Calculus

1232 Pages·2013·56.592 MB·English

by  Robert T. Smith, Roland B. Minton

Calculus
Fourth Edition
ROBERT T. SMITH
MillersvilleUniversityofPennsylvania
ROLAND B. MINTON
RoanokeCollege
CALCULUS,FOURTHEDITION
PublishedbyMcGraw-Hill,abusinessunitofTheMcGraw-HillCompanies,Inc.,1221AvenueoftheAmericas,
NewYork,NY10020.Copyright�c 2012byTheMcGraw-HillCompanies,Inc.Allrightsreserved.Previous
editions�c 2008,2002,and2000.Nopartofthispublicationmaybereproducedordistributedinanyformorby
anymeans,orstoredinadatabaseorretrievalsystem,withoutthepriorwrittenconsentofTheMcGraw-Hill
Companies,Inc.,including,butnotlimitedto,inanynetworkorotherelectronicstorageortransmission,or
broadcastfordistancelearning.
Someancillaries,includingelectronicandprintcomponents,maynotbeavailabletocustomersoutsidethe
UnitedStates.
Thisbookisprintedonacid-freepaper.
1234567890QVR/QVR10987654321
ISBN 978–0–07–338311–8
MHID 0–07–338311–2
VicePresident,Editor-in-Chief:MartyLange
VicePresident,EDP:KimberlyMeriwetherDavid
SeniorDirectorofDevelopment:KristineTibbetts
EditorialDirector:StewartK.Mattson
SponsoringEditor:JohnR.Osgood
DevelopmentalEditor:EveL.Lipton
MarketingManager:KevinM.Ernzen
LeadProjectManager:PeggyJ.Selle
SeniorBuyer:SandyLudovissy
LeadMediaProjectManager:JudiDavid
SeniorDesigner:LaurieB.Janssen
CoverDesigner:RonBissell
CoverImage:�c Gettyimages/GeorgeDieboldPhotography
SeniorPhotoResearchCoordinator:JohnC.Leland
Compositor:Aptara,Inc.
Typeface:10/12TimesRoman
Printer:Quad/Graphics
Allcreditsappearingonpageorattheendofthebookareconsideredtobeanextensionofthecopyrightpage.
LibraryofCongressCataloging-in-PublicationData
Smith,RobertT.(RobertThomas),1955-
Calculus/RobertT.Smith,RolandB.Minton.—4thed.
p. cm.
Includesindex.
ISBN978–0–07–338311–8—ISBN0–07–338311–2(hardcopy:alk.paper)
1.Transcendentalfunctions—Textbooks. 2.Calculus—Textbooks.
I.Minton,RolandB.,1956–II.Title.
QA353.S649 2012
515 .22—dc22 2010030314
www.mhhe.com
DEDICATION
ToPam,KatieandMichael
ToJan,KellyandGreg
Andinmemoryofourparents:
GeorgeandAnneSmith
and
PaulandMaryFrancesMinton
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�c 2006,andthreeearlierbooks
forMcGraw-HillHigherEducation.EarliereditionsofCalculushavebeentranslatedinto
Spanish,ChineseandKoreanandareinusearoundtheworld.
iv
Brief Table of Contents
..
CHAPTER 0 Preliminaries 1
..
CHAPTER 1 Limits and Continuity 47
..
CHAPTER 2 Differentiation 107
..
CHAPTER 3 Applications of Differentiation 173
..
CHAPTER 4 Integration 251
..
CHAPTER 5 Applications of the Definite Integral 315
..
CHAPTER 6 Exponentials, Logarithms and Other Transcendental
Functions 375
..
CHAPTER 7 Integration Techniques 421
..
CHAPTER 8 First-Order Differential Equations 491
..
CHAPTER 9 Infinite Series 531
..
CHAPTER 10 Parametric Equations and Polar Coordinates 625
..
CHAPTER 11 Vectors and the Geometry of Space 687
..
CHAPTER 12 Vector-Valued Functions 749
..
CHAPTER 13 Functions of Several Variables and Partial Differentiation 809
..
CHAPTER 14 Multiple Integrals 901
..
CHAPTER 15 Vector Calculus 977
..
CHAPTER 16 Second-Order Differential Equations 1073
..
APPENDIX A Proofs of Selected Theorems A-1
..
APPENDIX B Answers to Odd-Numbered Exercises A-13
v
Table of Contents
SeeingtheBeautyandPowerofMathematics xiii
ApplicationsIndex xxiv
..
CHAPTER 0 Preliminaries 1
0.1 The Real Numbers and the Cartesian Plane 2
. .
TheRealNumberSystemandInequalities TheCartesianPlane
0.2 Lines and Functions 9
. .
EquationsofLines Functions
0.3 Graphing Calculators and Computer Algebra Systems 21
0.4 Trigonometric Functions 27
0.5 Transformations of Functions 36
..
CHAPTER 1 Limits and Continuity 47
1.1 A Brief Preview of Calculus: Tangent Lines and the Length
of a Curve 47
1.2 The Concept of Limit 52
1.3 Computation of Limits 59
1.4 Continuity and Its Consequences 68
.
TheMethodofBisections
1.5 Limits Involving Infinity; Asymptotes 78
.
LimitsatInfinity
1.6 Formal Definition of the Limit 87
. .
ExploringtheDefinitionofLimitGraphically LimitsInvolvingInfinity
1.7 Limits and Loss-of-Significance Errors 98
.
ComputerRepresentationofRealNumbers
..
CHAPTER 2 Differentiation 107
2.1 Tangent Lines and Velocity 107
. .
TheGeneralCase Velocity
2.2 The Derivative 118
. .
AlternativeDerivativeNotations NumericalDifferentiation
vi
TableofContents vii
2.3 Computation of Derivatives: The Power Rule 127
. .
ThePowerRule GeneralDerivativeRules
. .
HigherOrderDerivatives Acceleration
2.4 The Product and Quotient Rules 135
. . .
ProductRule QuotientRule Applications
2.5 The Chain Rule 142
2.6 Derivatives of Trigonometric Functions 147
.
Applications
2.7 Implicit Differentiation 155
2.8 The Mean Value Theorem 162
..
CHAPTER 3 Applications of Differentiation 173
3.1 Linear Approximations and Newton’sMethod 174
. .
LinearApproximations Newton’sMethod
3.2 Maximum and Minimum Values 185
3.3 Increasing and Decreasing Functions 195
.
WhatYouSeeMayNotBeWhatYouGet
3.4 Concavity and the Second Derivative Test 203
3.5 Overview of Curve Sketching 212
3.6 Optimization 223
3.7 Related Rates 234
3.8 Rates of Change in Economics and the Sciences 239
..
CHAPTER 4 Integration 251
4.1 Antiderivatives 252
4.2 Sums and Sigma Notation 259
.
PrincipleofMathematicalInduction
4.3 Area 266
4.4 The Definite Integral 273
.
AverageValueofaFunction
4.5 The Fundamental Theorem of Calculus 284
4.6 Integration by Substitution 292
.
SubstitutioninDefiniteIntegrals
4.7 Numerical Integration 298
. .
Simpson’sRule ErrorBoundsforNumericalIntegration
..
CHAPTER 5 Applications of the Definite
Integral 315
5.1 Area Between Curves 315
5.2 Volume: Slicing, Disks and Washers 324
. . .
VolumesbySlicing TheMethodofDisks TheMethodofWashers
viii TableofContents
5.3 Volumes by Cylindrical Shells 338
5.4 Arc Length and Surface Area 345
. .
ArcLength SurfaceArea
5.5 Projectile Motion 352
5.6 Applications of Integration to Physics and Engineering 361
..
CHAPTER 6 Exponentials, Logarithms and Other
Transcendental Functions 375
6.1 The Natural Logarithm 375
.
LogarithmicDifferentiation
6.2 Inverse Functions 384
6.3 The Exponential Function 391
.
DerivativeoftheExponentialFunction
6.4 The Inverse Trigonometric Functions 399
6.5 The Calculus of the Inverse Trigonometric Functions 405
.
IntegralsInvolvingtheInverseTrigonometricFunctions
6.6 The Hyperbolic Functions 411
. .
TheInverseHyperbolicFunctions DerivationoftheCatenary
..
CHAPTER 7 Integration Techniques 421
7.1 Review of Formulas and Techniques 422
7.2 Integration by Parts 426
7.3 Trigonometric Techniques of Integration 433
.
IntegralsInvolvingPowersofTrigonometricFunctions
.
TrigonometricSubstitution
7.4 Integration of Rational Functions Using Partial Fractions 442
.
BriefSummaryofIntegrationTechniques
7.5 Integration Tables and Computer Algebra Systems 450
. .
UsingTablesofIntegrals IntegrationUsingaComputerAlgebraSystem
7.6 Indeterminate Forms and L’Hoˆpital’sRule 457
.
OtherIndeterminateForms
7.7 Improper Integrals 467
.
ImproperIntegralswithaDiscontinuousIntegrand
. .
ImproperIntegralswithanInfiniteLimitofIntegration AComparisonTest
7.8 Probability 479
..
CHAPTER 8 First-Order Differential
Equations 491
8.1 Modeling with Differential Equations 491
. .
GrowthandDecayProblems CompoundInterest
8.2 Separable Differential Equations 501
.
LogisticGrowth
TableofContents ix
8.3 Direction Fields and Euler’sMethod 510
8.4 Systems of First-Order Differential Equations 521
.
Predator-PreySystems
..
CHAPTER 9 Infinite Series 531
9.1 Sequences of Real Numbers 532
9.2 Infinite Series 544
9.3 The Integral Test and Comparison Tests 554
.
ComparisonTests
9.4 Alternating Series 565
.
EstimatingtheSumofanAlternatingSeries
9.5 Absolute Convergence and the Ratio Test 571
. . .
TheRatioTest TheRootTest SummaryofConvergenceTests
9.6 Power Series 579
9.7 Taylor Series 587
.
RepresentationofFunctionsasPowerSeries
.
ProofofTaylor’sTheorem
9.8 Applications of Taylor Series 599
.
TheBinomialSeries
9.9 Fourier Series 607
.
FunctionsofPeriodOtherThan2π
.
FourierSeriesandMusicSynthesizers
..
CHAPTER 10 Parametric Equations and
Polar Coordinates 625
10.1 Plane Curves and Parametric Equations 625
10.2 Calculus and Parametric Equations 634
10.3 Arc Length and Surface Area in Parametric Equations 641
10.4 Polar Coordinates 649
10.5 Calculus and Polar Coordinates 660
10.6 Conic Sections 668
. . .
Parabolas Ellipses Hyperbolas
10.7 Conic Sections in Polar Coordinates 677
..
CHAPTER 11 Vectors and the Geometry
of Space 687
11.1 Vectors in the Plane 688
11.2 Vectors in Space 697
.
VectorsinR3
11.3 The Dot Product 704
.
ComponentsandProjections
x TableofContents
11.4 The Cross Product 714
11.5 Lines and Planes in Space 726
.
PlanesinR3
11.6 Surfaces in Space 734
. . .
CylindricalSurfaces QuadricSurfaces AnApplication
..
CHAPTER 12 Vector-Valued Functions 749
12.1 Vector-Valued Functions 750
.
ArcLengthinR3
12.2 The Calculus of Vector-Valued Functions 758
12.3 Motion in Space 769
.
EquationsofMotion
12.4 Curvature 779
12.5 Tangent and Normal Vectors 786
. .
TangentialandNormalComponentsofAcceleration Kepler’sLaws
12.6 Parametric Surfaces 799
..
CHAPTER 13 Functions of Several Variables and
Partial Differentiation 809
13.1 Functions of Several Variables 809
13.2 Limits and Continuity 822
13.3 Partial Derivatives 833
13.4 Tangent Planes and Linear Approximations 844
.
IncrementsandDifferentials
13.5 The Chain Rule 854
.
ImplicitDifferentiation
13.6 The Gradient and Directional Derivatives 864
13.7 Extrema of Functions of Several Variables 874
.
ProofoftheSecondDerivativesTest
13.8 Constrained Optimization and Lagrange Multipliers 887
..
CHAPTER 14 Multiple Integrals 901
COEFFICENT OF RESTITUTION
RACKETS HELD BY VISE
BALL VELOCITY OF 385 M PA
FRAME BALL HITS FRAME
IN THIS AREA 14.1 Double Integrals 901
.
77474352245344554044404324534755432225 22333722424547552247262286172233270420 14.2 A.rDDeooauu,bbVlleeoIInnlutteemggrreaallssaoonvvdeerrCaGReenenecttraaenlrgRleoegfioMnsass 916
55 545152552355 524745445030 .
MomentsandCenterofMass
52 53 6665THROAT FIRGGSrrTeeaa StteeTrrR TTINhhaaGnn 34 14.3 Double Integrals in Polar Coordinates 926
FIRST STRING GGrreeaatteerr TThhaann 56
Prince Racket A M.F. MRoaocdk eSttandard 14.4 Surface Area 933
THROAT
14.5 Triple Integrals 938
.
MassandCenterofMass