(a)

Area of the circle is , Where is a radius.

(i)

Average rate of change if radius changes from to :

Average rate of change of with respect to is defined as on the interval .

Here , and .

Average rate of change in area is .

(ii)

Average rate of change if radius changes from to :

Average rate of change in area is

(iii)

Average rate of change if radius changes from to :

Average rate of change in area is

(b)

Instantaneous rate of change when :

Instantaneous rate of change of with respect to when  is defined as .

Area of the circle is .

Differentiate on each side with respect to .

.

Instantaneous rate of change of area at is .

.

(c) The circumference of the circle is , where is the radius of the circle.

consider .

Therefore rate of change of area with respect to its radius is equal to the its circumference.

If the radius is increased by an amount of .

Geometrical view of circle with an increased radius .

Therefore area of circle with increased radius of is .

If is small, then we can approximate .

Rate of change of area of new circle is with respect to radius can be considered as

 (a)

(i) Average rate of change if radius changes from to is .

(ii) Average rate of change if radius changes from to is .  

(iii) Average rate of change if radius changes from to is .

(b) Instantaneous rate of change when  is .

(c) The rate of change of area with respect to its radius is equal to the its circumference.