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# Calculus (6E 2007 ISBN 9780495011606) James Stewart

(Parte **1** de 6)

JAMES STEWART McMASTER UNIVERSITY

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Calculus, Sixth Edition James Stewart

Preface xi To the Studentxxii Diagnostic Tests xxiv

A PREVIEW OF CALCULUS2

FUNCTIONS AND MODELS10

Principles of Problem Solving 54

LIMITS 60

Problems Plus 110

1 CONTENTS

DERIV ATIVES 112

3.1Derivatives and Rates of Change113 Writing ProjectEarly Methods for Finding Tangents123

Review 196

Problems Plus 200

APPLICATIONS OF DIFFERENTIATION204

4.1Maximum and Minimum Values205 Applied ProjectThe Calculus of Rainbows213

Applied ProjectThe Shape of a Can268

Problems Plus 285 m=1 m=_1 m=0 iv |||| CONTENTS

INTEGRALS 288

Discovery ProjectArea Functions312

5.4Indefinite Integrals and the Net Change Theorem324 Writing ProjectNewton,Leibniz,and the Invention of Calculus332

Problems Plus 344

APPLICATIONS OF INTEGRATION346

Problems Plus 380

INVERSE FUNCTIONS: EXPONENTIAL, LOGARITHMIC , AND INVERSE TRIGONOMETRIC FUNCTIONS 384

7.1 Inverse Functions 385 Instructors may cover either Sections 7.2–7.4 or Sections 7.2*–7.4*. See the Preface .

vi |||| CONTENTS

7.6Inverse Trigonometric Functions454 Applied ProjectWhere To Sit at the Movies463

7.8Indeterminate Forms and L’Hospital’s Rule470 Writing ProjectThe Origins of L’Hospital’s Rule481

Review 482

Problems Plus 486

TECHNIQUES OF INTEGRATION488

8.6Integration Using Tables and Computer Algebra Systems525 Discovery ProjectPatterns in Integrals530

Problems Plus 557

FURTHER APPLICATIONS OF INTEGRATION560

9.1Arc Length561 Discovery ProjectArc Length Contest568

9.2Area of a Surface of Revolution568 Discovery ProjectRotating on a Slant574

9.3Applications to Physics and Engineering575 Discovery ProjectComplementary Coffee Cups586

Problems Plus 600

DIFFERENTIAL EQU ATIONS 602

10.3 Separable Equations 616 Applied ProjectHow Fast Does a Tank Drain?624

Applied ProjectWhich Is Faster,Going Up or Coming Down?626

10.4Models for Population Growth627 Applied ProjectCalculus and Baseball637

Problems Plus 654

PARAMETRIC EQUATIONS AND POLAR COORDINATES656

1.1Curves Defined by Parametric Equations657 Laboratory ProjectRunning Circles Around Circles665

1.2Calculus with Parametric Curves666 Laboratory ProjectBézier Curves675

Problems Plus 708

INFINITE SEQUENCES AND SERIES710

12.1 Sequences 711 Laboratory ProjectLogistic Sequences723

10 CONTENTS |||| vii

viii |||| CONTENTS

Writing ProjectHow Newton Discovered the Binomial Series784

Review 794

Problems Plus 797

VECTORS AND THE GEOMETRY OF SPACE800

13.5Equations of Lines and Planes830 Laboratory ProjectPutting 3D in Perspective840

13.6Cylinders and Quadric Surfaces840 Review 848

Problems Plus 851

VECTOR FUNCTIONS852

Review 885

Problems Plus 8

CONTENTS |||| ix

PARTIAL DERIV ATIVES 890

Discovery ProjectQuadratic Approximations and Critical Points969

15.8 Lagrange Multipliers 970 Applied ProjectRocket Science977

Applied Project Hydro-Turbine Optimization 979

Review 980

Problems Plus 984

MULTIPLE INTEGRALS 986

16.6 Triple Integrals 1026 Discovery ProjectVolumes of Hyperspheres1036

16.7Triple Integrals in Cylindrical Coordinates1036 Discovery ProjectThe Intersection of Three Cylinders1041

Problems Plus 1060

VECTOR CALCULUS1062

Review 1142

Problems Plus 1145

SECOND-ORDER DIFFERENTIAL EQUATIONS1146

APPENDIXES A1

A Numbers, Inequalities, and Absolute Values A2 BCoordinate Geometry and LinesA10 CGraphs of Second-Degree EquationsA16 D Trigonometry A24 ESigma NotationA34 FProofs of TheoremsA39 GComplex NumbersA48 HAnswers to Odd-Numbered ExercisesA57

INDEX A125

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.

The art of teaching,Mark Van Doren said,is the art of assisting discovery. I have tried to write a book that assists students in discovering calculus—both for its practical power and its surprising beauty. In this edition,as in the first five editions,I aim to convey to the student a sense of the utility of calculus and develop technical competence,but I also strive to give some appreciation for the intrinsic beauty of the subject. Newton undoubtedly experienced a sense of triumph when he made his great discoveries. I want students to share some of that excitement.

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

Focus on conceptual understanding.

I have tried to implement this goal through the Rule of Three:“Topics should be presented geometrically, numerically, and algebraically.” Visualization, numerical and graphical experimentation,and other approaches have changed how we teach conceptual reasoning in fundamental ways. More recently,the Rule of Three has been expanded to become the Rule of Fourby emphasizing the verbal,or descriptive,point of view as well.

In writing the sixth edition my premise has been that it is possible to achieve conceptual understanding and still retain the best traditions of traditional calculus. The book contains elements of reform,but within the context of a traditional curriculum.

ALTERNA TIVE VERSIONS I have written several other calculus textbooks that might be preferable for some instruc- tors. Most of them also come in single variable and multivariable versions.Calculus:Early Transcendentals,Sixth Edition,is similar to the present textbook except that the exponential,logarithmic,and inverse trigonometric functions are covered in the first semester.Essential Calculusis a much briefer book (800 pages),though it contains almost all of the topics in the present text. The relative brevity is achieved through briefer exposition of some topics and putting some features on the website.Essential Calculus:Early Transcendentalsresembles Essential Calculus,but the exponential,logarithmic,and inverse trigonometric functions are covered in Chapter 3.

xii||||PREFACECalculus:Concepts and Contexts,Third Edition,emphasizes conceptual understanding even more strongly than this book. The coverage of topics is not encyclopedic and the material on transcendental functions and on parametric equations is woven throughout the book instead of being treated in separate chapters. Calculus:Early Vectorsintroduces vectors and vector functions in the first semester and integrates them throughout the book. It is suitable for students taking Engineering and Physics courses concurrently with calculus.

Here are some of the changes for the sixth edition of Calculus.At the beginning of the book there are four diagnostic tests,in Basic Algebra,

Analytic Geometry,Functions,and Trigonometry. Answers are given and students who don’t do well are referred to where they should seek help (Appendixes,review sections of Chapter 1,and the website).In response to requests of several users,the material motivating the derivative is briefer:Sections 2.6 and 3.1 are combined into a single section called Derivatives and

Rates of Change.The section on Higher Derivatives in Chapter 3 has disappeared and that material is integrated into various sections in Chapters 2 and 3.Instructors who do not cover the chapter on differential equations have commented that the section on Exponential Growth and Decay was inconveniently located there. Accordingly,it is moved earlier in the book,to Chapter 7. This move precipitates a reorganization of Chapter 10.Sections 4.7 and 4.8 are merged into a single section,with a briefer treatment of opti- mization problems in business and economics.Sections 12.10 and 12.1 are 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 have decided to incorporate binomial series into 12.10.The material on cylindrical and spherical coordinates (formerly Section 13.7) is moved to Chapter 16,where it is introduced in the context of evaluating triple integrals.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.Many examples have been added or changed. For instance,Example 1 on page 143 was changed because students are often baffled when they see arbitrary constants in a problem and I wanted to give an example in which they occur. 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 my favor- ites:3.3.101,3.3.102,4.3.50,4.3.67,12.6.38,12.1.30,15.5.4,and 15.8.20–21.There are also some good new problems in the Problems Plus sections. See,for instance,Problems 2 and 1 on page 345,Problem 13 on page 382,and Problem 24 on page 799.The new project on page 586,Complementary Coffee Cups,comes from an article by Thomas Banchoff in which he wondered which of two coffee cups,whose convex and concave profiles fit together snugly,would hold more coffee.

PREFACE||||xiiiTools for Enriching Calculus (TEC) has been completely redesigned and is accessible on the Internet at w.stewartcalculus.com. It now includes what we call Visuals,brief animations of various figures in the text. In addition,there are now Visual,Modules, and Homework Hints for the multivariable chapters. See the description on page xv.The symbol has been placed beside examples (an average of three per section) for which there are videos of instructors explaining the example in more detail. This material is also available on DVD. See the description on page xxi.

Another type of exercise uses verbal description to test conceptual understanding (see

Exercises 2.5.8,3.1.54,4.3.51–52,and 8.8.67). I particularly value problems that combine and compare graphical,numerical,and algebraic approaches (see Exercises 3.7.23, 4.4.31–32, and 10.4.2).

GRADED EXERCISE SETSEach exercise set is carefully graded,progressing from basic conceptual exercises and skilldevelopment problems to more challenging problems involving applications and proofs.

REAL-WORLD DATAMy 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 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,Figure 1 in Section 1.1 (seismograms from the Northridge earthquake),Exercise 3.2.32 (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 Francisco power consumption). 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 2 in Section15.1). Partial derivatives are introduced in Section 15.3 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 15.4). Directional derivatives are introduced in Section 15.6 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 16.1). Vector fields are introduced in Section 16.1 by depictions of actual velocity vector fields showing San Francisco Bay wind patterns.

PROJECTSOne 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. I have included four kinds of projects:Applied Projectsinvolve applications that are designed to appeal to the imagination of students. The project after Section 10.3 asks whether a ball thrown upward takes longer to reach its maximum height or to fall back to its original height. (The answer might surprise you.) The project after Section 15.8 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. Laboratory xiv |||| PREFACE

Projectsinvolve technology; the one following Section 1.2 shows how to use Bézier curves to design shapes that represent letters for a laser printer. Writing Projectsask students to compare present-day methods with those of the founders of calculus—Fermat’s method for finding tangents,for instance. Suggested references are supplied. Discovery Projectsanticipate results to be discussed later or encourage discovery through pattern recognition (see the one following Section 8.6). Others explore aspects of geometry:tetrahedra (after Section 13.4),hyperspheres (after Section 16.6),and intersections of three cylinders (after Section 16.7). Additional projects can be found in theInstructor’s Guide (see,for instance,Group Exercise 5.1:Position from Samples).

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