This book combines traditional mainstream calculus with the most flexible approach to new ideas and calculator/computer technology. It contains superb problem sets and a fresh conceptual emphasis flavored by new technological possibilities.Chapter topics cover functions, graphs, and models; prelude to calculus; the derivative; additional applications of the derivative; the integral; applications of the integral; calculus of transcendental functions; techniques of integration; differential equations; polar coordinates and parametric curves; infinite series; vectors, curves, and surfaces in space; partial differentiation; multiple integrals; and vector calculus.For individuals interested in the study of calculus.

C. Henry Edwards is emeritus professor of mathematics at the University of Georgia. He earned his Ph.D. at the University of Tennessee in 1960, and recently retired after 40 years of classroom teaching (including calculus or differential equations almost every term) at the universities of Tennessee, Wisconsin, and Georgia, with a brief interlude at the Institute for Advanced Study (Princeton) as an Alfred P. Sloan Research Fellow. He has received numerous teaching awards, including the University of Georgia's honoratus medal in 1983 (for sustained excellence in honors teaching), its Josiah Meigs award in 1991 (the institution's highest award for teaching), and the 1997 statewide Georgia Regents award for research university faculty teaching excellence. His scholarly career has ranged from research and dissertation direction in topology to the history of mathematics to computing and technology in the teaching and applications of mathematics. In addition to being author or co-author of calculus, advanced calculus, linear algebra, and differential equations textbooks, he is well-known to calculus instructors as author of The Historical Development of the Calculus (Springer-Verlag, 1979). During the 1990s he served as a principal investigator on three NSF-supported projects: (1) A school mathematics project including Maple for beginning algebra students, (2) A Calculus-with-Mathematica program, and (3) A MATLAB-based computer lab project for numerical analysis and differential equations students.

David E. Penney, University of Georgia, completed his Ph.D. at Tulane University in 1965 (under the direction of Prof. L. Bruce Treybig) while teaching at the University of New Orleans. Earlier he had worked in experimental biophysics at Tulane University and the Veteran's Administration Hospital in New Orleans under the direction of Robert Dixon McAfee, where Dr. McAfee's research team's primary focus was on the active transport of sodium ions by biological membranes. Penney's primary contribution here was the development of a mathematical model (using simultaneous ordinary differential equations) for the metabolic phenomena regulating such transport, with potential future applications in kidney physiology, management of hypertension, and treatment of congestive heart failure. He also designed and constructed servomechanisms for the accurate monitoring of ion transport, a phenomenon involving the measurement of potentials in microvolts at impedances of millions of megohms. Penney began teaching calculus at Tulane in 1957 and taught that course almost every term with enthusiasm and distinction until his retirement at the end of the last millennium. During his tenure at the University of Georgia he received numerous University-wide teaching awards as well as directing several doctoral dissertations and seven undergraduate research projects. He is the author of research papers in number theory and topology and is the author or co-author of textbooks on calculus, computer programming, differential equations, linear algebra, and liberal arts mathematics.

About the Authors

xi

Preface

xiii

Functions, Graphs, and Models

1

(52)

Functions and Mathematical Modeling

2

(10)

Project: A Square Wading Pool

11

(1)

Graphs of Equations and Functions

12

(12)

Project: A Broken Tree

23

(1)

Polynomials and Algebraic Functions

24

(9)

Project: A Leaning Ladder

33

(1)

Transcendental Functions

33

(12)

Project: A Spherical Asteroid

45

(1)

Preview: What Is Calculus?

45

(8)

Review: Definitions and Concepts

48

(5)

Prelude to Calculus

53

(48)

Tangent Lines and Slope Predictors

54

(9)

Project: Numerical Slope Investigations

63

(1)

The Limit Concept

63

(12)

Project: Limits, Slopes, and Logarithms

74

(1)

More About Limits

75

(13)

Project: Numerical Epsilon-Delta Limit Investigations

87

(1)

The Concept of Continuity

88

(13)

Review: Definitions, Concepts, Results

99

(2)

The Derivative

101

(92)

The Derivative and Rates of Change

102

(13)

Basic Differentiation Rules

115

(11)

The Chain Rule

126

(7)

Derivatives of Algebraic Functions

133

(9)

Maxima and Minima of Functions on Closed Intervals

142

(10)

Project: When Is Your Coffee Cup Stablest?

150

(2)

Applied Optimization Problems

152

(13)

Derivatives of Trigonometric Functions

165

(11)

Successive Approximations and Newton's Method

176

(17)

Project: How Deep Does a Floating Ball Sink?

188

(1)

Review: Formulas, Concepts, Definitions

189

(4)

Additional Applications of the Derivative

193

(78)

Implicit Functions and Related Rates

194

(10)

Project: Investigating the Folium of Descartes

203

(1)

Increments, Differentials, and Linear Approximation

204

(8)

Increasing and Decreasing Functions and the Mean Value Theorem

212

(10)

The First Derivative Test and Applications

222

(10)

Project: Making a Candy Box With Lid

232

(1)

Simple Curve Sketching

232

(10)

Higher Derivatives and Concavity

242

(14)

Curve Sketching and Asymptotes

256

(15)

Project: Locating Special Points on Exotic Graphs

267

(1)

Review: Definitions, Concepts, Results

267

(4)

The Integral

271

(94)

Introduction

272

(1)

Antiderivatives and Initial Value Problems

272

(14)

Elementary Area Computations

286

(12)

Riemann Sums and the Integral

298

(10)

Project: Calculator/Computer Riemann Sums

307

(1)

Evaluation of Integrals

308

(10)

The Fundamental Theorem of Calculus

318

(10)

Integration by Substitution

328

(8)

Areas of Plane Regions

336

(10)

Numerical Integration

346

(19)

Project: Trapezoidal and Simpson Approximations

359

(2)

Review: Definitions, Concepts, Results

361

(4)

Applications of the Integral

365

(62)

Riemann Sum Approximations

366

(10)

Volumes by the Method of Cross Sections

376

(11)

Volumes by the Method of Cylindrical Shells

387

(9)

Project: Design Your Own Ring!

395

(1)

Are Length and Surface Area of Revolution

396

(9)

Force and Work

405

(11)

Centroids of Plane Regions and Curves

416

(11)

Review: Definitions, Concepts, Results

423

(4)

Calculus of Transcendental Functions

427

(62)

Exponential and Logarithmic Functions

428

(13)

Project: Discovering the Number e for Yourself

441

(1)

Indeterminate Forms and L'Hopital's Rule

441

(8)

More Indeterminate Forms

449

(6)

The Natural Logarithm as an Integral

455

(12)

Project: Natural Functional Equations

466

(1)

Inverse Trigonometric Functions

467

(10)

Hyperbolic Functions

477

(12)

Review: Formulas, Concepts, Definitions

485

(4)

Techniques of Integration

489

(56)

Introduction

490

(1)

Integral Tables and Simple Substitutions

490

(4)

Integration by Parts

494

(7)

Trigonometric Integrals

501

(7)

Rational Functions and Partial Fractions

508

(7)

Trigonometric Substitution

515

(6)

Integrals Involving Quadratic Polynomials

521

(5)

Improper Integrals

526

(19)

Summary

539

(6)

Differential Equations

545

(78)

Simple Equations and Models

546

(12)

Slope Fields and Euler's Method

558

(10)

Project: Computer-Assisted Slope Fields and Euler's Method

567

(1)

Separable Equations and Applications

568

(7)

Linear Equations and Applications

575

(12)

Population Models

587

(11)

Project: Predator-Prey Equations and Your Own Game Preserve

597

(1)

Linear Second-Order Equations

598

(9)

Mechanical Vibrations

607

(16)

Review: Definitions, Concepts, Results

618

(5)

Polar Coordinates and Parametric Curves

623

(58)

Analytic Geometry and the Conic Sections

624

(5)

Polar Coordinates

629

(9)

Area Computations in Polar Coordinates

638

(5)

Parametric Curves

643

(10)

Project: Trochoid Investigations

652

(1)

Integral Computations with Parametric Curves

653

(8)

Project: Moon Orbits and Race Tracks

660

(1)

Conic Sections and Applications

661

(20)

Review: Concepts and Definitions

679

(2)

Infinite Series

681

(90)

Introduction

682

(1)

Infinite Sequences

682

(9)

Project: Nested Radicals and Continued Fractions

691

(1)

Infinite Series and Convergence

691

(11)

Project: Numerical Summation and Geometric Series

701

(1)

Taylor Series and Taylor Polynomials

702

(13)

Project: Calculating Logarithms on a Deserted Island

715

(1)

The Integral Test

715

(7)

Project: The Number π, Once and for All

722

(1)

Comparison Tests for Positive-Term Series

722

(6)

Alternating Series and Absolute Convergence

728

(9)

Power Series

737

(13)

Power Series Computations

750

(8)

Project: Calculating Trigonometric Functions on a Deserted Island

758

(1)

Series Solutions of Differential Equations

758

(13)

Review: Definitions, Concepts, Results

767

(4)

Vectors, Curves, and Surfaces in Space

771

(78)

Vectors in the Plane

772

(6)

Three-Dimensional Vectors

778

(10)

The Cross Product of Vectors

788

(8)

Lines and Planes in Space

796

(7)

Curves and Motion in Space

803

(14)

Project: Does a Pitched Baseball Really Curve?

816

(1)

Curvature and Acceleration

817

(13)

Cylinders and Quadric Surfaces

830

(8)

Cylindrical and Spherical Coordinates

838

(11)

Review: Definitions, Concepts, Results

845

(4)

Partial Differentiation

849

(90)

Introduction

850

(1)

Functions of Several Variables

850

(10)

Limits and Continuity

860

(8)

Partial Derivatives

868

(10)

Multivariable Optimization Problems

878

(11)

Increments and Linear Approximation

889

(7)

The Multivariable Chain Rule

896

(11)

Directional Derivatives and the Gradient Vector

907

(11)

Lagrange Multipliers and Constrained Optimization

918

(9)

Project: Numerical Solution of Lagrange Multiplier Systems

927

(1)

Critical Points of Functions of Two Variables

927

(12)

Project: Critical Point Investigations

935

(1)

Review: Definitions, Concepts, Results

936

(3)

Multiple Integrals

939

(74)

Double Integrals

940

(7)

Project: Midpoint Sums Approximating Double Integrals

947

(1)

Double Integrals over More General Regions

947

(7)

Area and Volume by Double Integration

954

(7)

Double Integrals in Polar Coordinates

961

(7)

Applications of Double Integrals

968

(11)

Project: Optimal Design of Downhill Race-Car Wheels

978

(1)

Triple Integrals

979

(9)

Project: Archimedes' Floating Paraboloid

987

(1)

Integration in Cylindrical and Spherical Coordinates

988

(8)

Surface Area

996

(5)

Change of Variables in Multiple Integrals

1001

(12)

Review: Definitions, Concepts, Results

1009

(4)

Vector Calculus

1013

Vector Fields

1014

(5)

Line Integrals

1019

(11)

The Fundamental Theorem and Independence of Path

1030

(7)

Green's Theorem

1037

(10)

Surface Integrals

1047

(10)

Project: Surface Integrals and Rocket Nose Cones

1057

(1)

The Divergence Theorem

1057

(8)

Stokes' Theorem

1065

Review: Definitions, Concepts, Results

1072

APPENDICES

1

(44)

A: Real Numbers and Inequalities

1

(5)

B: The Coordinate Plane and Straight Lines

6

(7)

C: Review of Trigonometry

13

(6)

D: Proofs of the Limit Laws

19

(4)

E: The Completeness of the Real Number System

23

(5)

F: Existence of the Integral

28

(5)

G: Approximations and Riemann Sums

33

(3)

H: L'Hopital's Rule and Cauchy's Mean Value Theorem

36

(2)

I: Proof of Taylor's Formula

38

(1)

J: Conic Sections as Sections of a Cone

39

(1)

K: Proof of the Linear Approximation Theorem

40

(1)

L: Units of Measurements and Conversion Factors

41

(1)

M: Formulas from Algebra, Geometry, and Trigonometry

42

(2)

N: The Greek Alphabet

44

(1)

Answers to Odd-Numbered Problems

45

(54)

References For Further Study

99

Index

1

(1)

Table of Integrals

1

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Contemporary calculus instructors and students face traditional challenges as well as new ones that result from changes in the role and practice of mathematics by scientists and engineers in the world at large. As a consequence, this sixth edition of our calculus textbook is its most extensive revision since the first edition appeared in 1982.Two chapters of the fifth edition have been combined in a single more tightly organized one. An entirely new chapter now appears in the table of contents, and most of the remaining chapters have been extensively rewritten. About 125 of the book's over 750 worked examples are new for this edition and the 1825 figures in the text include 225 new computer-generated graphics. About 600 of its over 7000 problems are new, and these are augmented by 320 new conceptual discussion questions that now precede the problem sets. Moreover, 1050 new true/false questions are included in the Study Guides on the new CD-ROM that accompanies this edition. In summary, almost 2000 of these 8400-plus problems and questions are new, and the text discussion and explanations have undergone corresponding alteration and improvement. PRINCIPAL NEW FEATURESThe current revision of the text features More unified treatment oftranscendental functionsin Chapter 7, and A new Chapter 9 ondifferential equationsand applications.The new chapter on differential equations now appears immediately after Chapter 8 on techniques of integration. It includes both direction fields and Eider's method together with the more elementary symbolic methods (which exploit techniques from Chapter 8) and interesting applications of both first- and second-order equations. Chapter 11 (Infinite Series) now ends with a new section on power series solutions of differential equations, thus bringing full circle a unifying focus of second-semester calculus on elementary differential equations. NEW LEARNING RESOURCESConceptual Discussion Questions:The set of problems that concludes each section is now preceded by a briefConcepts: Questions and Discussionset consisting of several open-ended conceptual questions that can be used for either individual study or classroom discussion.The Text CD-ROM:The content of the new CD-ROM that accompanies this text is fully integrated with the textbook material, and is designed specifically for use hand-in-hand with study of the book itself. This CD-ROM features the following resources to support learning and teaching: Interactive True/False Study Guidesthat reinforce and encourage student reading of the text. Ten author-written questions for each section carefully guide students through the section, and students can request individual hints suggesting where in the section to look for needed information. Live Examplesfeature dynamic multimedia and computer algebra presentations--many accompanied by audio explanations--which enhance student intuition and understanding. These interactive examples expand upon many of the textbook's principal examples; students can change input data and conditions and then observe the resulting changes in step-by-step solutions and accompanying graphs and figures.Walkthrough videosdemonstrate how students can interact with these live examples. Homework Startersfor the principal types of computational problems in each textbook section, featuring both interactive presentations similar to the live examples and (Web-linked) voice-narrated videos of pencil-and-paper investigations illustrating typical initial steps in the solution of selected textbook problems. Computing Project Resourcessupport most of the almost three dozen projects that follow key sections in the text. For each such project marked in the text by a CD-ROM icon, more extended discussions illustrating Maple,Mathematica,MATLAB,

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