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ExponentialGrowthand Decay
LearningObjectives
A studentwill be able to:
- Applythe laws of exponentialand logarithmicfunctionsto a varietyof applications.
- Modelsituationsof growthand decayin a varietyof problems.
Whenthe rate of changein a substanceor populationis proportionalto the amountpresentat any time t,
we say that this substanceor populationis goingthrougheithera decayor a growth,dependingon the sign
of the constantof proportionality.
This kind of growthis calledexponentialgrowthand is characterizedby rapidgrowthor decay. For example,
a populationof bacteriamay increaseexponentiallywith time becausethe rate of changeof its population
is proportionalto its populationat a giveninstantof time (morebacteriamakemorebacteriaand fewer
bacteriamakefewerbacteria).The decompositionof a radioactivesubstanceis anotherexamplein which
the rate of decayis proportionalto the amountof the substanceat a giventime instant.In the business
world,the interestaddedto an investmenteachday, month,or year is proportionalto the amountpresent,
so this is also an exampleof exponentialgrowth.
Mathematically, the relationshipbetweenamount and time is a differentialequation:
Separatingvariables,
and integratingboth sides,