Chapter 11.1, Problem 21E

(a)

To determine

To find: The sketch for the curve over the given $t$ interval indicating the direction in which it is traced.

Answer to Problem 21E

The required sketch is shown in Figure 1

Explanation of Solution

Given Data:

The given equations are,

  $\begin{array}{l}x=2\sin t\\ y=\cos 2t\\ 1.5\le t\le 4.5\end{array}$

Calculation:

Consider the trigonometric equation as,

  $\cos 2t=1-2{\sin }^{2}t$

Substitute the values in the given equation then,

  $y=1-\frac{x^2}{2}$

The sketch for the above function is shown in Figure 1

  

Figure 1

(b)

To determine

To find: The identification for the requested points.

Answer to Problem 21E

The highest point on the graph is $\left(0,1\right)$ .

Explanation of Solution

Consider the graph is shown in Figure 1 the graph shows that the highest point is $\left(0,1\right)$ .

(c)

To determine

To find: The justification that you have found the requested point by analysing an appropriate derivative.

Answer to Problem 21E

The highest point on the graph is $\left(0,1\right)$ .

Explanation of Solution

To obtain the lowest point on the graph differentiate the given equation as,

  $\begin{array}{l}\frac{dy}{dt}=0\\ -\frac{1}{2}\left(2x\right)=0\\ x=0\end{array}$

Then, the value of $y$ is,

  $y=1$

Thus, highest point on the graph is $\left(0,1\right)$ .