Chapter 4.1, Problem 75E

To determine

To calculate: To find the value of t at which $\frac{dy}{dx}=0$ .

Answer to Problem 75E

The correct option is B.

Explanation of Solution

Given Information: The equations of the parametric curves are $x=t-\cos t\text{ and }y=-1+\sin t$

Calculation:

The derivative of the parametric curves with respect to t is,

  $\begin{array}{c}x=t-\cos t\\ \frac{dx}{dt}=1-\left(-\sin t\right)\\ =1+\sin t\\ y=-1+\sin t\\ \frac{dy}{dt}=\cos t\end{array}$

The value of $\frac{dy}{dx}$ is,

  $\begin{array}{c}\frac{dy}{dx}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}}\\ =\frac{\cos t}{1+\sin t}\end{array}$

The value of t where $\frac{dy}{dx}=0$ is,

  $\begin{array}{c}\frac{dy}{dx}=0\\ \frac{\cos t}{1+\sin t}=0\\ \cos t=0\\ t=\frac{\pi }{2},\frac{3\pi }{2}\end{array}$

Hence, the value oft at which $\frac{dy}{dx}=0$ is $t=\frac{\pi }{2}$ (Discarding $t=\frac{3\pi }{2}$ as that is not one of the options.)