January9,2009 11:08 m91-fm-et Sheetnumber4 Pagenumberiv cyanmagentayellowblack Achieve Positive Learning Outcomes W ileyPLUS combines robust course management tools with interactive teaching and learning resources all in one easy-to-use system. It has helped over half a million students and instructors achieve positive learning outcomes in their courses. WileyPLUS contains everything you and your students need— and nothing more, including: The entire textbook online—with dynamic links from homework to relevant sections. Students can use the online text and save up to half the cost of buying a new printed book. Automated assigning & grading of homework & quizzes. An interactive variety of ways to teach and learn the material. Instant feedback and help for students… available 24/7. “WileyPLUS helped me become more prepared. There were more resources available using WileyPLUS than just using a regular [printed] textbook, which helped out significantly. 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It’s all here: www.wileyplus.com — TechnIcAL SUPPOrT: A fully searchable knowledge base of FAQs and help documentation, available 24/7 Live chat with a trained member of our support staff during business hours A form to fill out and submit online to ask any question and get a quick response Instructor-only phone line during business hours: 1.877.586.0192 FAcULTy-Led TrAInIng ThrOUgh The WILey FAcULTy neTWOrk: register online: www.wherefacultyconnect.com Connect with your colleagues in a complimentary virtual seminar, with a personal mentor in your field, or at a live workshop to share best practices for teaching with technology. 1ST dAy OF cLASS…A nd beyOnd! resources you & your Students need to get Started & Use WileyPLUS from the first day forward. 2-Minute Tutorials on how to set up & maintain your WileyPLUS course User guides, links to technical support & training options WileyPLUS for Dummies: Instructors’ quick reference guide to using WileyPLUS Student tutorials & instruction on how to register, buy, and use WileyPLUS yOUr WileyPLUS AccOUnT MAnA ger: Your personal WileyPLUS connection for any assistance you need! SeT UP yOUr WileyPLUS cOUrSe In MInUT eS! Selected WileyPLUS courses with QuickStart contain pre-loaded assignments & presentations created by subject matter experts who are also experienced WileyPLUS users. Interested? See and try WileyPLUS in action! Details and Demo: www.wileyplus.com January9,2009 11:08 m91-fm-et Sheetnumber3 Pagenumberiii cyanmagentayellowblack 9 th EDITION CALCULUS EARLY TRANSCENDENTALS HOWARD ANTON DrexelUniversity IRL BIVENS DavidsonCollege STEPHEN DAVIS DavidsonCollege withcontributionsby Thomas Polaski WinthropUniversity JOHN WILEY & SONS, INC. January9,2009 11:08 m91-fm-et Sheetnumber4 Pagenumberiv cyanmagentayellowblack Publisher: LaurieRosatone AcquisitionsEditor: DavidDietz FreelanceDevelopmentalEditor: AnneScanlan-Rohrer MarketingManager: JaclynElkins AssociateEditor: MichaelShroff/WillArt EditorialAssistant: PamelaLashbrook FullServiceProductionManagement: CarolSawyer/ThePerfectProof SeniorProductionEditor: KenSantor SeniorDesigner: MadelynLesure AssociatePhotoEditor: SheenaGoldstein FreelanceIllustration: KarenHeyt CoverPhoto: ©EricSimonsen/GettyImages ThisbookwassetinLATEXbyTechsetters,Inc.,andprintedandboundbyR.R.Donnelley/JeffersonCity. The coverwasprintedbyR.R.Donnelley. Thisbookisprintedonacid-freepaper. Thepaperinthisbookwasmanufacturedbyamillwhoseforestmanagementprogramsincludesustainedyield harvestingofitstimberlands.Sustainedyieldharvestingprinciplesensurethatthenumbersoftreescuteachyear doesnotexceedtheamountofnewgrowth. Copyright©2009AntonTextbooks,Inc.Allrightsreserved. Nopartofthispublicationmaybereproduced,storedinaretrievalsystem,ortransmittedinanyformorby anymeans,electronic,mechanical,photocopying,recording,scanning,orotherwise,exceptaspermittedunder Sections107and108ofthe1976UnitedStatesCopyrightAct,withouteitherthepriorwrittenpermissionofthe Publisher,orauthorizationthroughpaymentoftheappropriateper-copyfeetotheCopyrightClearanceCenter, 222RosewoodDrive, Danvers, MA01923, (978)750-8400, fax(978)750-4470. RequeststothePublisher forpermissionshouldbeaddressedtothePermissionsDepartment,JohnWiley&Sons,Inc.,111RiverStreet, Hoboken,NJ07030,(201)748-6011,fax(201)748-6008,E-mail: [email protected] orforcustomerservice,call1(800)-CALL-WILEY(225-5945). ISBN978-0-470-18345-8 PrintedintheUnitedStatesofAmerica 10 9 8 7 6 5 4 3 2 January9,2009 11:08 m91-fm-et Sheetnumber5 Pagenumberv cyanmagentayellowblack AboutHOWARDANTON HowardAntonwrotetheoriginalversionofthistextandwastheauthorofthefirstsixeditions. He obtainedhisB.A.fromLehighUniversity,hisM.A.fromtheUniversityofIllinois,andhisPh.D. fromthePolytechnicUniversityofBrooklyn,allinmathematics. Intheearly1960sheworkedfor BurroughsCorporationandAvcoCorporationatCapeCanaveral,Florida,wherehewasinvolved withthemannedspaceprogram. In1968hejoinedtheMathematicsDepartmentatDrexel University,wherehetaughtfulltimeuntil1983. Sincethattimehehasbeenanadjunctprofessorat Drexelandhasdevotedthemajorityofhistimetotextbookwritingandactivitiesformathematical associations. Dr.AntonwaspresidentoftheEPADELSectionoftheMathematicalAssociationof America(MAA),servedontheBoardofGovernorsofthatorganization,andguidedthecreationof theStudentChaptersoftheMAA.Hehaspublishednumerousresearchpapersinfunctional analysis,approximationtheory,andtopology,aswellaspedagogicalpapers. Heisbestknownfor histextbooksinmathematics,whichareamongthemostwidelyusedintheworld. Thereare currentlymorethanonehundredversionsofhisbooks,includingtranslationsintoSpanish,Arabic, Portuguese,Italian,Indonesian,French,Japanese,Chinese,Hebrew,andGerman. 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 MAAProblemBookseriesandTheCollegeMathematicsJournalandisareviewerfor MathematicalReviews. Whenheisnotpursuingmathematics,ProfessorBivensenjoysjuggling, swimming,walking,andspendingtimewithhissonRobert. 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-treasurer. Heiscurrentlyafaculty consultantfortheEducationalTestingServiceAdvancedPlacementCalculusTest,aboardmember oftheNorthCarolinaAssociationofAdvancedPlacementMathematicsTeachers,andisactively involvedinnurturingmathematicallytalentedhighschoolstudentsthroughleadershipinthe CharlotteMathematicsClub. HewasformerlyNorthCarolinastatedirectorfortheMAA.For relaxation,heplaysbasketball,juggles,andtravels. ProfessorDavisandhiswifeElisabethhave threechildren,Laura,Anne,andJames,allformercalculusstudents. AboutTHOMASPOLASKI, ThomasW.PolaskireceivedhisB.S.fromFurmanUniversityandhisPh.D.inmathematicsfrom contributortotheninth DukeUniversity. HeiscurrentlyaprofessoratWinthropUniversity,wherehehastaught edition since1991. HewasnamedOutstandingJuniorProfessoratWinthropin1996. Hehaspublished articlesonmathematicspedagogyandstochasticprocessesandhasauthoredachapterina forthcominglinearalgebratextbook. ProfessorPolaskiisafrequentpresenteratmathematics meetings,givingtalksontopicsrangingfrommathematicalbiologytomathematicalmodelsfor baseball. HehasbeenanMAAVisitingLecturerandisareviewerforMathematicalReviews. ProfessorPolaskihasbeenareaderfortheAdvancedPlacementCalculusTestsformanyyears. In additiontocalculus,heenjoystravelandhiking. ProfessorPolaskiandhiswife,LeDayne,havea daughter,Kate,andliveinCharlotte,NorthCarolina. January9,2009 11:08 m91-fm-et 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 January9,2009 11:08 m91-fm-et Sheetnumber7 Pagenumbervii cyanmagentayellowblack PREFACE This ninth 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. However,thiseditionalsohasmanynewfeatures thatwehopewillattractnewusersandalsomotivatepastuserstotakeafreshlookatour work. Wehadtwomaingoalsforthisedition: • To make those adjustments to the order and content that would align the text more preciselywiththemostwidelyfollowedcalculusoutlines. • Toaddnewelementstothetextthatwouldprovideawiderrangeofteachingandlearning tools. Allofthechangeswerecarefullyreviewedbyanadvisorycommitteeofoutstandingteachers comprisedofbothusersandnonusersofthepreviousedition. Thechargeofthiscommittee wastoensurethatallchangesdidnotalterthoseaspectsofthetextthatattractedusersof the eighth edition and at the same time provide freshness to the new edition that would attractnewusers. Someofthemoresubstantivechangesaredescribedbelow. NEW FEATURES IN THIS EDITION New Elements in the Exercises Weaddednewtrue/falseexercises,newwriting exercises,andnewexercisetypesthatwererequestedbyreviewersoftheeighthedition. Making Connections We added this new element to the end of each chapter. A Making Connections exercise synthesizes concepts drawn across multiple sections of its chapterratherthanusingideasfromasinglesectionasisexpectedofaregularorreview exercise. ReorganizationofReviewMaterial Theprecalculusreviewmaterialthatwasin Chapter1oftheeightheditionformsChapter0oftheninthedition. Thebodyofmaterial inChapter1oftheeightheditionthatisnotgenerallyregardedasprecalculusreviewwas movedtoappropriatesectionsofthetextinthisedition. Thus,Chapter0focusesexclusively onthosepreliminarytopicsthatstudentsneedtostartthecalculuscourse. Parametric Equations Reorganized Intheeighthedition, parametricequations wereintroducedinthefirstchapterandpickedupagainlaterinthetext. Manyinstructors askedthatwereturntothetraditionalorganization,andwehavedoneso;thematerialon parametric equations is now first introduced and then discussed in detail in Section 10.1 (Parametric Curves). However, to support those instructors who want to continue the eighth edition path of giving an early exposure to parametric curves, we have provided Web materials (Web Appendix I) as well as self-contained exercise sets on the topic in Section6.4(LengthofaPlaneCurve)andSection6.5(AreaofaSurfaceofRevolution). vii January9,2009 11:08 m91-fm-et Sheetnumber8 Pagenumberviii cyanmagentayellowblack viii Preface Also,Section14.4(SurfaceArea;ParametricSurfaces)hasbeenreorganizedsosurfaces oftheformz=f(x,y)arediscussedbeforesurfacesdefinedparametrically. Differential Equations Reorganized We reordered and revised the chapter on differential equations so that instructors who cover only separable equations can do so without a forced diversion into general first-order equations and other unrelated topics. Thischaptercanbeskippedentirelybythosewhodonotcoverdifferentialequationsatall incalculus. New2DDiscussionofCentroidsandCenterofGravity Intheeighthedition andearlier,centroidsandcenterofgravitywerecoveredonlyinthreedimensions. Inthis edition we added a new section on that topic in Chapter 6 (Applications of the Definite Integral), socentroidsandcenterofgravitycannowbestudiedintwodimensions, asis commoninmanycalculuscourses. RelatedRatesandLocalLinearityReorganized Thesectionsonrelatedrates andlocallinearityweremovedtofollowthesectionsonimplicitdifferentiationandloga- rithmic,exponential,andinversetrigonometricfunctions,therebymakingarichervariety oftechniquesandfunctionsavailabletostudyrelatedratesandlocallinearity. RectilinearMotionReorganized Themoretechnicalaspectsofrectilinearmotion thatwerediscussedintheintroductorydiscussionofderivativesintheeightheditionhave been deferred so as not to distract from the primary task of developing the notion of the derivative. Thisalsoprovidesalessfragmenteddevelopmentofrectilinearmotion. OtherReorganization ThesectionGraphingFunctionsUsingCalculatorsandCom- puterAlgebraSystems,whichappearedinthetextbodyoftheeighthedition,isnowatext appendix(AppendixA),andthesectionsMathematicalModelsandSecond-OrderLinear HomogeneousDifferentialEquationsarenowpostedontheWebsitethatsupportsthetext. OTHER FEATURES Flexibility Thiseditionhasabuilt-inflexibilitythatisdesignedtoserveabroadspectrum ofcalculusphilosophies—fromtraditionalto“reform.” Technologycanbeemphasizedor not,andtheorderofmanytopicscanbepermutedfreelytoaccommodateeachinstructor’s specificneeds. Rigor Thechallengeofwritingagoodcalculusbookistostriketherightbalancebetween rigorandclarity. Ourgoalistopresentprecisemathematicstothefullestextentpossible inanintroductorytreatment. Whereclarityandrigorconflict,wechooseclarity;however, webelieveittobeimportantthatthestudentunderstandthedifferencebetweenacareful proof and an informal argument, so we have informed the reader when the arguments beingpresentedareinformalormotivational. Theoryinvolving(cid:2)-δ argumentsappearsin aseparatesectionsothatitcanbecoveredornot,aspreferredbytheinstructor. RuleofFour The“ruleoffour”referstopresentingconceptsfromtheverbal,algebraic, visual,andnumericalpointsofview. Inkeepingwithcurrentpedagogicalphilosophy,we usedthisapproachwheneverappropriate. Visualization Thiseditionmakesextensiveuseofmoderncomputergraphicstoclarify conceptsandtodevelopthestudent’sabilitytovisualizemathematicalobjects,particularly