In an exponential model, the decay rate is directly proportional to the quantity present. Thus, the decay rate is faster at the beginning of the process than it is later on.
Here is the differential equation (don't be scared, it's an elementary one as those things go):
dy/dt = ky the change of rate of the amt of stuff, y, with respect to time, t, is
directly proportional to y
dy/y = kdt mutiply both sides by dt and divide by y to "separate the variables"
ln y = kt + c integrate on the left with respect to y, on the right wrt t; c is the constant
of integration
y = e^(kt+c) rewrite the logarithmic expression as an exponential
e^c is the amount of "stuff" we start with. The constant of proportionality, k, depends on the circumstances -- drug, metabolism, etc in our case.
For exponential decay, k will be negative.
The same model applies to exponential growth, with k always positive.
All exponential functions are of the form y = Ae^kt. dy/dt = kAe^kt = kAy, i.e. the rate of change
of y wrt t is proportional to y.
BOYS! PLEASE BE NICE! No one knows everything about anything, but you all have valuable insights, so please continue to share and make an effort to keep your criticism constructive and not personal. THANK YOU!
Mary Rack
cg for mom 75/6
and
math instructor at Johnson County Community College, KS, USA
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