Exam 13: Vector Functions

A force with magnitude $14 \mathrm {~N}$ acts directly upward from the xy-plane on an object with mass 6 $\mathrm { kg }$ . The object starts at the origin with initial velocity $\mathbf { v } ( 0 ) = 7 \mathbf { i } - 2 \mathbf { j }$ . Find its position function. Select the correct answer.

Find the point of intersection of the tangent lines to the curve $\mathbf { r } ( t ) = \langle \sin \pi t , 7 \sin \pi t , \cos \pi t \rangle$ , at the points where $t = 0$ and $t = 0.5$ .

Let $\mathbf { r } ( t ) = \left\{ \sqrt { 2 - t } , \frac { \left( e ^ { t } - 2 \right) } { t } , \ln ( t + 1 ) \right\}$ Find the domain of $r$ .

At what point on the curve $x = t ^ { 3 } , y = 9 t , z = t ^ { 4 }$ is the normal plane parallel to the plane $3 x + 9 y - 4 z = 4 ?$

A projectile is fired from a height of $350 \mathrm { ft }$ with an initial speed of $200 \mathrm { ft } / \mathrm { sec }$ and an angle of elevation of $30 ^ { \circ }$ . a. What are the scalar tangential and normal components of acceleration of the projectile? b. What are the scalar tangential and normal components of acceleration of the projectile when the projectile is at its maximum height?

Find the velocity, acceleration, and speed of an object with position function $\mathbf { r } ( t ) = 3 t \mathbf { i } + \left( 6 - t ^ { 2 } \right) \mathbf { j }$ for $t = 1$ . Sketch the path of the object and its velocity and acceleration vectors.

Find the velocity of a particle that has the given acceleration and the given initial velocity. $\mathbf { a } ( t ) = 3 k , \mathbf { v } ( 0 ) = 12 \mathbf { i } - 7 \mathbf { j }$

Evaluate the integral. $\int \left( e ^ { 9 t } \mathbf { i } + 8 t \mathbf { j } + \ln t \mathbf { k } \right) d t$

Find the length of the curve $\mathbf { r } ( t ) = 2 t \mathbf { i } + t ^ { 2 } \mathbf { j } + \ln t \mathbf { k } , 1 \leq t \leq e ^ { 3 }$ Select the correct answer.

Find the limit $\lim _ { t \rightarrow 0 } \left[ \left( t ^ { 2 } + 3 \right) \mathbf { i } + \cos 2 t \mathbf { j } - 10 \mathbf { k } \right]$ Select the correct answer.

Find the length of the curve $\mathbf { r } ( t ) = - 2 t \mathbf { i } - t \mathbf { j } + t \mathbf { k } , - 2 \leq t \leq 1$ Select the correct answer.

Find $\mathbf { r } ^ { tt } ( t )$ for the function given. $\mathbf { r } ( t ) = 2 \mathbf { i } + \sin t \mathbf { j } + \cos t \mathbf { k }$

$\text { A particle moves with position function } \mathbf { r } ( t ) = 5 \cos t \mathbf { i } + 5 \sin t \mathbf { j } + 5 t \mathbf { k }$ $\text { Find the normal component of the acceleration vector. }$

Find the acceleration of a particle with the following position function. $\mathbf { r } ( t ) = \left\langle 2 t ^ { 2 } - 2,4 t \right)$

Find parametric equations for the tangent line to the curve with the given parametric equations at the specified point. $x = t ^ { 9 } , y = t ^ { 3 } , z = t ^ { 6 } ; ( 1,1,1 )$

Find the unit tangent and unit normal vectors $\mathrm { T } ( t )$ and $\mathrm { N } ( t )$ for the curve $C$ defined by $\mathbf { r } ( t ) = t \mathbf { i } + 2 t ^ { 2 } \mathbf { j }$ . Sketch the graph of $C$ , and show $\mathbf { T } ( t )$ and $N ( t )$ for $t = 1$ .