WPCg  IB.%XOV ݝ|{MK^ FʺcV9`9 !ksGqv[A*odű坚cJvXtw4<|gJŵ , H 9HxϷ5G  2YI9L<$gOB3i3/pEB5jGemQ XjVy ;@ `       M)  , ,  X5XX XX]XX X5 @  4X@  4  @@X]XXX]X]XXX] X]XXX]X]XXX]  @    1  A    (2M$ !  .,title  @    XXXC*jp`BookmanC XXXC*jp`BookmanC<g:paragraph -  ` ,..,     ` @ ` 0-/.setoff@      ` ,..,` !XXXC*jp`BookmanCXXXC*jp`BookmanC `     HP LaserJet III PostScriptPSCRIPT,,,,,,0 9DBWʌ wGTR3|xb(9 Z 6Times New Roman Regular X($ !     X5XX XX]XX X5   4X@  E1` (#&)\+- 0d247l9;>4E@@  ACalculusIProject:  DiscoveringtheDerivativeofanExponentialFunctionAnneLudingtonYoungaaDepartmentofMathematicalSciencesLoyolaCollegeinMarylandBaltimoreMD21210bbAbstract:Numericalestimatesofthederivativeofa^xareplotted.Exponentialcurvefittingtechniquesyieldthealgebraicformula.Thisbasicprojectleadstoextracreditproblems.* 4` (#&X*  4  @@X]XXX]X]XXX] X]XXX]X]XXX]  t   ` @,..,$ ` ` $@@      Iteachareformed,scientificcalculuscourseusingtheHughesHallet H  text.Asinmanysuchcourses,Iassignseveralprojectsduringthesemester.Theseprojectsrequirestudentstoworkinteamsoftwoorthree.Projectsarelongerandmorecomplicatedthanhomeworkproblemsorroutinelaboratoryexercises.Likealmostallworkinthecourse,projectsrequiretheuseoftechnology.Inaddition,theyhaveasignificantwritingcomponent.  ThefirstprojectthatIassigninCalculusIleadstotheformulaforthederivativeofanexponentialfunction.Itgivesstudentstheopportunitytoparticipateintheexcitingmathematicalprocessofmakingdiscoveriesandconjectures.IassigntheprojectaftercompletingChapter2.Atthispointstudentshavestudiedthemeaningofthederivativeandknowhowtoestimateitgraphicallyandnumerically.Theyhavenotyetlearnedanyalgebraicformulas.  Sincethemajorityoftheclassarefirstyearstudentsandthisistheirfirstproject,Idivideitintothreeparts.InPartIstudentsnumericallyestimatethederivativesoffourexponentialfunctions;e.g.,1.83x,2.83x, &! 3.83xand4.83x.InPartIItheywritealettertoaclassmateexplaining '# theirsolutionstoPartI.Finally,inPartIIItheywriteanewsletterarticlesummarizingtheirconclusionsand/orconjectures.TheactualprojectasIdistributeittomystudentsappearsattheendofthispaper. 4+&" Ї  TheprojectallowsstudentstoapplyandintegrateseveraldifferenttopicsthattheyhavestudiedinChapters1and2.FromChapter2theyknowhowtoestimatenumericallythederivativeatapoint.Byrepeatingthisprocessatseveralpoints,theygenerateatableforthederivativefunction.InChapter1studentslearnedhowtofitaformulatoatableand/oragraph.Inparticulartheystudiedthisprocessforanexponentialfunction.PartIoftheprojectrequiresputtingthesestepstogether.Thisissurprisinglydifficultforsomestudents.`   ݀IhavestudentsturnintheiranswerstoPart1beforetheyproceedtotheothertwoparts.Thisallowsmetopointoutmistakesand/ormakecomments.Forexample,usuallyonegroupwilltrytofitdifferenttypesoffunctionstothederivativesofthevariousfunctions;e.g.,alinetothefirst,aquadratictothesecondandanexponentialtothethird.Othergroupsmayusetheeaxformofanexponentialfunction;inthiscase,I  notethealternativeform.Stillotherstudentsmayapproximatethederivativeof1.83xwithsomethinglikec 1.8299x.Inthiscase,Ipoint   outthatroundedtotwodecimalplacesthederivativeisc 1.83x. B ̀PartIIoftheprojectgivesthestudentsatargetedwritingassignment.TheymustwritealettertoasickclassmateexplainingtheirsolutionstoPartI.Ipenalizestudentsiftheydonotwritealetteroriftheyfailtoincludewishesforaquickrecovery.Mostteamsaresuccessfulwiththispartoftheassignmentandstudentsseemtoenjoyit. @ InPartIIIstudentsmustwriteanewsletterarticlefortheirclassmatessummarizingtheteamsconclusionsand/orconjectures.WhatdoIexpect/hopethatstudentswillincludeintheirarticles?First,thatthederivativeofanexponentialfunctionseemstoanexponentialfunction.Second,thatthebaseseemstoremainthesame.Third,ifD(bx)=k bx, %  thenk=f'(0).Mostteamsdonotmakealloftheseobservations.Infact,manydonotevenmakethefirst.Becausetheydonotunderstandwhatitmeanstodrawconclusionsfromtheirobservations,somegroupsenduprepeatingtheirletters. 4+&" ЀItisperhapsnotsurprisingthatstudentshaveagreatdealofdifficultywithPartIII.Studentshavelittleornoexperiencemakingmathematicalconjectures.Evenwhentheyhavewrittenthederivativesof1.83xand2.83xask1 1.83xandk2 2.83x,respectively,theyare f unabletoconjecturethatthederivativeofbxisk bx. 8  ̀TheprojectleadstoseveraladditionalproblemswhichIofferasextracreditpossibilities.Eachoftheseproblemsisbasedontheassumptionthatthederivativeoff(x)=bxisf'(x)=f'(0) bx.TheseproblemsasI    distributethemtomystudentsappearattheendofthispaper.̀Inthefirstproblem,studentsaretoestimatethenumberBforwhichD(Bx)=Bx.Ofcourse,theresultisanestimatefore.Inthesecond,they H  aretoestimatek(b)whereD(bx)=k(b) bx.Thisleadstotheformula j  D(bx)=lnb bx.Finally,inthethirdproblem,studentsaretousethe  limitdefinitionofderivativetoalgebraicallyderiveD(bx)=f'(0) bx.  ̀Ifstudentscorrectlysolveanyoftheseadditionalproblems,Idistributetheirnewsletterstotheentireclass.Inanycase,Idiscussmysolutionstoallthreeproblems.Theyarethebasisofmydiscussionforthealgebraicapproachtotheexponentialfunction.  @@X]XXX]X]XXX] X]XXX]X]XXX]   @  @  @  @  @  @  @  @  @  @  @  @  @  @  @  @  @  @  @  @  @  @  @  @  @  @  @  @  @  @  @  @  @  @  @  @  @  @  @  @ STUDENTVERSIONOFTHE&  ԀPROJECT      PartI:̀Youandyourpartne'rhavebeenassigneda2digitnumber,gh.Useghtoformanewnumber,1.gh.Forexample,ifyournumberis83,thenyournewnumberis1.83. @ Nowconsiderthefunctionf(x)=(1.gh)x.(Intheexampleabove,the h$ functionwouldbef(x)=(1.83)x.)Estimatethederivativeoff(x)ateachof %  thefollowingpoints:   @` x=!1.5,!1,!0.5,0,0.5,1,1.5.  Recordyouranswersontheworksheet.Whenyouhavefinished,youwillhavegeneratedatable.Plotthesevalues.Findaformulathatfitsthedatainthetable.Addthegraphoftheformulatoyourplot. 4+&" ЀNow,repeattheabovedirectionsfor3otherfunctions: @ f(x)=(2.gh)x,f(x)=(3.gh)xandf(x)=(4.gh)x. " "ӀWhenyouhavefinished,turninyourworksheetsandaccompanyinggraphs.Iwillgradethem,verifyingcorrectanswersandindicatingmistakes.Theseworksheetsareduenolaterthandayd.(Theprojectwasdistributedondayd!5.)?1` (#&)\+- 0d247l9;>?@@  PartII:"$` (#&X$  ݀SusanisaLoyolastudentwhoistakingCalculusI.ShortlyafterherclasscompletedSection2.3,Susanwentskiingfortheweekend.Unfortunately,shewasinanaccidentandbrokeherleginseveralplaces.Althoughsheisrecovering,sheiscurrentlyintractioninthehospital.Happily,sheisabletoresumestudyingcalculus. @ WritealettertoSusandescribingyoursolutionforf(x)=(1.gh)x.That j  is,tellherhowyouestimatedthevaluesinthetableandhowyoufoundtheformula.Susansbeenthroughalotlately,soshesalittlefuzzyoncalculus.Youshouldtellhernotonlywhatyoudid,butalsoremindheraboutwhyyoudidit. @ Susan,ofcourse,doesnothaveaccesstoanycomputersoftwareinthehospital.Shedoeshoweverhavehergraphingcalculator.Inyourletter,giveherenoughinformationsothatshecanreproduceyourresultsandthendof(x)=(2.gh)xonherown. z  @ YourlettertoSusanshouldbenomorethan2pages.Itmustbetyped;complicatedequationsandthelikecanbeneatlyhandwritten.Yourletterisdueondayd+5.?1` (#&)\+- 0d247l9;>?@@  PartI& D II:%($` (#&X$  ݀Th'D$"(issemesterwearegoingtopublishourownin-classnewsletter.Innewsletterarticles,studentswillreporttheirdiscoveries.Attheendofanarticle,studentsmaymakeconjecturesand/oraskquestions.Newsletterarticlesmustmeetthefollowingcriteria:4 4 x1.Articlesmustbetyped.However,complicatedequationscanbeneatlyhandwritten.2.Graphs/tablescanappearinthebodyofthearticleorattheend.Theymustbeclearlylabeled. 4+&" 3.Articlesmustbeatleasthalfapageandnomorethan2pagesinlength.Thispagelimitincludesanyaccompanyinggraphs/tables.4! ` 4 !  Lookoveryourworksheets.Usetheresultstowriteanewsletterarticle.Yourarticleisdueondayd+7.!  STUDENTVERSIONOFEXTRACREDITPROBLEMS,̀BasedonyourworkforProject1,itappearsthatthederivativeofanexponentialfunctionisanexponentialfunctionwiththesamebase.Thatis,thederivativeoff(x)=bxisf'(x)=k bx.Nowifwesubstitutex=0,we T  get  f'(0)=k b0=k. H  H1` (#&)\+- 0d247l9;>` H@@  Thus,itappearsthatthederivativeoff(x)=bxisf'(x)=f'(0) bx. j  #.- ` ` (#&X-  ̀Hereare3additionalproblems.Forextracredit,dooneofthemandsummarizeyourresults/conclusionsinanewsletterstylearticle.! 4` !1.Assumethatthederivativeoff(x)=bxisf'(x)=f'(0) bx.Thuswemight d saythatthe nicestbaseforanexponentialfunctionisthenumberBforwhichf'(0)=1.FindanestimateforthatparticularbaseB.Thatis,estimatethenumberBforwhichthederivativeoff(x)=Bxisf'(x)=Bx. z 2.Assumethatthederivativeoff(x)=bxisf'(x)=k bx.Theconstantk  dependsonb.Thus,wecouldwritef'(x)=k(b) bx.Asmentionedabove, 0 k=k(b)=f'(0).Now,forf(x)=bx, !R Ѐ ` k(b)=f'(0) (b0.01!1)/0.01. $"t E1` (#&)\+- 0d247l9;>4E@@  Estimatek(b)=f'(0)fordifferentvaluesofb.Thatis,generateatablecontainingbandk(b).Plotthepointsofthetableandtrytofitaformulatothetable.Thenuseyourformulafork(b)tofindaformulaforthederivativeoff(x)=bx. &! 3* 4` (#&X*  E1` (#&)\+- 0d247l9;>4E@@  3.Forf(x)=bx,writedownthelimitdefinitionsoff'(x)andf'(0).Explain (@$  whyalgebraicallyitmakessensethatf'(x)=f'(0) bx. *b%! '5* 4` (#&X*