Q. 29 - 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.

$r\left(t\right)=(1+\sin t, 3-\cos 2t), for t\in [0,2\mathrm{\pi }]$

Step-by-Step Solution

Answer

The parametric equation of the vector valued functions $r\left(t\right)=(1+\sin t, 3-\cos 2t), for t\in [0,2\mathrm{\pi }]$ is $y=2x^2-4x+4$.

And the graph of the function is:

Step 1. Given Information.

The function:

$r\left(t\right)=(1+\sin t, 3-\cos 2t), for t\in [0,2\mathrm{\pi }]$

Step 2. Find the parametric equations.

The parametric equations for the given function is:

$$\begin{gathered} x=1+\sin t \\ y=3-\cos 2t \end{gathered}$$

We know that,

$\cos 2t=1-2{\sin }^{2}t$

From the given x, y,

$$\begin{gathered} x-1=\sin t \\ \cos 2t=3-y \end{gathered}$$

So, 

$$\begin{gathered} \cos 2t=1-2{\sin }^{2}t \\ 3-y=1-2{(x-1)}^2 \\ y=2x^2-4x+4 \end{gathered}$$

Step 3. Find the ordered pairs.

The ordered pairs of the function is:

t	x=1+sin t	y=3-cos2t	(x, y)
$0$	$1$	$2$	$(1,2)$
$\frac{\mathrm{\pi }}{2}$	$2$	$4$	$(2,4)$
$\mathrm{\pi }$	$1$	$2$	$(1,2)$
$\frac{3\mathrm{\pi }}{2}$	$0$	$4$	$(0,4)$
$2\mathrm{\pi }$	$1$	$2$	$(1,2)$

Step 4. Graph the function.

The graph of the function is: