Analysis of the Logistic Function P(t)

We will develop the results of this section as a series of exercises.

Exercise 1 : First, we note that the horizontal asymptotes are 0 and . Why?

Next, note that we can rewrite the differential equation as

.

Exercise 2 : From the above equation, we can see that the growth rate is at a maximum when . You should verify this by completing the following problem and explaining its relevance.

Problem : Let . Show that has a maximum when .

Exercise 3 : Show that implies that .

Exercise 4 : Show that ( ) is a point of inflection on the graph of .

Exercise 5 : Show that the maximum growth rate is .