Exam 10: Vector Functions

Find the curvature of the curve $y = e ^ { x } \text { at } x = 0$

If $\mathbf { r } ( t ) = t ^ { 2 } \mathbf { i } + 3 t \mathbf { j } + e ^ { t } \mathbf { k }$ , find the acceleration vector and the tangential component of the acceleration vector.

Is it possible for the velocity of a particle to be zero at the same time its acceleration is not zero? Explain.

Let . $\mathbf { r } ( t ) = \langle \cos 2 t , \sin 2 t , 3 \rangle$ . Compute the tangent vector $\mathbf { r } ^ { \prime } ( t )$ and show that it is always orthogonal to $\mathbf { r } ( t )$ .

A helix has radius 5 and height 6, and makes 4 revolutions. Find parametric equations of this helix. What is the arc length of the helix?

Floyd Thunderfoot is a punter for the Vikings. Today the Vikings are playing the Bears in the Metrodome. The Bears stop the Vikings at the Vikings' 40 yard line (line of scrimmage), and Floyd is called in to punt. Floyd needs to kick from 10 yards behind the line of scrimmage in order to get the punt off in time. If the ball has a hang time of 4 seconds and lands at the Bears' 10 yard line, at what angle did Floyd kick the ball, and at what speed? (Ignore air resistance.)

Let a (t), v (t), and r (t) denote the acceleration, velocity, and position at time t of an object moving in the xy-plane. Find r (t), given that $\mathbf { a } ( t ) = \left\langle e ^ { 2 t } + 2 t , e ^ { 2 t } - 3 \right\rangle \mathbf { v } ( 0 ) = \left( \frac { 3 } { 2 } , \frac { 7 } { 2 } \right) \text { and } \mathbf { r } ( 0 ) = \left( \frac { 5 } { 4 } , \frac { 9 } { 4 } \right)$

Identify the geometric object that is represented by parametric equations $\mathbf { r } ( t ) = \{ 3 \cos t , 3 \sin t , 5 \}$ .

Find the length of the curve $\mathbf { r } ( t ) = \langle 2 t , \sin t , \cos t \rangle , 0 \leq t \leq 2 \pi$

Let the position function of a particle be $\mathbf { r } ( t ) = \left\langle t ^ { 2 } , 2 t , e ^ { t } \right\rangle$ . Find the acceleration of the particle when t = 0.

Find a parametric representation for the surface consisting of that part of the hyperboloid $- x ^ { 2 } - y ^ { 2 } + z ^ { 2 } = 1 \text { that lies below the disk } \left\{ ( x , y ) \mid x ^ { 2 } + y ^ { 2 } \leq 4 \right\}$ .

Find $\mathbf { r } ( t ) \text { if } \mathbf { r } ^ { \prime } ( t ) = t ^ { 2 } \mathbf { i } + 4 t ^ { 3 } \mathbf { j } - t ^ { 2 } \mathbf { k } \text { and } \mathbf { r } ( 1 ) = \mathbf { i } + \mathbf { j } + 2 \mathbf { k }$

Are the two planes $\mathbf { r } _ { 1 } ( s , t ) = \langle 1 + s + t , s - t , 1 + 2 s \rangle \text { and } \mathbf { r } _ { 2 } ( s , t ) = \langle 2 + s + 2 t , 3 + t , s + 3 t \rangle$ parallel? Justify your answer.

Suppose a particle is moving in the xy-plane so that its position vector at time t is given by $\mathbf { r } ( t ) = \left\langle t ^ { 3 } - t , t - t ^ { 2 } \right\rangle$ . Find the velocity, speed, and acceleration of the particle at time t = 2.

Identify the geometric object that is represented by parametric equations $\mathbf { r } ( t , s ) = \langle t \cos s , t \sin s , t \rangle$ .

A cannon sits on top of a vertical tower 264 feet tall. It fires a cannonball at 80 ft/s. If the barrel of the cannon is elevated 30 degrees from the horizontal, find how far from the base of the tower the cannonball will land (assuming the ground around the tower is level).

Let the position function of a particle be $\mathbf { r } ( t ) = t \mathbf { i } + t ^ { 2 } \mathbf { j }$ . Find the tangential component of the acceleration vector when t = 1.

Find the curvature of the ellipse whose equation is given by $(\operatorname { acos } t , b \sin t , 0 ) \text { at } t = 0 \text { and } t = \frac { \pi } { 2 }$

Find $\lim _ { t \rightarrow \infty } \left[ \frac { t + 1 } { t - 1 } \mathbf { i } + \left( 1 + \frac { 1 } { t } \right) ^ { t } \mathbf { j } + \arctan t \mathbf { k } \right]$

A curve is given by the vector equation r (t) = (2 + cos t) i + (1 + sin t) j. Find a relation between x and y which has the same graph.