Table Of ContentROGAWSKI’S
CALCULUS for AP*
EARLY TRANSCENDENTALS
SECOND EDITION
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ROGAWSKI’S
CALCULUS for AP*
EARLY TRANSCENDENTALS
SECOND EDITION
JON ROGAWSKI
University of California, Los Angeles
RAY CANNON
Baylor University, TX
W. H. FREEMAN AND COMPANY
New York
To Julie
and
To the AP Teachers
This page intentionally left blank
CONTENTS
ROGAWSKI’ S CALCULUS for AP*
Early Transcendentals
Chapter 1 PRECALCULUS REVIEW 1 Chapter 4 APPLICATIONS OF THE DERIVATIVE 207
1.1 RealNumbers,Functions,andGraphs 1 4.1 LinearApproximationandApplications 207
1.2 LinearandQuadraticFunctions 13 4.2 ExtremeValues 215
1.3 TheBasicClassesofFunctions 21 4.3 TheMeanValueTheoremandMonotonicity 226
1.4 TrigonometricFunctions 25 4.4 TheShapeofaGraph 234
1.5 InverseFunctions 33 4.5 L’Hôpital’sRule 241
1.6 ExponentialandLogarithmicFunctions 43 4.6 GraphSketchingandAsymptotes 248
1.7 Technology:CalculatorsandComputers 51 4.7 AppliedOptimization 257
4.8 Newton’sMethod 269
4.9 Antiderivatives 275
PreparingfortheAPExam AP4-1
Chapter 2 LIMITS 59
Chapter 5 THE INTEGRAL 286
2.1 Limits,RatesofChange,andTangentLines 59
2.2 Limits:ANumericalandGraphicalApproach 67 5.1 ApproximatingandComputingArea 286
2.3 BasicLimitLaws 77 5.2 TheDefiniteIntegral 299
2.4 LimitsandContinuity 81 5.3 TheFundamentalTheoremofCalculus,PartI 309
2.5 EvaluatingLimitsAlgebraically 90 5.4 TheFundamentalTheoremofCalculus,PartII 316
2.6 TrigonometricLimits 95 5.5 NetChangeastheIntegralofaRate 322
2.7 LimitsatInfinity 100 5.6 SubstitutionMethod 328
2.8 IntermediateValueTheorem 106 5.7 FurtherTranscendentalFunctions 336
2.9 TheFormalDefinitionofaLimit 110 5.8 ExponentialGrowthandDecay 341
PreparingfortheAPExam AP2-1 PreparingfortheAPExam AP5-1
Chapter 6 APPLICATIONS OF THE INTEGRAL 357
Chapter 3 DIFFERENTIATION 120
6.1 AreaBetweenTwoCurves 357
6.2 SettingUpIntegrals:Volume,Density,AverageValue 365
3.1 DefinitionoftheDerivative 120 6.3 VolumesofRevolution 375
3.2 TheDerivativeasaFunction 129 6.4 TheMethodofCylindricalShells 384
3.3 ProductandQuotientRules 143 6.5 WorkandEnergy 391
3.4 RatesofChange 150
PreparingfortheAPExam AP6-1
3.5 HigherDerivatives 159
3.6 TrigonometricFunctions 165
3.7 TheChainRule 169 Chapter 7 TECHNIQUES OF INTEGRATION 400
3.8 DerivativesofInverseFunctions 178
3.9 DerivativesofGeneralExponentialandLogarithmic 7.1 IntegrationbyParts 400
Functions 182 7.2 TrigonometricIntegrals 405
3.10 ImplicitDifferentiation 188 7.3 TrigonometricSubstitution 413
3.11 RelatedRates 195 7.4 IntegralsInvolvingHyperbolicandInverseHyperbolic
PreparingfortheAPExam AP3-1 Functions 420
vi
CONTENTS CALCULUS vii
7.5 TheMethodofPartialFractions 426 Chapter 11 PARAMETRIC EQUATIONS, POLAR
7.6 ImproperIntegrals 436 COORDINATES, AND VECTOR
7.7 ProbabilityandIntegration 448
FUNCTIONS 607
7.8 NumericalIntegration 454
PreparingfortheAPExam AP7-1
11.1 ParametricEquations 607
11.2 ArcLengthandSpeed 620
11.3 PolarCoordinates 626
Chapter 8 FURTHER APPLICATIONS OF THE 11.4 Area,ArcLength,andSlopeinPolarCoordinates 634
INTEGRAL AND TAYLOR 11.5 VectorsinthePlane 641
POLYNOMIALS 467 11.6 DotProductandtheAnglebetweenTwoVectors 653
11.7 CalculusofVector-ValuedFunctions 660
8.1 ArcLengthandSurfaceArea 467 PreparingfortheAPExam AP11-1
8.2 FluidPressureandForce 474
8.3 CenterofMass 480
Chapter 12 DIFFERENTIATION IN SEVERAL
8.4 TaylorPolynomials 488
VARIABLES 672
PreparingfortheAPExam AP8-1
12.1 FunctionsofTwoorMoreVariables 672
12.2 LimitsandContinuityinSeveralVariables 684
Chapter 9 INTRODUCTION TO DIFFERENTIAL
12.3 PartialDerivatives 692
EQUATIONS 502
12.4 DifferentiabilityandTangentPlanes 703
12.5 TheGradientandDirectionalDerivatives 711
9.1 SolvingDifferentialEquations 502 12.6 TheChainRule 723
9.2 ModelsInvolving y(cid:2)=k(y−b) 511 12.7 OptimizationinSeveralVariables 731
9.3 GraphicalandNumericalMethods 516 12.8 LagrangeMultipliers:OptimizingwithaConstraint 745
9.4 TheLogisticEquation 524
9.5 First-OrderLinearEquations 528 APPENDICES A1
PreparingfortheAPExam AP9-1 A. TheLanguageofMathematics A1
B. PropertiesofRealNumbers A8
C. InductionandtheBinomialTheorem A13
Chapter 10 INFINITE SERIES 537 D. AdditionalProofs A18
ANSWERSTOODD-NUMBEREDEXERCISES A27
10.1 Sequences 537
10.2 SumminganInfiniteSeries 548 ANSWERSTOTHEODD-NUMBEREDPREPARINGFORTHE
10.3 ConvergenceofSerieswithPositiveTerms 559 APEXAMQUESTIONS A104
10.4 AbsoluteandConditionalConvergence 569
REFERENCES A109
10.5 TheRatioandRootTests 575
10.6 PowerSeries 579
PHOTOCREDITS A112
10.7 TaylorSeries 591
PreparingfortheAPExam AP10-1 INDEX I1
ABOUT JON ROGAWSKI
Asasuccessfulteacherformorethan30years,JonRogawskihaslistenedtoandlearned
muchfromhisownstudents.Thesevaluablelessonshavemadeanimpactonhisthinking,
hiswriting,andhisshapingofacalculustext.
JonRogawskireceivedhisundergraduateandmaster’sdegreesinmathematicssimul-
taneouslyfromYaleUniversity,andheearnedhisPh.D.inmathematicsfromPrinceton
University,wherehestudiedunderRobertLanglands.BeforejoiningtheDepartmentof
Mathematics at UCLAin 1986, where he is currently a full professor, he held teaching
andvisitingpositionsattheInstituteforAdvancedStudy,theUniversityofBonn,andthe
UniversityofParisatJussieuandatOrsay.
Jon’sareasofinterestarenumbertheory,automorphicforms,andharmonicanalysis
onsemisimplegroups.Hehaspublishednumerousresearcharticlesinleadingmathemat-
icsjournals,includingtheresearchmonographAutomorphicRepresentationsofUnitary
Groups in Three Variables (Princeton University Press). He is the recipient of a Sloan
FellowshipandaneditorofthePacificJournalofMathematicsandtheTransactionsof
theAMS.
Jonandhiswife,Julie,aphysicianinfamilypractice,havefourchildren.Theyrun
a busy household and, whenever possible, enjoy family vacations in the mountains of
California. Jon is a passionate classical music lover and plays the violin and classical
guitar.
ABOUT RAY CANNON
Ray Cannon received his B.A. in mathematics from the College of the Holy Cross and
his Ph.D. from Tulane University, studying under GailYoung. He served on the facul-
tiesofVanderbiltUniversity,theUniversityofNorthCarolina-ChapelHill,andStetson
UniversitybeforelandingatBaylorUniversity,wherehehastaughtsince1980.Hehas
spenttimeattheUniversityofMichiganasanONRPostdoctoralFellowandattheUnited
StatesMilitaryAcademyasavisitingprofessor.
Ray has long been interested in the articulation between high school and college
mathematicsandhasservedtheAPCalculusprograminavarietyofways:asaReader
oftheexams,asaTableLeader,asExamLeader(bothABandBC),and,finall ,through
fouryearsasChiefReader.HehasalsoservedontheCollegeBoard’sTestDevelopment
CommitteeforAPCalculus.RayisafrequentconsultantfortheCollegeBoard,present-
ingatworkshopsandleadingweek-longsummerinstitutes.Additionally,Rayservedon
MathematicalAssociation ofAmerica (MAA) committees concerned with the issue of
properplacementofstudentsinprecalculusandcalculuscourses.Rayhaswonnumerous
awardsforhisteachingandservice,includinguniversity-wideteachingawardsfromthe
UniversityofNorthCarolinaandBaylorUniversity.HewasnamedaPiperProfessorin
thestateofTexasin1997andhastwicebeengivenawardsbytheSouthwesternregion
oftheCollegeBoardforoutstandingcontributionstotheAdvancedPlacementProgram.
Rayandhiswife,Jo,havethreegrowndaughtersandenjoytravelingtogether.
