Applications of Calculus PDF
Preview Applications of Calculus
Philip 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
The list of books you might like
