HOWARD ANTON Drexel University IRL BIVENS Davidson College STEPHEN DAVIS Davidson ...

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FOR THE STUDENT
Calculusprovidesawayofviewingandanalyzingthephysi- theoreminthetextthatshowsthestatementtobetrueor
calworld. Aswithallmathematicscourses,calculusinvolves by finding a particular example in which the statement
equationsandformulas. However,ifyousuccessfullylearnto isnottrue.
use all the formulas and solve all of the problems in the text • Writingexercisesareintendedtotestyourabilitytoex-
butdonotmastertheunderlyingideas,youwillhavemissed plain mathematical ideas in words rather than relying
themostimportantpartofcalculus. Ifyoumastertheseideas, solelyonnumbersandsymbols. Allexercisesrequiring
you will have a widely applicable tool that goes far beyond writingshouldbeansweredincomplete,correctlypunc-
textbookexercises. tuated logical sentences—not with fragmented phrases
Beforestartingyourstudies,youmayfindithelpfultoleaf andformulas.
throughthistexttogetageneralfeelingforitsdifferentparts:
■ Each chapter ends with two additional sets of exercises:
■ The opening page of each chapter gives you an overview ChapterReviewExercises,which,asthenamesuggests,is
ofwhatthatchapterisabout,andtheopeningpageofeach aselectsetofexercisesthatprovideareviewofthemain
sectionwithinachaptergivesyouanoverviewofwhatthat concepts and techniques in the chapter, and Making Con-
section is about. To help you locate specific information, nections, in which exercises require you to draw on and
sectionsaresubdividedintotopicsthataremarkedwitha combinevariousideasdevelopedthroughoutthechapter.
boxlikethis . ■ Your instructor may choose to incorporate technology in
■ Each section ends with a set of exercises. The answers your calculus course. Exercises whose solution involves
tomostodd-numberedexercisesappearinthebackofthe theuseofsomekindoftechnologyaretaggedwithiconsto
book. Ifyoufindthatyouranswertoanexercisedoesnot alertyouandyourinstructor. Thoseexercisestaggedwith
matchthatinthebackofthebook,donotassumeimmedi- theicon requiregraphingtechnology—eitheragraphing
atelythatyoursisincorrect—theremaybemorethanone calculatororacomputerprogramthatcangraphequations.
w√aytoexpresstheanswer. For√example,ifyouransweris Those exercises tagged with the icon C require a com-
2/2 and the text answer is 1/ 2 , then both are correct puteralgebrasystem(CAS)suchasMathematica,Maple,
since your answer can be obtained by “rationalizing” the oravailableonsomegraphingcalculators.
textanswer. Ingeneral,ifyouranswerdoesnotmatchthat ■ At the end of the text you will find a set of four appen-
inthetext,thenyourbestfirststepistolookforanalgebraic dicescoveringvarioustopicssuchasadetailedreviewof
manipulationoratrigonometricidentitythatmighthelpyou trigonometry and graphing techniques using technology.
determineifthetwoanswersareequivalent. Iftheanswer Inside the front and back covers of the text you will find
isintheformofadecimalapproximation,thenyouranswer endpapersthatcontainusefulformulas.
mightdifferfromthatinthetextbecauseofadifferencein
■ Theideasinthistextwerecreatedbyrealpeoplewithin-
thenumberofdecimalplacesusedinthecomputations.
terestingpersonalitiesandbackgrounds. 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
AWordofEncouragement
inthesection;theyinvolverelativelylittlecomputation,
andhaveanswersprovidedattheendoftheexerciseset. As you work your way through this text you will find some
ideas that you understand immediately, some that you don’t
• The Focus on Concepts exercises, as their name sug-
understanduntilyouhavereadthemseveraltimes,andothers
gests,keyinonthemainideasinthesection.
thatyoudonotseemtounderstand,evenafterseveralreadings.
• True/False exercises focus on key ideas in a different
Donotbecomediscouraged—someideasareintrinsicallydif-
way.Youmustdecidewhetherthestatementistrueinall
ficult and take time to “percolate.” You may well find that a
possiblecircumstances,inwhichcaseyouwoulddeclare
hardideabecomesclearlaterwhenyouleastexpectit.
ittobe“true,”orwhethertherearesomecircumstances
inwhichitisnottrue,inwhichcaseyouwoulddeclare
WileyWebSiteforthisText
ittobe“false.” Ineachsuchexerciseyouareaskedto
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9
th
EDITION
CALCULUS
EARLY TRANSCENDENTALS
HOWARD ANTON DrexelUniversity
IRL BIVENS DavidsonCollege
STEPHEN DAVIS DavidsonCollege
withcontributionsby
Thomas Polaski WinthropUniversity
JOHN WILEY & SONS, INC.
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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.
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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
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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
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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