Q. 26 - Calculus

Question

sketch the parametric curve by eliminating the parameter 

$x=t+2,y=e^t,t\in \mathrm{ℝ}$

Step-by-Step Solution

Answer

The required graph is 

Step 1: Given information

The parametric curve is $x=t+2,y=e^t,t\in \mathrm{ℝ}$

Step 2: Calculation

$x=t+2.$Consider the parametric equations $x=t+2,y=e^{\text{'}},t\in \mathrm{ℝ}$

The objective is to sketch the parametric curve by eliminating the parameter.

Take the parametric curve $x=t+2.$

Add $-2$ on both sides of the equation.

$$\begin{gathered} x-2=t+2-2 \\ x-2=t+2-2 \end{gathered}$$

Thus.

$x-2=t$

Now substitute $t=x-2$ in the parametric equation $y=e^{\text{'}}$.

Then,

$y=e^{x-2}[\text{ since }t=x-2]$

Thus, the required equation after eliminating the parameter $t$ is $y=e^{r-2}$

To draw the graph of the equation assume width="122" style="max-width: none; vertical-align: -4px;" $x=-2,-1,0,1,2$.

Substitute width="51" style="max-width: none; vertical-align: -4px;" $x=-2$ in the equation $y=e^{x-2}.$

Then,

$$\begin{gathered} y=e^{-2-2} \\ y=e^{-4} \\ y=0.018 \\ (x,y)=(-2,0.018) \end{gathered}$$

Substitute width="51" style="max-width: none; vertical-align: -4px;" $x=-1$ in the equation width="63" style="max-width: none; vertical-align: -4px;" $y=e^{x-2}$.

Then,

width="131" style="max-width: none; vertical-align: -50px;" $$\begin{gathered} y=e^{-1-2} \\ y=e^{-3} \\ y=0.049 \\ (x,y)=(-1,0.049) \end{gathered}$$

Step 3: Further calculation

Substitute $x=0$in the equation $y=e^{x-2}$.

Then,

$$\begin{gathered} y=e^{0-2} \\ y=e^{-2} \\ y=0.13 \\ (x,y)=(0,0.13) \end{gathered}$$

Substitute $x=1$in the equation $y=e^{x-2}$Then,

$$\begin{gathered} y=e^{1-2} \\ y=e^{-1} \\ y=0.36 \\ (x,y)=(1,0.36) \end{gathered}$$

Substitute $x=2$in the equation$y=e^{x-2}$. Then,

$$\begin{gathered} y=e^{2-2} \\ y=e^0 \\ y=1 \\ (x,y)=(2,1) \end{gathered}$$

The graphical representation by using the points

 is as follows,

$(-2,0.018)(-1,0.049)(0,0.13)(1,0.36)(2,1)$

Therefore, the required equation after eliminating the parameter is $y=e^{x-2}$