Chapter 10.1, Problem 21E

$\text{(a)}$

To determine

To sketch: The graph of the given $t$ interval.

Answer to Problem 21E

The answer:

  

Explanation of Solution

Given information:

The equations are:

  $\begin{array}{c}x=2\sin t,\\ y=\cos (2t),1.5\le t\le 4.5\end{array}$

Calculation:

The follow graph of the curve based on the information provided in this exercise.

  

$\text{(b)}$

To determine

To find: The requested points.

Answer to Problem 21E

The answer: $(0,1)$

Explanation of Solution

Given information:

The equations are:

  $\begin{array}{c}x=2\sin t,\\ y=\cos (2t),1.5\le t\le 4.5\end{array}$

Calculation:

From part $(a)$ conclude that the lowest point of the curve is equal to $(0,1)$ .

Therefore, the required requested point $(0,1)$ .

$(c)$

To determine

To prove: The requested points by analyze a appropriate derivative.

Explanation of Solution

Given information:

The equations are:

  $\begin{array}{c}x=2\sin t,\\ y=\cos (2t),1.5\le t\le 4.5\end{array}$

Calculation:

To find the lowest point of the curve, to find the minimum of the function $y=\cos (2t)$ where,

  $1.5\le t\le 4.5$ .

Know that, $y$ has a relative minimum at the point $t=t_0$ if

  $\frac{dy}{dt}\left(t_0\right)=0$

To use the previous result,

  $\begin{array}{c}\frac{dy}{dt}=-\sin (2t)\cdot 2\\ =-2\sin (2t)\\ \Rightarrow \frac{dy}{dt}(\pi /2)=0\\ \frac{dy}{dt}(\pi )=0\end{array}$

To use the previous result, then

  $\begin{array}{c}x(\pi )=2\sin (\pi )\\ =2\cdot 0\\ =0\\ y(\pi )=\cos (2\pi )\\ =1\end{array}$

which means that the lowest point of the curve is equal to $(0,1)$ .