Q. 28 - 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.

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

Step-by-Step Solution

Answer

The parametric equation of the vector valued functions $r\left(t\right)=(\sin t, \cos 2t) for t\in [0,2\mathrm{\pi }]$ is $y=1-2x^2$.

And the graph of the function is:

Step 1. Given Information.

The function:

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

Step 2. Find the parametric equations.

The parametric equations for the given function is:

$$\begin{gathered} x=\sin t; \\ y=\cos 2t \end{gathered}$$

We know that,

$$\begin{gathered} \cos 2t=1-2s{\mathrm{in}}^{2 }t \\ y=1-2x^2 \end{gathered}$$

Step 3. Find the points.

Find the ordered pairs.

t	x=sin t	y=cos 2t	(x, y)
$0$	$0$	$1$	$(0,1)$
$\frac{\mathrm{\pi }}{2}$	$1$	$-1$	$(1,-1)$
$\mathrm{\pi }$	$0$	$1$	$(0,1)$
$\frac{3\mathrm{\pi }}{2}$	$-1$	$-1$	$(-1,-1)$
$2\mathrm{\pi }$	$0$	$1$	$(0,1)$

Step 4. Graph the function.

The graph of the function $r\left(t\right)$ is: