Calculus (3rd Edition)

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

Chapter 3 - Differentiation - 3.7 The Chain Rule - Exercises - Page 146: 78


$$2000\sqrt{3}\pi $$

Work Step by Step

Given $$P=R i^{2},\ \ \ \ i= \sin (4 \pi t)$$ and $R=1000$, $i(1/3)= -\dfrac{\sqrt{3}}{2}$ \begin{align*} \frac{dP}{dt}&= \frac{dP}{di}\frac{di}{dt}\\ &= (2Ri)( 4\pi \cos (4\pi t)) \end{align*} Hence \begin{align*} \frac{dP}{dt}\bigg|_{t=1/3}&= \frac{dP}{di}\frac{di}{dt}\\ &= (2Ri)( 4\pi \cos (4\pi t)) \\ &= \left(-2000\dfrac{\sqrt{3}}{2}\right) (-2\pi )\\ &= 2000\sqrt{3}\pi \end{align*}
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