Exercise 9

Consider the moving particle whose position at time $t$ in seconds is given by the vector-valued function $\vr$ defined by $\vr(t) = 5t \vi + 4\sin(3t) \vj + 4\cos(3t) \vk$. Use this function to answer each of the following questions.

1. Find the unit tangent vector, $\vT(t)$, to the space curve traced by $\vr(t)$ at time $t$. Write one sentence that explains what $\vT(t)$ tells us about the particle's motion.

2. Determine the speed of the particle moving along the space curve with the given parameterization.

3. Find the exact distance traveled by the particle on the time interval $[0,\pi/3]$.

4. Find the average velocity of the particle on the time interval $[0, \pi/3]$.

5. Determine the parameterization of the given curve with respect to arc length.