Chapter 4.1, Problem 71E

To determine

To find: the given statement is true or false and to justify the answer.

Answer to Problem 71E

The given statement is false because slope of the normal line to the curve $x=3\cos t,y=3\sin t$ at $t=\frac{\pi }{4}\text{is}1$ .

Explanation of Solution

Given information:

  $x=3\cos t,y=3\sin t$ at $t=\frac{\pi }{4}$

Calculation:

Slope of the given curve,

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

At $t=\frac{\pi }{4}$ ,

  $\begin{array}{l}\frac{dy}{dx}=-\cot \frac{\pi }{4}\\ \frac{dy}{dx}=-1\end{array}$

The slope of the tangent line is $-1$ .

The slope of the normal line is the negative reciprocal of the tangent line.

Slope of normal line is $-\frac{1}{1}=1$ .

Thus we can conclude that the given statement is false.