University Calculus: Early Transcendentals (3rd Edition)

Published by Pearson
ISBN 10: 0321999584
ISBN 13: 978-0-32199-958-0

Chapter 10 - Section 10.2 - Calculus with Parametric Curves - Exercises - Page 573: 43

Answer

a) $\dfrac{1}{2}$ b) $0$ c) $\dfrac{2\sqrt 3-1}{\sqrt 3-2}$

Work Step by Step

Take the derivative of the given equations and isolate the variables. $\dfrac{dy}{dt}=\cos \theta+2 \sin 2 \theta$ and $\dfrac{dx}{dt}=-\sin \theta+2 \cos 2 \theta$ Now, slope: $\dfrac{dy}{dx}=\dfrac{\cos \theta+2 \sin 2 \theta}{-\sin \theta+2 \cos 2 \theta}$ a) At $\theta =0$ Slope: $\dfrac{dy}{dx}=\dfrac{1}{2}$ b) At $\theta =\pi/2$ Slope: $\dfrac{dy}{dx}=0$ c) At $\theta =4\pi/3$ Slope: $\dfrac{dy}{dx}=\dfrac{\cos (4\pi/3)+2 \sin 2 (4\pi/3)}{-\sin (4\pi/3)+2 \cos 2 (4\pi/3)}=\dfrac{2\sqrt 3-1}{\sqrt 3-2}$
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