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4th Edition

James Stewart

Publisher: Cengage Learning

ISBN: 9781337687805

Chapter 3.8, Problem 13E

(a)

(i)

To determine

The average rate of change of the area of a circle with respect to its radius

Expert Solution

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

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

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

Therefore, the average rate of change of the area of a circle is

(ii)

To determine

The average rate of change of the area of a circle with respect to its radius

Expert Solution

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

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

Therefore, the average rate of change of the area of a circle is

(iii)

To determine

The average rate of change of the area of a circle with respect to its radius

Expert Solution

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

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

Therefore, the average rate of change of the area of a circle is

b)

To determine

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

Expert Solution

When *r* is 2 the instantaneous rate of change is

Calculate the instantaneous rate of change when

Differentiate the above area equation,

Substitute the value *r*.

Therefore, when the value of

c)

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.

Expert Solution

It is geometrically true. Hence, the resulting change in area

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.

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

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

Consider a circular ring of radius *r* and strip of thickness

Show that if the value of

Explain why it is geometrically true.

Approximate the resulting change in area

Write the expression for

The value of

Hence, it is proved.