Chapter 4.1, Problem 55E

To determine

To find: the lines that are tangent and normal to the curve.

Answer to Problem 55E

Tangent: $y=\pi x-\pi +2$

Normal: $y=-\frac{1}{\pi }x+\pi +2$

  

Explanation of Solution

Given information:

  $y=2\tan \left(\frac{\pi x}{4}\right)\text{at }x=1$

  $\begin{array}{l}\frac{dy}{dx}=\frac{d}{dx}2\tan \left(\frac{\pi x}{4}\right)\\ \frac{dy}{dx}=2{\sec }^2\frac{\pi }{4}\times \frac{\pi }{4}\\ \frac{dy}{dx}=\pi \end{array}$

At $x=1$ ,

  $\begin{array}{l}y=2\tan \left(\frac{\pi x}{4}\right)\\ y=2\tan \left(\frac{\pi }{4}\right)\\ y=2\end{array}$

By using the point slope form,

Tangent line is $y=\pi x-\pi +2$

Normal line is $y=-\frac{1}{\pi }x+\pi +2$ , by using the inverse slope.

Thus we can find the equation of the lines that are tangent and normal to the given curve.