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Exponential Models
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Modeling growth and decay with functions of the form Abx+c or Aebx+c, as well as the concept of half-life for exponential decay.
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Exponential models
SL AI 2.5
An exponential model represents quantities that multiply repetetively by a constant factor b. The basic form of an exponential is bx, but any exponential can be written in the form Abx+k.
The graph of an exponential model is a curve that approaches a horizontal asymptote at y=k on one side, and has a y-intercept at (0,A+k). Because of the asymptote on an exponential graph, exponential models are good at describing behaviors that level off over time.
Exponential growth
SL AI 2.5
Exponential growth describes quantities that increase by the same factor over a certain amount of time. Algebraically, exponential growth is modeled by functions of the form
f(t)=Abt+c,
where b>1. b is called the growth factor.
Note: Aekt is another model for exponential growth if the instantaneous growth rate, k, is positive.
Stewart EJ, Madden R, Paul G, Taddei F (2005), CC BY-SA 4.0
Exponential decay
SL AI 2.5
Exponential decay describes quantities that decrease by the same factor over a certain amount of time. Exponential decay is modeled by functions of the form
f(t)=Abt+c,
where 0<b<1. b is called the decay factor.
Note: Aekt is another model for exponential decay if the instantaneous growth rate, k, is negative.
Concept of half-life
AHL AI 2.9
For any quantity that decays exponentially, the half-life is the amount of time it takes for the quantity to halve in value.
Calculating half-life
AHL AI 2.9
From any exponential decay model of the form f(t)=Abkt (0<b<1), the half-life, or time for the value of f to reach half of its current value, is given by t1/2=−klogb2.
Most commonly, given an equation of the form f(t)=Aekt, the half life is given by −kln2.
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