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 - 10.1 Exercises - Page 665: 13

Answer

a) $y=\frac{1}{x}$ $0 \lt x \lt 1$ b) The arrow must point such that: x increases as t increases.
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Work Step by Step

a) First equation: $x=sin t$ Second equation: $y=csc t= \frac{1}{sin t}$ because by definition, $csc t = \frac{1}{sint}$ Replace $sin t$ with $x$ (from the first equation). We get $y=\frac{1}{x}$ $0 \lt t \lt \frac{π}{2}$ $sin(0)=0$ $sin(\frac{π}{2})=1$ Therefore, $0 \lt x \lt 1$ And because $y=\frac{1}{x}$ We have $y \gt \infty$ b) Draw graph of $y=\frac{1}{x}$, where $0 \lt x \lt 1$ There is a hole at (1 ,1) because $x=1$ is not in the domain and because $t=\frac{\pi}{2}$ is not in the domain. And $sin t$ increases as $t$ increases. Therefore, the arrow must point such that: x increases and y decreases as t increases.
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