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Chapter66.1ExponentialGrowthandDecayFunctionsExponentialGrowthandDecayFunctionsDefinitions:
Exponentialfunction:intheformof! = #$%where$ ≠ 1andmustbepositive.ExponentialGrowth:if# > 0#+,$ > 1then! = #$%isanexponentialgrowthfunctionExponentialDecay:if# > 0#+,0 < $ < 1then! = #$%isanexponentialdecayfunctionGrowth/DecayFactor:the“b”valueAsymptote:isalinethatagraphapproachesmoreandmoreclosely.(Doesnottouch!)
Example1:GraphingExponentialGrowthandDecayFunctionsTellwhethereachfunctionrepresentsexponentialgrowthorexponentialdecay.Thengraphthefunction.
a) ! = 2% b)! = /0
%
CriticalThinking:Yougoouttoeatatanicerestaurantandyoureceivethecheck.Thecheckisfor$78andyouwouldliketoleavea20%tip.Calculatethetotalofthecheckandtip.
ExponentialModelsExponentialGrowthModel:! = # 1 + 2 3 ExponentialDecayModel:! = # 1 − 2 3
***IMPORTANTNOTE!Whenusingapercentage“r”isalwayswrittenindecimalform!***
Example2:SolvingaReal-LifeProblemThevalueofacary(Inthousandsofdollars)canbeapproximatedbythemodel! = 25 0.85 3,wheretisthenumberofyearssincethecarwasnew.
a) Tellwhetherthemodelrepresentsexponentialgrowthordecay.
b) Identifytheannualpercentincreaseordecreaseinthevalueofacar.
c) Estimatewhenthevalueofthecarwillbe$8,000.Example3:WritinganExponentialModelIn2000,theworldpopulationwasabout6.09billion.Duringthenext13years,theworldpopulationincreasedbyabout1.18%eachyear.
a) Writeanexponentialgrowthmodelgivingthepopulationy(inbillions)tyearsafter2000.
b) Estimatetheworldpopulationby2005.
c) Estimatetheyearwhentheworldpopulationwas7billion.
Example4:RewritinganExponentialFunctionTheamounty(ingrams)oftheradioactiveisotopechromium-51remainingaftertdaysis! = 0.5 3/09,whereaistheinitialamount(ingrams).Whatpercentofthechromium-51decayseachday?CompoundInterest(Wealllovemoney!)
: = ; 1 + 2+
<3
Example5:FindingtheBalanceinanAccount.Youdeposit$9000inanaccountthatpays1.47%annualinterest.Findthebalanceafter3yearswhentheinterestiscompoundedquarterly.
Homework6,8,9-18(don’tgraph),19-21,24,29-35odd*,38,51
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