Calculus Early Transcendentals

2016·53.74 MB·English

by  Howard Anton

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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
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11
th
EDITION
CALCULUS
Carlos Fernandez/Getty Images
EARLY TRANSCENDENTALS
HOWARD ANTON DrexelUniversity
IRL BIVENS DavidsonCollege
STEPHEN DAVIS DavidsonCollege
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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.
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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
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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