Exam 13: Vector Functions

If $\mathbf { u } ( t ) = \mathbf { i } - 5 t ^ { 2 } \mathbf { j } + 4 t ^ { 4 } \mathbf { k }$ and $\mathbf { v } ( t ) = t \mathbf { i } - \cos t \mathbf { j } + \sin t \mathbf { k }$ , find $\frac { d } { d t } [ \mathbf { u } ( t ) \cdot \mathbf { v } ( t ) ]$ .

A force with magnitude $12$ N acts directly upward from the xy-plane on an object with mass $2$ kg. The object starts at the origin with initial velocity $\mathbf { v } ( 0 ) = 3 \mathbf { i } - 2 \mathbf { j }$ . Find its position function.

The curves $\mathbf { r } _ { 1 } ( t ) = \left\langle t , \mathrm { t } ^ { 4 } , \mathrm { t } ^ { 8 } \right\rangle$ and $\mathbf { r } _ { 2 } ( t ) = \langle \sin t , \sin 5 t , t \rangle$ intersects at the origin. Find their angle of intersection correct to the nearest degree.

If $\mathbf { r } ( t ) = \mathbf { i } + t \cos \pi t \mathbf { j } + 2 \sin \pi t\mathbf { k }$ , evaluate $\int _ { 0 } ^ { 1 } r ( t ) d t$ .

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

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

A projectile is fired from ground level with an initial speed of 1100 ft/sec and an angle of elevation of $60 ^ { \circ }$ A) Find the range of the projectile. B) What is the maximm height attained by the projectile? C) What is the speed of the projectile at impact? Round your answers to the nearest integer.

The figure shows a curve $C$ given by a vector function $\mathbf { r } ( t )$ . Choose the correct expression for $\mathbf { r } ^ { \prime } ( 4 )$ .

Find $\mathbf { r } ( t )$ satisfying the conditions for $\mathbf { r } ^ { \prime } ( t ) = 5 \mathbf { i } + 8 t \mathbf { j } - 6 t ^ { 2 } \mathbf { k }$ $\mathbf { r } ( 0 ) = \mathbf { i } + \mathbf { j }$

Find the unit tangent and unit normal vectors $\mathbf { T } ( t )$ and $\mathbf { N } ( t )$ for the curve C defined by $\mathbf { r } ( t ) = 5 \mathbf { i } + t \mathbf { j } + t ^ { 2 } \mathbf { k }$

Find the arc length function $s ( t )$ for the curve defined by $\mathbf { r } ( t ) = e ^ { 2 t } \cos t \mathbf { i } + e ^ { 2 t } \sin t \mathbf { j } + e ^ { 2 t } \mathbf { k }$ for $t \geq 0$ Then use this result to find a parametrization of C in terms of s.

Find the derivative $\frac { d } { d t } \left[ \mathbf { r } ( 6 t ) \cdot \mathbf { r } \left( t ^ { 4 } \right) \right]$

Use Simpson's Rule with 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.

Find the velocity and position vectors of an object with acceleration $\mathbf { a } ( t ) = 4 \mathbf { i } - 72 \mathbf { j } \mathbf { j } + ( 72 t + 4 ) \mathbf { k }$ initial velocity $\mathbf { v } ( 0 ) = \mathbf { i } + \mathbf { k }$ and initial position $\mathbf { r } ( 0 ) = \mathbf { j } + 3 \mathbf { k }$

Find the length of the curve $\mathbf { r } ( t ) = 8 t \mathbf { i } + 3 \cos t \mathbf { j } + 3 \sin t \mathbf { k }$ $0 \leq t \leq 2 \pi$

If $\mathbf { r } ( t ) = \left\langle t , t ^ { 9 } , t ^ { 11 } \right\rangle$ , find $\mathbf { r } ^ { \prime \prime } ( t )$ .

The position function of a particle is given by $\mathbf { r } ( t ) = \left\langle 5 t ^ { 2 } , 5 t , 5 t ^ { 2 } - 100 t \right\rangle$ When is the speed a minimum?

Find the speed of a particle with the given position function. $\mathbf { r } ( t ) = 5 \sqrt { 2 } t \mathbf { i } + e ^ { 5 t } \mathbf { j } - e ^ { - 5 t } \mathbf { k }$

Find the limit. $\lim _ { t \rightarrow 0 ^ { + } } \langle10 \cos t , 30 \sin t , 5 t \ln t\rangle$

Find the curvature of $y = x ^ { 4 }$ .