StewartCalcET8-01-04多媒體教學課件

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1FunctionsandModelsStewartCalcET8_01_04Copyright?CengageLearning.Allrightsreserved.

1.4ExponentialFunctionsStewartCalcET8_01_04ExponentialFunctionsThefunctionf

(x)=2xiscalledanexponentialfunctionbecausethevariable,x,istheexponent.Itshouldnotbeconfusedwiththepowerfunctiong

(x)=x2,inwhichthevariableisthebase.Ingeneral,anexponentialfunctionisafunctionoftheform

f

(x)=bxwherebisapositiveconstant.Let’srecallwhatthismeans.Ifx=n,apositiveinteger,thenStewartCalcET8_01_04ExponentialFunctionsIfx=0,thenb0=1,andifx=–n,wherenisapositiveinteger,thenIfxisarationalnumber,x=p

/q,wherepandqareintegersandq>0,thenButwhatisthemeaningofbxifxisanirrationalnumber?Forinstance,whatismeantbyor5

?StewartCalcET8_01_04ExponentialFunctionsTohelpusanswerthisquestionwefirstlookatthegraphofthefunctiony=2x,wherexisrational.ArepresentationofthisgraphisshowninFigure1.Wewanttoenlargethedomainof

y=2xtoincludebothrationaland

irrationalnumbers.Thereareholesinthegraphin

Figure1correspondingtoirrational

valuesofx.Wewanttofillintheholesbydefiningf

(x)=2x,where

x

,sothatf

isanincreasingfunction.Figure1Representationofy=2x,xrationalStewartCalcET8_01_04ExponentialFunctionsInparticular,sincetheirrationalnumbersatisfieswemusthaveandweknowwhat21.7and21.8meanbecause1.7and1.8arerationalnumbers.StewartCalcET8_01_04ExponentialFunctionsSimilarly,ifweusebetterapproximationsforweobtainbetterapproximationsforStewartCalcET8_01_04ExponentialFunctionsItcanbeshownthatthereisexactlyonenumberthatisgreaterthanallofthenumbers 21.7,21.73,21.732,21.7320,21.73205,…andlessthanallofthenumbers 21.8,21.74,21.733,21.7321,21.73206,…Wedefinetobethisnumber.Usingtheprecedingapproximationprocesswecancomputeitcorrecttosixdecimalplaces:StewartCalcET8_01_04ExponentialFunctionsSimilarly,wecandefine2x

(orbx,ifb>0)wherexisanyirrationalnumber.Figure2showshowalltheholesinFigure1havebeenfilledtocompletethegraphofthefunction

f

(x)=2x,x

Figure2y=2x,xrealFigure1Representationofy=2x,xrationalStewartCalcET8_01_04ExponentialFunctionsThegraphsofmembersofthefamilyoffunctionsy=bxareshowninFigure3forvariousvaluesofthebaseb.Figure3StewartCalcET8_01_04ExponentialFunctionsNoticethatallofthesegraphspassthroughthesame

point(0,1)becauseb0=1forb

0.Noticealsothatasthebasebgetslarger,theexponentialfunctiongrowsmorerapidly(forx>0).YoucanseefromFigure3thattherearebasicallythreekindsofexponentialfunctionsy=bx.If0<b<1,theexponentialfunctiondecreases;ifb=1,itisaconstant;andifb>1,itincreases.StewartCalcET8_01_04ExponentialFunctionsThesethreecasesareillustratedinFigure4.

Figure4(b)

y=1x(a)

y=bx,0<b<1(c)

y=bx,b>1StewartCalcET8_01_04ExponentialFunctionsObservethatifb

1,thentheexponentialfunctiony=bxhasdomainandrange(0,).

Noticealsothat,since(1/b)x=1/bx=b–x,

thegraphofy=(1/b)xisjustthereflectionofthegraphofy=bxaboutthey-axis.StewartCalcET8_01_04ExponentialFunctionsOnereasonfortheimportanceoftheexponentialfunctionliesinthefollowingproperties.Ifxandyarerationalnumbers,thentheselawsarewellknownfromelementaryalgebra.Itcanbeprovedthattheyremaintrueforarbitraryrealnumbersxandy.StewartCalcET8_01_04Example1Sketchthegraphofthefunctiony=3–2x

anddetermineitsdomainandrange.Solution:Firstwereflectthegraphofy=2x

[showninFigures2and5(a)]aboutthex-axistogetthegraphofy=–2x

in

Figure5(b).Figure5(a)

y=2x(b)

y=–2xFigure2y=2x,xrealStewartCalcET8_01_04Example1–SolutionThenweshiftthegraphofy=–2xupward3unitstoobtainthegraphofy=3–2x

in

Figure5(c).Thedomainisandtherangeis(,3).Figure5(c)

y=3–2xcont’dStewartCalcET8_01_04ApplicationsofExponentialFunctionsStewartCalcET8_01_04ApplicationsofExponentialFunctionsTheexponentialfunctionoccursveryfrequentlyinmathematicalmodelsofnatureandsociety.Hereweindicatebrieflyhowitarisesinthedescriptionofpopulationgrowthandradioactivedecay.Firstweconsiderapopulationofbacteriainahomogeneousnutrientmedium.Supposethatbysamplingthepopulationatcertainintervalsitisdeterminedthatthepopulationdoubleseveryhour.StewartCalcET8_01_04ApplicationsofExponentialFunctionsIfthenumberofbacteriaattimetisp

(t),wheretismeasuredinhours,andtheinitialpopulationisp(0)=1000,thenwehave

p(1)=2p(0)=2

1000

p(2)=2p(1)=22

1000

p(3)=2p(2)=23

1000Itseemsfromthispatternthat,ingeneral,

p

(t)=2t

1000=(1000)2tStewartCalcET8_01_04ApplicationsofExponentialFunctionsThispopulationfunctionisaconstantmultipleoftheexponentialfunctiony=2t,

soitexhibitstherapidgrowth.Underidealconditions(unlimitedspaceandnutritionandabsenceofdisease)thisexponentialgrowthistypicalofwhatactuallyoccursinnature.StewartCalcET8_01_04ApplicationsofExponentialFunctionsWhataboutthehumanpopulation?Table1showsdataforthepopulationoftheworldinthe20thcenturyandFigure8showsthecorrespondingscatterplot.Figure8ScatterplotforworldpopulationgrowthStewartCalcET8_01_04ApplicationsofExponentialFunctionsThepatternofthedatapointsinFigure8suggestsexponentialgrowth,soweuseagraphingcalculatorwithexponentialregressioncapabilitytoapplythemethodofleastsquaresandobtaintheexponentialmodel

P=(1436.53)?(1.01395)twheret=0correspondsto1900.

Figure9showsthegraphofthis

exponentialfunction

togetherwiththeoriginal

datapoints.Figure9ExponentialmodelforpopulationgrowthStewartCalcET8_01_04ApplicationsofExponentialFunctionsWeseethattheexponentialcurvefitsthedatareasonablywell.TheperiodofrelativelyslowpopulationgrowthisexplainedbythetwoworldwarsandtheGreatDepressionofthe1930s.StewartCalcET8_01_04Example3Thehalf-lifeofstrontium-90,90Sr,is25years.Thismeansthathalfofanygivenquantityof90Srwilldisintegratein25years.Ifasampleof90Srhasamassof24mg,findan

expressionforthemassm(t)thatremainsaftertyears.Findthemassremainingafter40years,correcttothenearestmilligram.(c)Useagraphingdevicetographm(t)andusethegraphtoestimatethetimerequiredforthemasstobereducedto5mg.StewartCalcET8_01_04Example3–Solution(a)Themassisinitially24mgandishalvedduringeach25-yearperiod,soStewartCalcET8_01_04Example3–SolutionFromthispattern,itappearsthatthemassremainingaftertyearsisThisisanexponentialfunctionwithbase(b)Themassthatremainsafter40yearsiscont’dStewartCalcET8_01_04Example3–Solution(c)Weuseagraphingcalculatororcomputertographthe

functioninFigure12.Wealsographthelinem=5andusethecursortoestimatethatm(t)=5whent

57.Sothemassofthesamplewillbereducedto5mgafterabout57years.cont’dFigure12StewartCalcET8_01_04TheNumbereStewartCalcET8_01_04TheNumbereOfallpossiblebasesforanexponentialfunction,thereisonethatismostconvenientforthepurposesofcalculus.Thechoiceofabasebisinfluencedbythewaythegraphofy=bxcrossesthey-axis.StewartCalcET8_01_04TheNumbereFigures13and14showthetangentlinestothegraphsof

y=2xandy=3xatthepoint(0,1).(Forpresentpurposes,youcanthinkofthetangentlinetoanexponentialgraphatapointasthelinethattouchesthegraphonlyatthatpoint.)Ifwemeasuretheslopesofthesetangentlinesat(0,1),wefindthatm

0.7fory=2xand

m

1.1fory=3x.Figure14Figure13StewartCalcET8_01_04TheNumbereItturnsout,thatsomeoftheformulasofcalculuswillbegreatlysimplifiedifwechoosethebasebsothattheslopeofthetangentlinetoy=bxat(0,1)isexactly1.

(SeeFigure15.)Figure15Thenaturalexponentialfunctioncrossesthey-axiswithaslopeof1.StewartCalcET8_01_04TheNumbereInfact,thereissuchanumberanditisdenotedby

thelettere.(ThisnotationwaschosenbytheSwissmathematicianLeonhardEulerin1727,probablybecauseitisthefirstletterofthewordexponential.)StewartCalcET8_01_04TheNumbereInviewofFigures13and14,itcomesasnosurprisethatthenumbereliesbetween2and3andthegraphofy=exliesbetweenthegraphsofy=2xandy=3x.

(Se