## Calculus: Early Transcendentals 8th Edition

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