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: 12

Answer

Cartesian Equation: $4x^2 + \frac 1 4 y^2 = 1$
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Work Step by Step

(a) $x = \frac 1 2 cos \theta \longrightarrow x^2 = (\frac 1 2 cos\theta)^2 = \frac 1 4 cos ^2 \theta \longrightarrow 4x^2 =cos^2 \theta$ $y = 2 sin \theta \longrightarrow y^2 = 4 sin^2 \theta \longrightarrow \frac 1 4 y^2 = sin^2 \theta$ Adding the equations: $4x^2 + \frac 1 4 y^2 = cos^2 \theta + sin^2 \theta$ $4x^2 + \frac 1 4 y^2 = 1$ (b) 1. Plot points determined by values for $\theta$ between $0$ and $\pi$. 2. Join them to produce a curve. 3. Draw the arrows indicating which direction the curve goes from $\theta = 0$ to $\theta = \pi$
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