems. An electronic version is available on the PowerLecture CD-ROM.
Instructor’s Guide for AP ® Calculus
by Douglas Shaw

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 multivariable calculus.
Solution Builder
www.cengage.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.

ISBN 0-495-56059-6

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, Multivariable

by Dan Clegg
ISBN 0-495-56056-1

Ancillaries for Instructors and Students
eBook Option
ISBN 0-495-56121-5

Whether you prefer a basic downloadable eBook or a premium multimedia eBook with search, highlighting, and note
taking capabilities as well as links to videos and simulations,
this new edition offers a range of eBook options to fit how you
want to read and interact with the content.

Includes worked-out solutions to all exercises in the text.
Printed Test Bank
by William Tomhave and Xuequi Zeng
ISBN 0-495-56123-1

Contains multiple-choice and short-answer test items that key
directly to the text.

xviii

Stewart Specialty Website
www.stewartcalculus.com
Contents: Algebra Review Additional Topics Drill
Web Links History of
exercises Challenge Problems
Mathematics Tools for Enriching Calculus (TEC)
N

N

N

N

N

N

|||| Electronic items

|||| Printed items

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Enhanced WebAssign
Instant feedback, grading precision, and ease of use are just
three reasons why WebAssign is the most widely used homework system in higher education. WebAssign’s homework
delivery 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: Concepts and Contexts—incorporating
exercises, examples, video skillbuilders and quizzes to promote
active learning and provide the immediate, relevant feedback
students want.

Student Solutions Manual, Multivariable
by Dan Clegg
ISBN 0-495-56055-3

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, Multivariable
by Philip B. Yasskin and Art Belmonte
ISBN 0-495-56058-8

The Brooks/Cole Mathematics Resource Center Website
www.cengage.com/math
When you adopt a Brooks/Cole, Cengage Learning 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 0-495-01492-3 (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 environment. Available for bundling with your Stewart
Calculus text at a special discount.

CalcLabs with Mathematica, Multivariable
by Selwyn Hollis
ISBN 0-495-82722-3

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, Second Edition
by Dennis Ebersole, Doris Schattschneider, Alicia Sevilla,
and Kay Somers
ISBN 0-495-01124-X

Written to improve algebra and problem-solving skills of
students 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 calculus courses that integrate the review of precalculus concepts or for individual use.

Student Resources
TEC 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.

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.

Study Guide, Multivariable
by Robert Burton and Dennis Garity
ISBN 0-495-56057-X

Contains key concepts, skills to master, a brief discussion of
the ideas of the section, and worked-out examples with tips
on how to find the solution.

|||| Electronic items

|||| Printed items

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xix


To the Student

Reading a calculus textbook is different from reading a newspaper 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 understand 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 J. Some exercises ask for a verbal 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 s2 Ϫ 1 and you
obtain 1͞(1 ϩ s2 ), 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 CAS is reserved for problems in which the full resources
of a computer algebra system (like Derive, Maple, Mathematica, 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 proportion 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 TEC and can be
accessed from www.stewartcalculus.com. It directs you to modules 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: 15. 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.
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

xx

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Infinite Sequences and Series

8

thomasmayerarchive.com

Infinite sequences and series were introduced briefly in A Preview of Calculus
in connection with Zeno’s paradoxes and the decimal representation of numbers.
Their importance in calculus stems from Newton’s idea of representing functions
as sums of infinite series. For instance, in finding areas he often integrated a
function by first expressing it as a series and then integrating each term of the
series. We will pursue his idea in Section 8.7 in order to integrate such functions
2
as eϪx . (Recall that we have previously been unable to do this.) Many of the
functions that arise in mathematical physics and chemistry, such as Bessel functions, are defined as sums of series, so it is important to be familiar with the
basic concepts of convergence of infinite sequences and series.
Physicists also use series in another way, as we will see in Section 8.8. In
studying fields as diverse as optics, special relativity, and electromagnetism, they
analyze phenomena by replacing a function with the first few terms in the series
that represents it.
553

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554

CHAPTER 8

INFINITE SEQUENCES AND SERIES

8.1 Sequences
A sequence can be thought of as a list of numbers written in a definite order:
a1 , a2 , a3 , a4 , . . . , an , . . .
The number a 1 is called the first term, a 2 is the second term, and in general a n is the nth
term. We will deal exclusively with infinite sequences and so each term a n will have a
successor a nϩ1 .
Notice that for every positive integer n there is a corresponding number a n and so a
sequence can be defined as a function whose domain is the set of positive integers. But we
usually write a n instead of the function notation f ͑n͒ for the value of the function at the
number n.
Notation: The sequence {a 1 , a 2 , a 3 , . . .} is also denoted by

͕a n ͖

ϱ

͕a n ͖ n෇1

or

EXAMPLE 1 Describing sequences Some sequences can be defined by giving a formula
for the nth term. In the following examples we give three descriptions of the sequence:
one by using the preceding notation, another by using the defining formula, and a third
by writing out the terms of the sequence. Notice that n doesn’t have to start at 1.

(a)

(b)
(c)
(d)

ͭ ͮ
ͭ
ͮ
ϱ

n
nϩ1

an ෇

n
nϩ1

an ෇

͑Ϫ1͒n͑n ϩ 1͒
3n

n෇1

͑Ϫ1͒n͑n ϩ 1͒
3n

{sn Ϫ 3 } ϱn෇3

a n ෇ sn Ϫ 3 , n ജ 3

ͭ ͮ

a n ෇ cos

n␲
cos
6

v

ϱ

n෇0

n␲
, nജ0
6

ͭ
ͭ

ͮ

1 2 3 4
n
, , , ,...,
,...
2 3 4 5

nϩ1

ͮ

2 3
4 5
͑Ϫ1͒n͑n ϩ 1͒
Ϫ , ,Ϫ ,
,...,
,...
3 9
27 81
3n

{0, 1, s2 , s3 , . . . , sn Ϫ 3 , . . .}

ͭ

1,

n␲
s3 1
, , 0, . . . , cos
,...
2 2
6

ͮ

EXAMPLE 2 Find a formula for the general term a n of the sequence

ͭ

ͮ

3
4
5
6
7
,Ϫ ,
,Ϫ
,
,...
5
25 125
625 3125

assuming that the pattern of the first few terms continues.
SOLUTION We are given that

a1 ෇

3
5

a2 ෇ Ϫ

4
25

a3 ෇

5
125

a4 ෇ Ϫ

6
625

a5 ෇

7
3125

Notice that the numerators of these fractions start with 3 and increase by 1 whenever we
go to the next term. The second term has numerator 4, the third term has numerator 5; in
general, the nth term will have numerator n ϩ 2. The denominators are the powers of 5,

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SECTION 8.1

SEQUENCES

555

so a n has denominator 5 n. The signs of the terms are alternately positive and negative,
so we need to multiply by a power of Ϫ1. In Example 1(b) the factor ͑Ϫ1͒ n meant we
started with a negative term. Here we want to start with a positive term and so we use
͑Ϫ1͒ nϪ1 or ͑Ϫ1͒ nϩ1. Therefore
a n ෇ ͑Ϫ1͒ nϪ1

nϩ2
5n

EXAMPLE 3 Here are some sequences that don’t have simple defining equations.

(a) The sequence ͕pn ͖, where pn is the population of the world as of January 1 in the
year n.
(b) If we let a n be the digit in the nth decimal place of the number e, then ͕a n ͖ is a welldefined sequence whose first few terms are
͕7, 1, 8, 2, 8, 1, 8, 2, 8, 4, 5, . . .͖
(c) The Fibonacci sequence ͕ fn ͖ is defined recursively by the conditions
f1 ෇ 1

f2 ෇ 1

fn ෇ fnϪ1 ϩ fnϪ2

nജ3

Each term is the sum of the two preceding terms. The first few terms are
͕1, 1, 2, 3, 5, 8, 13, 21, . . .͖
This sequence arose when the 13th-century Italian mathematician known as Fibonacci
solved a problem concerning the breeding of rabbits (see Exercise 47).
a¡

a™ a£

1
2

0

A sequence such as the one in Example 1(a), a n ෇ n͑͞n ϩ 1͒, can be pictured either by
plotting its terms on a number line, as in Figure 1, or by plotting its graph, as in Figure 2.
Note that, since a sequence is a function whose domain is the set of positive integers, its
graph consists of isolated points with coordinates

a¢
1

FIGURE 1

͑1, a1 ͒

an

͑2, a2 ͒

͑3, a3 ͒

...

͑n, a n ͒

...

From Figure 1 or Figure 2 it appears that the terms of the sequence a n ෇ n͑͞n ϩ 1͒ are
approaching 1 as n becomes large. In fact, the difference

1

1Ϫ

7

a¶= 8
0

1 2 3 4 5 6 7

n

n
1
෇
nϩ1
nϩ1

can be made as small as we like by taking n sufficiently large. We indicate this by writing
lim a n ෇ lim

FIGURE 2

nlϱ

nlϱ

n
෇1
nϩ1

In general, the notation
lim a n ෇ L

nlϱ

means that the terms of the sequence ͕a n ͖ approach L as n becomes large. Notice that the
following definition of the limit of a sequence is very similar to the definition of a limit of
a function at infinity given in Section 2.5.

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556

CHAPTER 8

INFINITE SEQUENCES AND SERIES

1

Definition A sequence ͕a n ͖ has the limit L and we write

lim a n ෇ L

nlϱ

A more precise definition of the limit of a
sequence is given in Appendix D.

or

a n l L as n l ϱ

if we can make the terms a n as close to L as we like by taking n sufficiently large.
If lim n l ϱ a n exists, we say the sequence converges (or is convergent). Otherwise,
we say the sequence diverges (or is divergent).
Figure 3 illustrates Definition 1 by showing the graphs of two sequences that have the
limit L.
an

an

L

L

FIGURE 3

Graphs of two
sequences with
lim an= L

0

0

n

n

n  `

If you compare Definition 1 with Definition 2.5.4 you will see that the only difference
between lim n l ϱ a n ෇ L and lim x l ϱ f ͑x͒ ෇ L is that n is required to be an integer. Thus
we have the following theorem, which is illustrated by Figure 4.
2 Theorem If lim x l ϱ f ͑x͒ ෇ L and f ͑n͒ ෇ a n when n is an integer, then
lim n l ϱ a n ෇ L.

y

y=ƒ

L

0

FIGURE 4

x

1 2 3 4

In particular, since we know from Section 2.5 that lim x l ϱ ͑1͞x r ͒ ෇ 0 when r Ͼ 0, we
have
3

lim

nlϱ

1
෇0
nr

if r Ͼ 0

If an becomes large as n becomes large, we use the notation
lim a n ෇ ϱ

nlϱ

In this case the sequence ͕a n ͖ is divergent, but in a special way. We say that ͕a n ͖ diverges
to ϱ.
The Limit Laws given in Section 2.3 also hold for the limits of sequences and their
proofs are similar.

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