(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.