Chapter 1.7, Problem 22E

(a)

To determine

To find: The correct parametric curve for the given graph.

Answer to Problem 22E

The parametric curve for the graph is curve (III.) as the parametric curve is the loop between $x=1$ and $x=2$ .

Explanation of Solution

Given:

The given graph is shown in Figure 1

  

Figure 1

Calculation:

From the graph the range of the variable $x$ is,

  $1\le x\le 2$

The range for the $t$ for the variable $x$ is,

  $0\le t\le 1$

From the graph the range of the variable $y$ is,

  $-1\le y\le 1$

The range for the $t$ for the variable $y$ is,

  $0\le t\le 1$

This shows that the parametric curve is the loop between $x=1$ and $x=2$ that start and end at the same point $\left(1,0\right)$ .

Thus, the parametric curve for the graph is curve (III.)

(b)

To determine

To find: The correct parametric curve for the given graph.

Answer to Problem 22E

The correct parametric curve is (I) as the range of $x$ is from $2$ to $-2$ .

Explanation of Solution

Given:

The given graph is shown in Figure 1

  

Figure 1

Calculation:

The given curve start and end at the $\left(0,0\right)$ .

From the graph the range of the variable $x$ is,

  $-2\le x\le 2$

The range for the $t$ for the variable $x$ is,

  $0\le t\le 1$

From the graph the range of the variable $y$ is,

  $-2\le y\le 2$

The range for the $t$ for the variable $y$ is,

  $0\le t\le 1$

This shows that both the graph are periodic as the range of $x$ is from $2$ to $-2$ six times. This, shows that for the given graph the correct parametric curve is curve (I)