The logistic equation is often used to model population growth

(see Section 4.1.1); the number, N, in the population grows as a function of time according to N(t) =

N0 1 + e

−kt .

a. Choose nice simple values for N0 and k (put them both equal to 1, for example) and plot N(t). Why does it make sense to plot for t < 0 as well as for t > 0?

b. Calculate and plot N 0

(t).

c. Show that N 0

(t) = e

−tN 2

. Hence, show that a growing population will grow slowly when the population is low.

d. Show that N 00(t) = e

−tN 2

(2e

−tN −1) and hence show that N

00 = 0 when t = 0. As you will learn in Chapter 17, the maximum of N 0

is at a place where N 00 = 0. Use this fact to determine when the maximal rate of growth occurs