Calculus: Applications and Technology, 3rd Edition

769 Pages·2004·7.6 MB·English

by  Edmond C. Tomastik

Calculus
Applications
and
Technology
T H I R D E D I T I O N
This page intentionally left blank
Calculus
Applications
and
Technology
T H I R D E D I T I O N
Edmond C. Tomastik
University of Connecticut
With Interactive Illustrations by
Hu Hohn, Massachusetts School of Art
Jean Marie McDill, California Polytechnic State University, San Luis Obispo
Agnes Rash, St. Joseph’s University
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StudentEdition: ISBN0-534-46496-3 Mexico
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Instructor’sEdition: ISBN0-534-46498-X Paraninfo
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An Overview of Third
Edition Changes
1. Inthisneweditionwehavefollowedageneralphilosophyofdividingthematerial
intosmaller,moremanageablesections. Thishasresultedinanincreaseinthe
number of sections. We think this makes it easier for the instructor and the
student,givesmoreflexibility,andcreatesabetterflowofmaterial.
2. Toaddtotheflexibility,manysectionsnowhaveenrichmentsubsections. Ma-
terialinsuchenrichmentsubsectionsisnotneededinthesubsequenttext(except
possiblyinlaterenrichmentsubsections). Nowinstructorscaneasilytailorthe
materialinthetexttoteachacourseatdifferentlevels.
3. Thethirdeditionhasevenmorereferencedreal-lifeexamples. Itisimportantto
realizethatthemathematicalmodelspresentedinthesereferencedexamplesare
modelscreatedbytheexpertsintheirfieldsandpublishedinrefereedjournals. So
notonlyisthedatainthesereferencedexamplesrealdata,butthemathematical
modelsbasedonthisrealdatahavebeencreatedbyexpertsintheirfields(and
notbyus).
4. Mathematicalmodelingisstressedinthisedition. Mathematicalmodelingisan
attempttodescribesomepartoftherealworldinmathematicalterms. Already
atthebeginningofSection1.2wedescribethethreestepsinmathematicalmod-
eling: formulation, mathematical manipulation, and evaluation. We return to
thisthemeoften. Forexample,inSection5.6onoptimizationandmodelingwe
giveasix-stepprocedureformathematicalmodelingspecificallyusefulinopti-
mization. Essentiallyeverysectionhasexamplesandexercisesinmathematical
modeling.
5. Thiseditionalsoincludesmanymoreopportunitiestomodelbycurvefitting.
Inthiskindofmodelingwehaveasetofdataconnectingtwovariables,x and
y,andgraphedinthexy-plane. Wethentrytofindafunctiony =f(x)whose
graph comes as close as possible to the data. This material is found in a new
Chapter2andcanbeskippedwithoutanylossofcontinuityintheremainderof
thetext. Curve-fittingexercisesareclearlymarkedassuch.
6. The text is now technology independent. Graphing calculators or computers
workjustaswellwiththetext.
7. Adiskwithinteractiveillustrationsisnowincludedwitheachtext. Thesein-
teractiveillustrationsprovidethestudentandinstructorwithwonderfuldemon-
strationsofmanyoftheimportantideasinthecalculus. Theyappearinevery
chapter. These demonstrations and explorations are highlighted in the text at
appropriatetimes. Theyprovideanextraordinarymeansofobtainingdeepand
clearinsightsintotheimportantconcepts. Weareextremelyexcitedtopresent
theseinthisformat.
v
vi AnOverviewofThirdEditionChanges
Chapter1. Functions. Thischapternowcontainsfivesections: 1.1,Functions;
1.2,MathematicalModels;1.3,ExponentialModels;1.4,CombinationsofFunctions;
and1.5,Logarithms. Thematerialthatcoveredmodelingwithleastsquareshasall
beenmovedtoanewChapter2. Mostofthematerialinthesectionsonquadraticsand
specialfunctionshasbeenmovedtotheReviewAppendix. Ageometricdefinition
ofcontinuitynowappearsinthefirstsection.
Chapter2. ModelingwithLeastSquares. Thisisanewchapterandplacesall
thematerialonleastsquaresthatwasoriginallyinChapter1intothisnewchapter.
Instructorswhowishcanignorethematerialinthischapter.
Chapter 3. Limits and the Derivative. This chapter has been substantially
revised. The material on the limit definition of continuity is now an “enrichment”
subsectionofthefirstsectiononlimitsandisnotneededintheremainderofthetext.
Thematerialonlimitsatinfinityhasbeenmovedtoalaterchapter. Thesectiononrates
ofchangenowhasmoreexamplesofaverageratesofchange. Moreemphasisisput
oninterpretationsofratesofchangeandonunits. Theoldsectiononderivativeshas
beenmadeintotwosections,thefirstonderivativesandthesecondonlocallinearity.
Thenewsectiononderivativeshasmoreemphasisongraphingthederivativegiven
thefunctionandalsooninterpretations. Thesectiononlocallinearitynowincludes
marginalanalysisandtheeconomicinterpretationofthederivativesofcost,revenue,
andprofits. Thislattermaterialwasformerlyinalaterchapter.
Chapter4. RulesforDerivatives. Thischapternowincludesmore“intuitive,”
thatis,geometricalandnumerical,sketchesofanumberofproofs,theformalproofs
beinggiveninenrichmentsubsections. Thus,ageometricalsketchoftheproofforthe
derivativeofaconstanttimesafunctionisgiven,andnumericalevidencefortheproof
forthederivativeofthesumoftwofunctionsisgiven. Theformalproofsofthese,
togetherwiththeproofofthederivativeoftheproduct,areinoptionalsubsections.
More geometrical insight has been added to the chain rule, and more emphasis is
putondeterminingunits. Themoredifficultproofsintheexponentialandlogarithm
sectionhavebeenplacedinanenrichmentsubsection. Elasticityofdemandnowhas
it’sownsection. Theintroductorymaterialonelasticityhasbeenrewrittentomake
thetopicmoretransparent. Thelastsectiononapplicationsonrenewableresources
hasbeenupdatedwithtimelynewmaterial.
Chapter5. CurveSketchingandOptimization. Thischapterhasbeenexten-
sivelyreorganized. Thesecondsectiononthesecondderivativenowcontainsonly
materialspecifictoconcavityandthesecondderivativetestandismuchshorterand
muchmoremanageable. Thematerialonadditionalcurvesketchingthatwasprevi-
ouslyinthissectionhasbeengivenitsownsection,Section5.4. Limitsatinfinityare
nowdiscussedinSection3,havingbeenmovedfromanearlierchapter. Itisinthis
chapterthatthismaterialisactuallyused, soitseemsappropriatethatitbelocated
here. The old section on optimization has been split into two sections, the first on
absoluteextremaandthesecondonoptimizationandmathematicalmodeling.Anew
sectiononthelogisticcurvehasbeencreatedfrommaterialfoundscatteredinvarious
sections. Withitsownsection, newmaterialhasbeenaddedtogivethisimportant
model its proper due (although instructors can omit this material without effecting
theflowofthetext).
Chapter6. Integration. Thesectiononsubstitutionhasbeenrefocusedtohave
a more intuitive as opposed to formal approach and is now more easily accessible.
To the third section, on distance traveled, more examples of Riemann sums have
beenadded,andtakingthelimitasn→∞ispostponeduntilthenextsection. The
sectiononthedefiniteintegralnowcontainssomepropertiesofintegralsthatwere
notfoundinthelastedition. Thesectiononthefundamentaltheoremofcalculushas
AnOverviewofThirdEditionChanges vii
beenextensivelyrewritten,withadi(cid:1)fferentproofofthefundamentaltheoremgiven.
x
Wefirstshowthatthederivativeof f(t)dt isf(x)usingageometricargument
a
usingthenewpropert(cid:1)iesofintegralsthatwereincludedintheprevioussectionand
b
thenproceedtoprove f(t)dt =F(b)−F(a),whereF isanantiderivative. The
a
moreformalproofisgiveninanenrichmentsubsection. Finally,anewSection6.7
has been created to include the various applications of the integral that had been
scatteredinprevioussections.
Chapter7. AdditionalTopicsinIntegration. Theinteractiveillustrationsin
thenumericalintegrationsectionyieldconsiderableinsightintothesubject. Students
canmovefromonemethodtoanotherandchooseanynandseethegraphsandthe
numericalanswersimmediately.
Chapter 8. Functions of Several Variables. Graphing in several variables
andvisualizingthegeometricinterpretationofpartialderivativesisalwaysdifficult.
Thereareseveralinteractiveillustrationsinthischapterthatareextremelyhelpfulin
thisregard.
Chapter9. TheTrigonometricFunctions. Thischaptercoversanintroduction
tothetrigonometricfunctions,includingdifferentiationandintegration.
Chapter 10. Taylor Polynomials and Infinite Series. This chapter covers
Taylor polynomials and infinite series. Sections 10.1, 10.2, and 10.7 constitute a
subchapteronTaylorpolynomials. Section10.7iswrittensothatthereadercango
fromSection10.2directlytoSection10.7.
Chapter11. ProbabilityandCalculus. Thischapterisonprobability. Section
11.1isabriefreviewofdiscreteprobability. Section11.2considerscontinuousprob-
abilitydensityfunctionsandSection11.3presentstheexpectedvalueandvariance
ofthesefunctions. Section11.4coversthenormaldistribution.
Chapter 12. Differential Equations. This chapter is a brief introduction to
differentialequationsandincludesthetechniqueofseparationofvariables,approx-
imate solutions using Euler’s method, some qualitative analysis, and mathematical
problemsinvolvingtheharvestingofarenewablenaturalresource.
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Preface
Calculus: Applications and Technology is designed to be used in a one- or two-
semestercalculuscourseaimedatstudentsmajoringinbusiness,management,eco-
nomics,orthelifeorsocialsciences. Thetextiswrittenforastudentwithtwoyears
of high school algebra. A wide range of topics is included, giving the instructor
considerableflexibilityindesigningacourse.
Since the text uses technology as a major tool, the reader is required to use a
computer or a graphing calculator. The Student’s Suite CD with the text, gives all
thedetails,inuserfriendlyterms,neededtousethetechnologyinconjunctionwith
thetext. Thistext,togetherwiththeaccompanyingStudent’sSuiteCD,constitutesa
completelyorganized,self-contained,user-friendlysetofmaterial,evenforstudents
withoutanyknowledgeofcomputersorgraphingcalculators.
Philosophy
The writing of this text has been guided by four basic principles, all of which are
consistentwiththecallbynationalmathematicsorganizationsforreformincalculus
teachingandlearning.
1. TheRuleofFour: Whereappropriate,everytopicshouldbepresentedgraph-
ically,numerically,algebraically,andverbally.
2. Technology: Incorporatetechnologyintothecalculusinstruction.
3. The Way ofArchimedes: Formal definitions and procedures should evolve
fromtheinvestigationofpracticalproblems.
4. Teaching Method: Teach calculus using the investigative, exploratory ap-
proach.
TheRuleofFour. Byalwaysbringinggraphicalandnumerical,aswellasalge-
braic,viewpointstobearoneachtopic,thetextpresentsaconceptualunderstanding
ofthecalculusthatisdeepandusefulinaccommodatingdiverseapplications. Some-
timesaproblemisdonealgebraically,thensupportednumericallyand/orgraphically
(withagrapher). Sometimesaproblemisdonenumericallyand/orgraphically(with
agrapher),thenconfirmedalgebraically. Othertimesaproblemisdonenumerically
orgraphicallybecausethealgebraistootime-consumingorimpossible.
Technology. Technology permits more time to be spent on concepts, problem
solving,andapplications. Thetechnologyisusedtoassistthestudenttothinkabout
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