Chapter 1.7, Problem 35E

(a)

To determine

To find: The comparison for the curves by the parametric equations and the whey in which they differ.

Explanation of Solution

Given:

The given parametric equation is $x=t^3$ and $y=t^2$ .

Calculation:

The given parametric equation is,

  $x=t^6$

Then,

  $\begin{array}{l}x=t^3\\ t=x^{\frac{2}{3}}\end{array}$

The given equation for the parametric curve is,

  $y=t^2$

Then,

  $\begin{array}{c}y=t^2\\ ={\left(x^{\frac{1}{3}}\right)}^2\\ =x^{\frac{2}{3}}\end{array}$

The graph for the equation $y=x^{\frac{2}{3}}$ is shown in Figure 1

  

Figure 1

(b)

To determine

To find: The comparison for the curves by the parametric equations and the whey in which they differ.

Answer to Problem 35E

The graph for the equations $x=t^6$ and $y=t^4$ only the portion of $x\ge 0$ is generated.

Explanation of Solution

Given:

The given parametric equation is $x=t^6$ and $y=t^4$ .

Calculation:

The given parametric equation is,

  $x=t^6$

Then,

  $\begin{array}{l}x=t^6\\ t=x^{\frac{1}{6}}\end{array}$

The given equation for the parametric curve is,

  $y=t^4$

Then,

  $\begin{array}{c}y=t^4\\ ={\left(x^{\frac{1}{6}}\right)}^4\\ =x^{\frac{2}{3}}\end{array}$

The graph for the given parametric equations is shown in Figure 2

  

Figure 2

The graph from the parametric equations $x=t^6$ and $y=t^4$ only the portion of $x\ge 0$ is generated.

(c)

To determine

To find: The comparison for the curves by the parametric equations and the whey in which they differ.

Answer to Problem 35E

The graph for the equations $x=e^{-3t}$ and $y=e^{-2t}$ only the portion of $x\ge 0$ is generated.

Explanation of Solution

Given:

The given parametric equation is $x=e^{-3t}$ and $y=e^{-2t}$ .

Calculation:

The given parametric equation is,

  $x=e^{-3t}$

Then,

  $\begin{array}{l}\ln x=-3t\ln e\\ t=\frac{-1}{3}\ln x\end{array}$

The given equation for the parametric curve is,

  $y=e^{-2t}$

Then,

  $\begin{array}{c}y=e^{-2\left(\frac{-1}{3}\ln x\right)}\\ =e^{\ln x^{\frac{2}{3}}}\\ =x^{\frac{2}{3}}\end{array}$

The graph for the given parametric equations is shown in Figure 3

  

Figure 3

The graph for the given parametric equations $x=e^{-3t}$ and $y=e^{-2t}$ only the part of the graph $x>0$ is generated.