CALCULUS SET FREE Calculus Set Free: Infinitesimals to the Rescue Charles Bryan Dawson UniversityProfessorofMathematics,UnionUniversity,USA 3 3 GreatClarendonStreet,Oxford,OX26DP, UnitedKingdom OxfordUniversityPressisadepartmentoftheUniversityofOxford. 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Contents PrefacefortheStudent ix PrefacefortheInstructor xi Acknowledgments xv Review 1 0.1 AlgebraReview,PartI 3 0.2 AlgebraReview,PartII 15 0.3 TrigonometryReview 35 0.4 FunctionsReview,PartI 59 0.5 FunctionsReview,PartII 83 0.6 AvoidingCommonErrors 99 I Hyperreals, Limits, and Continuity 133 1.0 Motivation 135 1.1 Infinitesimals 139 1.2 Approximation 153 1.3 HyperrealsandFunctions 167 1.4 Limits,PartI 177 1.5 Limits,PartII 191 1.6 Continuity,PartI 209 1.7 Continuity,PartII 223 1.8 Slope,Velocity,andRatesofChange 243 II Derivatives 259 2.1 TheDerivative 261 2.2 DerivativeRules 275 2.3 TangentLinesRevisited 291 2.4 DerivativesofTrigonometricFunctions 307 2.5 ChainRule 319 2.6 ImplicitDifferentiation 331 2.7 RatesofChange:MotionandMarginals 343 2.8 RelatedRates:PythagoreanRelationships 353 2.9 RelatedRates:Non-PythagoreanRelationships 367 III Applications of the Derivative 383 3.1 AbsoluteExtrema 385 3.2 MeanValueTheorem 401 vi Contents 3.3 LocalExtrema 413 3.4 Concavity 429 3.5 CurveSketching:Polynomials 443 3.6 LimitsatInfinity 463 3.7 CurveSketching:GeneralFunctions 479 3.8 Optimization 495 3.9 Newton’sMethod 515 IV Integration 527 4.1 Antiderivatives 529 4.2 FiniteSums 543 4.3 AreasandSums 565 4.4 DefiniteIntegral 581 4.5 FundamentalTheoremofCalculus 597 4.6 SubstitutionforIndefiniteIntegrals 609 4.7 SubstitutionforDefiniteIntegrals 619 4.8 NumericalIntegration,PartI 627 4.9 NumericalIntegration,PartII 643 4.10 InitialValueProblemsandNetChange 659 V Transcendental Functions 671 5.1 Logarithms,PartI 673 5.2 Logarithms,PartII 687 5.3 InverseFunctions 699 5.4 Exponentials 717 5.5 GeneralExponentials 731 5.6 GeneralLogarithms 743 5.7 ExponentialGrowthandDecay 761 5.8 InverseTrigonometricFunctions 777 5.9 HyperbolicandInverseHyperbolicFunctions 795 5.10 ComparingRatesofGrowth 811 5.11 LimitswithTranscendentalFunctions:L’Hospital’sRule,PartI 827 5.12 L’Hospital’sRule,PartII:MoreIndeterminateForms 839 5.13 FunctionswithoutEnd 853 VI Applications of Integration 867 6.1 AreabetweenCurves 869 6.2 Volumes,PartI 887 6.3 Volumes,PartII 907 6.4 ShellMethodforVolumes 919 6.5 Work,PartI 935 Contents vii 6.6 Work,PartII 949 6.7 AverageValueofaFunction 957 VII Techniques of Integration 967 7.1 AlgebraforIntegration 969 7.2 IntegrationbyParts 981 7.3 TrigonometricIntegrals 995 7.4 TrigonometricSubstitution 1009 7.5 PartialFractions,PartI 1021 7.6 PartialFractions,PartII 1037 7.7 OtherTechniquesofIntegration 1051 7.8 StrategyforIntegration 1065 7.9 TablesofIntegralsandUseofTechnology 1077 7.10 TypeIImproperIntegrals 1091 7.11 TypeIIImproperIntegrals 1107 VIII Alternate Representations: Parametric and Polar Curves 1119 8.1 ParametricEquations 1121 8.2 TangentstoParametricCurves 1137 8.3 PolarCoordinates 1149 8.4 TangentstoPolarCurves 1167 8.5 ConicSections 1179 8.6 ConicSectionsinPolarCoordinates 1203 IX Additional Applications of Integration 1217 9.1 ArcLength 1219 9.2 AreasandLengthsinPolarCoordinates 1233 9.3 SurfaceArea 1249 9.4 LengthsandSurfaceAreaswithParametricCurves 1263 9.5 HydrostaticPressureandForce 1271 9.6 CentersofMass 1285 9.7 ApplicationstoEconomics 1297 9.8 LogisticGrowth 1313 X Sequences and Series 1327 10.1 Sequences 1329 10.2 SequenceLimits 1343 10.3 InfiniteSeries 1357 10.4 IntegralTest 1377 10.5 ComparisonTests 1391 10.6 AlternatingSeries 1411 10.7 RatioandRootTests 1425 viii Contents 10.8 StrategyforTestingSeries 1443 10.9 PowerSeries 1455 10.10TaylorandMaclaurinSeries 1471 Index 1495 AnswerstoOdd-numberedExercises 1509 Preface for the Student Formany,thestudyofcalculusisseenasariteofpassage—toconquercalculusistopassthroughthegate- way to the sciences, engineering, mathematics, business, economics, technology, and many other fields. Forsome,thestudyofcalculusisindicativeofachievement,ahallmarkofaqualityeducation.Afewcan’t wait to study calculus, their curiosity overflowing with enthusiasm. Yet others see calculus as an annoy- ance,somethingtotolerateinpursuitofmoreimportantormoreinterestingsubjects.Thisbookisforall ofyou. Whateveryourreasonforstudyingcalculus,itismyhopethatthistextfacilitatesnotjustthemastering oftechnicalskillsandtheunderstandingofmathematicalconcepts,butalsotraininginthinkinginapatient, systematic,disciplined,andlogicalmanner.Althoughtechnicalskillscanbeusefulforsomestudentsin theircareers,andtheunderstandingofmathematicalconceptscanbeofusetoevenmore,thehabitsof mindcreatedbycarefulthinkingcanbeofusetoeveryone,atanytime,inanyplace. Preparation for success Ifyouhavelearnedtodriveacar,thenyoumayrecallhowdrivingtookmuchconsciousthoughtatfirst; but later, with practice, driving became much more of a background task. The same is true of addition andmultiplicationfacts;thetask2+4takesverylittlementalenergy.Thisisthehallmarkofdeeplearning: whenataskhasbeenlearnedthoroughly,thenitcanbeperformedaccuratelywithlittleeffort. Success in calculus is much easier if basic algebraic and trigonometric skills have been learned this deeply.Ifthequadraticformulaandlawsofexponentscanbeappliedaccuratelyupondemand,thenthe mindisfree toconcentrateontheconcepts athand. Ifnot, theninstead ofjuggling threenewconcepts consciously,adozenormoredistractingitemsthatmustberelearnedcompetewiththenewconceptsfor mentalenergy,hamperingone’slearningofthenewmaterial. Mentalskill-buildingismuchthesameasphysicalskill-building;ittakestimeandconsistentefforton thepartofthelearner.Liftingweightsseveraltimesperweekforamonthisamuchmoreeffectivestrategy thanwaitinguntilthenightbeforetheskillstesttotrytocramtheentiremonth’srepsintooneevening’s workout.Body-buildingsimplydoesnotworkthatway,andneitherdoeslearningmathematics. Onefinalbitofadvice:learnfromfailure.Everyonemakeserrors,eventextbookauthorswithdecades ofexperienceinthesubject.Nooneisperfect.Butwhenyoumakeanerror,makesureyouunderstand whyitwasanerror,whyadifferentapproachmustbeused,andhowtoavoidmakingthesameerrorin thefuture.Thereisoftensomethingtobelearnedfromyourerrors.Mistakesarenottobefeared,butto beusedtoyouradvantage! Features of this textbook What makes this textbook different? The most obvious answer is that it uses infinitesimals, which are infinitelysmallnumbersthatyoumightnothaveencounteredinyourpreviouscourses.