licensed to: [email protected] ca l c u l u...

Licensed to: [email protected]

s i n g l e v a r i a b l e

C A L C U L U Se a r l y t r a n s c e n d e n t a l s

V o l u m e 1 : C h a p t e r s 1 – 6

Copyright 2005 Thomson Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.


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A B O U T T H E C O V E R

The art on the cover was created by Bill Ralph, a mathematician who uses mod-

ern mathematics to produce visual representations of “dynamical systems.”

Examples of dynamical systems in nature include the weather, blood pressure,

the motions of the planets, and other phenomena that involve continual change.

Such systems, which tend to be unpredictable and even chaotic at times, are mod-

eled mathematically using the concepts of composition and iteration of functions.

The process of creating the cover art starts with a photograph of a violin. The

color values at each point on the photograph are then converted into numbers

and a particular function is evaluated at each of those numbers giving a new

number at each point of the photograph. The same function is then evaluated at

each of these new numbers. Repeating this process produces a sequence of num-

bers called iterates of the function. The original photograph is then “repainted”

using colors determined by certain properties of this sequence of iterates and the

mathematical concept of “dimension.” The final image is the result of mingling

photographic reality with the complex behavior of a dynamical system.



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f i f t h e d i t i o n

V o l u m e 1 : C h a p t e r s 1 – 6

J A M E S S T E W A R T

McMaster University



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Preface ix

To the Student xviii

A Preview of Calculus 2

|||| 1 Functions and Models 10

1.1 Four Ways to Represent a Function 11

1.2 Mathematical Models: A Catalog of Essential Functions 25

1.3 New Functions from Old Functions 38

1.4 Graphing Calculators and Computers 48

1.5 Exponential Functions 55

1.6 Inverse Functions and Logarithms 63

Review 77

Principles of Problem Solving 80

|||| 2 Limits and Derivatives 86

2.1 The Tangent and Velocity Problems 87

2.2 The Limit of a Function 92

2.3 Calculating Limits Using the Limit Laws 104

2.4 The Precise DeÞnition of a Limit 114

2.5 Continuity 124

2.6 Limits at InÞnity; Horizontal Asymptotes 135

2.7 Tangents, Velocities, and Other Rates of Change 149

v

C O N T E N T S



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vi ❙ ❙ ❙ ❙ CONTENTS

2.8 Derivatives 158

Writing Project ■ Early Methods for Finding Tangents 164

2.9 The Derivative as a Function 165

Review 176

Problems Plus 180

|||| 3 Differentiation Rules 182

3.1 Derivatives of Polynomials and Exponential Functions 183

3.2 The Product and Quotient Rules 192

3.3 Rates of Change in the Natural and Social Sciences 199

3.4 Derivatives of Trigonometric Functions 211

3.5 The Chain Rule 217

3.6 Implicit Differentiation 227

3.7 Higher Derivatives 236

Applied Project ■ Where Should a Pilot Start Descent? 243

Applied Project ■ Building a Better Roller Coaster 243

3.8 Derivatives of Logarithmic Functions 244

3.9 Hyperbolic Functions 250

3.10 Related Rates 256

3.11 Linear Approximations and Differentials 262

Laboratory Project ■ Taylor Polynomials 269

Review 270

Problems Plus 274

|||| 4 Applications of Differentiation 278

4.1 Maximum and Minimum Values 279

Applied Project ■ The Calculus of Rainbows 288

4.2 The Mean Value Theorem 290

4.3 How Derivatives Affect the Shape of a Graph 296

4.4 Indeterminate Forms and LÕHospitalÕ s Rule 307

Writing Project ■ The Origins of L’Hospital’s Rule 315

4.5 Summary of Curve Sketching 316

4.6 Graphing with Calculus and Calculators 324



CONTENTS ❙ ❙ ❙ ❙ vii

4.7 Optimization Problems 331

Applied Project ■ The Shape of a Can 341

4.8 Applications to Business and Economics 342

4.9 NewtonÕs Method 347

4.10 Antiderivatives 353

Review 361

Problems Plus 365

|||| 5 Integrals 368

5.1 Areas and Distances 369

5.2 The DeÞnite Inte gral 380

Discovery Project ■ Area Functions 393

5.3 The Fundamental Theorem of Calculus 394

5.4 IndeÞnite Inte grals and the Net Change Theorem 405

Writing Project ■ Newton, Leibniz, and the Invention of Calculus 413

5.5 The Substitution Rule 414

5.6 The Logarithm DeÞned as an Inte gral 422

Review 430

Problems Plus 434

|||| 6 Applications of Integration 436

6.1 Areas between Curves 437

6.2 Volumes 444

6.3 Volumes by Cylindrical Shells 455

6.4 Work 460

6.5 Average Value of a Function 464

Applied Project ■ Where to Sit at the Movies 468

Review 468

Problems Plus 470



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viii ❙ ❙ ❙ ❙ CONTENTS

|||| Appendixes A1

A Numbers, Inequalities, and Absolute Values A2

B Coordinate Geometry and Lines A10

C Graphs of Second-Degree Equations A16

D Trigonometry A24

E Sigma Notation A34

F Proofs of Theorems A39

G Complex Numbers A46

H Answers to Odd-Numbered Exercises A55

|||| Index A87



The art of teaching, Mark Van Doren said, is the art of assisting discovery. I have tried towrite a book that assists students in discovering calculusÑboth for its practical po wer andits surprising beauty. In this edition, as in the Þrst four editions, I aim to convey to the stu-dent a sense of the utility of calculus and develop technical competence, but I also striveto give some appreciation for the intrinsic beauty of the subject. Newton undoubtedlyexperienced a sense of triumph when he made his great discoveries. I want students toshare some of that excitement.

The emphasis is on understanding concepts. I think that nearly everybody agrees thatthis should be the primary goal of calculus instruction. In fact, the impetus for the currentcalculus reform movement came from the Tulane Conference in 1986, which formulatedas their Þrst recommendation:

Focus on conceptual understanding.

I have tried to implement this goal through the Rule of Three: Ò Topics should be pre-sented geometrically, numerically, and algebraically.Ó Visualization, numerical and graph-ical experimentation, and other approaches have changed how we teach conceptual reasoning in fundamental ways. More recently, the Rule of Three has been expanded tobecome the Rule of Four by emphasizing the verbal, or descriptive, point of view as well.

In writing the Þfth edition my premise has been that it is possible to achie ve concep-tual understanding and still retain the best traditions of traditional calculus. The book con-tains elements of reform, but within the context of a traditional curriculum. (Instructorswho prefer a more streamlined curriculum should look at my book Calculus: Concepts and

Contexts, Second Edition.)

|||| W h a t ’s N e w i n t h e F i f t h E d i t i o n

By way of preparing to write the Þfth edition of this te xt, I spent a year teaching calculusfrom the fourth edition at the University of Toronto. I listened carefully to my studentsÕquestions and my colleaguesÕ suggestions. And as I prepared each lecture I sometimes real-ized that an additional example was needed, or a sentence could be clariÞed, or a sectioncould use a few more exercises of a certain type. In addition, I paid attention to the sug-gestions sent to me by many users and to the comments of the reviewers.

P R E F A C E

ix

A great discovery solves a great problem but there is a grain of discovery in the

solution of any problem. Your problem may be modest; but if it challenges your

curiosity and brings into play your inventive faculties, and if you solve it by your

own means, you may experience the tension and enjoy the triumph of discovery.

GEORGE POLYA



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The structure of Single Variable Calculus, Early Transcendentals, Fifth Edition, remainslargely unchanged, but there are hundreds of improvements, small and large:

■ The review of inverse trigonometric functions has been moved from an appendix toSection 1.6.

■ New phrases and margin notes have been added to clarify the exposition.

■ A number of pieces of art have been redrawn.

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

■ Examples have been added. For instance, I added the new Example 1 in Section 5.3(page 394) because students have a tough time grasping the idea of a function deÞnedby an integral with a variable limit of integration. I think it helps to look at Example 1before considering the Fundamental Theorem of Calculus.

■ Extra steps have been provided in some of the existing examples.

■ More than 25% of the exercises in each chapter are new. Here are a few of myfavorites:

IÕve also added new problems to the Problems Plus sections. See, for instance,Problems 20 and 21 on page 277.

■ One new project has been added (see page 243). It asks students to design a rollercoaster so the track is smooth at transition points.

■ A CD called Tools for Enriching Calculus (TEC ) is available for use with the Þfthedition. See the description on page xi.

■ Conscious of the need to control the size of the book, IÕve put additional topics (withexercises) on the revamped web site www.stewartcalculus.com (see the description onpage xii) rather than in the text itself. These include the new topics Fourier Series andFormulas for the Remainder Term in Taylor Series, as well as topics that appeared inprevious editions: Review of Basic Algebra, Rotation of Axes, and Lies My Calcula-tor and Computer Told Me.

|||| F e a t u r e s

C o n c e p t u a l E x e r c i s e s The most important way to foster conceptual understanding is through the problems thatwe assign. To that end I have devised various types of new problems. Some exercise setsbegin with requests to explain the meanings of the basic concepts of the section. (See, forinstance, the Þrst fe w exercises in Sections 2.2, 2.5, and 2.7.) Similarly, all the review sec-tions begin with a Concept Check and a True-False Quiz. Other exercises test conceptual understanding through graphs or tables (see Exercises 2.8.1—3,2.9.35—38,and 3.7.1—4).

Another type of exercise uses verbal description to test conceptual understanding (seeExercises 2.5.8, 2.9.48, and 4.3.59—60). I particularly value problems that combine andcompare graphical, numerical, and algebraic approaches (see Exercises 2.6.35—36 and3.3.23).

G r a d e d E x e r c i s e S e t s Each exercise set is carefully graded, progressing from basic conceptual exercises and skill-development problems to more challenging problems involving applications and proofs.

Exercise Page Exercise Page

2.8.34 164 3.9.55 256

4.4.74 315 5.4.52 412

x ❙ ❙ ❙ ❙ PREFACE


Licensed to: [email protected] e a l - W o r l d D a t a My assistants and I spent a great deal of time looking in libraries, contacting companies

and government agencies, and searching the Internet for interesting real-world data to intro-duce, motivate, and illustrate the concepts of calculus. As a result, many of the examplesand exercises deal with functions deÞned by such numerical data or graphs. See, forinstance, Figures 1, 11, and 12 in Section 1.1 (seismograms from the Northridge earth-quake), Exercise 2.9.36 (percentage of the population under age 18), Exercise 5.1.14(velocity of the space shuttle Endeavour), and Figure 4 in Section 5.4 (San Franciscopower consumption).

P r o j e c t s One way of involving students and making them active learners is to have them work (per-haps in groups) on extended projects that give a feeling of substantial accomplishmentwhen completed. I have included four kinds of projects: Applied Projects involve applica-tions that are designed to appeal to the imagination of students. Laboratory Projects

involve technology. Writing Projects ask students to compare present-day methods withthose of the founders of calculus. Suggested references are supplied. Discovery Projects

anticipate results to be discussed later or encourage discovery through pattern recognition.Additional projects can be found in the InstructorÕs Guide (see, for instance, Group Exer-cise 5.1: Position from Samples) and also in the CalcLabs supplements.

P r o b l e m S o l v i n g Students usually have difÞculties with problems for which there is no single well-deÞnedprocedure for obtaining the answer. I think nobody has improved very much on GeorgePolyaÕs four-stage problem-solving strategy and, accordingly, I have included a version ofhis problem-solving principles following Chapter 1. They are applied, both explicitly andimplicitly, throughout the book. After the other chapters I have placed sections calledProblems Plus, which feature examples of how to tackle challenging calculus problems. Inselecting the varied problems for these sections I kept in mind the following advice fromDavid Hilbert: ÒA mathematical problem should be difÞcult in order to entice us, yet notinaccessible lest it mock our efforts.Ó When I put these challenging problems on assign-ments and tests I grade them in a different way. Here I reward a student signiÞ-cantly for ideas toward a solution and for recognizing which problem-solving principlesare relevant.

T e c h n o l o g y The availability of technology makes it not less important but more important to clearlyunderstand the concepts that underlie the images on the screen. But, when properly used,graphing calculators and computers are powerful tools for discovering and understandingthose concepts. This textbook can be used either with or without technology and I use twospecial symbols to indicate clearly when a particular type of machine is required. The icon

; indicates an exercise that deÞnitely requires the use of such technology , but that is notto say that it canÕt be used on the other exercises as well. The symbol is reserved forproblems in which the full resources of a computer algebra system (like Derive, Maple,Mathematica, or the TI-89/92) are required. But technology doesnÕt make pencil and paperobsolete. Hand calculation and sketches are often preferable to technology for illustratingand reinforcing some concepts. Both instructors and students need to develop the abilityto decide where the hand or the machine is appropriate.

T o o l s f o r E n r i c h i n g™ C a l c u l u s The CD-ROM called TEC is a companion to the text and is intended to enrich and com-plement its contents. Developed by Harvey Keynes at the University of Minnesota and DanClegg at Palomar College, TEC uses a discovery and exploratory approach. In sections ofthe book where technology is particularly approprimate, marginal icons direct stu-dents toTEC modules that provide a laboratory environment in which they can explore the topic indifferent ways and at different levels. Instructors can choose to become involved at sever-al different levels, ranging from simply encouraging students to use the modules for inde-pendent exploration, to assigning speciÞc e xercises from those included with each module,or to creating additional exercises, labs, and projects that make use of the modules.

CAS

PREFACE ❙ ❙ ❙ ❙ xi



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xii ❙ ❙ ❙ ❙ PREFACE

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 hintsare usually presented in the form of questions and try to imitate an effective teaching assis-tant by functioning as a silent tutor. They are constructed so as not to reveal any more ofthe actual solution than is minimally necessary to make further progress.

W e b S i t e : w w w . s t e w a r t c a l c u l u s . c o m This site has been renovated and now includes the following.

■ Algebra Review, with tutorial

■ Additional Topics (complete with exercise sets): Fourier Series, Formulas for theRemainder Term in Taylor Series, Rotation of Axes, Lies My Calculator and Computer Told Me

■ Drill exercises that appeared in previous editions, together with their solutions

■ Problems Plus from prior editions

■ Links, for particular topics, to outside web resources

■ History of Mathematics, with links to the better historical web sites

■ Downloadable versions of CalcLabs for Derive and TI graphing calculators

|||| C o n t e n t

This volume consists of the Þrst six chapters of Single Variable Calculus: Early Transcen-

dentals, Fifth Edition.

A P r e v i e w o f C a l c u l u s The book begins with an overview of the subject and includes a list of questions to moti-vate the study of calculus.

1 ■ F u n c t i o n s a n d M o d e l s From the beginning, multiple representations of functions are stressed: verbal, numerical,visual, and algebraic. A discussion of mathematical models leads to a review of the standardfunctions, including exponential and logarithmic functions, from these four points of view.

2 ■ L i m i t s a n d D e r i v a t i v e s The material on limits is motivated by a prior discussion of the tangent and velocity prob-lems. Limits are treated from descriptive, graphical, numerical, and algebraic points ofview. Section 2.4, on the precise ∑-∂ deÞnition of a limit, is an optional section. Sections2.8 and 2.9 deal with derivatives (especially with functions deÞned graphically and numer-ically) before the differentiation rules are covered in Chapter 3. Here the examples andexercises explore the meanings of derivatives in various contexts.

3 ■ D i f f e r e n t i a t i o n R u l e s All the basic functions, including exponential, logarithmic, and inverse trigonometricfunctions, are differentiated here. When derivatives are computed in applied situations, stu-dents are asked to explain their meanings.

4 ■ A p p l i c a t i o n s o f D i f f e r e n t i a t i o n The basic facts concerning extreme values and shapes of curves are deduced from theMean Value Theorem. Graphing with technology emphasizes the interaction between cal-culus and calculators and the analysis of families of curves. Some substantial optimizationproblems are provided, including an explanation of why you need to raise your head 42¡to see the top of a rainbow.

5 ■ I n t e g r a l s The area problem and the distance problem serve to motivate the deÞnite inte gral, withsigma notation introduced as needed. (Full coverage of sigma notation is provided inAppendix E.) Emphasis is placed on explaining the meanings of integrals in various con-texts and on estimating their values from graphs and tables.


Licensed to: [email protected] ■ A p p l i c a t i o n s o f I n t e g r a t i o n Here I present the applications of integrationÑarea, volume, work, average valueÑthat

can reasonably be done without specialized techniques of integration. General methods areemphasized. The goal is for students to be able to divide a quantity into small pieces, esti-mate with Riemann sums, and recognize the limit as an integral.

|||| A n c i l l a r i e s

Single Variable Calculus, Early Transcendentals, Fifth Edition, is supported by a completeset of ancillaries developed under my direction. Each piece has been designed to enhancestudent understanding and to facilitate creative instruction. The tables on pages xvi—xviidescribe each of these ancillaries.

|||| A c k n o w l e d g m e n t s

The preparation of this and previous editions has involved much time spent reading thereasoned (but sometimes contradictory) advice from a large number of astute reviewers. I greatly appreciate the time they spent to understand my motivation for the approach taken.I have learned something from each of them.

F i f t h E d i t i o n R e v i e w e r s

PREFACE ❙ ❙ ❙ ❙ xiii

Martina Bode,Northwestern University

Philip L. Bowers,Florida State University

Scott Chapman,Trinity University

Charles N. Curtis,Missouri Southern State College

Elias Deeba,University of Houston—Downtown

Greg Dresden,Washington and Lee University

Martin Erickson,Truman State University

Laurene V. Fausett,Georgia Southern University

Norman Feldman,Sonoma State University

Jos� D. Flores,The University of

South Dakota

Howard B. Hamilton,California State University, Sacramento

Gary W. Harrison,College of Charleston

Randall R. Holmes,Auburn University

James F. Hurley,University of Connecticut

Matthew A. Isom,Arizona State University

Zsuzsanna M. Kadas,St. MichaelÕs College

Frederick W. Keene,Pasadena City College

Robert L. Kelley,University of Miami

John C. Lawlor,University of Vermont

Christopher C. Leary,State University of New York

at Geneseo

Gerald Y. Matsumoto,American River College

Michael Monta�o,Riverside Community College

Hussain S. Nur,California State University, Fresno

Mike Penna,Indiana University—Purdue

University Indianapolis

John Ringland,State University of New York at Buffalo

E. Arthur Robinson, Jr.,The George Washington University

Rob RootLafayette College

Teresa Morgan Smith,Blinn College

Donald W. Solomon,University of Wisconsin—Milwaukee

Kristin Stoley,Blinn College

Paul Xavier Uhlig,St. MaryÕs University, San Antonio

Dennis H. Wortman,University of Massachusetts, Boston

Xian Wu,University of South Carolina

P r e v i o u s E d i t i o n R e v i e w e r s

B. D. Aggarwala,University of Calgary

John Alberghini,Manchester Community College

Michael Albert,Carnegie-Mellon University

Daniel Anderson,University of Iowa

Donna J. Bailey,Northeast Missouri State University

Wayne Barber,Chemeketa Community College

Neil Berger,University of Illinois, Chicago

David Berman,University of New Orleans

Richard Biggs,University of Western Ontario



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xiv ❙ ❙ ❙ ❙ PREFACE

Robert Blumenthal,Oglethorpe University

Barbara Bohannon,Hofstra University

Jay Bourland,Colorado State University

Stephen W. Brady,Wichita State University

Michael Breen,Tennessee Technological University

Stephen BrownRobert N. Bryan,

University of Western Ontario

David Buchthal,University of Akron

Jorge Cassio,Miami-Dade Community College

Jack Ceder,University of California,

Santa Barbara

James Choike,Oklahoma State University

Barbara Cortzen,DePaul University

Carl Cowen,Purdue University

Philip S. Crooke,Vanderbilt University

Daniel Cyphert,Armstrong State College

Robert DahlinGregory J. Davis,

University of Wisconsin—Green Bay

Daniel DiMaria,Suffolk Community College

Seymour Ditor,University of Western Ontario

Daniel Drucker,Wayne State University

Kenn Dunn,Dalhousie University

Dennis Dunninger,Michigan State University

Bruce Edwards,University of Florida

David Ellis,San Francisco State University

John Ellison,Grove City College

Garret Etgen,University of Houston

Theodore G. Faticoni,Fordham University

Newman Fisher,San Francisco State University

William Francis,Michigan Technological University

James T. Franklin,Valencia Community College, East

Stanley Friedlander,Bronx Community College

Patrick Gallagher,Columbia University—New York

Paul Garrett,University of Minnesota—Minneapolis

Frederick Gass,Miami University of Ohio

Bruce Gilligan,University of Regina

Matthias K. Gobbert,University of Maryland,

Baltimore County

Gerald Goff,Oklahoma State University

Stuart Goldenberg,California Polytechnic State University

John A. Graham,Buckingham Browne &

Nichols School

Richard Grassl,University of New Mexico

Michael Gregory,University of North Dakota

Charles Groetsch,University of Cincinnati

Salim M. Ha�dar,Grand Valley State University

D. W. Hall,Michigan State University

Robert L. Hall,University of Wisconsin—Milwaukee

Darel Hardy,Colorado State University

Melvin Hausner,New York University/Courant Institute

Curtis Herink,Mercer University

Russell Herman,University of North Carolina

at Wilmington

Allen Hesse,Rochester Community College

Gerald Janusz,University of Illinois at

Urbana-Champaign

John H. Jenkins,Embry-Riddle Aeronautical University,

Prescott Campus

Clement Jeske,University of Wisconsin, Platteville

Carl Jockusch,University of Illinois at

Urbana-Champaign

Jan E. H. Johansson,University of Vermont

Jerry Johnson,Oklahoma State University

Matt KaufmanMatthias Kawski,

Arizona State University

Virgil Kowalik,Texas A&I University

Kevin Kreider,University of Akron

Leonard Krop,DePaul University

Mark Krusemeyer,Carleton College

David Leeming,University of Victoria

Sam Lesseig,Northeast Missouri State University

Phil Locke,University of Maine

Joan McCarter,Arizona State University

Phil McCartney,Northern Kentucky University

Igor Malyshev,San Jose State University

Larry MansÞeld,Queens College

Mary Martin,Colgate University

Nathaniel F. G. Martin,University of Virginia

Tom Metzger,University of Pittsburgh

Teri Jo Murphy,University of Oklahoma

Richard Nowakowski,Dalhousie University

Wayne N. Palmer,Utica College

Vincent Panico,University of the PaciÞc

F. J. Papp,University of Michigan—

Dearborn

Mark Pinsky,Northwestern University

Lothar Redlin,The Pennsylvania State University

Tom Rishel,Cornell University



PREFACE ❙ ❙ ❙ ❙ xv

In addition, I would like to thank George Bergman, Stuart Goldenberg, Emile LeBlanc,Gerald Leibowitz, Charles Pugh, Marina Ratner, Peter Rosenthal, and Alan Weinstein fortheir suggestions; Dan Clegg for his research in libraries and on the Internet; Arnold Goodfor his treatment of optimization problems with implicit differentiation; Al Shenk andDennis Zill for permission to use exercises from their calculus texts; COMAP for permis-sion to use project material; George Bergman, David Bleecker, Dan Clegg, Victor Kaftal,Anthony Lam, Jamie Lawson, Ira Rosenholtz, Lowell Smylie, and Larry Wallen for ideas forexercises; Dan Drucker for the roller derby project; Tom Farmer, Fred Gass, John Ramsay,Larry Riddle, and Philip StrafÞn for ideas for projects; Dan Anderson and Dan Drucker forsolving the new exercises; and Jeff Cole and Dan Clegg for their careful preparation andproofreading of the answer manuscript. IÕm grateful to Jeff Cole for suggesting ways toimprove the exercises. Dan Clegg acted as my assistant throughout; he proofread, madesuggestions, and contributed many of the new exercises.

In addition, I thank those who have contributed to past editions: Ed Barbeau, FredBrauer, Andy Bulman-Fleming, Bob Burton, Tom DiCiccio, Garret Etgen, Chris Fisher,Gene Hecht, Harvey Keynes, Kevin Kreider, E. L. Koh, Zdislav Kovarik, David Leep,Lothar Redlin, Carl Riehm, Doug Shaw, and Saleem Watson.

I also thank Stephanie Kuhns, Kathi Townes, and Brian Betsill of TECHarts for theirproduction services and the following Brooks/Cole staff: Kirk Bomont, editorial produc-tion project manager; Karin Sandberg, Stephanie Taylor, and Bryan Vann, marketing team;Stacy Green, assistant editor, and Jessica Zimmerman, editorial assistant; Earl Perry, tech-nology project manager, and Jessica Reed, print/media buyer. They have all done an out-standing job.

I have been very fortunate to have worked with some of the best mathematics editors inthe business over the past two decades: Ron Munro, Harry Campbell, Craig Barth, JeremyHayhurst, 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 writethe books that I want to write.

JAMES STEWART

Joel W. Robbin,University of Wisconsin—Madison

Richard Rockwell,PaciÞc Union Colle ge

Richard Ruedemann,Arizona State University

David Ryeburn,Simon Fraser University

Richard St. Andre,Central Michigan University

Ricardo Salinas,San Antonio College

Robert Schmidt,South Dakota State University

Eric Schreiner,Western Michigan University

Mihr J. Shah,Kent State University—Trumbull

Theodore Shifrin,University of Georgia

Wayne Skrapek,University of Saskatchewan

Larry Small,Los Angeles Pierce College

William Smith,University of

North Carolina

Edward Spitznagel,Washington University

Joseph Stampßi,Indiana University

M. B. Tavakoli,Chaffey College

Stan Ver Nooy,University of Oregon

Andrei Verona,California State University—

Los Angeles

Russell C. Walker,Carnegie Mellon University

William L. Walton,McCallie School

Jack Weiner,University of Guelph

Alan Weinstein,University of California, Berkeley

Theodore W. Wilcox,Rochester Institute of Technology

Steven Willard,University of Alberta

Robert Wilson,University of Wisconsin—Madison

Jerome Wolbert,University of Michigan—

Ann Arbor

Mary Wright,Southern Illinois University—

Carbondale

Paul M. Wright,Austin Community College



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xvi

|||| Electronic items |||| Printed items

|||| A n c i l l a r i e s f o r I n s t r u c t o r s

Instructor’s Resource CD-ROMISBN 0-534-39340-3

Contains Electronic InstructorÕs Guide, Electronic Solutions,

BCA Testing, Instructions for BCA Homework, and electronic

transparencies (CalcLink).

■ http:// bca.brookscole.com

Tools for Enriching Calculus CD-ROM by Harvey B. Keynes, James Stewart, and Dan Clegg

ISBN 0-534-39731-X

TEC provides a laboratory environment in which students can

explore selected topics. TEC also includes homework hints for

representative exercises.

Instructor’s Guideby Douglas Shaw, Harvey B. Keynes, and James Stewart

ISBN 0-534-39334-9

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 InstructorÕs Resource CD-ROM.

Instructor’s Guide for AP ® Calculusby Douglas Shaw and Robert Gerver, contributing author

ISBN 0-534-39341-1

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 Early Transcendentals by Daniel Anderson, Jeffery A. Cole, and Daniel Drucker

ISBN 0-534-39332-2

Includes worked-out solutions to all exercises in the text.

Printed Test ItemsISBN 0-534-39336-5

Contains multiple-choice and short-answer test items that key

directly to the text.

Brooks/Cole Assessment (BCA) Testing ISBN 0-534-39335-7

Available online or on CD-ROM, browser-based BCA is fully

integrated text-speciÞc testing , tutorial, and course manage-

ment software. With no need for plug-ins or downloads, BCA

offers algorithmically generated problem values and machine-

graded free response mathematics.■ http:// bca.brookscole.com

Text-Specific VideosISBN 0-534-39325-X

Text-speciÞc videotape sets, available at no charge to adopters,

consisting of one tape per text chapter. Each tape features a

10- to 20-minute problem-solving lesson for each section of

the chapter.

Transparencies by James Stewart

Single Variable ISBN 0-534-39337-3

Full-color, large-scale sheets of reproductions of material from

the text.

|||| A n c i l l a r i e s f o r I n s t r u c t o r s a n d S t u d e n t s

Stewart Specialty Web Sitewww.stewartcalculus.com

Contents: Algebra Review with tutorial ■ Additional Topics

■ Drill exercises ■ Problems Plus ■ Web Links ■ History

of Mathematics ■ Downloadable versions of CalcLabs for

Derive and TI graphing calculators

BCA Homework

ISBN 0-534-39380-2

BCA Homework allows instructors to assign machine-gradable

homework problems that help students identify where they need

additional help. That assistance is available through worked-

out solutions that guide students through the steps of problem

solving, or via live online tutoring at vMentor. The tutors at

this online service will skillfully guide students through a prob-

lem, using unique two-way audio and whiteboard features.

■ http:// bca.brookscole.com

The Brooks/Cole Mathematics Resource Center Web Sitehttp://mathematics.brookscole.com

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 Web site features everything from



xvii

|||| Electronic items |||| Printed items

book-speciÞc r esources to newsgroups. ItÕs a great way to make

teaching and learning an interactive and intriguing experience.

WebTutor™ on WebCTISBN 0-534-39328-4

Lecture notes, discussion threads, and quizzes on WebCT.

Journey Through Calculusby Bill Ralph in conjunction with James Stewart

Student Version ISBN 0-534-26220-1

Instructor’s Version ISBN 0-534-36823-9

A Calculus CD-ROM that brings together activities, tutorials,

testing, a computer algebra system, and calculus content into

one uniÞed en vironment.

|||| S t u d e n t R e s o u r c e s

Tools for Enriching™ Calculus CD-ROM by Harvey B. Keynes, James Stewart, and Dan Clegg

ISBN 0-534-39731-X

TEC provides a laboratory environment in which students can

explore selected topics. TEC also includes homework hints for

representative exercises.

Interactive Video SkillBuilder CD-ROM ISBN 0-534-39326-8

Think of it as portable ofÞce hour s! 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. Also included is MathCue Tutorial software.

This dual-platform software presents and scores problems and

tutors students by displaying annotated, step-by-step solutions.

Problem sets may be customized.

Study Guide

Single Variable Early Transcendentals by Richard St. Andre

ISBN 0-534-39331-4

Contains a short list of key concepts, a short list of skills to

master, a brief introduction to the ideas of the section, an elab-

oration 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 Early Transcendentalsby Daniel Anderson, Jeffery A. Cole, and Daniel Drucker

ISBN 0-534-39333-0

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 Variableby Philip Yasskin, Albert Boggess, David Barrow,

Maurice Rahe, Jeffery Morgan, Michael Stecher,

Art Belmonte, and Kirby Smith

ISBN 0-534-39370-5

CalcLabs with Mathematica

Single Variableby Selwyn Hollis

ISBN 0-534-39371-3

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 Calculusby Dennis Ebersole, Doris Schattschneider,

Alicia Sevilla, and Kay Somers

ISBN 0-534-26592-8

Written to improve algebra and problem-solving skills of

students taking a calculus course, every chapter in this com-

panion is keyed to a calculus topic, providing conceptual

background and speciÞc alg ebra techniques needed to under-

stand and solve calculus problems related to that topic. It is

designed for calculus courses that integrate the review of pre-

calculus concepts or for individual use.

Linear Algebra for Calculusby 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.



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xviii

T O T H E S T U D E N T

Reading a calculus textbook is different from reading a news-

paper or a novel, or even a physics book. DonÕt 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 deÞnitions to see the e xact meanings of the terms. And

before you read each example, I suggest that you cover up the

solution and try solving the problem yourself. YouÕll 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 logi-

cally. Learn to write the solutions of the exercises in a con-

nected, step-by-step fashion with explanatory sentencesÑ 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 H. 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 donÕt

worry that you havenÕt found the deÞniti ve 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, donÕt immediately assume youÕre wrong. For example,

if the answer given in the back of the book is and you

obtain , then youÕre right and rationalizing the

denominator will show that the answers are equivalent.

The icon; indicates an exercise that deÞnitely 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 doesnÕt mean that graphing devices canÕt be used to check

your work on the other exercises as well. The symbol is

reserved for problems in which the full resources of a com-

puter algebra system (like Derive, Maple, Mathematica, or the

TI-89/92) are required.

CAS

1�(1 � s2 )s2 � 1

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.

The icon indicates a reference to the CD-ROM Journey

Through Calculus. The symbols in the margin refer you to the

location in Journey where a concept is introduced through an

interactive exploration or animation.

The CD-ROM Tools for Enriching Calculus (see inside

front cover for availability) is referred to by means of the

symbol . It directs you to modules in which you can explore

aspects of calculus for which the computer is particularly use-

ful. 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 man-

ner with pencil and paper to work out the details. If a particular

hint doesnÕt enable you to solve the problem, you can click to

reveal the next hint.

The CD-ROM Interactive Video Skillbuilder (see inside front

cover for availability) contains videos of instructors explaining

two or three of the examples 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 Þnish the course. Because you will lik ely forget some

of the speciÞc details of calculus, the book will serve as a use-

ful 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

23.





V o l u m e 1 : C h a p t e r s 1 – 6



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A Preview of Calculus

By the time you finish this course, you will

be able to explain the formation and location

of rainbows, compute the force exerted by

water on a dam, analyze the population

cycles of predators and prey, and calculate

the escape velocity of a rocket.

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Licensed to: [email protected] is fundamentally different from the mathematics

that you have studied previously. Calculus is less static and

more dynamic. It is concerned with change and motion; it

deals with quantities that approach other quantities. For

that reason it may be useful to have an overview of the sub-

ject before beginning its intensive study. Here we give a glimpse of some of the

main ideas of calculus by showing how the concept of a limit arises when we

attempt to solve a variety of problems.

T h e A r e a P r o b l e m

The origins of calculus go back at least 2500 years to the ancient Greeks, who found areasusing the Òmethod of e xhaustion.Ó They knew how to Þnd the area of any polygon bydividing it into triangles as in Figure 1 and adding the areas of these triangles.

It is a much more difÞcult problem to Þnd the area of a curv ed Þgure. The Greekmethod of exhaustion was to inscribe polygons in the Þgure and circumscribe polygonsabout the Þgure and then let the number of sides of the polygons increase. Figure 2 illus-trates this process for the special case of a circle with inscribed regular polygons.

Let be the area of the inscribed polygon with sides. As increases, it appears thatbecomes closer and closer to the area of the circle. We say that the area of the circle is

the limit of the areas of the inscribed polygons, and we write

The Greeks themselves did not use limits explicitly. However, by indirect reasoning,Eudoxus (Þfth century B.C.) used exhaustion to prove the familiar formula for the area ofa circle:

We will use a similar idea in Chapter 5 to Þnd areas of re gions of the type shown inFigure 3. We will approximate the desired area by areas of rectangles (as in Figure 4),let the width of the rectangles decrease, and then calculate as the limit of these sums ofareas of rectangles.

A

A

A � �r 2.

A � limnl�

An

An

nnAn

A¡™ ��A¶ ��AßA∞A¢A£

FIGURE 2

A

3

FIGURE 3

1n

10 x

y

(1, 1)

10 x

y

(1, 1)

14

12

34

0 x

y

1

(1, 1)

FIGURE 4

10 x

y

y=≈

A

(1, 1)

The Preview Module is a numerical

and pictorial investigation of the

approximation of the area of a circle

by inscribed and circumscribed polygons.

FIGURE 1

A=A¡+A™+A£+A¢+A∞

A¡

A™

A£ A¢

A∞



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4 � � � � A PREVIEW OF CALCULUS

The area problem is the central problem in the branch of calculus called integral cal-

culus. The techniques that we will develop in Chapter 5 for Þnding areas will also enableus to compute the volume of a solid, the length of a curve, the force of water against a dam,the mass and center of gravity of a rod, and the work done in pumping water out of a tank.

T h e T a n g e n t P r o b l e m

Consider the problem of trying to Þnd an equation of the tangent line to a curve withequation at a given point . (We will give a precise deÞnition of a tangent line inChapter 2. For now you can think of it as a line that touches the curve at as in Figure 5.)Since we know that the point lies on the tangent line, we can Þnd the equation of if weknow its slope . The problem is that we need two points to compute the slope and weknow only one point, , on . To get around the problem we Þrst Þnd an approximation to

by taking a nearby point on the curve and computing the slope of the secant line. From Figure 6 we see that

Now imagine that moves along the curve toward as in Figure 7. You can see thatthe secant line rotates and approaches the tangent line as its limiting position. This meansthat the slope of the secant line becomes closer and closer to the slope of the tan-gent line. We write

and we say that is the limit of as approaches along the curve. Since approachesas approaches , we could also use Equation 1 to write

SpeciÞc e xamples of this procedure will be given in Chapter 2.The tangent problem has given rise to the branch of calculus called differential calcu-

lus, which was not invented until more than 2000 years after integral calculus. The mainideas behind differential calculus are due to the French mathematician Pierre Fermat(1601—1665) and were developed by the English mathematicians John Wallis(1616—1703),Isaac Barrow (1630—1677),and Isaac Newton (1642—1727) and the Germanmathematician Gottfried Leibniz (1646—1716).

The two branches of calculus and their chief problems, the area problem and the tan-gent problem, appear to be very different, but it turns out that there is a very close con-nection between them. The tangent problem and the area problem are inverse problems ina sense that will be described in Chapter 5.

V e l o c i t y

When we look at the speedometer of a car and read that the car is traveling at 48 mi�h,what does that information indicate to us? We know that if the velocity remains constant,then after an hour we will have traveled 48 mi. But if the velocity of the car varies, whatdoes it mean to say that the velocity at a given instant is 48 mi�h?

m � limxl a

f �x� � f �a�

x � a2

PQa

xPQmPQm

m � limQlP

mPQ

mmPQ

PQ

mPQ �

f �x� � f �a�

x � a1

PQ

mPQQm

tP

m

tP

P

Py � f �x�

t

0

y

x

P

y=ƒ

t

P

Q

t

0 x

y

y

0 xa x

ƒ-f(a)P{a,�f(a)}

x-a

t

Q{x, ƒ}

FIGURE 5

The tangent line at P

FIGURE 6

The secant line PQ

FIGURE 7

Secant lines approaching thetangent line

Is it possible to fill a circle with rectangles?

Try it for yourself.

Resources / Module 1

/ Area

/ Rectangles in Circles



A PREVIEW OF CALCULUS � � � � 5

In order to analyze this question, letÕs examine the motion of a car that travels along astraight road and assume that we can measure the distance traveled by the car (in feet) atl-second intervals as in the following chart:

As a Þrst step to ward Þnding the v elocity after 2 seconds have elapsed, we Þnd the a ver-age velocity during the time interval :

Similarly, the average velocity in the time interval is

We have the feeling that the velocity at the instant � 2 canÕt be much different from theaverage velocity during a short time interval starting at . So letÕs imagine that the dis-tance traveled has been measured at 0.l-second time intervals as in the following chart:

Then we can compute, for instance, the average velocity over the time interval :

The results of such calculations are shown in the following chart:

The average velocities over successively smaller intervals appear to be getting closer toa number near 10, and so we expect that the velocity at exactly is about 10 ft�s. InChapter 2 we will deÞne the instantaneous v elocity of a moving object as the limitingvalue of the average velocities over smaller and smaller time intervals.

In Figure 8 we show a graphical representation of the motion of the car by plotting thedistance traveled as a function of time. If we write , then is the number of feettraveled after seconds. The average velocity in the time interval is

average velocity �

distance traveled

time elapsed�

f �t� � f �2�

t � 2

�2, t�t

f �t�d � f �t�

t � 2


16.80 � 10.00

2.5 � 2� 13.6 ft�s

�2, 2.5�

t � 2t


25 � 10

3 � 2� 15 ft�s

2 � t � 3

� 16.5 ft�s

�43 � 10

4 � 2


distance traveled

time elapsed

2 � t � 4

FIGURE 8

t

d

0 1 2 3 4 5

10

20

P{2,�f(2)}

Q{ t,�f(t)}

t � Time elapsed (s) 0 1 2 3 4 5

d � Distance (ft) 0 2 10 25 43 78

t 2.0 2.1 2.2 2.3 2.4 2.5

d 10.00 11.02 12.16 13.45 14.96 16.80

Time interval

Average velocity (ft�s) 15.0 13.6 12.4 11.5 10.8 10.2

�2, 2.1��2, 2.2��2, 2.3��2, 2.4��2, 2.5��2, 3�



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which is the same as the slope of the secant line in Figure 8. The velocity when is the limiting value of this average velocity as approaches 2; that is,

and we recognize from Equation 2 that this is the same as the slope of the tangent line tothe curve at .

Thus, when we solve the tangent problem in differential calculus, we are also solvingproblems concerning velocities. The same techniques also enable us to solve problemsinvolving rates of change in all of the natural and social sciences.

T h e L i m i t o f a S e q u e n c e

In the Þfth century B.C. the Greek philosopher Zeno of Elea posed four problems, nowknown as ZenoÕs paradoxes, that were intended to challenge some of the ideas concerningspace and time that were held in his day. ZenoÕs second paradox concerns a race betweenthe Greek hero Achilles and a tortoise that has been given a head start. Zeno argued, as fol-lows, that Achilles could never pass the tortoise: Suppose that Achilles starts at position

and the tortoise starts at position . (See Figure 9.) When Achilles reaches the point, the tortoise is farther ahead at position . When Achilles reaches , the tor-

toise is at . This process continues indeÞnitely and so it appears that the tortoise willalways be ahead! But this deÞes common sense.

One way of explaining this paradox is with the idea of a sequence. The successive posi-tions of Achilles or the successive positions of the tortoise form what is known as a sequence.

In general, a sequence is a set of numbers written in a deÞnite order . For instance,the sequence

can be described by giving the following formula for the th term:

We can visualize this sequence by plotting its terms on a number line as in Figure 10(a)or by drawing its graph as in Figure 10(b). Observe from either picture that the terms ofthe sequence are becoming closer and closer to 0 as increases. In fact, we canÞnd terms as small as we please by making large enough. We say that the limit of thesequence is 0, and we indicate this by writing

In general, the notation

lim nl�

an � L

lim nl�

1

n� 0

n

nan � 1�n

an �

1

n

n

{1, 12 , 13 , 1

4 , 15 , . . .}

�an�

�t1, t2, t3, . . .��a1, a2, a3, . . .�

Achilles

tortoise

a¡ a™ a£ a¢ a∞

t¡ t™ t£ t¢

. . .

. . .FIGURE 9

t3

a3 � t2t2a2 � t1

t1a1

P

v � lim tl 2

f �t� � f �2�

t � 2

t

t � 2vPQ

1

n1 2 3 4 5 6 7 8

FIGURE 10

10

a¡a™a£a¢

(a)

(b)




is used if the terms approach the number as becomes large. This means that the num-bers can be made as close as we like to the number by taking sufÞciently lar ge.

The concept of the limit of a sequence occurs whenever we use the decimal represen-tation of a real number. For instance, if

then

The terms in this sequence are rational approximations to .LetÕs return to ZenoÕs paradox. The successive positions of Achilles and the tortoise

form sequences and , where for all . It can be shown that both sequenceshave the same limit:

It is precisely at this point that Achilles overtakes the tortoise.

T h e S u m o f a S e r i e s

Another of ZenoÕs paradoxes, as passed on to us by Aristotle, is the following: ÒA manstanding in a room cannot walk to the wall. In order to do so, he would Þrst ha ve to go halfthe distance, then half the remaining distance, and then again half of what still remains.This process can always be continued and can never be ended.Ó (See Figure 11.)

Of course, we know that the man can actually reach the wall, so this suggests that per-haps the total distance can be expressed as the sum of inÞnitely man y smaller distances asfollows:

1 �

1

2�

1

4�

1

8�

1

16� � � � �

1

2n� � � �3

12

14

18

116FIGURE 11

p

limnl�

an � p � limnl�

tn

nan � tn�tn��an�

�

limn l

�

an � �

��

a7 � 3.1415926

a6 � 3.141592

a5 � 3.14159

a4 � 3.1415

a3 � 3.141

a2 � 3.14

a1 � 3.1

nLan

nLan

Watch a movie of Zeno’s attempt to reach

the wall.

Resources / Module 1

/ Introduction

/ Zeno’s Paradox



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Zeno was arguing that it doesnÕt make sense to add inÞnitely man y numbers together. Butthere are other situations in which we implicitly use inÞnite sums. F or instance, in decimalnotation, the symbol means

and so, in some sense, it must be true that

More generally, if denotes the nth digit in the decimal representation of a number, then

Therefore, some inÞnite sums, or inÞnite series as the y are called, have a meaning. But wemust deÞne carefully what the sum of an inÞnite series is.

Returning to the series in Equation 3, we denote by the sum of the Þrst terms of theseries. Thus

Observe that as we add more and more terms, the partial sums become closer and closerto 1. In fact, it can be shown that by taking large enough (that is, by adding sufÞcientlymany terms of the series), we can make the partial sum as close as we please to the number 1. It therefore seems reasonable to say that the sum of the inÞnite series is 1 andto write

1

2�

1

4�

1

8� � � � �

1

2n� � � � � 1

sn

n

s16 �

1

2�

1

4� � � � �

1

216� 0.99998474

� � �

s10 �12 �

14 � � � � �

11024 � 0.99902344

� � �

s7 �12 �

14 �

18 �

116 �

132 �

164 �

1128 � 0.9921875

s6 �12 �

14 �

18 �

116 �

132 �

164 � 0.984375

s5 �12 �

14 �

18 �

116 �

132 � 0.96875

s4 �12 �

14 �

18 �

116 � 0.9375

s3 �12 �

14 �

18 � 0.875

s2 �12 �

14 � 0.75

s1 �12 � 0.5

nsn

0.d1d2 d3d4 . . . �

d1

10�

d2

102�

d3

103� � � � �

dn

10n� � � �

dn

3

10�

3

100�

3

1000�

3

10,000� � � � �

1

3

3

10�

3

100�

3

1000�

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0.3 � 0.3333 . . .




In other words, the reason the sum of the series is 1 is that

In Chapter 11 we will discuss these ideas further. We will then use NewtonÕs idea ofcombining inÞnite series with dif ferential and integral calculus.

S u m m a r y

We have seen that the concept of a limit arises in trying to Þnd the area of a re gion, theslope of a tangent to a curve, the velocity of a car, or the sum of an inÞnite series. In eachcase the common theme is the calculation of a quantity as the limit of other, easily calcu-lated quantities. It is this basic idea of a limit that sets calculus apart from other areas ofmathematics. In fact, we could deÞne calculus as the part of mathematics that deals withlimits.

Sir Isaac Newton invented his version of calculus in order to explain the motion of theplanets around the Sun. Today calculus is used in calculating the orbits of satellites andspacecraft, in predicting population sizes, in estimating how fast coffee prices rise, in fore-casting weather, in measuring the cardiac output of the heart, in calculating life insurancepremiums, and in a great variety of other areas. We will explore some of these uses of cal-culus in this book.

In order to convey a sense of the power of the subject, we end this preview with a listof some of the questions that you will be able to answer using calculus:

1. How can we explain the fact, illustrated in Figure 12, that the angle of elevationfrom an observer up to the highest point in a rainbow is 42¡? (See page 288.)

2. How can we explain the shapes of cans on supermarket shelves? (See page 341.)

3. Where is the best place to sit in a movie theater? (See page 468.)

4. How far away from an airport should a pilot start descent? (See page 243.)

lim nl�

sn � 1

rays from Sun

observer

rays from Sun

42°

138°

FIGURE 12


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