October15,2011 13:47 flast Sheetnumber3 Pagenumberxx cyanmagentayellowblack October10,2011 15:10 ffirs Sheetnumber3 Pagenumberiii cyanmagentayellowblack 10 th EDITION CALCULUS DavidHenderson/GettyImages EARLY TRANSCENDENTALS HOWARD ANTON DrexelUniversity IRL BIVENS DavidsonCollege STEPHEN DAVIS DavidsonCollege JOHN WILEY & SONS, INC. 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ISBN978-0-470-64769-1 PrintedintheUnitedStatesofAmerica 10 9 8 7 6 5 4 3 2 1 October10,2011 15:10 ffirs Sheetnumber5 Pagenumberv cyanmagentayellowblack AboutHOWARDANTON HowardAntonobtainedhisB.A.fromLehighUniversity,hisM.A.fromtheUniversityofIllinois, andhisPh.D.fromthePolytechnicUniversityofBrooklyn,allinmathematics. Intheearly1960she workedforBurroughsCorporationandAvcoCorporationatCapeCanaveral,Florida,wherehewas involvedwiththemannedspaceprogram. In1968hejoinedtheMathematicsDepartmentatDrexel University,wherehetaughtfulltimeuntil1983. SincethattimehehasbeenanEmeritusProfessor atDrexelandhasdevotedthemajorityofhistimetotextbookwritingandactivitiesformathematical associations. Dr.AntonwaspresidentoftheEPADELsectionoftheMathematicalAssociationof America(MAA),servedontheBoardofGovernorsofthatorganization,andguidedthecreationof thestudentchaptersoftheMAA.Hehaspublishednumerousresearchpapersinfunctionalanalysis, approximationtheory,andtopology,aswellaspedagogicalpapers. Heisbestknownforhis textbooksinmathematics,whichareamongthemostwidelyusedintheworld. Therearecurrently morethanonehundredversionsofhisbooks,includingtranslationsintoSpanish,Arabic, Portuguese,Italian,Indonesian,French,Japanese,Chinese,Hebrew,andGerman. Histextbookin linearalgebrahaswonboththeTextbookExcellenceAwardandtheMcGuffeyAwardfromthe TextbookAuthor’sAssociation. Forrelaxation,Dr.Antonenjoystravelingandphotography. AboutIRLBIVENS IrlC.Bivens,recipientoftheGeorgePolyaAwardandtheMertenM.HassePrizeforExpository WritinginMathematics,receivedhisA.B.fromPfeifferCollegeandhisPh.D.fromtheUniversity ofNorthCarolinaatChapelHill,bothinmathematics. Since1982,hehastaughtatDavidson College,wherehecurrentlyholdsthepositionofprofessorofmathematics.Atypicalacademicyear seeshimteachingcoursesincalculus,topology,andgeometry. Dr.Bivensalsoenjoysmathematical history,andhisannualHistoryofMathematicsseminarisaperennialfavoritewithDavidson mathematicsmajors. Hehaspublishednumerousarticlesonundergraduatemathematics,aswellas researchpapersinhisspecialty,differentialgeometry. Hehasservedontheeditorialboardsofthe MAAProblemBookseries,theMAADolcianiMathematicalExpositionsseriesandTheCollege MathematicsJournal.Whenheisnotpursuingmathematics,ProfessorBivensenjoysreading, juggling,swimming,andwalking. AboutSTEPHENDAVIS StephenL.DavisreceivedhisB.A.fromLindenwoodCollegeandhisPh.D.fromRutgers Universityinmathematics. HavingpreviouslytaughtatRutgersUniversityandOhioState University,Dr. DaviscametoDavidsonCollegein1981,whereheiscurrentlyaprofessorof mathematics. Heregularlyteachescalculus,linearalgebra,abstractalgebra,andcomputerscience. Asabbaticalin1995–1996tookhimtoSwarthmoreCollegeasavisitingassociateprofessor. ProfessorDavishaspublishednumerousarticlesoncalculusreformandtesting,aswellasresearch papersonfinitegrouptheory,hisspecialty. ProfessorDavishasheldseveralofficesinthe SoutheasternsectionoftheMAA,includingchairandsecretary-treasurerandhasservedonthe MAABoardofGovernors. HeiscurrentlyafacultyconsultantfortheEducationalTestingService forthegradingoftheAdvancedPlacementCalculusExam,webmasterfortheNorthCarolina AssociationofAdvancedPlacementMathematicsTeachers,andisactivelyinvolvedinnurturing mathematicallytalentedhighschoolstudentsthroughleadershipintheCharlotteMathematicsClub. Forrelaxation,heplaysbasketball,juggles,andtravels. ProfessorDavisandhiswifeElisabethhave threechildren,Laura,Anne,andJames,allformercalculusstudents. October10,2011 15:10 ffirs Sheetnumber6 Pagenumbervi cyanmagentayellowblack To my wife Pat and my children: Brian, David, and Lauren In Memory of my mother Shirley my father Benjamin my thesis advisor and inspiration, George Bachman my benefactor in my time of need, Stephen Girard (1750–1831) —HA To my son Robert —IB To my wife Elisabeth my children: Laura, Anne, and James —SD September30,2011 17:46 fpref Sheetnumber1 Pagenumbervii cyanmagentayellowblack PREFACE This tenth edition of Calculus maintains those aspects of previous editions that have led totheseries’success—wecontinuetostriveforstudentcomprehensionwithoutsacrificing mathematical accuracy, and the exercise sets are carefully constructed to avoid unhappy surprisesthatcanderailacalculusclass. Allofthechangestothetentheditionwerecarefullyreviewedbyoutstandingteachers comprisedofbothusersandnonusersofthepreviousedition. Thechargeofthiscommittee wastoensurethatallchangesdidnotalterthoseaspectsofthetextthatattractedusersof thenintheditionandatthesametimeprovidefreshnesstotheneweditionthatwouldattract newusers. NEW TO THIS EDITION • ExercisesetshavebeenmodifiedtocorrespondmorecloselytoquestionsinWileyPLUS. Inaddition,moreWileyPLUSquestionsnowcorrespondtospecificexercisesinthetext. • Newappliedexerciseshavebeenaddedtothebookandexistingappliedexerciseshave beenupdated. • Whereappropriate,additionalskill/practiceexerciseswereadded. OTHER FEATURES Flexibility Thiseditionhasabuilt-inflexibilitythatisdesignedtoserveabroadspectrum ofcalculusphilosophies—fromtraditionalto“reform.” Technologycanbeemphasizedor not,andtheorderofmanytopicscanbepermutedfreelytoaccommodateeachinstructor’s specificneeds. Rigor Thechallengeofwritingagoodcalculusbookistostriketherightbalancebetween rigorandclarity. Ourgoalistopresentprecisemathematicstothefullestextentpossible inanintroductorytreatment. Whereclarityandrigorconflict,wechooseclarity;however, webelieveittobeimportantthatthestudentunderstandthedifferencebetweenacareful proofandaninformalargument,sowehaveinformedthereaderwhentheargumentsbeing presentedareinformalormotivational. Theoryinvolving(cid:2)-δargumentsappearsinseparate sectionssothattheycanbecoveredornot,aspreferredbytheinstructor. RuleofFour The“ruleoffour”referstopresentingconceptsfromtheverbal,algebraic, visual,andnumericalpointsofview. Inkeepingwithcurrentpedagogicalphilosophy,we usedthisapproachwheneverappropriate. Visualization Thiseditionmakesextensiveuseofmoderncomputergraphicstoclarify conceptsandtodevelopthestudent’sabilitytovisualizemathematicalobjects,particularly thosein3-space. Forthosestudentswhoareworkingwithgraphingtechnology,thereare vii September30,2011 17:46 fpref Sheetnumber2 Pagenumberviii cyanmagentayellowblack viii Preface many exercises that are designed to develop the student’s ability to generate and analyze mathematicalcurvesandsurfaces. Quick Check Exercises Eachexercisesetbeginswithapproximatelyfiveexercises (answers included) that are designed to provide students with an immediate assessment of whether they have mastered key ideas from the section. They require a minimum of computationandareansweredbyfillingintheblanks. FocusonConceptsExercises Eachexercisesetcontainsaclearlyidentifiedgroup ofproblemsthatfocusonthemainideasofthesection. TechnologyExercises Mostsectionsincludeexercisesthataredesignedtobesolved using either a graphing calculator or a computer algebra system such as Mathematica, Maple,ortheopensourceprogramSage. Theseexercisesaremarkedwithaniconforeasy identification. Applicability of Calculus One of the primary goals of this text is to link calculus to the real world and the student’s own experience. This theme is carried through in the examplesandexercises. Career Preparation Thistextiswrittenatamathematicallevelthatwillpreparestu- dentsforawidevarietyofcareersthatrequireasoundmathematicsbackground,including engineering,thevarioussciences,andbusiness. Trigonometry Review Deficiencies in trigonometry plague many students, so we haveincludedasubstantialtrigonometryreviewinAppendixB. Appendix on Polynomial Equations Becausemanycalculusstudentsareweak insolvingpolynomialequations,wehaveincludedanappendix(AppendixC)thatreviews theFactorTheorem,theRemainderTheorem,andproceduresforfindingrationalroots. Principles of Integral Evaluation The traditional Techniques of Integration is entitled “Principles of Integral Evaluation” to reflect its more modern approach to the material. Thechapteremphasizesgeneralmethodsandtheroleoftechnologyratherthan specifictricksforevaluatingcomplicatedorobscureintegrals. Historical Notes The biographies and historical notes have been a hallmark of this textfromitsfirsteditionandhavebeenmaintained. Allofthebiographicalmaterialshave beendistilledfromstandardsourceswiththegoalofcapturingandbringingtolifeforthe studentthepersonalitiesofhistory’sgreatestmathematicians. Margin Notes and Warnings Theseappearinthemarginsthroughoutthetextto clarifyorexpandonthetextexpositionortoalertthereadertosomepitfall.