Question

Instantaneous rate of change?

Answer 1

(a).

The function is P(x) = 20x - 0.01x² - 100.

Substitute x = 20 in the above equation.

P(20) = 20(20) - 0.01(20)² - 100

P(20) = 400 - 4 - 100

P(20) = $296.

The profit for 20 trees is $296.

Answer 2

(b).

The function is P(x) = 20x - 0.01x² - 100.

Calculate the average change in the interval [a, b] by using formula: [f(b) - f(a)] / (b - a).

Substitute a = 11 and b = 16 in the above formula.

Average change = [P(16) - P(11)] / (16 - 11)

= {[20(16) - 0.01(16)² - 100] - [20(11) - 0.01(11)² - 100]} / 5

= {217.44 - 118.79} / 5

= 98.65 / 5

= 19.73

The Average change in profit sales from 11 to 16 tress is $19.73 per tree.

Answer 3

(c).

The function is P(x) = 20x - 0.01x² - 100.

To find the instantaneous rate of change, differentiate with respect to x.

P'(x) = 20 - 0.02x

Substitute x = 20 in the above equation.

P'(20) = 20 - 0.02(20)

P'(20) = 19.6

The instantaneous rate of change of profit at sales level 20 tress is $19.6 per tree.