Chapter 4.1, Problem 51E

To determine

To calculate:the first derivative of the function.

Answer to Problem 51E

Value of $\frac{ds}{dt}$ at $\theta =\frac{3\pi }{2}$ is $5$ .

Explanation of Solution

Given information:

1. The given function is as follows

  $s=\cos \theta$ .

2. Need to evaluate $\frac{ds}{dt}$ when $\theta =\frac{3\pi }{2}$ and $\frac{d\theta }{dt}=5$ .

Formula used:

Chain rule: $\frac{ds}{dt}=\frac{ds}{d\theta }\cdot \frac{d\theta }{dt}$ .

Calculation:

Give equation is

  $s=\cos \theta$

  $\frac{ds}{d\theta }=\frac{d}{d\theta }\cos \theta$

  $=-\sin \theta$

when $\theta =\frac{3\pi }{2}$ , $\frac{ds}{d\theta }=-\sin \left(\frac{3\pi }{2}\right)=1$ and $\frac{d\theta }{dt}=5$

Apply chain rule

  $\frac{ds}{dt}=\frac{ds}{d\theta }\cdot \frac{d\theta }{dt}$

  $=1\cdot 5$

  $=5$

Hence value of $\frac{ds}{dt}$ at $\theta =\frac{3\pi }{2}$ is $\underset{\_}{5}$ .