Calculus - Early Transcendentals (International Student Version)

1316 Pages·2013·21.602 MB·English

by  Howard Anton, Irl Bivens, Stephen Davis

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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
(cid:2) (cid:2)
un+1 au
1. undu= +C 10. audu= +C
n+1 lna
(cid:2) (cid:2)
du
2. =ln|u|+C 11. lnudu=ulnu−u+C
u
(cid:2) (cid:2)
3. eudu=eu+C 12. cotudu=ln|sinu|+C
(cid:2) (cid:2)
4. sinudu=−cosu+C secudu=ln|secu+tanu|+C
13. (cid:4) (cid:5)
(cid:2) =ln|tan 1π+1u |+C
5. cosudu=sinu+C (cid:2) 4 2
(cid:2) cscudu=ln|cscu−cotu|+C
14.
6. tanudu=ln|secu|+C =ln|tan1u|+C
(cid:2) (cid:3) (cid:2) 2 (cid:3)
7. sin−1udu=usin−1u+ 1−u2+C 15. cot−1udu=ucot−1u+ln 1+u2+C
(cid:2) (cid:3) (cid:2) (cid:3)
8. cos−1udu=ucos−1u− 1−u2+C 16. sec−1udu=usec−1u−ln|u+ u2−1|+C
(cid:2) (cid:3) (cid:2) (cid:3)
9. tan−1udu=utan−1u−ln 1+u2+C 17. csc−1udu=ucsc−1u+ln|u+ u2−1|+C
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RECIPROCALSOFBASICFUNCTIONS
(cid:2) (cid:2)
1 1
18. du=tanu∓secu+C 22. du= 1(u∓ln|sinu±cosu|)+C
(cid:2) 1±sinu (cid:2) 1±cotu 2
1 1
19. du=−cotu±cscu+C 23. du=u+cotu∓cscu+C
(cid:2) 1±cosu (cid:2) 1±secu
1 1
20. du= 1(u±ln|cosu±sinu|)+C 24. du=u−tanu±secu+C
(cid:2) 1±tanu 2 (cid:2) 1±cscu
1 1
21. du=ln|tanu|+C 25. du=u−ln(1±eu)+C
sinucosu 1±eu
POWERSOFTRIGONOMETRICFUNCTIONS
(cid:2) (cid:2)
26. sin2udu= 1u−1sin2u+C 32. cot2udu=−cotu−u+C
(cid:2) 2 4 (cid:2)
27. cos2udu= 1u+1sin2u+C 33. sec2udu=tanu+C
(cid:2) 2 4 (cid:2)
28. tan2udu=tanu−u+C 34. csc2udu=−cotu+C
(cid:2) (cid:2) (cid:2) (cid:2)
29. sinnudu=−1sinn−1ucosu+n−1 sinn−2udu 35. cotnudu=− 1 cotn−1u− cotn−2udu
(cid:2) n n (cid:2) (cid:2) n−1 (cid:2)
30. cosnudu= 1cosn−1usinu+n−1 cosn−2udu 36. secnudu= 1 secn−2utanu+n−2 secn−2udu
(cid:2) n (cid:2) n (cid:2) n−1 n−1 (cid:2)
31. tannudu= 1 tann−1u− tann−2udu 37. cscnudu=− 1 cscn−2ucotu+n−2 cscn−2udu
n−1 n−1 n−1
PRODUCTSOFTRIGONOMETRICFUNCTIONS
(cid:2) (cid:2)
sin(m+n)u sin(m−n)u cos(m+n)u cos(m−n)u
38. sinmusinnudu=− + +C 40. sinmucosnudu=− − +C
(cid:2) 2(m+n) 2(m−n) (cid:2) 2(m+n) 2(m−n)(cid:2)
39. cosmucosnudu= si2n((mm++nn))u+si2n((mm−−nn))u+C 41. sinmucosnudu=−sinm−m1u+consn+1u+mm+−n1 sinm−2ucosnudu
(cid:2)
= sinm+1ucosn−1u+ n−1 sinmucosn−2udu
m+n m+n
PRODUCTSOFTRIGONOMETRICANDEXPONENTIALFUNCTIONS
(cid:2) (cid:2)
eau eau
42. eausinbudu= (asinbu−bcosbu)+C 43. eaucosbudu= (acosbu+bsinbu)+C
a2+b2 a2+b2
POWERSOFu MULTIPLYINGORDIVIDINGBASICFUNCTIONS
(cid:2) (cid:2)
44. usinudu=sinu−ucosu+C 51. ueudu=eu(u−1)+C
(cid:2) (cid:2) (cid:2)
45. ucosudu=cosu+usinu+C 52. uneudu=uneu−n un−1eudu
(cid:2) (cid:2) (cid:2)
46. u2sinudu=2usinu+(2−u2)cosu+C 53. unaudu= unau − n un−1audu+C
(cid:2) (cid:2) lna lna (cid:2)
47. u2cosudu=2ucosu+(u2−2)sinu+C 54. eudu =− eu + 1 eudu
(cid:2) (cid:2) (cid:2) un (n−1)un−1 n−1 (cid:2) un−1
48. unsinudu=−uncosu+n un−1cosudu 55. audu =− au + lna audu
(cid:2) (cid:2) (cid:2) un (n−1)un−1 n−1 un−1
49. uncosudu=unsinu−n un−1sinudu 56. du =ln|lnu|+C
(cid:2) ulnu
un+1
50. unlnudu= [(n+1)lnu−1]+C
(n+1)2
POLYNOMIALSMULTIPLYINGBASICFUNCTIONS
(cid:2)
57. p(u)eaudu= 1p(u)eau− 1 p(cid:4)(u)eau+ 1 p(cid:4)(cid:4)(u)eau−··· [signsalternate:+−+−···]
(cid:2) a a2 a3
58. p(u)sinaudu=−1p(u)cosau+ 1 p(cid:4)(u)sinau+ 1 p(cid:4)(cid:4)(u)cosau−··· [signsalternateinpairsafterfirstterm:++−−++−−···]
(cid:2) a a2 a3
59. p(u)cosaudu= 1p(u)sinau+ 1 p(cid:4)(u)cosau− 1 p(cid:4)(cid:4)(u)sinau−··· [signsalternateinpairs:++−−++−−···]
a a2 a3
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10
th
EDITION
CALCULUS
EARLY TRANSCENDENTALS
International Student Version
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.
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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
January2,2012 18:35 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
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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.