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Calculus
EARLY TRANSCENDENTAL FUNCTIONS
Fourth Edition
ROBERT T. SMITH
MillersvilleUniversityofPennsylvania
ROLAND B. MINTON
RoanokeCollege
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CALCULUS:EARLYTRANSCENDENTALFUNCTIONS,FOURTHEDITION
PublishedbyMcGraw-Hill,abusinessunitofTheMcGraw-HillCompanies,Inc.,1221AvenueoftheAmericas,
NewYork,NY10020.Copyright(cid:2)c 2012byTheMcGraw-HillCompanies,Inc.Allrightsreserved.Previous
editions(cid:2)c 2007,2002,and2000.Nopartofthispublicationmaybereproducedordistributedinanyformorby
anymeans,orstoredinadatabaseorretrievalsystem,withoutthepriorwrittenconsentofTheMcGraw-Hill
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UnitedStates.
Thisbookisprintedonacid-freepaper.
1234567890QVR/QVR10987654321
ISBN 978–0–07–353232–5
MHID 0–07–353232–0
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LibraryofCongressCataloging-in-PublicationData
Smith,RobertT.
Calculus:earlytranscendentalfunctions/RobertT.Smith,RolandB.Minton.—4thed.
p. cm.
Includesindex.
ISBN978–0–07–353232–5—ISBN0–07–353232–0(hardcopy:alk.paper)
1.Calculus—Textbooks. I.Minton,RolandB.,1956–II.Title.
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DEDICATION
ToPam,KatieandMichael
ToJan,KellyandGreg
Andinmemoryofourparents:
GeorgeandAnneSmith
and
PaulandMaryFrancesMinton
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About the Authors
Robert T. Smith is Professor of Mathematics and Dean of the School of Science and
MathematicsatMillersvilleUniversityofPennsylvania,wherehehasbeenafacultymember
since 1987. Prior to that, he was on the faculty at Virginia Tech. He earned his Ph.D. in
mathematicsfromtheUniversityofDelawarein1982.
ProfessorSmith’smathematicalinterestsareintheapplicationofmathematicstoprob-
lemsinengineeringandthephysicalsciences.Hehaspublishedanumberofresearcharticles
ontheapplicationsofpartialdifferentialequationsaswellasoncomputationalproblemsin
x-raytomography.HeisamemberoftheAmericanMathematicalSociety,theMathematical
AssociationofAmerica,andtheSocietyforIndustrialandAppliedMathematics.
ProfessorSmithlivesinLancaster,Pennsylvania,withhiswifePam,hisdaughterKatie
andhissonMichael.Hisongoingextracurriculargoalistolearntoplaygolfwellenough
tonotcomeinlastinhisannualmathematicians/statisticianstournament.
RolandB.MintonisProfessorofMathematicsandChairoftheDepartmentofMathemat-
ics,ComputerScienceandPhysicsatRoanokeCollege,wherehehastaughtsince1986.
Prior to that, he was on the faculty at Virginia Tech. He earned his Ph.D. from Clemson
Universityin1982.HeistherecipientofRoanokeCollegeawardsforteachingexcellence
andprofessionalachievement,aswellasthe2005VirginiaOutstandingFacultyAwardand
the2008GeorgePolyaAwardformathematicsexposition.
ProfessorMinton’scurrentresearchprogramisinthemathematicsofgolf,especially
the analysis of ShotLink statistics. He has published articles on various aspects of sports
science,andco-authoredwithTimPenningsanarticleonPennings’dogElvisandhisability
to solve calculus problems. He is co-author of a technical monograph on control theory.
Hehassupervisednumerousindependentstudiesandheldworkshopsforlocalhighschool
teachers.HeisanactivememberoftheMathematicalAssociationofAmerica.
ProfessorMintonlivesinSalem,Virginia,withhiswifeJanandoccasionallywithhis
daughter Kelly and son Greg when they visit. He enjoys playing golf when time permits
and watching sports events even when time doesn’t permit. Jan also teaches at Roanoke
Collegeandisveryactiveinmathematicseducation.
InadditiontoCalculus:EarlyTranscendentalFunctions,ProfessorsSmithandMinton
arealsocoauthorsofCalculus:ConceptsandConnections(cid:2)c 2006,andthreeearlierbooks
forMcGraw-HillHigherEducation.EarliereditionsofCalculushavebeentranslatedinto
Spanish,ChineseandKoreanandareinusearoundtheworld.
iv
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Brief Table of Contents
..
CHAPTER 0 Preliminaries 1
..
CHAPTER 1 Limits and Continuity 65
..
CHAPTER 2 Differentiation 125
..
CHAPTER 3 Applications of Differentiation 211
..
CHAPTER 4 Integration 299
..
CHAPTER 5 Applications of the Definite Integral 377
..
CHAPTER 6 Integration Techniques 447
..
CHAPTER 7 First-Order Differential Equations 499
..
CHAPTER 8 Infinite Series 539
..
CHAPTER 9 Parametric Equations and Polar Coordinates 633
..
CHAPTER 10 Vectors and the Geometry of Space 695
..
CHAPTER 11 Vector-Valued Functions 757
..
CHAPTER 12 Functions of Several Variables and Partial Differentiation 817
..
CHAPTER 13 Multiple Integrals 909
..
CHAPTER 14 Vector Calculus 985
..
CHAPTER 15 Second-Order Differential Equations 1081
..
APPENDIX A Proofs of Selected Theorems A-1
..
APPENDIX B Answers to Odd-Numbered Exercises A-13
v
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Table of Contents
SeeingtheBeautyandPowerofMathematics xiii
ApplicationIndex xxiv
..
CHAPTER 0 Preliminaries 1
0.1 Polynomials and Rational Functions 2
. . .
TheRealNumberSystemandInequalities EquationsofLines Functions
0.2 Graphing Calculators and Computer Algebra Systems 19
0.3 Inverse Functions 26
0.4 Trigonometric and Inverse Trigonometric Functions 31
.
TheInverseTrigonometricFunctions
0.5 Exponential and Logarithmic Functions 42
. .
HyperbolicFunctions FittingaCurvetoData
0.6 Transformations of Functions 53
..
CHAPTER 1 Limits and Continuity 65
1.1 A Brief Preview of Calculus: Tangent Lines and the Length
of a Curve 65
1.2 The Concept of Limit 70
1.3 Computation of Limits 77
1.4 Continuity and Its Consequences 86
.
TheMethodofBisections
1.5 Limits Involving Infinity; Asymptotes 96
.
LimitsatInfinity
1.6 Formal Definition of the Limit 106
. .
ExploringtheDefinitionofLimitGraphically LimitsInvolvingInfinity
1.7 Limits and Loss-of-Significance Errors 117
.
ComputerRepresentationofRealNumbers
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TableofContents vii
..
CHAPTER 2 Differentiation 125
2.1 Tangent Lines and Velocity 125
. .
TheGeneralCase Velocity
2.2 The Derivative 136
. .
AlternativeDerivativeNotations NumericalDifferentiation
2.3 Computation of Derivatives: The Power Rule 145
. .
ThePowerRule GeneralDerivativeRules
. .
HigherOrderDerivatives Acceleration
2.4 The Product and Quotient Rules 153
. . .
ProductRule QuotientRule Applications
2.5 The Chain Rule 160
2.6 Derivatives of Trigonometric Functions 167
.
Applications
2.7 Derivatives of Exponential and Logarithmic
Functions 175
.
DerivativesofExponentialFunctions
.
DerivativeoftheNaturalLogarithm
.
LogarithmicDifferentiation
2.8 Implicit Differentiation and Inverse Trigonometric
Functions 183
.
DerivativesoftheInverseTrigonometricFunctions
2.9 The Hyperbolic Functions 192
.
TheInverseHyperbolicFunctions
2.10 The Mean Value Theorem 198
..
CHAPTER 3 Applications of
Differentiation 211
3.1 Linear Approximations and Newton’sMethod 212
. .
LinearApproximations Newton’sMethod
3.2 Indeterminate Forms and l’Hoˆpital’sRule 223
.
OtherIndeterminateForms
3.3 Maximum and Minimum Values 232
3.4 Increasing and Decreasing Functions 243
.
WhatYouSeeMayNotBeWhatYouGet
3.5 Concavity and the Second Derivative Test 250
3.6 Overview of Curve Sketching 259
3.7 Optimization 269
3.8 Related Rates 279
3.9 Rates of Change in Economics and the Sciences 285
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viii TableofContents
..
CHAPTER 4 Integration 299
4.1 Antiderivatives 300
4.2 Sums and Sigma Notation 309
.
PrincipleofMathematicalInduction
4.3 Area 316
4.4 The Definite Integral 323
.
AverageValueofaFunction
4.5 The Fundamental Theorem of Calculus 334
4.6 Integration by Substitution 343
.
SubstitutioninDefiniteIntegrals
4.7 Numerical Integration 352
. .
Simpson’sRule ErrorBoundsforNumericalIntegration
4.8 The Natural Logarithm as an Integral 364
.
TheExponentialFunctionastheInverseoftheNaturalLogarithm
..
CHAPTER 5 Applications of the Definite
Integral 377
5.1 Area Between Curves 377
5.2 Volume: Slicing, Disks and Washers 386
. . .
VolumesbySlicing TheMethodofDisks TheMethodofWashers
5.3 Volumes by Cylindrical Shells 400
5.4 Arc Length and Surface Area 407
. .
ArcLength SurfaceArea
5.5 Projectile Motion 414
5.6 Applications of Integration to Physics and Engineering 424
5.7 Probability 435
..
CHAPTER 6 Integration Techniques 447
6.1 Review of Formulas and Techniques 448
6.2 Integration by Parts 452
6.3 Trigonometric Techniques of Integration 459
.
IntegralsInvolvingPowersofTrigonometricFunctions
.
TrigonometricSubstitution
6.4 Integration of Rational Functions Using Partial Fractions 468
.
BriefSummaryofIntegrationTechniques
6.5 Integration Tables and Computer Algebra Systems 476
. .
UsingTablesofIntegrals IntegrationUsingaComputerAlgebraSystem
6.6 Improper Integrals 483
.
ImproperIntegralswithaDiscontinuousIntegrand
.
ImproperIntegralswithanInfiniteLimitofIntegration
.
AComparisonTest
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TableofContents ix
..
CHAPTER 7 First-Order Differential
Equations 499
7.1 Modeling with Differential Equations 499
. .
GrowthandDecayProblems CompoundInterest
7.2 Separable Differential Equations 509
.
LogisticGrowth
7.3 Direction Fields and Euler’sMethod 518
7.4 Systems of First-Order Differential Equations 529
.
Predator-PreySystems
..
CHAPTER 8 Infinite Series 539
8.1 Sequences of Real Numbers 540
8.2 Infinite Series 552
8.3 The Integral Test and Comparison Tests 562
.
ComparisonTests
8.4 Alternating Series 573
.
EstimatingtheSumofanAlternatingSeries
8.5 Absolute Convergence and the Ratio Test 579
. .
TheRatioTest TheRootTest
.
SummaryofConvergenceTests
8.6 Power Series 587
8.7 Taylor Series 595
.
RepresentationofFunctionsasPowerSeries
.
ProofofTaylor’sTheorem
8.8 Applications of Taylor Series 607
.
TheBinomialSeries
8.9 Fourier Series 615
.
FunctionsofPeriodOtherThan2π
.
FourierSeriesandMusicSynthesizers
..
CHAPTER 9 Parametric Equations and Polar
Coordinates 633
9.1 Plane Curves and Parametric Equations 633
9.2 Calculus and Parametric Equations 642
9.3 Arc Length and Surface Area in Parametric
Equations 649
9.4 Polar Coordinates 657
9.5 Calculus and Polar Coordinates 668
9.6 Conic Sections 676
. . .
Parabolas Ellipses Hyperbolas
9.7 Conic Sections in Polar Coordinates 685
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