Chapter 11.1, Problem 19E

(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 19E

The required sketch is shown in Figure 1

Explanation of Solution

Given Data:

The given equations are,

  $\begin{array}{l}x=2\sin t\\ y=\cos t\\ 0\le t\le \pi \end{array}$

Calculation:

Consider the trigonometric equation as,

  ${\sin }^{2}t+{\cos }^{2}t=1$

Substitute the values in the given equation then,

  $\begin{array}{l}{\left(\frac{x}{2}\right)}^2+y^2=1\\ \frac{x^2}{4}+\frac{y^2}{1}=1\left(0\le x\le 2\right)\end{array}$

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 19E

The lowest point on the graph is $\left(2,0\right)$ .

Explanation of Solution

Consider the graph is shown in Figure 1 the graph shows that the lowest point is $\left(2,0\right)$ .

(c)

To determine

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

Answer to Problem 19E

The lowest point on the graph is $\left(2,0\right)$ .

Explanation of Solution

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

  $\begin{array}{l}\frac{dx}{dt}=0\\ 2\cos t=0\\ t=\frac{\pi }{2}\end{array}$

Then,

  $x=2\sin \frac{\pi }{2}$

Then, the value of $y$ is,

  $\begin{array}{c}y=\cos \frac{\pi }{2}\\ =0\end{array}$

Thus, lowest point on the graph is $\left(2,0\right)$ .