Chapter 10.7, Problem 16AYU

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.

$x=e^t$, $y=e^{-t}$; $t\ge 0$

To determine

To find:

a. The rectangular equation of the curve.

Answer to Problem 16AYU

$xy=1,\left\{\text{log}x\ge 0\right\}$

Explanation of Solution

Given:

$x=e^t,y=e^{-t};t\ge 0$

Calculation:

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

$xy=1,t\ge 0$

$xy=1\left\{\text{log}x\ge 0\right\}$

To determine

To graph:

b. $x=e^t,y=e^{-t};t\ge 0$, verify with the graph of rectangular equation of the curve.

Answer to Problem 16AYU

Explanation of Solution

Given:

$x=e^t,y=e^{-t};t\ge 0$

Graph:

The graph of $x=e^t,y=e^{-t};t\ge 0$ is plotted.

The graph of $xy=1\left\{\text{log}x\ge 0\right\}$ is plotted.

Graph 1 represents the graph of $x=e^t,y=e^{-t}; t\ge 0$ 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 $xy=1\left\{\text{log}x\ge 0\right\}$.

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