Calculus

1314 Pages·2012·22.83 MB·English

by  Howard Anton

October 15, 2011 13:09 fmend Sheet number 1 Page number 2 cyan magenta yellow black
GEOMETRY FORMULAS
A=area, S=lateralsurfacearea, V =volume, h=height, B=areaofbase, r=radius, l=slantheight, C=circumference, s=arclength
Parallelogram Triangle Trapezoid Circle Sector
a
r
h h h s
u
b b b
r
A = 1 r2u, s = ru
A = bh A = 1 bh A = 1 (a + b)h A = pr2, C = 2pr 2
2 2 (u in radians)
Right Circular Cylinder Right Circular Cone Any Cylinder or Prism with Parallel Bases Sphere
h l h h r
h
r r
B B
V = pr2h, S = 2prh V = 1 pr2h, S = prl V = Bh V = 4 pr3, S = 4pr2
3 3
ALGEBRA FORMULAS
THEQUADRATIC
FORMULA THEBINOMIALFORMULA
Teqhueastioolnutaioxn2s+ofbtxhe+quca=dra0tiacre (x+y)n=xn+nxn−1y+n(1n·−21)xn−2y2+n(n−1·12)(·n3−2)xn−3y3+···+nxyn−1+yn
√
x= −b± b2−4ac (x−y)n=xn−nxn−1y+n(n−1)xn−2y2−n(n−1)(n−2)xn−3y3+···±nxyn−1∓yn
2a 1·2 1·2·3
TABLE OF INTEGRALS
BASICFUNCTIONS
! !
un+1 au
1. undu= +C 10. audu= +C
n+1 lna
! !
du
2. =ln|u|+C 11. lnudu=ulnu−u+C
u
! !
3. eudu=eu+C 12. cotudu=ln|sinu|+C
! !
4. sinudu=−cosu+C secudu=ln|secu+tanu|+C
13. (cid:10) (cid:11)
! =ln|tan 1π+1u |+C
5. cosudu=sinu+C ! 4 2
! cscudu=ln|cscu−cotu|+C
14.
6. tanudu=ln|secu|+C =ln|tan1u|+C
! (cid:4) ! 2 (cid:4)
7. sin−1udu=usin−1u+ 1−u2+C 15. cot−1udu=ucot−1u+ln 1+u2+C
! (cid:4) ! (cid:4)
8. cos−1udu=ucos−1u− 1−u2+C 16. sec−1udu=usec−1u−ln|u+ u2−1|+C
! (cid:4) ! (cid:4)
9. tan−1udu=utan−1u−ln 1+u2+C 17. csc−1udu=ucsc−1u+ln|u+ u2−1|+C
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RECIPROCALSOFBASICFUNCTIONS
! !
1 1
18. du=tanu∓secu+C 22. du= 1(u∓ln|sinu±cosu|)+C
! 1±sinu ! 1±cotu 2
1 1
19. du=−cotu±cscu+C 23. du=u+cotu∓cscu+C
! 1±cosu ! 1±secu
1 1
20. du= 1(u±ln|cosu±sinu|)+C 24. du=u−tanu±secu+C
! 1±tanu 2 ! 1±cscu
1 1
21. du=ln|tanu|+C 25. du=u−ln(1±eu)+C
sinucosu 1±eu
POWERSOFTRIGONOMETRICFUNCTIONS
! !
26. sin2udu= 1u−1sin2u+C 32. cot2udu=−cotu−u+C
! 2 4 !
27. cos2udu= 1u+1sin2u+C 33. sec2udu=tanu+C
! 2 4 !
28. tan2udu=tanu−u+C 34. csc2udu=−cotu+C
! ! ! !
29. sinnudu=−1sinn−1ucosu+n−1 sinn−2udu 35. cotnudu=− 1 cotn−1u− cotn−2udu
! n n ! ! n−1 !
30. cosnudu= 1cosn−1usinu+n−1 cosn−2udu 36. secnudu= 1 secn−2utanu+n−2 secn−2udu
! n ! n ! n−1 n−1 !
31. tannudu= 1 tann−1u− tann−2udu 37. cscnudu=− 1 cscn−2ucotu+n−2 cscn−2udu
n−1 n−1 n−1
PRODUCTSOFTRIGONOMETRICFUNCTIONS
! !
sin(m+n)u sin(m−n)u cos(m+n)u cos(m−n)u
38. sinmusinnudu=− + +C 40. sinmucosnudu=− − +C
! 2(m+n) 2(m−n) ! 2(m+n) 2(m−n)!
39. cosmucosnudu= si2n((mm++nn))u+si2n((mm−−nn))u+C 41. sinmucosnudu=−sinm−m1u+consn+1u+mm+−n1 sinm−2ucosnudu
!
= sinm+1ucosn−1u+ n−1 sinmucosn−2udu
m+n m+n
PRODUCTSOFTRIGONOMETRICANDEXPONENTIALFUNCTIONS
! !
eau eau
42. eausinbudu= (asinbu−bcosbu)+C 43. eaucosbudu= (acosbu+bsinbu)+C
a2+b2 a2+b2
POWERSOFu MULTIPLYINGORDIVIDINGBASICFUNCTIONS
! !
44. usinudu=sinu−ucosu+C 51. ueudu=eu(u−1)+C
! ! !
45. ucosudu=cosu+usinu+C 52. uneudu=uneu−n un−1eudu
! ! !
46. u2sinudu=2usinu+(2−u2)cosu+C 53. unaudu= unau − n un−1audu+C
! ! lna lna !
47. u2cosudu=2ucosu+(u2−2)sinu+C 54. eudu =− eu + 1 eudu
! ! ! un (n−1)un−1 n−1 ! un−1
48. unsinudu=−uncosu+n un−1cosudu 55. audu =− au + lna audu
! ! ! un (n−1)un−1 n−1 un−1
49. uncosudu=unsinu−n un−1sinudu 56. du =ln|lnu|+C
! ulnu
un+1
50. unlnudu= [(n+1)lnu−1]+C
(n+1)2
POLYNOMIALSMULTIPLYINGBASICFUNCTIONS
!
57. p(u)eaudu= 1p(u)eau− 1 p(cid:5)(u)eau+ 1 p(cid:5)(cid:5)(u)eau−··· [signsalternate:+−+−···]
! a a2 a3
58. p(u)sinaudu=−1p(u)cosau+ 1 p(cid:5)(u)sinau+ 1 p(cid:5)(cid:5)(u)cosau−··· [signsalternateinpairsafterfirstterm:++−−++−−···]
! a a2 a3
59. p(u)cosaudu= 1p(u)sinau+ 1 p(cid:5)(u)cosau− 1 p(cid:5)(cid:5)(u)sinau−··· [signsalternateinpairs:++−−++−−···]
a a2 a3
October28,2011 17:19 for-the-student Sheetnumber1 Pagenumberii cyanmagentayellowblack
FOR THE STUDENT
Calculusprovidesawayofviewingandanalyzingthephysi- by finding a particular example in which the statement
calworld. Aswithallmathematicscourses,calculusinvolves isnottrue.
equationsandformulas. However,ifyousuccessfullylearnto • Writingexercisesareintendedtotestyourabilitytoex-
use all the formulas and solve all of the problems in the text
plain mathematical ideas in words rather than relying
butdonotmastertheunderlyingideas,youwillhavemissed
solelyonnumbersandsymbols. Allexercisesrequiring
themostimportantpartofcalculus. Ifyoumastertheseideas,
writingshouldbeansweredincomplete,correctlypunc-
you will have a widely applicable tool that goes far beyond
tuated logical sentences—not with fragmented phrases
textbookexercises.
andformulas.
Beforestartingyourstudies,youmayfindithelpfultoleaf
■ Each chapter ends with two additional sets of exercises:
throughthistexttogetageneralfeelingforitsdifferentparts:
ChapterReviewExercises,which,asthenamesuggests,is
■ The opening page of each chapter gives you an overview a select set of exercises that provide a review of the main
ofwhatthatchapterisabout,andtheopeningpageofeach concepts and techniques in the chapter, and Making Con-
sectionwithinachaptergivesyouanoverviewofwhatthat nections, in which exercises require you to draw on and
section is about. To help you locate specific information, combinevariousideasdevelopedthroughoutthechapter.
sections are subdivided into topics that are marked with a
■ Your instructor may choose to incorporate technology in
boxlikethis .
your calculus course. Exercises whose solution involves
■ Each section ends with a set of exercises. The answers theuseofsomekindoftechnologyaretaggedwithiconsto
tomostodd-numberedexercisesappearinthebackofthe alertyouandyourinstructor. Thoseexercisestaggedwith
book. Ifyoufindthatyouranswertoanexercisedoesnot theicon requiregraphingtechnology—eitheragraphing
matchthatinthebackofthebook,donotassumeimmedi- calculatororacomputerprogramthatcangraphequations.
atelythatyoursisincorrect—theremaybemorethanone Those exercises tagged with the icon C require a com-
w√aytoexpresstheanswer. For√example,ifyouransweris puteralgebrasystem(CAS)suchasMathematica,Maple,
2/2 and the text answer is 1/ 2 , then both are correct oravailableonsomegraphingcalculators.
since your answer can be obtained by “rationalizing” the
■ At the end of the text you will find a set of four appen-
textanswer. Ingeneral,ifyouranswerdoesnotmatchthat
dices covering various topics such as a detailed review of
inthetext,thenyourbestfirststepistolookforanalgebraic
trigonometry and graphing techniques using technology.
manipulationoratrigonometricidentitythatmighthelpyou
Inside the front and back covers of the text you will find
determineifthetwoanswersareequivalent. Iftheanswer
endpapersthatcontainusefulformulas.
isintheformofadecimalapproximation,thenyouranswer
■ Theideasinthistextwerecreatedbyrealpeoplewithin-
mightdifferfromthatinthetextbecauseofadifferencein
terestingpersonalitiesandbackgrounds. Picturesandbio-
thenumberofdecimalplacesusedinthecomputations.
graphicalsketchesofmanyofthesepeopleappearthrough-
■ The section exercises include regular exercises and four
outthebook.
special categories: Quick Check, Focus on Concepts,
True/False,andWriting. ■ Notesinthemarginareintendedtoclarifyorcommenton
importantpointsinthetext.
• TheQuickCheckexercisesareintendedtogiveyouquick
feedbackonwhetheryouunderstandthekeyideasinthe AWordofEncouragement
section; they involve relatively little computation, and
As you work your way through this text you will find some
haveanswersprovidedattheendoftheexerciseset.
ideas that you understand immediately, some that you don’t
• TheFocusonConceptsexercises,astheirnamesuggests,
understanduntilyouhavereadthemseveraltimes,andothers
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
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10
th
EDITION
CALCULUS
DavidHenderson/GettyImages
HOWARD ANTON DrexelUniversity
IRL BIVENS DavidsonCollege
STEPHEN DAVIS DavidsonCollege
JOHN WILEY & SONS, INC.
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AboutHOWARDANTON HowardAntonobtainedhisB.A.fromLehighUniversity,hisM.A.fromtheUniversityofIllinois,
andhisPh.D.fromthePolytechnicUniversityofBrooklyn,allinmathematics. Intheearly1960she
workedforBurroughsCorporationandAvcoCorporationatCapeCanaveral,Florida,wherehewas
involvedwiththemannedspaceprogram. In1968hejoinedtheMathematicsDepartmentatDrexel
University,wherehetaughtfulltimeuntil1983. SincethattimehehasbeenanEmeritusProfessor
atDrexelandhasdevotedthemajorityofhistimetotextbookwritingandactivitiesformathematical
associations. Dr.AntonwaspresidentoftheEPADELsectionoftheMathematicalAssociationof
America(MAA),servedontheBoardofGovernorsofthatorganization,andguidedthecreationof
thestudentchaptersoftheMAA.Hehaspublishednumerousresearchpapersinfunctionalanalysis,
approximationtheory,andtopology,aswellaspedagogicalpapers. Heisbestknownforhis
textbooksinmathematics,whichareamongthemostwidelyusedintheworld. Therearecurrently
morethanonehundredversionsofhisbooks,includingtranslationsintoSpanish,Arabic,
Portuguese,Italian,Indonesian,French,Japanese,Chinese,Hebrew,andGerman. Histextbookin
linearalgebrahaswonboththeTextbookExcellenceAwardandtheMcGuffeyAwardfromthe
TextbookAuthor’sAssociation. 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
MAAProblemBookseries,theMAADolcianiMathematicalExpositionsseriesandTheCollege
MathematicsJournal.Whenheisnotpursuingmathematics,ProfessorBivensenjoysreading,
juggling,swimming,andwalking.
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-treasurerandhasservedonthe
MAABoardofGovernors. HeiscurrentlyafacultyconsultantfortheEducationalTestingService
forthegradingoftheAdvancedPlacementCalculusExam,webmasterfortheNorthCarolina
AssociationofAdvancedPlacementMathematicsTeachers,andisactivelyinvolvedinnurturing
mathematicallytalentedhighschoolstudentsthroughleadershipintheCharlotteMathematicsClub.
Forrelaxation,heplaysbasketball,juggles,andtravels. ProfessorDavisandhiswifeElisabethhave
threechildren,Laura,Anne,andJames,allformercalculusstudents.
November14,2011 18:25 ffirs 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
November14,2011 18:28 fpref Sheetnumber1 Pagenumbervii cyanmagentayellowblack
PREFACE
This tenth 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.
Allofthechangestothetentheditionwerecarefullyreviewedbyoutstandingteachers
comprisedofbothusersandnonusersofthepreviousedition. Thechargeofthiscommittee
wastoensurethatallchangesdidnotalterthoseaspectsofthetextthatattractedusersof
thenintheditionandatthesametimeprovidefreshnesstotheneweditionthatwouldattract
newusers.
NEW TO THIS EDITION
• ExercisesetshavebeenmodifiedtocorrespondmorecloselytoquestionsinWileyPLUS.
Inaddition,moreWileyPLUSquestionsnowcorrespondtospecificexercisesinthetext.
• Newappliedexerciseshavebeenaddedtothebookandexistingappliedexerciseshave
beenupdated.
• Whereappropriate,additionalskill/practiceexerciseswereadded.
OTHER FEATURES
Flexibility Thiseditionhasabuilt-inflexibilitythatisdesignedtoserveabroadspectrum
ofcalculusphilosophies—fromtraditionalto“reform.” Technologycanbeemphasizedor
not,andtheorderofmanytopicscanbepermutedfreelytoaccommodateeachinstructor’s
specificneeds.
Rigor Thechallengeofwritingagoodcalculusbookistostriketherightbalancebetween
rigorandclarity. Ourgoalistopresentprecisemathematicstothefullestextentpossible
inanintroductorytreatment. Whereclarityandrigorconflict,wechooseclarity;however,
webelieveittobeimportantthatthestudentunderstandthedifferencebetweenacareful
proofandaninformalargument,sowehaveinformedthereaderwhentheargumentsbeing
presentedareinformalormotivational. Theoryinvolving(cid:2)-δargumentsappearsinseparate
sectionssothattheycanbecoveredornot,aspreferredbytheinstructor.
RuleofFour The“ruleoffour”referstopresentingconceptsfromtheverbal,algebraic,
visual,andnumericalpointsofview. Inkeepingwithcurrentpedagogicalphilosophy,we
usedthisapproachwheneverappropriate.
Visualization Thiseditionmakesextensiveuseofmoderncomputergraphicstoclarify
conceptsandtodevelopthestudent’sabilitytovisualizemathematicalobjects,particularly
thosein3-space. Forthosestudentswhoareworkingwithgraphingtechnology,thereare
vii
November14,2011 18:28 fpref Sheetnumber2 Pagenumberviii cyanmagentayellowblack
viii Preface
many exercises that are designed to develop the student’s ability to generate and analyze
mathematicalcurvesandsurfaces.
Quick Check Exercises Eachexercisesetbeginswithapproximatelyfiveexercises
(answers included) that are designed to provide students with an immediate assessment
of whether they have mastered key ideas from the section. They require a minimum of
computationandareansweredbyfillingintheblanks.
FocusonConceptsExercises Eachexercisesetcontainsaclearlyidentifiedgroup
ofproblemsthatfocusonthemainideasofthesection.
TechnologyExercises Mostsectionsincludeexercisesthataredesignedtobesolved
using either a graphing calculator or a computer algebra system such as Mathematica,
Maple,ortheopensourceprogramSage. Theseexercisesaremarkedwithaniconforeasy
identification.
Applicability of Calculus One of the primary goals of this text is to link calculus
to the real world and the student’s own experience. This theme is carried through in the
examplesandexercises.
Career Preparation Thistextiswrittenatamathematicallevelthatwillpreparestu-
dentsforawidevarietyofcareersthatrequireasoundmathematicsbackground,including
engineering,thevarioussciences,andbusiness.
Trigonometry Review Deficiencies in trigonometry plague many students, so we
haveincludedasubstantialtrigonometryreviewinAppendixB.
Appendix on Polynomial Equations Becausemanycalculusstudentsareweak
insolvingpolynomialequations,wehaveincludedanappendix(AppendixC)thatreviews
theFactorTheorem,theRemainderTheorem,andproceduresforfindingrationalroots.
Principles of Integral Evaluation The traditional Techniques of Integration is
entitled “Principles of Integral Evaluation” to reflect its more modern approach to the
material. Thechapteremphasizesgeneralmethodsandtheroleoftechnologyratherthan
specifictricksforevaluatingcomplicatedorobscureintegrals.
Historical Notes The biographies and historical notes have been a hallmark of this
textfromitsfirsteditionandhavebeenmaintained. Allofthebiographicalmaterialshave
beendistilledfromstandardsourceswiththegoalofcapturingandbringingtolifeforthe
studentthepersonalitiesofhistory’sgreatestmathematicians.
Margin Notes and Warnings Theseappearinthemarginsthroughoutthetextto
clarifyorexpandonthetextexpositionortoalertthereadertosomepitfall.