Calculus of a Single Variable

910 Pages·1997·27.616 MB·English

by  Larson, Hostetler, Edwards

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Succeed in Calt‘ulus
Use a Wide range of valuable resources to excel in calculus.
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Editor in Chief, Mathematics: Charles Hartford
Managing Editor: Cathy Cantin
SeniorAssociate Editor: Maureen Brooks
Associate Editor: Michael Richards
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No part ofthiswork may be reproduced or transmitted in any form orby any means,
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Printed in the USA.
Library ofCongress Catalog Card Number: 97-72981
ISBN: 0-395-88578-7
89—VH—01 00
CONTENTS V
Contems
A Word from the Authors (Preface) ix
Index of Applications xxxi
Chapter P Preparation for Calculus 1
d
P-1 GriIPhS and Models 3
P.2 Linear Models and Rates of Change 11

P.3 Functions and Their Graphs 20 E

R4 Fitting Models to Data 31 E

Review Exercises 37 E;
E .:J:— 1% ti{- i f

40 120 200 280 360
Day(0<——>December21)
Chapter 1 Limits and Their Properties
1.1 A Preview of Calculus 41 y
1.2 Finding Limits Graphically and Numerically 47
1.3 Evaluating Limits Analytically 56
1.4 Continuity and One-Sided Limits 67
lim'f(x)atf(c)
1.5 Infinite Limits 79
Review Exercises 87 }
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Chapter 2 Differentiation 89
2.1 The Derivative and the Tangent Line Problem 91 ,=—5x22+4x2+5
2.2 Basic Differentiation Rules and Rates of Change 102 9‘ +1) ,, _e .

2.3 The Product and Quotient Rules and Higher-Order
Derivatives 114
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2.4 The Chain Rule 124 .....
2.5 Implicit Differentiation 134
2.6 Related Rates 141 M _4":
Review Exercises 150 y= 5x4
Vi CONTENTS
Chapter 3 Applications of Differentiation
3.1 Extrema on an Interval 155
3.2 Rolle's Theorem and the Mean Value Theorem 163
3.3 Increasing and Decreasing Functions and the
First Derivative Test 169
3.4 Concavity and the Second Derivative Test 179
3.5 Limits at Infinity 187
3.6 A Summary of Curve Sketching 196
3.7 Optimization Problems 205 ¥ X
3.8 Newton's Method 215 f’(x)'< 0 if’(x) = 0 ' f’(x) > 0-
3.9 Differentials 221 ‘
3.10 Business and Economics Applications 228
Review Exercises 235
\,
Chapter 4 Integration 239
4.1 Antiderivatives and Indefinite Integration 241 {
4.2 Area 252
4.3 Riemann Sums and Definite Integrals 264
4.4 The Fundamental Theorem of Calculus 274
4.5 Integration by Substitutibn 287

4.6 Numerical Integration 299
Review Exercises 306 /

309
5.1 The Natural Logarithmic Function and Differentiation 311 y
5.2 The Natural Logarithmic Function and Integration 321 1.25 8 7 h

5.3 Inverse Functions 329
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5.4 Exponential Functions: Differentiation and Integration 338 am
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5.5 Bases Other than e and Applications 348 (in 1.15
5.6 Differential Equations: Growth and Decay 358 re
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5.7 Differential Equations: Separation of Variables 366 cu
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5.8 Inverse Trigonometric Functions and Differentiation 377 ht
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5.9 Inverse Trigonometric Functions and Integration 385 e
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5.10 Hyperbolic Functions 392
Review Exercises 402
CONTENTS Vii
Chapter 6 Applications of Integration
6.] Area of a Region Between Two Curves 407 y
6.2 Volume: The Disc Method 416
6.3 Volume: The Shell Method 427
6.4 Arc Length and Surfaces of Revolution 435
6.5 Work 445
6.6 Moments, Centers of Mass, and Centroids 454
6.7 Fluid Pressure and Fluid Force 465
Review Exercises 471

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Chapter 7 Integration Techniques, L'Hfipital's Rule, and Improper Integrals 473
7.1 Basic Integration Rules 475
7.2 Integration by Parts 481
7.3 Trigonometric Integrals 490
7.4 Trigonometric Substitution 499
7.5 Partial Fractions 508

7.6 Integration by Tables and Other Integration Techniques 517
7.7 Indeterminate Forms and L'H6pital’s Rule 523

7.8 Improper Integrals 533

Review Exercises 542
Chapter 8 Infinite Series
8.] Sequences 547
8.2 Series and Convergence 558
8.3 The Integral Test and p-Series 568
8.4 Comparisons of Series 574
8.5 Alternating Series 581
8.6 The Ratio and Root Tests 588
8.7 Taylor Polynomials and Approximations 596
8.8 Power Series 606
8.9 Representation of Functions by Power Series 615
8.10 Taylor and Maclaurin Series 622
Review Exercises 632
viii CONTENTS
Chapter 9 Conics, Parametric Equations, and Polar Coordinates

9.1 Conics and Calculus 637 Vertical y
9.2 Plane Curves and Parametric Equations 652 supportingcable
(60,20)
9.3 Parametric Equations and Calculus 662
9.4 Polar Coordinates and Polar Graphs 671
9.5 Area and Arc Length in Polar Coordinates 681
9.6 Polar Equations of Conics and Kepler’s Laws 689
Review Exercises 696
Appendix A Precalculus Review A1
A] Real Numbers and the Real Line A1
A2 The Cartesian Plane A10
A.3 Review of Trigonometu't Functions A17
Appendix B Proofs of Selected Theorems A28
Appendix C Basic Differentiation Rules for Elementary Functions A44
Appendix D Integration Tables A45
Appendix E Rotation and the General Second-Degree Equation A51
Appendix F Complex Numbers A57
Answers to Odd-Numbered Exercises A69
Index A159
PREFACE
A wordfiom the authors. ..
WelcometoCalculusofaSingleVariable, SixthEdition! We areexcitedabout
the Sixth Edition and hope youwillbe too afteryou hear aboutwhywewrote
it, what’s new about it, and how it will carry you and your calculus students
into the twenty-first century!
Reform in Mathematics Education
Asyouknow, thecurrentreformmovementinmatheducationstartedabout 15
years ago and has involved all levels ofmathematics education from kinder—
garten through college. There may be some who say reform was not needed.
However, thevastmajority ofmath educators agree thatreformwas essential.
In fact, the new math of the 19603, 19705, and early 19805 was a national
disaster. Itwas far too abstract and far too removed from the real—life applica-
tions that were the actual foundation of mathematics. The result was evident
to everyone—math phobia, falling test scores, high drop-out rates, and a gen-
eral sense that students were not learning to be creative problem solvers.
So what were the proposed solutions? There were many—more real-life
connections, more incorporation oftechnology, curriculum revisions, and the
development ofalternative forms ofteaching, assessment, and learning.
Where This Text Stands with Respect to Reform
Where does the Sixth Edition stand with respect to reform? This is one ofthe
“The text has definitely most common questions we are asked. Our answer: The text has definitely
benefitedfrom reform. benefited from reform. More than that, over the years our calculus text has
actuallyledtheway in developingmany innovativelearningtechniques. From
More than that, over the
its first edition, the text stressed the importance ofgraphical learning—much
years our calculus text more than other texts in use in the late 1970s and early 19805. This text was
has actually ledthe way one ofthe first to incorporate computer-generated art—both two-dimensional
and three-dimensional—to aid in the visualization of complex mathematical
in developing many
concepts.
innovative learning We have always paid careful attention to the presentation—using precise
techniques.” mathematical language, innovativefull-colordesignsforemphasis andclarity,
and alevel ofexpositionthat appeals to students—to create aneffectiveteach-
ing and learning tool. Although difficult to quantify, this feature has been
praised by thousands of students and their instructors over the past 20 years.
With each edition, we have continued to incorporate the best strategies for
teaching calculus, using the pedagogy we have developed as a result of our
teaching experiences, as well as many suggestions from thoughtfulusers.
This Sixth Edition might best be described as fitting midway between
texts thatdefinethemselves as traditional and thosethatareconsideredreform
texts. Our approachis like that ofatraditional text in thatwe firmlybelieve in
the importance ofcarefully developed theory, correct statements oftheorems,
inclusion ofproofs, and mastery oftraditional calculus skills. We have found
no evidencethatitissomehowpossible to applycalculusinreal-lifesituations
withoutfirstbeing able to understand and “do” calculus.