The fish population increases according to the logistic curve is , where  is measured in months.

(a)

Graph:

Graph the function .

(b) What is the limiting size of the fish population.

The limiting size of the fish population is .

.

The limiting population is .

(c) At what rates is the fish population changing at the end of  month and at the end of  months.

Apply derivative on each side with respect to .

.

The rate of change of fish population at the end of  month.

Substitute  in .

.

The rate of change of fish population at the end of  month is .

The rate of change of fish population at the end of  months.

Substitute  in .

.

The rate of change of fish population at the end of  months is .

(d) After how many months is the population increasing most rapidly.

Again apply derivative on each side with respect to .

.

The maximum value of  occurs at .

Apply log on each side.

.

The population increasing most rapidly after  months.

(a)

Graph:

Graph the function .

(b) The limiting population is .

(c)

The rate of change of fish population at the end of  month is .

The rate of change of fish population at the end of  months is .

(d) The population increasing most rapidly after  months.