Exploring Differential Equations & Exponentials
by Mike Martin
This page generates data for an exponential function and explores the continuous rate of change using average rates of change. The exponential function may be written as
The instantaneous rate of change is estimated from the following approximation
The fourth column of the generated table of data below gives approximations to the instantaneous rate of change. The fifth column in the data below gives approximations to what is called the per capita production rate. The per capita production rate is defined as the instantaneous rate of change divided by the amount or population. This can be expressed symbolically as
Upon investigation, the per capita production rate appears to be constant. What happens to the per capita production rate as Δt gets smaller, approaching zero? Upon close examination, you'll find that the calculated constant in the fifth column approaches ln(b) as Δt goes to zero. Note that T, a positive parameter, represents the maximum value of the time interval in which the approximations are made; that is, we consider data on the time interval of [0, T].
Enter in the values of the non-negative parameters (described above), then press the "Evaluate" button.