Calculus (3rd Edition)

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

Chapter 12 - Parametric Equations, Polar Coordinates, and Conic Sections - Chapter Review Exercises - Page 638: 8


The arrows indicate the direction of motion.

Work Step by Step

We have $x\left( t \right) = 1 + \cos t$ and $y\left( t \right) = \sin 2t$. For the interval $0 \le \theta \le 2\pi $, we evaluate several points in rectangular coordinates corresponding to $\theta = 0,\frac{\pi }{4},\frac{\pi }{2},...,2\pi $ and list them on the following table: $\begin{array}{*{20}{c}} t&{\left( {x,y} \right)}\\ 0&{\left( {2,0} \right)}\\ {\frac{\pi }{4}}&{\left( {1.71,1} \right)}\\ {\frac{\pi }{2}}&{\left( {1,0} \right)}\\ {\frac{{3\pi }}{4}}&{\left( {0.293, - 1} \right)}\\ \pi &{\left( {0,0} \right)}\\ {\frac{{5\pi }}{4}}&{\left( {0.293,1} \right)}\\ {\frac{{3\pi }}{2}}&{\left( {1,0} \right)}\\ {\frac{{7\pi }}{4}}&{\left( {1.71, - 1} \right)}\\ {2\pi }&{\left( {2,0} \right)} \end{array}$ Then we plot the points in rectangular coordinates and sketch the curve by connecting these points. The arrows indicate the direction of motion as are shown in the figure.
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