Q. 32 - 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.

$r\left(t\right)=({\cos }^{2}t, 4int, t) for t\in [0,2\mathrm{\pi }]$

Step-by-Step Solution

Answer

The parametric equation of the vector valued functions $r\left(t\right)=({\cos }^{2}t, 4int, t) for t\in [0,2\mathrm{\pi }]$ is $x\left(t\right)={\cos }^{2}t; y\left(t\right)=4\sin t; z\left(t\right)=t$.

And the graph of the function is:

Step 1. Given Information.

The function:

$r\left(t\right)=({\cos }^{2}t, 4int, t) for t\in [0,2\mathrm{\pi }]$

Step 2. Find the parametric equations.

The parametric equations for the given function is:

$$\begin{gathered} x\left(t\right)={\cos }^{2}t; \\ y\left(t\right)=4\sin t; \\ z\left(t\right)=t \end{gathered}$$

t	x(t)	y(t)	z(t)	(x, y, z)
$0$	$1$	$0$	$0$	$(1,0,0)$
$\frac{\mathrm{\pi }}{2}$	$0$	$4$	$\frac{\mathrm{\pi }}{2}$	$(0,4,\frac{\mathrm{\pi }}{2})$
$\mathrm{\pi }$	$1$	$0$	$\mathrm{\pi }$	$(1,0,\mathrm{\pi })$
$\frac{3\mathrm{\pi }}{2}$	$0$	$-4$	$\frac{3\mathrm{\pi }}{2}$	$(0,-4,\frac{3\mathrm{\pi }}{2})$
$2\mathrm{\pi }$	$1$	$0$	$2\mathrm{\pi }$	$(1,0,2\mathrm{\pi })$

Step 3. Graph the function.

So the graph of the function is: