Chapter 1.7, Problem 36E

(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 $x=t$ and $y=t^{-2}$ .

Calculation:

Consider the given equation is,

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

The variable y can take any of the real value for all the value of $t$ .

The function $y=t^{-2}$ also follows the same and it can take up any values. That is why the curve is above the x axis and the is present on both side of the y axis.

The value of $y$ is infinite for all the values $t=0$ and the curve touches the $y$ axis in the long run of the $y$ axis.

The graph for the parametric equations $x=\cos t$ and $y={\sec }^{2}t$ is shown below.

The graph 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 36E

The required parametric curve is shown in Figure 2 and it is seen that the value of $y$ is infinite for $t=\frac{\pi }{2}$ .

Explanation of Solution

Given:

The given parametric equation is $x=\cos t$ and $y={\sec }^{2}t$ .

Calculation:

The given parametric equation is,

  $\begin{array}{l}x=\cos t\\ y={\sec }^{2}t\end{array}$

For $x=\cos t$ the variable $x$ can take any values that is between $-1$ to $1$ and that includes both the 1 and -1 for all the values of $t$ . It is given that $y={\sec }^{2}t$ and it also follows that the variable $y$ can take up any values for $t$ .

From the given parametric equation it is observed that the value of $y$ is infinite for $t=\frac{\pi }{2}$ and the curve seems to be touching the $y$ axis in the long run of the $y$ axis.

The diagram for the graph of the given parametric equation is shown below.

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

  

Figure 2

(c)

To determine

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

Answer to Problem 36E

The graph is shown in Figure 3 and it is same as the graph for first set of the parametric equations.

Explanation of Solution

Given:

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

Calculation:

The given parametric equation is,

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

The variable $x$ can take up any positive values for the variable $t$ and same is the case for variable $y$ can also take up any positive value for the variable $t$ .

The curve lies in the first quadrant. This is clear from the parametric equations that value of $x$ and $y$ is never zero and the curve seems to touch the axes in the long run.

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

  

Figure 3

The graph is same as the graph for first set of the parametric equations.