Calculus: Early Transcendentals 8th Edition

Published by Cengage Learning
ISBN 10: 1285741552
ISBN 13: 978-1-28574-155-0

Chapter 10 - Section 10.2 - Calculus with Parametric Curves - 10.2 Exercises - Page 655: 27


(a) $\frac{dy}{dx} = \frac{dsin~\theta}{r-d~cos~\theta}$ (b) The trochoid does not have a vertical tangent.

Work Step by Step

(a) We can find $\frac{dy}{dx}$: $\frac{dy}{dx} = \frac{\frac{dy}{d\theta}}{\frac{dx}{d\theta}} = \frac{dsin~\theta}{r-d~cos~\theta}$ (b) Note that $\frac{r}{d} \gt 1$ when $r \gt d$ We can try to find values of $\theta$ such that the denominator of $\frac{dy}{dx}$ is $0$ $r-d~cos~\theta = 0$ $d~cos~\theta = r$ $cos~\theta = \frac{r}{d} \gt 1$ There are no values of $\theta$ such that $cos~\theta \gt 1$ Therefore, the trochoid does not have a vertical tangent.
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