Chapter 11, Problem 14RE

a.

To determine

To find the points at which the tangent to the curve is horizontal.

Answer to Problem 14RE

The point doesn’t exist.

Explanation of Solution

Given:

The given equations are $x=4\cos t,y=9\sin t$

Calculation:

Let us find out the slope

  $\begin{array}{l}\frac{dy}{dx}=0\\ \frac{\frac{dy}{dt}}{\frac{dx}{dt}}=\frac{9\cos t}{-4\sin t}=0\\ -\frac{9}{4}\cot t=0\\ t=\infty \end{array}$

b.

To determine

To find the points at which the tangent to the curve is vertical.

Answer to Problem 14RE

  $(4,0)$

Explanation of Solution

Given:

The given equations are $x=4\cos t,y=9\sin t$

Calculation:

  $\begin{array}{l}\frac{dx}{dy}=0\\ \frac{\frac{dx}{dt}}{\frac{dy}{dt}}=\frac{-4\sin t}{9\cos t}=0\\ -\frac{4}{9}\tan t=0\\ t=0\end{array}$

Therefore, points are $x=4,y=0$