Question

In problem 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.

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

Short answer

The answer is$y=\frac{1}{x}.$

Step-by-step solution

Step 1. Given information.

$x=e^t,y=e^{-t}$is given.

Step 2. Calculation.

Graph the curve defined by the parametric equations -

$t$	$x$	$y$	$(x,y)$
$0$	$1$	$1$	$(1,1)$
1	$2.718$	$0.36$	$(2.718,0.36)$

We find the rectangular equation by eliminating the parameter t from the parametric equations.

$$\begin{gathered} x=e^t \\ e=x^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$t$}\right.} \end{gathered}$$

Put the value of e-

$$\begin{gathered} y=e^{-t} \\ y={\left(x^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$t$}\right.}\right)}^{-t}y=x^{-1} \\ y=\frac{1}{x} \end{gathered}$$

Thus, the equation is$y=\frac{1}{x}.$