Exponential functions show up in lots of applications, ranging from financial calculations to heat transfer to bacterial growth. In this lab, we will work with exponential functions to model the concentration of a drug in a patient. Before we introduce the model, we need some general background on models of growth and decay.

In exponential growth and decay problems, the independent variable is
almost always *t*, representing time. Another convention is to always
write the exponential functions in terms of the natural constant
*e*. That is, the general function describing exponential growth is
written

where *A* is a constant and *k* is a positive constant. The general
form for exponential decay is

where *A* is a constant and *k* is a positive constant. The reason
these conventions are used is that these forms arise naturally in
solving problems that involve exponential growth and decay. The
constant *A* is the value of the function at *t*=0. The constant *k*
is called the growth rate in exponential growth and the decay rate in
exponential decay. In a process that can be modeled by exponential
functions, the rate constant *k* depends only on the process and the
conditions under which it is carried out.

For example, suppose that
growth of a population of bacteria can be modeled by an exponential
function. By running an experiment in which the number of bacteria are
counted as a function of time, a value of *k* can be determined. The
crucial fact is that the value of *k* depends on the environmental
conditions of the experiment, but does not depend on the initial
amount of bacteria present.
That is, the same value of *k* can be used to model the growth of
the same bacteria in other experiments, as long as the environmental
conditions are the same as in the original experiment. Put more
simply, if you do another experiment with the same bacteria under the
same conditions, only the value of *A*, the number of bacteria present
initially, changes.

Fri Jan 31 14:58:36 EST 1997