Question

a) Find the average rate of change of the area of a circle with respect to its radius r as r changes from 4 to each of the following.

(i) 4 to 5 _______

(ii) 4 to 4.5 _______

(iii) 4 to 4.1 _______


(b) Find the instantaneous rate of change when r = 4.

A'(4) = ______

Answer 1

Area = π x radius²
dA/dr = 2πr
Average rate of change = (rate of change at one instant + rate of change at other instant)/2
= (2πr₁ + 2πr₂)/2
= (r₁ + r₂)π
Ai) 9π
Aii) 8.5π
Aiii) 8.1π

B) Instantaneous change:
(dA/dr) at (r = 4) = 2π(4)
= 8π