Multivariable Calculus, 7th Edition

Published by Brooks Cole
ISBN 10: 0-53849-787-4
ISBN 13: 978-0-53849-787-9

Chapter 10 - Parametric Equations and Polar Coordinates - 10.3 Exercises - Page 687: 47

Answer

See image:

Work Step by Step

The curve seems to be sinusiodal, with period $\pi$ ...$ (\sin 2x)$, moved to the right by $\displaystyle \frac{\pi}{2}\qquad...\qquad [\sin(2x-\frac{\pi}{2})]$ the horizontal axis is raised up by $1.5$ units $\displaystyle \qquad...\qquad [1.5+\sin(2x-\frac{\pi}{2})]$ The graph oscillates between 0.5 and 2, so we set amplitude =$\displaystyle \frac{2-0.5}{2}=0.75$ so we define $r=f(\theta))=1.5+0.75\displaystyle \sin(2\theta-\frac{\pi}{2})$ Now, build a table of values for $\theta=0,\pi/12,\pi/6,...$ (start with 0, add $\pi/12 $ to next $\theta$) and for each $\theta$ calculate x coordinate = $ f(\theta)\cdot\cos\theta,\qquad y=f(\theta)\cdot\sin\theta$ Plot the points and join with a smooth curve. (Work done in geogebra.)
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