Chapter 1, Problem 31RE

(a)

To determine

To find: The sketch for the curve represented by the parametric equations and then indicate the direction in which the curve is traced as the value of $t$ increases.

Explanation of Solution

Given:

The given parametric equations is $x=e^t$ and $y=\sqrt{t}$ .

The range of the parameter $t$ is $0\le t\le 1$ .

Calculation:

Consider the given parametric equations are,

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

The table for the value of $x$ and $y$ for different value of the parameter is shown in Table 1

Table 1

$t$	$x$	$y$
0	1	0
1	2.718281828	1
2	7.389056099	1.414213562
3	20.08553692	1.732050808
4	54.59815003	2
5	148.413591	2.236067977
6	403.4284935	2.44945743

The plot for the above points on the coordinate plane is shown in Figure 1

  

Figure 1

(b)

To determine

To find: The Cartesian equation of the curve.

Answer to Problem 31RE

The required Cartesian equation is $y=\sqrt{\ln x}$ .

Explanation of Solution

Given:

The given parametric equations is $x=e^t$ and $y=\sqrt{t}$ .

The range of the parameter $t$ is $0\le t\le 1$ .

Calculation:

Consider the given parametric equations are,

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

Take natural log on both the sides for the equation $x=e^t$ as,

  $\begin{array}{l}\ln x=\ln e^t\\ \ln x=t\end{array}$

Consider the equation $y=\sqrt{t}$ as,

  $y=\sqrt{\ln x}$

Thus, the required Cartesian equation is $y=\sqrt{\ln x}$ .