exponential decay - michigan technological university指数衰减-密歇根理工大学

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1、ExponentialDecayAugust18,2008MelanieHarmalaPre-CalculusStatementofProblem:Howcanweusewhatweknowaboutexponentialgrowthanddecaytosolveproblemsinvolvingdecayofradioactiveelements?Wewillusetheformulaswehavelearnedforexponentialgrowthandapplythemtoexponentia

2、ldecay,specificallyinthedecayofradioactiveelements.StudentswillapplyknowledgegainedattheAlgebraIIlevelandinthepreviousseverallessonsonexponentialfunctions.Wewillalsodiscussthediscoveryoftherelevanceofeanditsuseinsolvingproblemsinvolvingcontinuousgrowtho

3、rdecay.IntendedOutcomes:§Explainthepartsoff(x)=a.bxwithrespecttoexponentialdecay§Knowthate=lim(1+1/n)nforn®¥§Knowandusetherelationshipbetweenkandhalflife§UseA=Ao.ekttosolveexponentialdecayproblemsMaterials:§Paper§Pencil§Scientificcalculator§Dice§Microso

4、ftExcelforenteringandgraphingdataSafetyConsiderations:notapplicableLessonandActivities:Teachercommentsandquestionsareinitalics.Possibleintendedstudentresponsesareunderlined.Day1We’vetalkedaboutexponentialfunctionsanddonemanyexamplesofexponentialgrowth.T

5、odaywe’llfocusonexponentialdecay.Howdoyouthinkexponentialdecaydiffersfromexponentialgrowth?Howisitsimilar?Whatisthebasicformofanexponentialequation?Somethingisgettingsmallerinsteadofgrowingbigger.Weusethesameequation,butthebaseislessthanone.F(x)=a.bx.If

6、weusetheequationf(x)=a.bx,whatchangeswhenwe’retalkingaboutdecayinsteadofgrowth?Thebasebwouldbebetween0and1.Thebaseisgreaterthan1forexponentialgrowth.Themostcommonapplicationofexponentialdecayisthedecayofcertainradioactiveelements.Withradioactiveelements

7、,werefertoitshalf-life.Half-lifereferstotheamountoftimeittakesforhalfofaradioactiveelementtodecay.DoesanyonerememberwhatequationyouusedinAlgebraIItosolveproblemsinvolvinghalf-life?Whatequationdidweuseforasituationsuchascompoundinginterest?Howcouldwechan

8、geittomodeldecayinsteadofgrowth?A=Ao(1+r)n.Insteadof(1+r)use(1–r)orsomethingsimilar.Ifwearetalkingabouthalf-life,whatishappeningtothequantity(1+r)?Remember,ifwe’reaccumulatinginterestat3%,theexpressionwouldbe(1+.03).We’relosingha