#### To determine

The average rate of change of the area of a circle with respect to its radius r as r change from 2 to 3.

#### Answer

The average rate of change of the area of a circle is 5π_.

#### Explanation

Expression for the area of the circle is given as below.

A(r)=πr2

Calculate the average rate of change of the area of a circle, when the radius changes from 2 to 3.

A(r)=A(3)−A(2)3−2A(r)=9π−4π1A(r)=5π

Therefore, the average rate of change of the area of a circle is 5π_.

#### To determine

The average rate of change of the area of a circle with respect to its radius r as r change from 2 to 2.5.

#### Answer

The average rate of change of the area of a circle is 4.5π_.

#### Explanation

Calculate the average rate of change of the area of a circle, when the radius changes from 2 to 2.5.

A(r)=A(2.5)−A(2)2.5−2A(r)=6.25π−4π0.5A(r)=4.5π

Therefore, the average rate of change of the area of a circle is 4.5π_.

#### To determine

The average rate of change of the area of a circle with respect to its radius r as r change from 2 to 2.1.

#### Answer

The average rate of change of the area of a circle is 4.1π_

#### Explanation

Calculate the average rate of change of the area of a circle, when the radius changes from 2 to 2.1.

A(r)=A(2.1)−A(2)2.1−2A(r)=4.41π−4π0.1A(r)=4.1π

Therefore, the average rate of change of the area of a circle is 4.1π_.

#### To determine

The instantaneous rate of change when *r* = 2.

#### Answer

When *r* is 2 the instantaneous rate of change is 4π_.

#### Explanation

Calculate the instantaneous rate of change when r is 2 using the below equation.

A(r)=πr2

Differentiate the above area equation,

A'(r)=2πr (1)

Substitute the value 2 for *r*.

A'(2)=2π(2)A'(2)=4π

Therefore, when the value of r is 2 the instantaneous rate of change is 4π_.

#### To determine

**To show:** The rate of change of the area of a circle with respect to its radius is equal to the circumference of the circle and why it is geometrically true.

#### Answer

It is geometrically true. Hence, the resulting change in area ΔA if change in radius Δr is small is ΔAΔr≃2πr.

#### Explanation

Show that the rate of change of the area of a circle with respect to its radius is equal to the circumference of the circle.

Circumference of the circle is given as below.

c(r)=2πr (2)

First derivative of the area of the circle is given as below from the equation (1).

A'(r)=2πr

Compare both the equations (1) and (2).

c(r)=A'(r)=2πr

Consider a circular ring of radius *r* and strip of thickness Δr along the radial direction representing the change in the radius for the circle as shown in figure (1).

Show that if the value of Δr is small, then the change in the circle is approximately equal to its circumference.

Explain why it is geometrically true.

Approximate the resulting change in area ΔA, if the value of Δr is very small.

Write the expression for ΔA in algebraically manner.

ΔA=Area of ring with increased radius - Area of the ring ΔA=A(r+Δr)−A(r)ΔA=π(r+Δr)2−πr2ΔA=2πr(Δr)+π(Δr)2

The value of Δr is very small. Therefore, the above equation will become as below.

ΔA=2πr(Δr)ΔAΔr≃2πr

Hence, it is proved.