Chapter 1.7, Problem 26E

(a)

To determine

To find: The correct match for the given parametric equations.

Answer to Problem 26E

The correct match is V.

Explanation of Solution

Given:

The given equation is,

  $\begin{array}{l}x=t^4-t+1\\ y=t^2\end{array}$

Calculation:

Consider the given equation,

  $\begin{array}{l}x=t^4-t+1\\ y=t^2\end{array}$

Eliminate $t$ from the given parametric equation as,

  $\begin{array}{l}y=t^2\\ t=\sqrt{y}\end{array}$

Then,

  $\begin{array}{c}x={\left(\sqrt{y}\right)}^4-\left(\sqrt{y}\right)+1\\ =y^2-\sqrt{y}+1\end{array}$

The graph for the given equation is shown in Figure 1

Figure 1

(b)

To determine

To find: The correct match for the given parametric equations.

Answer to Problem 26E

The correct match is I.

Explanation of Solution

Given:

The given equation is,

  $\begin{array}{l}x=t^2-2t\\ y=\sqrt{t}\end{array}$

Calculation:

Consider the given equation,

  $\begin{array}{l}x=t^2-2t\\ y=\sqrt{t}\end{array}$

Eliminate $t$ from the given parametric equation as,

  $\begin{array}{l}y=\sqrt{t}\\ t=y^2\end{array}$

Then,

  $\begin{array}{c}x={\left(y^2\right)}^2-2y^2\\ =y^4-2y^2\end{array}$

The graph for the given equation is shown in Figure 2

Figure 2

(c)

To determine

To find: The correct match for the given parametric equations.

Answer to Problem 26E

The correct match is II.

Explanation of Solution

Given:

The given equation is,

  $\begin{array}{l}x=\sin 2t\\ y=\sin \left(t+\sin 2t\right)\end{array}$

Calculation:

Consider the given equation,

  $\begin{array}{l}x=\sin 2t\\ y=\sin \left(t+\sin 2t\right)\end{array}$

The graph is the intersection of the sine curves.

The graph for the given equation is shown in Figure 3

Figure 3

(d)

To determine

To find: The correct match for the given parametric equations.

Answer to Problem 26E

The correct match is VI.

Explanation of Solution

Given:

The given equation is,

  $\begin{array}{l}x=\cos 5t\\ y=\sin 2t\end{array}$

Calculation:

Consider the given equation,

  $\begin{array}{l}x=\cos 5t\\ y=\sin 2t\end{array}$

The graph is the intersection of the sine curves and the cosine curves.

The graph for the given equation is shown in Figure 4

Figure 4

(e)

To determine

To find: The correct match for the given parametric equations.

Answer to Problem 26E

The correct match is IV.

Explanation of Solution

Given:

The given equation is,

  $\begin{array}{l}x=t+\sin 4t\\ y=t^2+\cos 3t\end{array}$

Calculation:

Consider the given equation,

  $\begin{array}{l}x=t+\sin 4t\\ y=t^2+\cos 3t\end{array}$

The graph is the intersection of the sine curves and the cosine curves.

The graph for the given equation is shown in Figure 5

Figure 5

(f)

To determine

To find: The correct match for the given parametric equations.

Answer to Problem 26E

The correct match is III.

Explanation of Solution

Given:

The given equation is,

  $\begin{array}{l}x=\frac{\sin 2t}{4+t^2}\\ y=\frac{\cos 2t}{4+t^2}\end{array}$

Calculation:

Consider the given equation,

  $\begin{array}{l}x=\frac{\sin 2t}{4+t^2}\\ y=\frac{\cos 2t}{4+t^2}\end{array}$

Then,

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

This is the equation of the circle.

The graph for the given equation is shown in Figure 6

Figure 6