Chapter 10.7, Problem 15AYU

To determine

The graph of the curve of a parametric equations $x=2e^t,y=1+e^t;t\ge 0$, show its orientation. Find the rectangular equation of the curve.

Answer to Problem 15AYU

Solution:

The graph of the parametric equations $x=2e^t,y=1+e^t;t\ge 0$ is

The rectangular equation of the parametric equation $x=2e^t,y=1+e^t;t\ge 0$ is $x-2y=-2$

Explanation of Solution

Given information:

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

Explanation:

To graph the given parametric equation, plug some values of $t$ in the given equation and find few points on the curve.

The interval for $t$ is $-\infty <t<\infty$

$\begin{array}{cccc}t& x\left(t\right)=2e^t& y\left(t\right)=1+e^t& \left(x,y\right)\\ 0& 2& 2& \left(2,2\right)\\ 1& 5.44& 3.72& \left(5.44,3.72\right)\\ 2& 14.78& 8.39& \left(14.78,8.39\right)\\ 3& 40.17& 21.09& \left(40.17,21.09\right)\end{array}$

The orientation is the curve traced out in a certain direction by the corresponding succession of points $\left(x,y\right)$.

The arrow shows the orientation along the curve as $t$ varies from $t\ge 0$.

As $t$ takes values from $t\ge 0$, the corresponding orientation is from $\left(2,2\right)$ to $\left(40.17,21.09\right)$

Use points in the table to sketch the curve.

The graph of the parametric equation $x=2e^t,y=1+e^t;t\ge 0$ is

Now, let $x=2e^t$ and $y=1+e^t$

$\Rightarrow \frac{x}{2}=e^t$ and $y-1=e^t$

$\Rightarrow \frac{x}{2}=y-1$

$\Rightarrow x=2\left(y-1\right)$

$\Rightarrow x-2y=-2$

The rectangular equation of the parametric equation $x=2e^t,y=1+e^t;t\ge 0$ is $x-2y=-2$.