Getting the Most Out of the

GettingtheMostOutoftheFundamentalTheoremaby=f(t)xDanKennedyBaylorSchoolChattanooga,[e-mail]ustopicstoteachintheolddayswasRiemannsums.Theywerehardtodraw,hardtocompute,and(manyfelt)totallyunnecessary.Thatwaswhymostofusquicklymovedontoantiderivatives,whichishowwewantedstudentstodointegrals.Needlesstosay,whenwecametotheFundamentalTheorem,studentsfoundittobethegreatestanticlimaxinthecourse.Integrationanddifferentiationarereverseoperations?Well,duh.ThenalongcametheTIgraphingcalculators.UsingtheintegralutilityintheCALCmenu,studentscouldactuallyseeanintegralaccumulatingvaluefromlefttorightalongthex-axis,justasalimitofRiemannsumswoulddo:Sonowwecandoallkindsofsummingproblemsbeforeweevenmentionanantiderivative.Historically,that’swhatscientistshadtodobeforecalculus.Here’swhyitmatteredtothem:mi/hrhr14402060v(t)=40d=120mimi/hrhr14402060v(t)d=120miThecalculuspioneersknewthattheareawouldstillyielddistance,butwhatwastheconnectiontotangentlines?Andwasthereaneasywaytofindtheseirregularly-shapedareas?SincethetimeofArchimedes,scientistshadbeenfindingareasofirregularly-shapedregionsbydividingthemintoregularly-shapedregions.ThatiswhatRiemannsumsareallabout.2.0332812.0082482.000329Withgraphingcalculators,studentscanfindthesesumswithoutthetedium.Theycanalsoimaginethetediumofdoingthesesumsbyhand!Bestofall,theycanactuallyseethelimitingcase:Andthecalculatorshowsthethinrectanglesaccumulatingfromlefttoright–idealforunderstandingtheFTC!aby=f(t)Letusconsiderapositivecontinuousfunctionfdefinedon[a,b].Chooseanarbitraryxin[a,b].xEachchoiceofxdeterminesauniqueareafromatox,denotedasusualby()xaftdtaby=f(t)xThusisafunctionofxon[a,b].Whatisthederivativeofthisfunction?()xaftdt00()()()lim()limxhxxaaahxhxhftdtftdtdftdtdxhftdthaby=f(t)xx+h00()()()lim()limxhxxaaahxhxhftdtftdtdftdtdxhftdth()fxSo()().xadftdtfxdxButthatisonlyhalfthestory.Nowthatweknowthatisanantiderivativeoff,weknowthatitdiffersfromanyantiderivativeoffbyaconstant.Thatis,ifFisanyantiderivativeoff,()xaftdt()().xaftdtFxCTofindC,wecanplugina:()()0()()aaftdtFaCFaCCFaSo()()().xaftdtFxFaNowpluginb:()()().baftdtFbFaThiswastheFTC.Thiswastheresultthatchangedtheworld.2.000329Now,insteadofwastingafullafternoonjusttogetanapproximationoftheareaunderonearchofthesinecurve,youcouldfindoneantiderivative,plugintwonumbers,andsubtract!0sin()cos()cos()cos(0)2.0xdxxSince2000,theAPCalculusTestDevelopmentCommitteehasbeenemphasizingaconceptualunderstandingofthedefiniteintegral,resultinginthese“new”problemtypes:FunctionsdefinedasintegralsAccumulationProblemsIntegralsfromTablesFinding,givenandInterpretingtheDefiniteIntegral()fb()fa()fxProblemoftheDay#29:Suppose31()2xftdtxxk.(a)Findf(x).(b)Findk.312()2)()32xddftdtxxkdxdxfxx(a)BytheFundamentalTheorem,(b)Pluginx=1:131()12(1)011ftdtkkkQuickAside:MyFirstDayontheAPCalculusTestDevelopmentCommitteeIf2()fxx,whichofthefollowingcouldbethegraphof1()xyftdt?(A)(B)(C)(D)(E)Herewastheproblem(1987):Thisproblemhadbeenchecked:1.bytheauthorwhohadwrittenit;2.bythecommitteethathadokayedit;3.bythecommitteethathadokayeditforapre-test;4.bytheETStestdevelopmentspecialists;5.byourcommittee,reviewingthefinalformofthecollegepre-test.MycolleaguesweretwoproblemsfurtherintothetestwhenIaskedifwecouldgobackforanotherlook.Theproposedkeywas(B).Thatis,WhileeveryonewasconcentratingontheFundamentalTheoremapplication,theyhadmissedthehidden“initialcondition”thatymustequalzerowhenx=1!If2()fxx,thegraphof1()xyftdtcouldbe(B)Here’s1995/BC-6:12-1-2-3123450Letfbeafunctionwhosedomainistheclosedinterval[0,5].Thegraphoffisshownabove.Let320()().xhxftdt(a)Findthedomainofh.(b)Find(2).h(c)Atwhatxish(x)aminimum?Showtheanalysisthatleadstoyourconclusion.(a)Thedomainofhisallxforwhichisdefined:320()xftdt035642xx(b)AlittleChainRule:3201()()32213(2)(4)22xdxhxftdtfdxhf12-1-2-3123450Letfbeafunctionwhosedomainistheclosedinterval[0,5].Thegraphoffisshownabove.Let320()().xhxftdt(a)Findthedomainofh.(b)Find(2).h(c)Atwhatxish(x)aminimum?Showtheanalysisthatleadstoyourconclusion.(c)Sinceispositivefrom-6to-1andnegativefrom-1to4,theminimumoccursatanendpoint.Bycomparingareas,h(4)h(-6)=0,sotheminimumoccursatx=4.320()xftdtThis“areacomparison”genreofproblemwasprettycommonintheearlygraphingcalculatordays.AccumulationProblemsPerhapsthemostgroundbreakingchangeinthe1998APCourseDescriptionwasthedecisionnottolistthe“applicationsofintegration”thatastudentshouldknow.Instead,studentswouldbeexpectedtohaveenoughfamiliaritywith“accumulation”problemstomodelthemwithintegralsinfreshsituations.Theexamssincethenhaveprovidedanabundanceoffreshsituations!Allstudentsshouldknowhowtointerpretthefollowingapplicationsasaccumulations:Areas(sumsofrectangles)Volumes(sumsofregular-shapedslices)Displacements(sumsofv(t)∙∆t)Averagevalues(Integrals/intervals)BC:Arclengths(sumsofhypotenuses)BC:Polarareas(sumsofsectors)ProblemoftheDay#27:Findtheaveragelengthofallchordsofacircleofradiusrthatareperpendiculartoagivendiame