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Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

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BuyFindarrow_forward

Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

The cost (in dollars) of producing x units of a certain commodity is C(x) = 5000 + 10x + 0.05x2.

(a) Find the average rate of change of C with respect to x when the production level is changed

(i) from x = 100 to x = 105

(ii) from x= 100 to x = 101

(b) Find the instantaneous rate of change of C with respect to x when x = 100. (This is called the marginal cost. Its significance will be explained in Section 3.7.)

(a)

To determine

To find: The average rate of change of C(x) when x is changed for the respected intervals.

Explanation

Given:

Let C(x) be the cost (in dollars) of producing x units of a commodity.

The given cost function C(x)=5000+10x+0.05x2.

Formula used:

The average rate of change of C with respect to x on the interval [x1,x2] is,

ΔCΔx=C(x2)C(x1)x2x1 (1)

Calculation:

Section (i)

Obtain the average rate of change of C(x) when x=100 to x=105.

Substitute 100 for x1 and 105 for x2 in equation (1),

ΔCΔx=C(105)C(100)105100=(5000+10(105)+0.05(105)2)(5000+10(100)+0.05(100)2)105100=(5000+1050+551.25)(5000+1000+500)5=6601

(b)

To determine

To estimate: The instantaneous rate of change of C with respect to x when x=100.

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