Exam 12: Vector-Valued Functions

Suppose the two particles travel along the space curves $\mathbf { r } ( t ) = t ^ { 2 } \mathbf { i } + ( 8 t - 16 ) \mathbf { j } + t ^ { 2 } \mathbf { k }$ and $\mathbf { u } ( t ) = ( 3 t + 4 ) \mathbf { i } + t ^ { 2 } \mathbf { j } + ( 5 t - 4 ) \mathbf { k }$ . A collision will occur at the point of intersection $P$ if both particles are at $P$ at the same time. Find the point of collision.

Find the indefinite integral below. $\int \left( 5 e ^ { 5 t } \mathbf { i } - 4 \sin 4 t \mathbf { j } + 6 \cos 3 t \mathbf { k } \right) d t$ Do not include an arbitrary constant vector.

Determine the range of a projectile fired at a height of 5 feet above the ground with an initial velocity of 1000 feet per second and at an angle of $45 ^ { \circ } \text { above the horizontal. Use the model }$ for projectile motion, assuming there is no air resistance. Round your answer to three decimal places.

$\text { Find } \mathbf { r } ^ { t } ( t ) \text { given the following vector function. }$ $\mathbf { r } ( t ) = \frac { 4 } { t ^ { 2 } } \mathbf { i } - 6 t ^ { 4 } \mathbf { j } + \frac { t ^ { 5 } } { 6 } \mathbf { k }$

Find a vector-valued function, using the given parameter, to represent the intersection of the surfaces given below. Surfaces Parameter z=+,z=81 x=98\pit

Evaluate the limit given below. $\lim _ { t \rightarrow 4 } \left( 9 t \mathbf { i } + \frac { t ^ { 2 } - 16 } { t ^ { 2 } - 4 t } \mathbf { j } + \frac { 9 } { t } \mathbf { k } \right)$

Use the given acceleration function and initial conditions to find the position at time t = 1. $\mathbf { a } ( t ) = 10 \mathbf { i } + 6 \mathbf { j } + 8 \mathbf { k } , \quad \mathbf { v } ( 0 ) = \mathbf { 0 } , \quad \mathbf { r } ( 0 ) = 5 \mathbf { j }$

A baseball is hit 3 feet above the ground at 80 feet per second and at an angle of $\text { of } 45 ^ { \circ }$ with respect to the ground. Find the range. Round your answer to one decimal place.

A baseball player at second base throws a ball 87 feet to the player at first base. The ball is thrown 5 feet above the ground with an initial velocity of 50 miles per hour and at an angle of Above the horizontal. At which height does the player at first base catch the ball? Round your Answer to three decimal places.

Find the indefinite integral below. $\int \left( \frac { - 6 } { t ^ { 3 } } \mathbf { i } + 25 t ^ { 4 } \mathbf { j } + 4 t ^ { - 1 / 5 } \mathbf { k } \right) d t$ Do not include an arbitrary constant vector.

Find the open interval on which the curve given by the vector-valued function $r ( \theta ) = ( 6 \theta + 6 \sin \theta ) \mathbf { i } + ( 8 - 6 \cos \theta ) \mathbf { j }$ is smooth.

Represent the following curve by a vector-valued function. $\frac { x ^ { 2 } } { 121 } + \frac { y ^ { 2 } } { 64 } = 1 , x \geq 0$

$\text { The position vector } r ( t ) = \langle 7 t , 6 \cos t , 6 \sin t \rangle \text { describes the path of an object moving in }$ space. Find the speed $s ( t )$ of the object.

$\text { The position vector } \mathbf { r } ( t ) = \left( 4 \ln t , \frac { 2 } { t } , 6 t ^ { 9 } \right) \text { describes the path of an object moving in }$ space. Find the velocity $\mathbf { v } ( t )$ of the object.

Find a vector-valued function, using the given parameter, to represent the intersection of the surfaces given below. Surfaces Parameter ++=36,x+y=6 x=3+3t

Suppose the outer edge of a playground slide is in the shape of a helix of radius 10.5 meters. The slide has a height of 2 meters and makes one complete revolution from top to bottom. Find a vector-valued function for the helix.

$\text { Find } a _ { \mathbf { T } } \text { at time } t = 1 \text { for the plane curve } \mathbf { r } ( t ) = 6 t ^ { 2 } \mathbf { i } + 5 t \mathbf { j } \text {. Round your answer to }$ three decimal places.

Find the point on the curve given below at which the curvature K is zero. $y = 10 x ^ { 3 } + 31 x ^ { 2 } + 5 x$

Find the unit tangent vector $\mathrm { T } ( t ) \text { for the line tangent to the space curve }$ $\mathbf { r } ( t ) = \langle 12 \cos t , 12 \sin t , 3 )$ at point $p ( 6 \sqrt { 2 } , 6 \sqrt { 2 } , 3 )$ .

Sketch the curve represented by the vector-valued function $\mathbf { r } ( t ) = ( 4 - t ) \mathbf { i } + \sqrt { t } \mathbf { j } \text { and }$ give the orientation of the curve.