(cid:0)(cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:4)(cid:7) (cid:8)(cid:9) (cid:10)(cid:11)(cid:8)(cid:12) (cid:8)(cid:12)(cid:13)(cid:14)(cid:14) (cid:15)(cid:4)(cid:16) (cid:17)(cid:18)(cid:2)(cid:2)(cid:19) (cid:20)(cid:5)(cid:21)(cid:3)(cid:2)(cid:4) (cid:8) (cid:22)(cid:6)(cid:23)(cid:2) (cid:20)(cid:5)(cid:21)(cid:3)(cid:2)(cid:4) (cid:24) (cid:7)(cid:6)(cid:20) (cid:21)(cid:6)(cid:23)(cid:2)(cid:20)(cid:19)(cid:6) (cid:7)(cid:2)(cid:26)(cid:26)(cid:27)(cid:28) (cid:3)(cid:26)(cid:6) (cid:29) FOR THE STUDENT Calculusprovidesawayofviewingandanalyzingthephysi- Writingexercisesareintendedtotestyourabilitytoex- • calworld.Aswithallmathematicscourses,calculusinvolves plain mathematical ideas in words rather than relying equationsandformulas.However,ifyousuccessfullylearnto solelyonnumbersandsymbols. Allexercisesrequiring use all the formulasand solveall ofthe problemsin the text writingshouldbeansweredincomplete,correctlypunc- butdonotmastertheunderlyingideas,youwillhavemissed tuated logical sentences—not with fragmented phrases themostimportantpartofcalculus. Ifyoumastertheseideas, andformulas. you will have a widely applicable tool that goes far beyond Each chapter ends with two additional sets of exercises: textbookexercises. ChapterReviewExercises,which,asthenamesuggests,is Beforestartingyourstudies,youmayfindithelpfultoleaf a select set of exercises that provide a review of the main throughthistexttogetageneralfeelingforitsdifferentparts: concepts and techniques in the chapter, and Making Con- The opening page of each chapter gives you an overview nections, in which exercises require you to draw on and ofwhatthatchapterisabout,andtheopeningpageofeach combinevariousideasdevelopedthroughoutthechapter. sectionwithinachaptergivesyouanoverviewofwhatthat Your instructor may choose to incorporate technology in section is about. To help you locate specific information, your calculus course. Exercises whose solution involves sections are subdividedinto topics that are marked with a theuseofsomekindoftechnologyaretaggedwithiconsto boxlikethis . alertyouandyourinstructor. Thoseexercisestaggedwith Each section ends with a set of exercises. The answers theicon requiregraphingtechnology—eitheragraphing to most odd-numberedexercisesappear in the back of the calculatororacomputerprogramthatcangraphequations. book. Ifyoufindthatyouranswertoanexercisedoesnot Those exercises tagged with the icon C require a com- matchthatinthebackofthebook,donotassumeimmedi- puter algebrasystem (CAS) such as Mathematica, Maple, ately that yoursis incorrect—theremay be morethan one oravailableonsomegraphingcalculators. way to express the answer. For example, if your answer Inside the front and back covers of the text you will find is √2/2 and the text answer is 1/√2 , then both are cor- endpapersthatcontainusefulformulas. rect since your answer can be obtained by “rationalizing” Five summary appendices are included at the end of the thetextanswer. Ingeneral,ifyouranswerdoesnotmatch text, surveying key definitions and facts from trigonome- thatinthetext,thenyourbestfirststepistolookforanal- try,generalfunctions,andinversefunctions.Moredetailed gebraicmanipulationoratrigonometricidentitythatmight versions of these appendices, as well as eleven additional help you determine if the two answers are equivalent. If appendices that supply further support and/or extend the theanswerisintheformofadecimalapproximation,then text,areavailableinwebversions. your answer might differ from that in the text because of a difference in the number of decimal places used in the The ideas in this text were created by real people with computations. interestingpersonalitiesandbackgrounds.Picturesandbio- The section exercises include regular exercises and four graphicalsketchesofmanyofthesepeopleappearthrough- special categories: Quick Check, Focus on Concepts, outthebook. True/False,andWriting. Notesinthemarginareintendedtoclarifyorcommenton The Quick Check exercises are intended to give you importantpointsinthetext. • quickfeedbackonwhetheryouunderstandthekeyideas A Wordof Encouragement inthesection;theyinvolverelativelylittlecomputation, andhaveanswersprovidedattheendoftheexerciseset. As you work your way through this text you will find some The Focus on Concepts exercises, as their name sug- ideas that you understand immediately, some that you don’t • gests,keyinonthemainideasinthesection. understand until you have read them several times, and oth- ers that you do not seem to understand, even after several True/False exercises focus on key ideas in a different • readings. Do not become discouraged—some ideas are in- way. You must decide whether the statement is true in trinsically difficult and take time to “percolate.” You may allpossiblecircumstances,inwhichcaseyouwouldde- wellfindthatahardideabecomesclearlaterwhenyouleast clareit to be“true,” or whetherthereare somecircum- expectit. stancesinwhichitisnottrue,inwhichcaseyouwould declare it to be “false.” In each such exercise you are asked to “Explain your answer.” You might do this by WebSitesfor this Text notingatheoreminthetextthatshowsthestatementto www.antontextbooks.com be true or byfindinga particularexamplein which the statementisnottrue. www.wiley.com/go/global/anton (cid:0)(cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:4)(cid:7) (cid:8)(cid:9) (cid:10)(cid:11)(cid:8)(cid:12) (cid:8)(cid:12)(cid:13)(cid:14)(cid:14) (cid:15)(cid:4)(cid:16) (cid:17)(cid:18)(cid:2)(cid:2)(cid:19) (cid:20)(cid:5)(cid:21)(cid:3)(cid:2)(cid:4) (cid:10) (cid:22)(cid:6)(cid:23)(cid:2) (cid:20)(cid:5)(cid:21)(cid:3)(cid:2)(cid:4) (cid:24)(cid:24) (cid:7)(cid:6)(cid:20) (cid:21)(cid:6)(cid:23)(cid:2)(cid:20)(cid:19)(cid:6) (cid:7)(cid:2)(cid:26)(cid:26)(cid:27)(cid:28) (cid:3)(cid:26)(cid:6) (cid:29) (cid:0)(cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:4)(cid:7) (cid:8)(cid:9) (cid:10)(cid:11)(cid:8)(cid:12) (cid:8)(cid:12)(cid:13)(cid:14)(cid:14) (cid:15)(cid:4)(cid:16) (cid:17)(cid:18)(cid:2)(cid:2)(cid:19) (cid:20)(cid:5)(cid:21)(cid:3)(cid:2)(cid:4) (cid:30) (cid:22)(cid:6)(cid:23)(cid:2) (cid:20)(cid:5)(cid:21)(cid:3)(cid:2)(cid:4) (cid:24)(cid:24)(cid:24) (cid:7)(cid:6)(cid:20) (cid:21)(cid:6)(cid:23)(cid:2)(cid:20)(cid:19)(cid:6) (cid:7)(cid:2)(cid:26)(cid:26)(cid:27)(cid:28) (cid:3)(cid:26)(cid:6) (cid:29) iii (cid:0)(cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:4)(cid:7) (cid:8)(cid:9) (cid:10)(cid:11)(cid:8)(cid:12) (cid:8)(cid:12)(cid:13)(cid:14)(cid:14) (cid:15)(cid:4)(cid:16) (cid:17)(cid:18)(cid:2)(cid:2)(cid:19) (cid:20)(cid:5)(cid:21)(cid:3)(cid:2)(cid:4) (cid:31) (cid:22)(cid:6)(cid:23)(cid:2) (cid:20)(cid:5)(cid:21)(cid:3)(cid:2)(cid:4) (cid:24) (cid:7)(cid:6)(cid:20) (cid:21)(cid:6)(cid:23)(cid:2)(cid:20)(cid:19)(cid:6) (cid:7)(cid:2)(cid:26)(cid:26)(cid:27)(cid:28) (cid:3)(cid:26)(cid:6) (cid:29) (cid:0)(cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:4)(cid:7) (cid:8)(cid:9) (cid:10)(cid:11)(cid:8)(cid:12) (cid:8)(cid:12)(cid:13)(cid:14)(cid:14) (cid:15)(cid:4)(cid:16) (cid:17)(cid:18)(cid:2)(cid:2)(cid:19) (cid:20)(cid:5)(cid:21)(cid:3)(cid:2)(cid:4) (cid:14) (cid:22)(cid:6)(cid:23)(cid:2) (cid:20)(cid:5)(cid:21)(cid:3)(cid:2)(cid:4) (cid:7)(cid:6)(cid:20) (cid:21)(cid:6)(cid:23)(cid:2)(cid:20)(cid:19)(cid:6) (cid:7)(cid:2)(cid:26)(cid:26)(cid:27)(cid:28) (cid:3)(cid:26)(cid:6) (cid:29) 11 th EDITION CALCULUS Carlos Fernandez/Getty Images EARLY TRANSCENDENTALS HOWARD ANTON DrexelUniversity IRL BIVENS DavidsonCollege STEPHEN DAVIS DavidsonCollege (cid:0)(cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:4)(cid:7) (cid:8)(cid:9) (cid:10)(cid:11)(cid:8)(cid:12) (cid:8)(cid:12)(cid:13)(cid:14)(cid:14) (cid:15)(cid:4)(cid:16) (cid:17)(cid:18)(cid:2)(cid:2)(cid:19) (cid:20)(cid:5)(cid:21)(cid:3)(cid:2)(cid:4) (cid:12) (cid:22)(cid:6)(cid:23)(cid:2) (cid:20)(cid:5)(cid:21)(cid:3)(cid:2)(cid:4) (cid:24) (cid:7)(cid:6)(cid:20) (cid:21)(cid:6)(cid:23)(cid:2)(cid:20)(cid:19)(cid:6) (cid:7)(cid:2)(cid:26)(cid:26)(cid:27)(cid:28) (cid:3)(cid:26)(cid:6) (cid:29) VicePresident&Director: LaurieRosatone AcquisitionsEditor: ShannonCorliss SponsoringEditor: JenniferBrady FreelanceProjectEditor: AnneScanlan-Rohrer MarketingManager: JohnLaVaccaIII SeniorProductDesigner: DavidDietz ProductDesignManager: ThomasKulesa MarketSolutionsAssistant: RyannDannelly ProductionServices: CarolSawyer/ThePerfectProof SeniorContentManager: KarolineLuciano/ValerieZaborski SeniorProductionEditor: LauraAbrams SeniorDesigner: MaureenEide PhotoEditor: JamesRussiello CoverPhoto: CarlosFernandez/GettyImages SunriseIcon: AlexSava/iStockphoto ThisbookwassetinLATEXbyMPSLimited, andprintedandboundbyCourierKendallville. 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Copyright(cid:2)c 2015AntonTextbooks,Inc.Allrightsreserved. Nopartofthispublication maybereproduced, storedinaretrieval system, ortransmitted inanyformorby anymeans,electronic,mechanical,photocopying,recording,scanning,orotherwise,exceptaspermittedunder Sections107and108ofthe1976UnitedStatesCopyrightAct,withouteitherthepriorwrittenpermissionofthe Publisher,orauthorizationthroughpaymentoftheappropriateper-copyfeetotheCopyrightClearanceCenter, 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 750-4470. Requests to the Publisher 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). ISBN978-1-118-88382-2 PrintedintheUnitedStatesofAmerica 10 9 8 7 6 5 4 3 2 1 (cid:0)(cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:4)(cid:7) (cid:8)(cid:9) (cid:10)(cid:11)(cid:8)(cid:12) (cid:8)(cid:12)(cid:13)(cid:14)(cid:14) (cid:15)(cid:4)(cid:16) (cid:17)(cid:18)(cid:2)(cid:2)(cid:19) (cid:20)(cid:5)(cid:21)(cid:3)(cid:2)(cid:4) ! (cid:22)(cid:6)(cid:23)(cid:2) (cid:20)(cid:5)(cid:21)(cid:3)(cid:2)(cid:4) (cid:24)(cid:24) (cid:7)(cid:6)(cid:20) (cid:21)(cid:6)(cid:23)(cid:2)(cid:20)(cid:19)(cid:6) (cid:7)(cid:2)(cid:26)(cid:26)(cid:27)(cid:28) (cid:3)(cid:26)(cid:6) (cid:29) AboutHOWARDANTON HowardAntonobtainedhisB.A.fromLehighUniversity,hisM.A.fromtheUniversityofIllinois, andhisPh.D.fromthePolytechnicUniversityofBrooklyn,allinmathematics.Intheearly1960s heworkedforBurroughsCorporationandAvcoCorporationatCapeCanaveral,Florida,wherehe wasinvolvedwiththemannedspaceprogram.In1968hejoinedtheMathematicsDepartmentat DrexelUniversity,wherehetaughtfulltimeuntil1983.SincethattimehehasbeenanEmeritus ProfessoratDrexelandhasdevotedthemajorityofhistimetotextbookwritingandactivitiesfor mathematicalassociations.Dr.AntonwaspresidentoftheEPADELsectionoftheMathematical AssociationofAmerica(MAA),servedontheBoardofGovernorsofthatorganization,andguided thecreationofthestudentchaptersoftheMAA.Hehaspublishednumerousresearchpapersin functionalanalysis,approximationtheory,andtopology,aswellaspedagogicalpapers.Heisbest knownforhistextbooksinmathematics,whichareamongthemostwidelyusedintheworld.There arecurrentlymorethanonehundredversionsofhisbooks,includingtranslationsintoSpanish, Arabic,Portuguese,Italian,Indonesian,French,Japanese,Chinese,Hebrew,andGerman.His textbookinlinearalgebrahaswonboththeTextbookExcellenceAwardandtheMcGuffeyAward fromtheTextbookAuthor’sAssociation.Forrelaxation,Dr.Antonenjoystravelingand photography. AboutIRLBIVENS IrlC.Bivens,arecipientoftheGeorgePolyaAwardandtheMertenM.HassePrizeforExpository WritinginMathematics,receivedhisA.B.fromPfeifferCollegeandhisPh.D.fromtheUniversity ofNorthCarolinaatChapelHill,bothinmathematics. Since1982hehastaughtatDavidson College,wherehecurrentlyholdsthepositionofprofessorofmathematics. Atypicalacademic yearseeshimteachingcoursesincalculus,complexanalysis,andthehistoryofmathematics. He haspublishednumerousarticlesonundergraduatemathematics,aswellasresearchpapersin differentialgeometry. PastservicetotheMAAincludesmembershipontheeditorialboardsofThe CollegeMathematicsJournal,theProblemBookseries,DolcianiMathematicalExpositions,along withserviceonawardscommitteesforexpositorywriting.Forrecreation,heenjoysswimming, juggling,andtheexplorationofmathematicallegerdemain. AboutSTEPHENDAVIS StephenL.DavisreceivedhisB.A.fromLindenwoodCollegeandhisPh.D.fromRutgers Universityinmathematics.HavingpreviouslytaughtatRutgersUniversityandOhioState University,Dr.DavisjoinedthefacultyatDavidsonCollegein1981,whereheiscurrentlya professorofmathematics.Heregularlyteachescalculus,linearalgebra,abstractalgebra,and computerscience.ProfessorDavishaspublishednumerousarticlesoncalculusreformandtesting, aswellasresearchpapersonfinitegrouptheory.ProfessorDavishasheldseveralofficesinthe SoutheasternsectionoftheMAA,includingchairandsecretary-treasurer,andhasrepresentedthe sectionontheMAABoardofGovernors.HehasbeeninvolvedinthegradingoftheAdvanced PlacementCalculusExamsince1988andbeganafour-yeartermasChiefReaderin2015.Heis alsowebmasterfortheNorthCarolinaAssociationofAdvancedPlacementMathematicsTeachers andisactivelyinvolvedinnurturingmathematicallytalentedhighschoolstudentsthrough leadershipintheCharlotteMathematicsClub.Forrelaxation,heplaysbasketball,travels,and occasionallyjuggles.ProfessorDavisandhiswifeElisabethhavethreechildren,Laura,Anne,and James,allformercalculusstudents. (cid:0)(cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:4)(cid:7) (cid:8)(cid:9) (cid:10)(cid:11)(cid:8)(cid:12) (cid:8)(cid:12)(cid:13)(cid:14)(cid:14) (cid:15)(cid:4)(cid:16) (cid:17)(cid:18)(cid:2)(cid:2)(cid:19) (cid:20)(cid:5)(cid:21)(cid:3)(cid:2)(cid:4) " (cid:22)(cid:6)(cid:23)(cid:2) (cid:20)(cid:5)(cid:21)(cid:3)(cid:2)(cid:4) (cid:24)(cid:24)(cid:24) (cid:7)(cid:6)(cid:20) (cid:21)(cid:6)(cid:23)(cid:2)(cid:20)(cid:19)(cid:6) (cid:7)(cid:2)(cid:26)(cid:26)(cid:27)(cid:28) (cid:3)(cid:26)(cid:6) (cid:29) To my wife Patand my children: Brian,David,and Lauren In Memoryof my motherShirley my fatherBenjamin my thesisadvisorandinspiration,GeorgeBachman my benefactorin my time of need,StephenGirard (1750–1831) —HA To my sonRobert —IB To my wife Elisabeth my children: Laura,Anne,and James —SD (cid:0)(cid:2)(cid:3) (cid:5) (cid:6)(cid:7)(cid:8) (cid:7)(cid:9)(cid:6)(cid:10) (cid:6)(cid:11)(cid:12)(cid:11)(cid:7) (cid:13)(cid:14)(cid:3)(cid:15)(cid:13) (cid:16)(cid:5)(cid:15)(cid:15)(cid:17) (cid:18)(cid:19)(cid:20)(cid:21)(cid:15)(cid:3) (cid:6) (cid:22)(cid:2)(cid:23)(cid:15) (cid:18)(cid:19)(cid:20)(cid:21)(cid:15)(cid:3) (cid:24)(cid:25) (cid:26)(cid:2)(cid:18) (cid:20)(cid:2)(cid:23)(cid:15)(cid:18)(cid:17)(cid:2) (cid:26)(cid:15)(cid:27)(cid:27)(cid:28)(cid:29) (cid:21)(cid:27)(cid:2) (cid:30) PREFACE ThiseleventheditionofCalculusmaintainsthoseaspectsofpreviouseditionsthathaveled totheseries’success—wecontinuetostriveforstudentcomprehensionwithoutsacrificing mathematicalaccuracy, and the exercise sets are carefully constructed to avoid unhappy surprisesthatcanderailacalculusclass. Allofthechangestotheeleventheditionwerecarefullyreviewedbyoutstandingteach- erscomprisedofbothusersandnonusersofthepreviousedition. Thechargeofthiscom- mittee was to ensure that all changesdid notalter those aspects of the text thatattracted users of the tenth edition and at the same time provide freshness to the new edition that wouldattractnewusers. NEW TO THIS EDITION (cid:129) Mostofthepre-calculusmaterialinthe10theditionChapter0hasbeenmovedtoAp- pendices,andtheremainingChapter0materialismergedintoChapter1. (cid:129) Someproseinotherareasofthetexthasbeentightenedtoenhanceclarityandstudent understanding. (cid:129) Newappliedexerciseshavebeenaddedtothebookandsomeexistingappliedexercises havebeenupdated. OTHER FEATURES Flexibility Thiseditionhasabuilt-inflexibilitythatisdesignedtoserveabroadspec- trumof calculusphilosophies—fromtraditionalto “reform.” Technologycan be empha- sized or not, and the orderof manytopics can be permutedfreely to accommodateeach instructor’sspecificneeds. Rigor The challengeof writing a goodcalculusbookis to strike the rightbalancebe- tween rigor and clarity. Our goal is to present precise mathematics to the fullest extent possibleinanintroductorytreatment. Whereclarityandrigorconflict,wechooseclarity; however,webelieveittobeimportantthatthestudentunderstandthedifferencebetween acarefulproofandaninformalargument,sowehaveinformedthereaderwhentheargu- mentsbeingpresentedareinformalor motivational. Theoryinvolvingǫ-δ argumentsap- pearsinseparatesectionssothattheycanbecoveredornot,aspreferredbytheinstructor. RuleofFour The“ruleoffour”referstopresentingconceptsfromtheverbal,algebraic, visual,andnumericalpointsofview. Inkeepingwithcurrentpedagogicalphilosophy,we usedthisapproachwheneverappropriate. Visualization Thiseditionmakesextensiveuseofmoderncomputergraphicstoclarify conceptsandtodevelopthestudent’sabilitytovisualizemathematicalobjects,particularly ix