Q. 33 - 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.

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

Step-by-Step Solution

Answer

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

And the graph of the function is:

Step 1. Given Information.

The function:

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

Step 2. Find the parametric equations.

The parametric equations for the given function is:

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

We know that,

$$\begin{gathered} \sin 2t=2\sin t \cos t \\ {\sin }^{2}2t=4{\sin }^{2}t {\cos }^{2}t \end{gathered}$$
Substitute

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

$y^2=4(1-x)x$

Step 3. Graph the function.

So the graph of the function is: