Notes on the Exponential Function: Growth and Decay : A Differential Equation

Many quantities that change over time demonstrate exponential growth or exponential decay. For example, populations grow exponentially as long as there are enough resources to carry the population as its numbers increase. Continually compounded bank balances grow exponentially at a rate that depends on the interest rate. On the other hand, radioactive materials tend to degrade over time; otherwise known as half-life. A characteristic that all of these natural processes possess is that their rate of change depends on the size of the population, or the amount of material, that’s present at any given time. In the case of populations, bacteria, rabbits, antelope, or otherwise, the rate of growth increases as the size of the population increases. In the case of radioactive decay, the rate at which the radioactive material dissipates, decreases as the amount of material decreases.

These phenomenon, as well as many others, can be expressed in the language of calculus as:

What’s being expressed here is the idea that the rate at which f(t) is changing (that’s df/dt), is proportional to the value of f(t) itself. This is a differential equation — an equation that expresses a relationship between a function and its derivatives.

The exponential function: We want to know what the equation for f(t) is. What we know is that the rate at which f(t) is changing, at time t, is proportional to it’s value at time t. We can gain clarity by examining the simplest function there is that has this characteristic. Namely, e^x.