Q. 31 - Calculus

Question

Find parametric equations for each of the vector-valued functions in Exercises 26–34, and sketch the graphs of the functions, indicating the direction for increasing values of t.

$r\left(t\right)=(t,t^2,t^3) for t\in [0,2]$

Step-by-Step Solution

Answer

The parametric equation of the vector valued functions $r\left(t\right)=(t,t^2,t^3) for t\in [0,2]$ is $x\left(t\right)=t; y\left(t\right)=t^2; z\left(t\right)=t^3$.

And the graph of the function is:

Step 1. Given Information.

The function:

$r\left(t\right)=(t,t^2,t^3) for t\in [0,2]$

Step 2. Find the parametric equations.

The parametric equations for the given function is:

$$\begin{gathered} x\left(t\right)=t; \\ y\left(t\right)=t^2; \\ z\left(t\right)=t^3 \end{gathered}$$

t	x(t)	y(t)	z(t)	(x,y,z)
$0$	$0$	$0$	$0$	$(0,0,0)$
$1$	$1$	$1$	$1$	$(1,1,1)$
$2$	$2$	$4$	$8$	$(2,4,8)$

Step 3. Graph the function.

So the graph of the function is: