Chapter 9, Problem 42CLT

a.

To determine

Calculate the set of parametric equation using the rectangular equation by using the parameters $t=x$ .

Answer to Problem 42CLT

The two equations are $x=t,y=\frac{e^{2t}}{e^{2t}+1}$ .

Explanation of Solution

Given:

It is given in the question that the equation is $y=\frac{e^{2x}}{e^{2x}+1}$ .

Concept Used:

In this, use the concept of putting the value of parameters in the given equation and get the two parametric equation.

Calculation: Here, the equation is $y=\frac{e^{2x}}{e^{2x}+1}$ .

Let $x=t$ ,

Substitute to the x and y :

So,

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

Conclusion:

The two equation is $x=t,y=\frac{e^{2t}}{e^{2t}+1}$ .

b.

To determine

Calculate the set of parametric equation using the rectangular equation by using the parameters $t=x/2$ .

Answer to Problem 42CLT

The parametric equations are $x=2t,y=\frac{e^{4t}}{e^{4t}+1}$ .

Explanation of Solution

Given: It is given in the question that the equation is $y=\frac{e^{2x}}{e^{2x}+1}$ .

Concept Used:

In this, use the concept of putting the value of parameters in the given equation and get the two parametric equation.

Calculation: Here, the equation is $y=\frac{e^{2x}}{e^{2x}+1}$ .

Let,

  $\begin{array}{l}t=x/2\\ x=2t\end{array}$

Substitute to the x and y :

So,

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

Conclusion:

The equations are $x=2t,y=\frac{e^{4t}}{e^{4t}+1}$ .