Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 3 - Differentiation - 3.4 Rates of Change - Exercises - Page 130: 20


$$-0.1312 ~W/\Omega,\ \ \ -0.06086~W/\Omega$$

Work Step by Step

We are given $$P=\frac{2.25 R }{(R+0.5)^{2}}$$ We find the derivative as follows: \begin{aligned} \frac{d P}{d R} &=\frac{(R+0.5)^{2} \frac{d}{d R}(2.25 R)-(2.25 R) \frac{d}{d R}(R+0.5)^{2}}{(R+0.5)^{4}} \\ &=\frac{(R+0.5)^{2}(2.25)-2(2.25 R) (R+0.5)}{(R+0.5)^{4}} \\ &=\frac{(R+0.5)(2.25 R+1.125-4.5 R)}{(R+0.5)^{3}}\\ &=\frac{1.125-2.25 R}{(R+0.5)^{3}} \end{aligned} Now, we plug in the given values: \begin{align*} \left.\frac{d P}{d R}\right|_{R=3}&=\frac{1.125-2.25(3)}{((3)+0.5)^{3}}=-0.1312~W/\Omega \\ \frac{d P}{d R}\bigg |_{R=5}&=\frac{1.125-2.25(5)}{((5)+0.5)^{3}}=-0.06086~W/\Omega \end{align*}
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