Chapter 1.7, Problem 13E

(a)

To determine

To find:The Cartesian equation of the curve.

Explanation of Solution

Given: The parametric equations are $x=e^{2t}$ , $y=t+1$

Consider the expression.

  $\begin{array}{c}x=e^{2t}\\ \ln x=2t\\ t=\frac{1}{2}\ln x\end{array}$

And,

  $\begin{array}{c}y=t+1\\ =\frac{1}{2}\ln x+1\end{array}$

Therefore, the equation in Cartesian form is $y=\frac{1}{2}\ln x+1$ .

(b)

To determine

To sketch:The curve by using the parametric equation.

Explanation of Solution

Given: The parametric equations are $x=e^{2t}$ , $y=t+1$ .

The sketch of the graph is shown in figure below.

  

Figure (1)

Therefore, the required graph is shown in figure (1).