Chapter 11.3, Problem 9QR

To determine

To find: the point on the curve where the slop is undefined.

Answer to Problem 9QR

The pints on the curve where the slop is undefined are $\left(3,0\right)\text{and}\left(-3,0\right)$ .

Explanation of Solution

Given information:

  $\begin{array}{l}x=3\cos t\\ y=5\sin t\\ 0\le t\le 2\pi \end{array}$

Formula used:

  $du/dv=\frac{du/dx}{dv/dx}$

Calculation:

Given that theexpression of $x$ is $3\cos t$ for $0\le t\le 2\pi$ .

Take differentiation of $x$ with respect to $t$ .

  $\begin{array}{c}dx/dt=\frac{d}{dt}3\cos t\\ =-3\sin t\end{array}$

Similarly, the given expression of $y$ is $5\sin t$ for $0\le t\le 2\pi$ .

Take differentiation of $y$ with respect to $t$ .

  $\begin{array}{c}dy/dt=\frac{d}{dt}5\sin t\\ =5\cos t\end{array}$

Now, calculate $dy/dx$ . Use the formula $du/dv=\frac{du/dx}{dv/dx}$ .

  $\begin{array}{c}dy/dx=\frac{dy/dt}{dx/dt}\\ =\frac{5\cos t}{-3\sin t}\\ =-\frac{5}{3}\cot t\end{array}$

Given that the slop of the curve is undefined.

  $\begin{array}{c}dy/dx=\infty \\ -\frac{5}{3}\cot t=\infty \\ \cot t=\infty \\ Thereforet=0,\pi ,2\pi \end{array}$

Substitute the value of $t$ in the expression of $x$ and $y$ .

  $\begin{array}{c}x\left(0\right)=3\cos \left(0\right)\\ =3\\ x\left(\pi \right)=3\cos \left(\pi \right)\\ =-3\\ x\left(2\pi \right)=3\cos \left(2\pi \right)\\ =3\end{array}$

Similarly,

  $\begin{array}{c}y\left(0\right)=5\sin \left(0\right)\\ =0\\ y\left(\pi \right)=5\sin \left(\pi \right)\\ =0\\ y\left(2\pi \right)=5\sin \left(2\pi \right)\\ =0\end{array}$

Conclusion:

Hence, the required pints on the curve where the slop is undefined are $\left(3,0\right)\text{and}\left(-3,0\right)$ .