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.