AP Calculus AB Book - Early Transcendentals/Single Variables

1319 Pages·2010·20.7 MB·English

December30,2008 11:00 M91-for-the-student Sheetnumber1 Pagenumberii cyanmagentayellowblack
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
“Explain your answer.” You might do this by noting a www.wiley.com/college/anton
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. Very helpful...and very easy to use.”
— Student Victoria Cazorla,
Dutchess County Community College
See and try
WileyPLUS in action!
Details and Demo:
www.wileyplus.com
Wiley is committed to making your entire WileyPLUS experience
productive & enjoyable by providing the help, resources, and
personal support you & your students need, when you need it.
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, [phone], fax[phone]. RequeststothePublisher
forpermissionshouldbeaddressedtothePermissionsDepartment,JohnWiley&Sons,Inc.,111RiverStreet,
Hoboken,NJ07030,[phone],fax[phone],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