Calculus 8th Edition

Published by Cengage
ISBN 10: 1285740629
ISBN 13: 978-1-28574-062-1

Chapter 2 - Derivatives - 2.7 Rates of Change in the Natural and Social Sciences - 2.7 Exercises - Page 180: 29

Answer

(a) $C(x)$ = $2000+3x+0.01x^{2}+0.0002x^{3}$ $C'(x)$ = $3+0.02x+0.0006x^{2}$ (b) $C'(100)$ = $3+0.02(100)+0.0006(100)^{2}$ = $11$ $dollars/pair$ $C'(100)$ is the rate at which the cost is increasing as the 100th pair of jeans is produced. It predicts the (approximate) cost of the 101st pair. (c) The cost of manufacturing the 101st pair of jeans is $C(101)-C(100)$ = $2611.07-2600$ $\approx$ $11.07$ $dollars$ This is close to the marginal cost from part (b).

Work Step by Step

(a) $C(x)$ = $2000+3x+0.01x^{2}+0.0002x^{3}$ $C'(x)$ = $3+0.02x+0.0006x^{2}$ (b) $C'(100)$ = $3+0.02(100)+0.0006(100)^{2}$ = $11$ $dollars/pair$ $C'(100)$ is the rate at which the cost is increasing as the 100th pair of jeans is produced. It predicts the (approximate) cost of the 101st pair. (c) The cost of manufacturing the 101st pair of jeans is $C(101)-C(100)$ = $2611.07-2600$ $\approx$ $11.07$ $dollars$ This is close to the marginal cost from part (b).
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