   Chapter 3.7, Problem 13E ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

#### Solutions

Chapter
Section ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

# (a) Find the average rate of change of the area of a circle with respect to its radius r as r changes from(i) 2 to 3(ii) 2 to 2.5(iii) 2 to 2.1(b) Find the instantaneous rate of change when r = 2.(c) Show that the rate of change of the area of a circle with respect to its radius {at any r) is equal to the circumference of the circle. Try to explain geometrically why this is true by drawing a circle whose radius is increased by an amount ∆r. How can you approximate the resulting change in area ∆A if ∆r is small?

(a)

(i)

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.

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

(ii)

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.

(iii)

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.

b)

To determine

The instantaneous rate of change when r = 2.

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.

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