Q. 34 - 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.

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

Step-by-Step Solution

Answer

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

And the graph of the function is:

Step 1. Given Information.

The function:

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

Step 2. Find the parametric equations.

The parametric equations for the given function is:

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

Step 3. Graph the function.

So the graph of the function  is: