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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621

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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621
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?

To determine

Part (a):

To find: The average rate of change of the area of a circle when r changes from

(i) 2 to 3

(ii) 2 to 2.5

 (iii) 2 to 2.1

Explanation

1) Concept:

Use concept of average rate of change

Ar

2) Formula:

I. Area of circle: Ar= πr2

II. The average rate of change of the area of a circle with respect to radius r over [r1, r2]

 Ar =Ar2 - Ar1r2 - r1

3) Given:

(i) 2 to 3

(ii) 2 to 2.5

 (iii) 2 to 2.1

4) Calculation:

(i)  By using average rate of change formula,

Ar =Ar2 - Ar1r2 - r1

Ar =πr22 - πr12r2 - r1

Substitute r2=3 and r1=2

Ar =π32-π223-2=9π-4π=5π

Thus,

Ar= 5π

(ii)  By using average rate of change formula,

Ar =Ar2 - Ar1r2 - r1

Ar =πr22 - πr12r2 - r1

Substitute r2=2

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