Calculus: Single and Multivariable--Instructor's Manual

1244 Pages·2012·27.62 MB·English

by  Hughes-Hallett, Deborah;McCallum, William G;Gleason, Andrew M

C a l c u l u s
6 Edition Single & Multivariable
th
Hughes-Hallett Gleason McCallum et al.

CALCULUS
SixthEdition

CALCULUS
SixthEdition
ProducedbytheCalculusConsortiumandinitiallyfundedbyaNationalScienceFoundationGrant.
DeborahHughes-Hallett WilliamG.McCallum AndrewM.Gleason
UniversityofArizona UniversityofArizona HarvardUniversity
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HarvardUniversityExtension UniversityofArizona UniversityofArizona
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PREFACE
Calculusisoneofthegreatestachievementsofthehumanintellect.Inspiredbyproblemsinastronomy,
NewtonandLeibnizdevelopedtheideasofcalculus300yearsago.Sincethen,eachcenturyhasdemonstrated
the power of calculus to illuminate questions in mathematics, the physical sciences, engineering, and the
socialandbiologicalsciences.
Calculushasbeensosuccessfulbothbecauseitscentraltheme—change—ispivotaltoananalysisofthe
naturalworldandbecauseofitsextraordinarypowertoreducecomplicatedproblemstosimpleprocedures.
Thereinliesthedangerinteachingcalculus:itispossibletoteachthesubjectasnothingbutprocedures—
thereby losing sight of both the mathematics and of its practical value.This edition of Calculus continues
ourefforttopromotecoursesinwhichunderstandingandcomputationreinforceeachother.
Mathematical ThinkingSupported byTheoryandModeling
Thefirststageinthedevelopmentofmathematicalthinkingistheacquisitionofaclearintuitivepictureofthe
centralideas.Inthenextstage,thestudentlearnstoreasonwiththeintuitiveideasinplainEnglish.Afterthis
foundationhasbeenlaid,thereisachoiceofdirection.Allstudentsbenefitfromboththeoryandmodeling,
butthebalancemaydifferfordifferentgroups.Somestudents,suchasmathematicsmajors,mayprefermore
theory,whileothersmayprefermoremodeling.Forinstructorswishingtoemphasizetheconnectionbetween
calculusandotherfields,thetextincludes:
• Avarietyofproblemsfromthephysicalsciencesandengineering.
• Examplesfromthebiologicalsciencesandeconomics.
• Modelsfromthehealthsciencesandofpopulationgrowth.
• Newproblemsonsustainability.
• NewcasestudiesonmedicinebyDavidE.Sloane,MD.
Originofthe Text
Fromthebeginning,thistextbookgrewoutofacommunityofmathematicsinstructorseagertofindeffective
ways for students to learn calculus. This Sixth Edition of Calculus reflects the many voices of users at
researchuniversities,four-yearcolleges,communitycolleges,andsecondaryschools.Theirinputandthatof
ourpartnerdisciplines,engineeringandthenaturalandsocialsciences,continuetoshapeourwork.
ActiveLearning: GoodProblems
As instructors ourselves, we know that interactive classrooms and well-crafted problems promote student
learning.Sinceitsinception,thehallmarkofourtexthasbeenitsinnovativeandengagingproblems.These
problemsprobestudentunderstandinginwaysoftentakenforgranted.Praisedfortheircreativityandvariety,
theinfluenceoftheseproblemshasextendedfarbeyondtheusersofourtextbook.
The Sixth Edition continues this tradition. Under our approach, which we called the “Rule of Four,”
ideasarepresentedgraphically,numerically,symbolically,andverbally,therebyencouragingstudentswith
avarietyoflearningstylestoexpandtheirknowledge.Thiseditionexpandsthetypesofproblemsavailable:
• NewStrengthenYourUnderstanding problemsatthe endofeverysection.These problemsaskstu-
dentstoreflectonwhattheyhavelearnedbydeciding“Whatiswrong?”withastatementandto“Give
anexample”ofanidea.
• ConcepTests promote active learning in the classroom. These can be used with or without clickers
(personalresponsesystems),andhavebeenshowntodramaticallyimprovestudentlearning.Available
inabookoronthewebatwww.wiley.com/college/hughes-hallett.
v
vi Preface
• ClassWorksheetsallowinstructorstoengagestudentsinindividualorgroupclass-work.Samplesare
availableintheInstructor’sManual,andallareonthewebatwww.wiley.com/college/hughes-hallett.
• UpdatedDataandModels.Forexample,Section11.7followsthecurrentdebateonPeakOilProduc-
tion, underscoring the importance of mathematics in understanding the world’s economic and social
problems.
• Projectsattheendofeachchapterprovideopportunitiesforasustainedinvestigation,oftenusingskills
fromdifferentpartsofthecourse.
• DrillExercisesbuildstudentskillandconfidence.
• OnlineProblemsavailableinWileyPLUSorWeBWorK,forexample.Manyproblemsarerandomized,
providingstudentswithexpandedopportunitiesforpracticewithimmediatefeedback.
SymbolicManipulationandTechnology
Tousecalculuseffectively,studentsneedskillinbothsymbolicmanipulationandtheuseoftechnology.The
balancebetweenthetwomayvary,dependingontheneedsofthestudentsandthewishesoftheinstructor.
Thebookisadaptabletomanydifferentcombinations.
Thebookdoesnotrequireanyspecificsoftwareortechnology.Ithasbeenusedwithgraphingcalcula-
tors,graphingsoftware,andcomputeralgebrasystems.Anytechnologywiththeabilitytographfunctions
andperformnumericalintegrationwillsuffice.Studentsareexpectedtousetheirownjudgmenttodetermine
wheretechnologyisuseful.
Content
This content represents our vision of how calculus can be taught. It is flexible enough to accommodate
individualcourseneedsandrequirements.Topicscaneasilybeaddedordeleted,ortheorderchanged.
Changestothe textinthe SixthEditionarein italics. Inallchapters,manynewproblemswere added
andotherswereupdated.
Chapter1:ALibraryofFunctions
This chapter introduces all the elementary functions to be used in the book. Although the functions are
probablyfamiliar,thegraphical,numerical,verbal,andmodelingapproachtothemmaybenew.Weintroduce
exponentialfunctionsattheearliestpossiblestage,sincetheyarefundamentaltotheunderstandingofreal-
worldprocesses.The chapterconcludeswith a section onlimits, allowingfora discussionofcontinuityat
a pointand onan interval.The sectionon limits is flexibleenoughto allow for a briefintroductionbefore
derivativesorforamoreextensivetreatment.
Chapter2:KeyConcept:TheDerivative
The purposeof this chapter is to give the studenta practicalunderstandingof the definitionof the deriva-
tive and its interpretationas an instantaneousrate of change.The powerrule is introduced;otherrules are
introducedinChapter3.
Chapter3:Short-CutstoDifferentiation
ThederivativesofallthefunctionsinChapter1areintroduced,aswellastherulesfordifferentiatingprod-
ucts;quotients;andcomposite,inverse,hyperbolic,andimplicitlydefinedfunctions.
Chapter4:UsingtheDerivative
The aim of this chapter is to enable the student to use the derivative in solving problems, including opti-
mization,graphing,rates,parametricequations,andindeterminateforms.Itisnotnecessarytocoverallthe
sectionsinthischapter.
Toincreaseaccesstooptimization,manysectionsofthischapterhavebeenstreamlined.Optimization
andModelingarenowinSection4.3,followedbyFamiliesofFunctionsandModelinginSection4.4.Upper
andlowerboundshavebeenmovedtoSection4.2,andgeometricoptimizationisnowcombinedwithOpti-
mizationandModeling.Section4.8onParametricEquationsislinkedto AppendixD, allowingdiscussion
ofvelocityasavector.
Preface vii
Chapter5:KeyConcept:TheDefiniteIntegral
Thepurposeofthischapteristogivethestudentapracticalunderstandingofthedefiniteintegralasalimit
of Riemann sums and to bring out the connection between the derivative and the definite integral in the
FundamentalTheoremofCalculus.
Section5.3nowincludestheapplicationoftheFundamentalTheoremofCalculustothecomputationof
definiteintegrals.TheuseofintegralstofindaveragesisnowinSection5.4.
Chapter6:ConstructingAntiderivatives
This chapter focuses on going backward from a derivative to the original function, first graphically and
numerically,thenanalytically.ItintroducestheSecondFundamentalTheoremofCalculusandtheconcept
ofadifferentialequation.
Section6.3onDifferentialEquationsandMotioncontainsthematerialfromtheformerSection6.5.
Chapter7:Integration
This chapter includes several techniques of integration, including substitution, parts, partial fractions, and
trigonometricsubstitutions;othersareincludedinthetableofintegrals.Therearediscussionsofnumerical
methodsandofimproperintegrals.
Section7.4nowincludestheuseoftrianglestohelpstudentsvisualizeatrigonometricsubstitution.The
twoformersectionsonnumericalmethodshavebeencombinedintoSection7.5.
Chapter8:UsingtheDefiniteIntegral
This chapter emphasizesthe idea of subdividinga quantity to produceRiemann sums which, in the limit,
yieldadefiniteintegral.Itshowshowtheintegralisusedingeometry,physics,economics,andprobability;
polarcoordinatesareintroduced.Itisnotnecessarytocoverallthesectionsinthischapter.
Chapter9:SequencesandSeries
Thischapterfocusesonsequences,seriesofconstants,andconvergence.Itincludestheintegral,ratio,com-
parison,limitcomparison,andalternatingseriestests.Italsointroducesgeometricseriesandgeneralpower
series,includingtheirintervalsofconvergence.
Chapter10:ApproximatingFunctions
ThischapterintroducesTaylorSeriesandFourierSeriesusingtheideaofapproximatingfunctionsbysimpler
functions.
Chapter11:DifferentialEquations
Thischapterintroducesdifferentialequations.Theemphasisisonqualitativesolutions,modeling,andinter-
pretation.
Section11.7onLogisticModels(formerlyonpopulationmodels)hasbeenrewrittenaroundthethought-
provoking predictions of peak oil production. This section encourages students to use the skills learned
earlierinthecoursetoanalyzeaproblemofglobalimportance.Sections11.10and11.11onSecondOrder
DifferentialEquationsarenowonthewebatwww.wiley.com/college/hughes-hallett.
Chapter12:FunctionsofSeveralVariables
Thischapterintroducesfunctionsofmanyvariablesfromseveralpointsofview,usingsurfacegraphs,con-
tour diagrams, and tables. We assume throughout that functions of two or more variables are defined on
regionswithpiecewisesmoothboundaries.Weconcludewithasectiononcontinuity.
Chapter13:AFundamentalTool:Vectors
Thischapterintroducesvectorsgeometricallyandalgebraicallyanddiscussesthedotandcrossproduct.
viii Preface
Chapter14:DifferentiatingFunctionsofSeveralVariables
Partial derivatives, directional derivatives, gradients, and local linearity are introduced. The chapter also
discusseshigherorderpartialderivatives,quadraticTaylorapproximations,anddifferentiability.
Chapter15:Optimization
Theideasofthepreviouschapterareappliedtooptimizationproblems,bothconstrainedandunconstrained.
Chapter16:IntegratingFunctionsofSeveralVariables
ThischapterdiscussesdoubleandtripleintegralsinCartesian,polar,cylindrical,andsphericalcoordinates.
TheformerSection16.7hasbeenmovedtothenewChapter21.
Chapter17:ParameterizationandVectorFields
Thischapterdiscussesparameterizedcurvesandmotion,vectorfieldsandflowlines.
TheformerSection17.5hasbeenmovedtothenewChapter21.
Chapter18:LineIntegrals
Thischapterintroduceslineintegralsandshowshowtocalculatethemusingparameterizations.Conservative
fields, gradientfields, the FundamentalTheorem of Calculus for Line Integrals, and Green’s Theorem are
discussed.
Chapter19:FluxIntegralsandDivergence
This chapter introduces flux integrals and shows how to calculate them over surface graphs, portions of
cylinders,andportionsofspheres.Thedivergenceisintroducedanditsrelationshiptofluxintegralsdiscussed
intheDivergenceTheorem.
ThisnewchaptercombinesSections19.1and19.2withSections20.1and20.2fromthefifthedition
Chapter20:TheCurlandStokes’Theorem
ThepurposeofthischapteristogivestudentsapracticalunderstandingofthecurlandofStokes’Theorem
andtolayouttherelationshipbetweenthetheoremsofvectorcalculus.
ThischapterconsistsofSections20.3–20.5fromthefifthedition.
Chapter21:Parameters,Coordinates,andIntegrals
Thisnewchaptercoversparameterizedsurfaces,thechangeofvariableformulainadoubleortripleintegral,
andfluxthoughaparameterizedsurface.
Appendices
There are appendices on roots, accuracy, and bounds; complex numbers; Newton’s Method; and determi-
nants.
Projects
TherearenewprojectsinChapter12:“Heathrow”;Chapter19:“SolidAngle”;andChapter20:“Magnetic
fieldgeneratedbyacurrentinawire”.
ChoiceofPaths: LeanorExpanded
Forthosewhoprefertheleantopiclistofearliereditions,wehavekeptclearthemainconceptualpaths.For
example,
• TheKeyConceptchaptersonthederivativeandthedefiniteintegral(Chapters2and5)canbecovered
attheoutsetofthecourse,rightafterChapter1.