Chapter 10.7, Problem 15AYU

In Problems 7-26, graph the curve whose parametric equations are given and show its orientation. Find the rectangular equation of each curve. Verify your graph using a graphing utility.

$x=2e^t$, $y=1+e^t$; $t\ge 0$

To determine

To find:

a. The rectangular equation of the curve.

Answer to Problem 15AYU

$2y-x-2=0\left\{\text{log}\left(\frac{x}{2}\right)\ge 0\right\}$

Explanation of Solution

Given:

$x=2e^t$, $y=1+e^t;t\ge 0$

Calculation:

$x=2e^t$, $y=1+e^t;t\ge 0$

$y=1+\frac{x}{2},t\ge 0$

$2y-x-2=0\left\{\text{log}\left(\frac{x}{2}\right)\ge 0\right\}$

To determine

To graph:

b. $x=2t-4, y=4t²;-\infty \le t\ge \infty$, verify with the graph of rectangular equation of the curve.

Answer to Problem 15AYU

Explanation of Solution

Given:

$x=2t-4, y=4t²;-\infty \le t\ge \infty$

Graph:

The graph of $x=2e^t$, $y=1+e^t;t\ge 0$ is plotted.

The graph of $2y-x-2=0\left\{\text{log}\left(\frac{x}{2}\right)\ge 0\right\}$ is plotted.

Graph 1 represents the graph of $x=2e^t$, $y=1+e^t;t\ge 0$ plotted in parametric mode and the orientation is shown.

The rectangular equation of the curve is calculated. Graph 2 represents the curve of the rectangular equation $2y-x-2=0\left\{\text{log}\left(\frac{x}{2}\right)\ge 0\right\}$.

Both the equation represents the same graph and orientation. Hence it is verified.