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: 20

Answer

This equation describes a parabola that is concave upward, and has its vertex at $(0,0)$.

Work Step by Step

- Knowing that $rcos(\theta) = x$ and $rsin(\theta) = y$: $r = tan(\theta)sec(\theta)$ $r = \frac {sin(\theta)}{cos(\theta)} \frac 1 {cos(\theta)}$ $r = \frac {sin(\theta)}{cos^2(\theta)}$ $rcos^2(\theta) = sin(\theta)$ - Multiply both sides by $r$: $r^2cos^2(\theta) = rsin(\theta)$ $x^2 = y$ - This is a simple quadratic equation, so it describes a parabola. - Since the coefficient of $x^2$ is positive (1), the parabola is concave upward. - The parabola has its vertex at $(0,0)$, since that is the point with the lowest $y$ value.
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