Table Of ContentPhilip Straffin, Editor
Volume 3
Resources for Calculus
A. Wayne Roberts,
Project Director
MAA Notes Number 29
The Mathematical Association of America
APPLICATIONS OF CALCULUS
Philip Straffin, Editor
ResourcesforCalculusCollection
Volume3
APPLICATIONS OF CALCULUS
PhilipStraffin, Editor
AProjectof
TheAssociatedColleges oftheMidwestand
TheGreatLakes Colleges Association
Writers for thisVolume
ClarkBenson
NationalSecurityAgency
Paul Campbell
BeloitCollege
Kevin Hastings
KnoxCollege
StevenJanke
Colorado College
WalterMeyer
AdelphiUniversity
Thomas Moore
GrinnellCollege
PhilipStraffin
BeloitCollege
SupportedbytheNationalScienceFoundation
A.WayneRoberts,ProjectDirector
MAANotes Volume29
PublishedandDistributedby
TheMathematicalAssociationofAmerica
MAA Notes and Reports Series
The MAA Notes and Reports Series, started in 1982, addresses a broad range of topics and themes of interest to all who
are involved with undergraduate mathematics. The volumesin this series are readable, informative, and useful, and help the
mathematical communitykeepup with developmentsof importanceto mathematics.
Editorial Board
TinaStraley,Editor
Nancy Baxter-Hastings SylviaT.Bozeman
DonaldW. Bushaw SheldonP. Gordon
Stephen BMaurer RogerB.Nelsen
MaryR.Parker Barbara E.Reynolds (Sr)
DonaldG.Saari AnitaE.Solow
PhilipD.Straffin
MAA Notes
11. Keys to Improved Instruction by Teaching Assistants and Part-Time Instructors, Committee on TeachingAssistants and
Part-Time Instructors, Bettye Anne Case, Editor.
13. Reshaping CollegeMathematics, Committee on the Undergraduate Program inMathematics, LynnA.Steen, Editor.
14. Mathematical Writing,byDonaldE. Knuth,TracyLarrabee, and Paul M. Roberts.
16. Using WritingtoTeachMathematics, Andrew Sterrett,Editor.
17. Priming theCalculus Pump: Innovations and Resources,Committee on Calculus Reform and the First Two Years, a sub
comittee oftheCommitteeon theUndergraduateProgramin Mathematics, Thomas W. Tucker, Editor.
18. Models forUndergraduateResearch inMathematics,Lester Senechal, Editor.
19. VisualizationinTeachingandLearning Mathematics,Committee on Computers inMathematics Education, Steve Cunning-
ham and WalterS.Zimmermann, Editors.
20. The LaboratoryApproach toTeachingCalculus,L. CarlLeinbach etal., Editors.
21. PerspectivesonContemporaryStatistics, David C. Hoaglin and DavidS.Moore, Editors.
22. HeedingtheCall for Change: SuggestionsforCurricularAction, LynnA.Steen, Editor.
24. SymbolicComputationin Undergraduate MathematicsEducation, Zaven A.Karian, Editor.
25. The Conceptof Function: Aspectsof EpistemologyandPedagogy,Guershon HarelandEd Dubinsky, Editors.
26. Statistics fortheTwenty-FirstCentury,Florenceand Sheldon Gordon, Editors.
27. ResourcesforCalculusCollection, Volume1:Learningby Discovery: A Lab Manual forCalculus,Anita E. Solow,Editor.
28. ResourcesforCalculus Collection, Volume2: CalculusProblems fora NewCentury,Robert Fraga, Editor.
29. ResourcesforCalculus Collection, Volume3:ApplicationsofCalculus, Philip Straffin, Editor.
30. ResourcesforCalculusCollection, Volume4:ProblemsforStudent Investigation,Michael B.Jackson andJohn R.Ramsay,
Editors.
31. ResourcesforCalculus Collection, Volume5:ReadingsforCalculus, Underwood Dudley, Editor.
32. EssaysinHumanisticMathematics, Alvin White,Editor.
33. Research Issuesin Undergraduate Mathematics Learning:Preliminary Analyses and Results, James J. Kaput and Ed Du
binsky,Editors.
34. InEves' Circles,Joby Milo Anthony, Editor.
35. You're theProfessor,What Next? IdeasandResourcesforPreparing CollegeTeachers, The Committee on Preparationfor
College Teaching, Bettye Anne Case, Editor.
36. PreparingforaNewCalculus: Conference Proceedings,Anita E. Solow, Editor.
37. A PracticalGuide to Cooperative Learning in CollegiateMathematics, Nancy L. Hagelgans, Barbara E. Reynolds, SDS,
KeithSchwingendorf, Draga Vidakovic, EdDubinsky,Mazen Shahin, G.Joseph Wimbish,Jr.
38. ModelsThatWork:CaseStudies in EffectiveUndergraduateMathematics Programs,Alan C.Tucker, Editor.
39. Calculus:TheDynamicsofChange, CUPMSubcommitteeon Calculus Reform andthe FirstTwoYears, A.WayneRoberts,
Editor.
40. VitaMathematica:Historical Research andIntegrationwithTeaching,RonaldCalinger; Editor.
41. GeometryTurnedOn: DynamicSoftware inLearning,Teaching,and Research,JamesR.KingandDorisSchattschneider;
Editors.
42. Resources for Teaching Linear Algebra, DavidCarlson, Charles R. Johnson, David C. Lay, A. DuanePorter; Ann E.
Watkins, William Watkins, Editors.
43. Student AssessmentinCalculus: A ReportoftheNSFWorkingGrouponAssessmentinCalculus,AlanSchoenfeld, Editor.
44. ReadingsinCooperativeLearningforUndergraduateMathematics,EdDubinsky, DavidMathews, andBarbaraE. Reynolds,
Editors.
45. ConfrontingtheCore Curriculum: ConsideringChangein theUndergraduateMathematics Major,JohnA.Dossey, Editor.
46. Womenin Mathematics:Scaling the Heights,DeborahNolan, Editor.
47. ExemplaryProgramsin Introductory CollegeMathematics:InnovativeProgramsUsing Technology, SusanLenker; Editor.
48. Writingin theTeachingandLearning ofMathematics,JohnMeierandThomas Rishel.
49. Assessment PracticesinUndergraduateMathematics,BonnieGold, Editor.
50. RevolutionsinDifferentialEquations: ExploringODEswith Modem Technology,MichaelJ.Kallaher; Editor.
51. Using HistorytoTeachMathematics: AnInternationalPerspective, VictorKatz, Editor.
52. TeachingStatistics: Resources forUndergraduateInstructors,Thomas 1.Moore, Editor.
53. GeometryatWork:Papersin Applied Geometry,Catherine A.Gorini, Editor.
54. TeachingFirst: AGuide forNew Mathematicians, Thomas W. Rishel.
55. Cooperative Learning in Undergraduate Mathematics:Issues That Matter and Strategies That Work,Elizabeth C. Rogers,
Barbara E.Reynolds, NeilA. Davidson, andAnthonyD.Thomas, Editors.
56. Changing Calculus: AReportonEvaluation EffortsandNational Impactfrom 1988to 1998,SusanL. Ganter.
Thesevolumescan beordered from:
MAAService Center
P.O.Box91112
Washington,DC 20090-1112
800-331-1MAA FAX:301-206-9789
Teachers may reproduce these modules for their students and they may be modified to suit particular classroom
needs. However, the modules remain the property of the Mathematical Association of America and the collection
may not be used for commercial gain.
SixthPrinting
@1999 bytheMathematicalAssociationofAmerica
ISBN0-88385-085-0
LibraryofCongressCatalogNumber92-62281
PrintedintheUnitedStatesofAmerica
CurrentPrinting
10 9 8 7 6
INTRODUCTION
RESOURCES FOR CALCULUS COLLECTION
Beginning with aconferenceatTulaneUniversity inJanuary, 1986,theredevelopedinthemathe
maticscommunity asense thatcalculuswasnotbeingtaughtinawaybefitting asubjectthatwas
atoncetheculminationofthesecondarymathematicscurriculumandthegatewaytocollegiatesci
enceandmathematics.Fartoomanyofthestudentswhostartedthecoursewerefailingtocomplete
itwithagradeofCorbetter,andperhapsworse,anembarrassingnumberwhodidcompleteitpro
fessed eithernottounderstanditornottolikeit,orboth. Formoststudents itwas notasatisfying
culminationoftheirsecondarypreparation,anditwasnotagatewaytofuture work. Itwasanexit
Muchofthedifficulty hadtodowith thedelivery system: classes thatwere toolarge, senior
faculty who hadlargely deserted thecourse, andteaching assistants whose timeandinterestwere
focused ontheirowngraduatework. Otherdifficulties camefrom wellintentionedeffortstopack
intothecourse allthetopicsdemandedbytheincreasingnumberofdisciplines requiring calculus
of theirstudents. It was acknowledged, however, thatifthecourse hadindeed become ablurfor
students, itjustmightbebecause those choosing thetopics tobepresented andthemethods for
presentingthemhadnotkepttheirgoalsinfocus.
Itwastotheselatterconcernsthatwerespondedindesigningourproject. Weagreedthatthere
oughttobeanopportunityforstudentstodiscoverinsteadofalwaysbeingtold. Weagreedthatthe
availability ofcalculators andcomputersnotonlycalled forexercisesthatwould notberendered
trivialbysuchtechnology, butwould infactdirectattention more toideas thanto techniques. It
seemedtousthatthereshouldbeexplanationsofapplicationsofcalculusthatwereself-contained,
andbothaccessibleandrelevanttostudents. Wewerepersuadedthatcalculusstudentsshould,like
students in anyothercollege course, have someassignments thatcalled forlibrary work, some
pondering, someimagination,andaboveall,aclearlyreasonedandwrittenconclusion. Finally,we
cametobelieve thatthere should beavailable tostudents some collateral readings thatwould set
calculusinanintellectualcontext.
Wereasonedthattheachievementofthesegoalscalledfortheavailabilityofnewmaterials,and
thattheuncertainty ofjustwhatmightwork, coupledwiththenumberofpeopletryingtoaddress
thedifficulties, calledforalargecollectionofmaterials from which individuals couldselect. Our
goalwas todevelopsuchmaterials, andtoencourage peopletousethem inanywaytheysawfit.
Inthisspirit, andwiththehelpoftheNoteseditorandcommitteeoftheMathematicalAssociation
ofAmerica, wehaveproduced five volumes ofmaterialsthatare,withtheexceptionofvolume V
where wedonotholdoriginalcopyrights,meanttobeinthepublicdomain.
We expectthatsomeofthese materials maybecopied directly andhanded toanentire class,
whileothersmaybegiventoasinglestudentorgroupofstudents.Somewillprovide abasisfrom
whichlocaladaptationscanbedeveloped. Wewillbepleasedifauthorsaskforpermission, which
weexpecttobegenerousingranting,toincorporateourmaterialsintotextsorlaboratorymanuals.
We hopethatinallofthese ways, indeed inanywayshortofreproducing substantial segmentsto
vii
viii
APPLICATIONSOFCALCULUS
sellforprofit, ourmaterial willbe usedtogreatlyexpand ideasabouthowthecalculus mightbe
taught.
ThoughI asProjectDirectorneverentertained theideathatwecouldwriteasingletextthat
wouldbeacceptabletoall26schoolsintheproject,itwasclearthatsomecommonnotionoftopics
essential to any calculus coursewould be necessary to give us direction. The task of forging a
commonsyllabus wasmanaged byAndy Sterrettwithatactandefficiency thatwasinstructiveto
usall, andtheproductofthiswork,anannotatedcoresyllabus, appears asanappendix inVolume
1.Someoftheothervolumes refertothissyllabustoindicatewhere,inacourse, certainmaterials
mightbeused.
Thisprojectwassituated intwoconsortiaofliberalarts colleges, notbecause weintendedto
developmaterials forthisspecific audience, butbecause ourschools providealargereservoirof
classroom teachers wholavishon calculus thesameattention agraduate faculty mightgiveto its
introductory analysis course. Ourschools, intheirtotality, wereequipped withmostvarieties of
computerlabs,andweincludedinourconsortiamanypeoplewhohadbecomenationalleadersin
theuseofcomputeralgebrasystems.
Wealsofeltthatourcampusesgaveusthecapabilitytotestmaterialsintheclassroom.Thesize
ofourschoolsenablesustoimplementanewideawithoutcuttingthrough theredtapeofalarger
institution, andwecanjustasquicklyreverseourselveswhenitisapparentthatwhatwearedoing
isnotworking. We arepracticedingoinginbothdirections. Continual testingofthematerials we
weredeveloping wasseenasanintegralpartofourproject, anactivity thatGeorgeAndrews, with
thetitleofProjectEvaluator,keptbeforeusthroughouttheproject.
Thevalueofourcontributions willnowbejudgedbythelargermathematicalcommunity, but
I wasrightinthinkingthatI couldfind inourconsortia thegreatabundance of talentnecessary
foranundertakingofthismagnitude. AnitaSolowbroughttotheprojectabackgroundofeditorial
workandquicklybecamenotonlyoneoftheeditorsofourpublications,butalsoapersontowhom
Iturnedforadviceregarding theprojectasawhole. PhilStraffin, drawingonhisassociation with
UMAP,wasanidealpersontoeditacollectionofapplications,andwasanotherpersonwhobrought
editorialexperiencetoourproject.Woody Dudley cametotheprojectasawriterwellknownfor
hiswittyandincisivecommentaryonmathematicalliterature,andwasanidealchoicetoassemble
acollectionofreadings.
Ourtwoeditorsleastexperiencedinmathematicalexposition,BobFragaandMicJackson,both
justifiedtheconfidenceweplacedinthem.Theybroughttotheprojectanenthusiasmandfreshness
from whichweallbenefited,andtheywereableatallpointsintheprojecttodrawuponanexcellent
corpsofgifted andexperiencedwriters. When,inthelastmonths oftheproject,MicJacksontook
anoverseasassignmentonanEarlhamprogram, itwaspossibletomoveJohnRamsay intoMic's
positionpreciselybecauseof theexcellent working relationship thathadexistedonthesewriting
teams.
Theentireteamoffive editors,projectevaluatorandsyllabuscoordinatorworked together as
a harmonious teamoverthe five yearduration of thisproject. Eachmember, in tum,developed
a groupof writers, readers, andclassroom usersas necessary to complete thetask. I believe my
chiefcontributionwastoidentifyandbringthesetalentedpeopletogether, andtoseethattheywere
supported bothfinancially andbythehumanresources available intheschools thatmakeuptwo
remarkableconsortia.
A.WayneRoberts
MacalesterCollege
1993
THE FIVE VOLUMES OF THE
RESOURCES FOR CALCULUS COLLECTION
1. Learning byDiscovery: ALab Manual for Calculus
Anita E.Solow, editor
Theavailability ofelectronic aidsforcalculatingmakesitpossibleforstudents, ledbygoodques
tions andsuggestedexperiments, todiscoverforthemselves numerous ideasonceaccessible only
onthebasisoftheoreticalconsiderations.Thiscollectionprovidesquestions andsuggestionson26
differenttopics.Developedtobeindependentof anyparticularhardwareorsoftware,thesemate
rialscanbethebasisofformalcomputerlabsorhomeworkassignments. Althoughdesignedtobe
donewiththehelpofacomputeralgebrasystem,mostofthelabscanbesuccessfullydonewitha
graphingcalculator.
2. Calculus Problems for a NewCentury
Robert Fraga, editor
Studentsstillneeddrillproblemstohelpthemmasterideasandtogivethemasenseofprogressin
theirstudies. Acalculatorcanbe usedinmanycases,however, torendertrivialalistoftraditional
exercises. Thiscollection, organizedbytopicscommonlygroupedinsectionsofatraditionaltext,
seekstoprovideexercisesthatwillaccomplishthepurposesmentionedabove,evenforthestudent
makingintelligentuseoftechnology.
3. Applications of Calculus
Philip Straffin, editor
Everyoneagreesthatthereshouldbeavailablesomeself-containedexamplesofapplicationsofthe
calculusthataretractable, relevant, andinterestingto students. Heretheyare,18innumber, in a
form tobe consultedby ateacherwanting to enricha course, to be handedoutto a classif it is
deemedappropriatetotakeadayortwoofclasstimeforagoodapplication,orto behandedto an
individualstudentwithinterestsnotbeingcoveredinclass.
4. Problems for Student Investigation
Michael B.Jackson andJohn R.Ramsay, editors
Calculusstudentsshouldbeexpectedtoworkonproblemsthatrequireimagination;outsidereading
andconsultation,cooperation,andcoherentwriting.Theyshouldworkonopen-endedproblemsthat
Ix
