Exam 13: Vector Functions

A projectile is fired with an initial speed of $700 \mathrm {~m} / \mathrm { s }$ and angle of elevation $60 ^ { \circ }$ . Find the range of the projectile.

Sketch the curve of the vector function $\mathbf { r } ( t ) = 3 t \mathbf { i } + ( 2 t + 3 ) \mathbf { j } , - 1 \leq t \leq 2$ , and indicate the orientation of the curve.

Find the domain of the vector function $\mathbf { r } ( t ) = 8 t \mathbf { i } + \frac { 1 } { t - 4 } \mathbf { j }$ . Select the correct answer.

Find the limit $\lim _ { t \rightarrow 0 } \left[ \left( t ^ { 2 } + 3 \right) \mathbf { i } + \cos 5 t \mathbf { j } - 6 \mathbf { k } \right]$ . Select the correct answer.

Let $C$ be a smooth curve defined by $\mathbf { r } ( t ) = 7 \mathbf { i } + t \mathbf { j } + 4 t ^ { 2 } \mathbf { k }$ , and let $\mathrm { T } ( t )$ and $\mathrm { N } ( t )$ be the unit tangent vector and unit normal vector to $C$ corresponding to $t$ . The plane determined by $\mathbf { T }$ and $\mathbf { N }$ is called the osculating plane. Find an equation of the osculating plane of the curve described by $\mathbf { r } ( t )$ at $t = 1$ .

Find a vector function describing the curve of intersection of the cylinder $x ^ { 2 } + y ^ { 2 } = 25$ and the plane $x + y + 6 z = 7$ .

Find $\mathbf { r } ^ { \prime } ( t )$ for the function given. $\mathbf { r } ( t ) = 4 \mathbf { i } + \sin t \mathbf { j } + \cos t \mathbf { k }$

Use Simpson's Rule with $\mathrm { n } = 4$ to estimate the length of the arc of the curve with equations $x = \sqrt { t } , y = \frac { 4 } { t } , z = t ^ { 2 } + 1$ , from $( 1,4,2 )$ to $( 2,1,17 )$ . Round your answer to four decimal places. Select the correct answer.

Find $\mathbf { r } ^ { \prime \prime } ( t )$ for the function given. $\mathbf { r } ( t ) = 8 \mathbf { i } + \sin t \mathbf { j } + \cos t \mathbf { k }$

Find parametric equations for the tangent line to the curve with parametric equations $x = 9 t$ , $y = 7 t ^ { 2 } , z = 4 t ^ { 3 }$ at the point with $t = 1$ . Select the correct answer.

Find the length of the curve $\mathbf { r } ( t ) = 2 t \mathbf { i } + t ^ { 2 } \mathbf { j } + \ln t \mathbf { k } , 1 \leq t \leq e ^ { 6 }$ . Select the correct answer.

What force is required so that a particle of mass $m$ has the following position function? $r ( t ) = 5 t ^ { 3 } \mathbf { i } + 9 t ^ { 2 } \mathbf { j } + 7 t ^ { 3 } \mathbf { k }$

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

Find parametric equations for the tangent line to the curve with the given parametric equations at the specified point. $x = t ^ { 11 } , y = t ^ { 5 } , z = t ^ { 6 } ; ( 1,1,1 )$

Find the position vector of a particle that has the given acceleration and the given initial velocity and position. $\mathbf { a } ( t ) = - 4 \mathbf { k } , \mathbf { v } ( 0 ) = \mathbf { i } + \mathbf { j } - 20 \mathbf { k } , \mathbf { r } ( 0 ) = 4 \mathbf { i } + 9 \mathbf { j }$

A mortar shell is fired with a muzzle speed of $250 \mathrm { ft } / \mathrm { sec }$ . Find the angle of elevation of the mortar if the shell strikes a target located $1100 \mathrm { ft }$ away. Round your answer to 2 decimal places.

A particle moves with position function $\mathbf { r } ( t ) = \left( 21 t - 7 t ^ { 3 } - 5 \right) \mathbf { i } + 21 t ^ { 2 } \mathbf { j }$ Find the tangential component of the acceleration vector.

The curvature of the curve given by the vector function $r$ is $k ( t ) = \frac { \left| \mathbf { r } ^ { t } ( t ) \times \mathbf { r } ^ { tt } ( t ) \right| } { \left| \mathbf { r } ^ {t } ( t ) \right| ^ { 3 } }$ Use the formula to find the curvature of $\mathbf { r } ( t ) = \left( \sqrt { 19 } t , e ^ { t } , e ^ { - t } \right\}$ at the point $( 0,1,1 )$ .