Exponential Functions 指数函数

1.3ExponentialFunctionsGregKelly,HanfordHighSchool,Richland,WashingtonPhotobyVickieKelly,2008AcadiaNationalPark,MaineAlthoughsomeoftoday’slectureisfromthebook,someofitisnot.Youmusttakenotestobesuccessfulincalculus.WewillbeusingtheTI-89calculatorinthisclass.YoumayuseeithertheTI-89TitaniumortheolderTI-89.TI-89TitaniumThepicturesinthelectureswillusuallyillustratetheolderTI-89.AlthoughthebuttonsontheTitaniumEditionaredifferentshapesandcolors,theyareinthesamepositionsandhavethesamefunctions.TI-89If$100isinvestedfor4yearsat5.5%interest,compoundedannually,theendingamountis:4100123.881.055OntheTI-89:1001.055^4ENTERAttheendofeachyear,interestispaidontheamountintheaccountandaddedbackintotheaccount,sotheamountofincreasegetslargereachyear.Thisisanexampleofanexponentialfunction:xfaxexponentbaseGraphforina[-5,5]by[-2,5]window:xya2,3,5aMODEGraph…….FUNCTIONDisplayDigits…FLOAT6Angle…….RADIANENTERY=12^yxx23^yxx35^yxxWINDOWGraphforina[-5,5]by[-2,5]window:xya2,3,5aY=12^yxx23^yxx35^yxxWINDOWGRAPHGraphforina[-5,5]by[-2,5]window:xya2,3,5aWhereis?235xxx0,xWhereis?235xxx,0xWhereis?235xxx0xGraphforina[-5,5]by[-2,5]window:xya2,3,5aWhereis?235xxx0,xWhereis?235xxx,0xWhereis?235xxx0xWhatisthedomain?,Whatistherange?0,Populationgrowthcanoftenbemodeledwithanexponentialfunction:Ratio:502349361.0176511150231.01751.01761.02461.0175WorldPopulation:19864936million1987502319885111198952011990532919915422Theworldpopulationinanyyearisabout1.018timesthepreviousyear.in2010:1954221.018P7609.7About7.6billionpeople.Nineteenyearspast1991.Radioactivedecaycanalsobemodeledwithanexponentialfunction:Supposeyoustartwith5gramsofaradioactivesubstancethathasahalf-lifeof20days.Whenwilltherebeonlyonegramleft?After20days:1552240days:251542tdays:20152tyInPre-Calcyousolvedthisusinglogs.Todaywearegoingtosolveitgraphicallyforpractice.Y=11yx25(1/2)^(/20)yxxWINDOWGRAPHWINDOWGRAPHUpperboundandlowerboundarex-values.F5Math5IntersectionUsethearrowkeystoselectafirstcurve,secondcurve,lowerboundandupperbound,andpressENTEReachtime.46daysManyreal-lifephenomenacanbemodeledbyanexponentialfunctionwithbase,where.e2.718281828eecanbeapproximatedby:11xfxxAs,xfxeGraph:y=(1+1/x)^xina[-10,10]by[-5,10]window.Use“trace”toinvestigatethefunction.TblSetWecanhavethecalculatorconstructatabletoinvestigatehowthisfunctionbehavesasxgetsmuchlarger.tblStart…….1000tbl………..1000ENTERENTERENTERTABLEMovetothey1columnandscrolldowntowatchtheyvalueapproache.pTheTI-89hastheexponentialgrowthanddecaymodelbuiltinasanexponentialregressionequation.Aregressionequationstartswiththepointsandfindstheequation.U.S.Population:188018901900191019201930194019501960197050.2million63.076.092.0105.7122.8131.7151.3179.3203.3Tosimplify,letrepresent1880,represent1890,etc.0x1x0,1,2,3,4,5,6,7,8,9L1ENTER2nd{0,1,2,3,4,5,6,7,8,92nd}STOalphaL1ENTER50.2,63,76,92,105.7,122.8,131.7,151.3,179.3,203.3L2ExpRegL1,L2ENTER2ndMATH63StatisticsRegressions2ExpRegalphaL1alphaL2ENTERDoneThecalculatorshouldreturn:,(UppercaseLusedforclarity.)50.2,63,76,92,105.7,122.8,131.7,151.3,179.3,203.3L2ShowStatENTER2ndMATH68StatisticsShowStatENTERThecalculatorgivesyouanequationandconstants:xyab55.0542581.160626abExpRegL1,L2ENTER2ndMATH63StatisticsRegressions2ExpRegalphaL1alphaL2ENTERDoneThecalculatorshouldreturn:,Wecanusethecalculatortoplotthenewcurvealongwiththeoriginalpoints:Y=y1=regeq(x)2ndVAR-LINKregeqx)Plot1ENTERENTERWINDOWPlot1ENTERENTERWINDOWGRAPHWINDOWGRAPHWhatdoesthisequationpredictforthepopulationin1990?F3TraceThisletsusseevaluesforthedistinctpoints.Movestotheline.Thisletsustracealongtheline.11ENTEREntersanx-valueof11.Whatdoesthisequationpredictforthepopulationin1990?11ENTEREntersanx-valueof11.In1990,thepopulationwaspredictedtobe283.4million.Thisisanoverestimateof33million,or13%.Whymightthisbe?Tofindtheannualrateofgrowth:Sinceweused10yearintervalswithb=1.160626:101.160631r101.160631r0.015ror1.5%p