Exam 12: Vector-Valued Functions

Because of a storm, ground controllers instruct the pilot of a plane flying at an altitude of 1.6 miles to make a $90^{\circ}$ turn and climb to an altitude of 1.8 miles. The model for the path of the plane during this maneuver is $\mathbf { r } ( t ) = \langle 4 \cos 4 \pi t , 4 \sin 4 \pi t , 1.6 + 1.6 t \rangle , 0 \leq t \leq \frac { 1 } { 8 } \text { where } t \text { is the }$ time in hours and is the distance in miles. Determine the speed of the plane. Round your answer to The nearest integer.

A particle moves in the yz-plane along the curve represented by the vector-valued function $\mathbf { r } ( t ) = ( 4 \cos t ) \mathbf { j } + ( 9 \sin t ) \mathbf { k }$ . Find the minimum value of $\left\| \mathbf { r } ^ {t } \right\|$

Evaluate the limit given below. $\lim _ { t \rightarrow 2 } \left( e ^ { - 3 t } i + \frac { 2 t ^ { 2 } } { t ^ { 2 } + 6 t } j + \frac { 2 } { t } k \right)$

Find the unit tangent vector to the curve given below at the specified point. $\mathbf { r } ( 2 ) = 5 \cos t \mathbf { i } + 4 \sin t \mathbf { j } , t = \frac { 2 \pi } { 3 }$

Evaluate the definite integral below. $\int _ { 0 } ^ { 2 } \left( - 6 t ^ { 2 } \mathbf { i } + 10 t ^ { 4 } \mathbf { j } + 15 t ^ { 4 } \mathbf { k } \right) d t$

Find the curvature K of the curve given below. $\mathbf { r } ( t ) = t \mathbf { i } + 2 t ^ { 2 } \mathbf { j } + 2 t \mathbf { k }$

Find the principle unit normal vector to the curve given below at the specified point. $\mathbf { r } ( t ) = t \mathbf { i } + \frac { 4 } { t } \mathbf { j } , \quad t = 3$

$D _ { t } [ 2 \mathbf { r } ( t ) - 5 \mathbf { u } ( t ) ] \text { given the following }$ Use the properties of the derivative to find vector-valued functions. (t)=2t+6+2 (t)=2+4+3

Find the curvature of the plane curve $y = 3 x ^ { 2 } + 4 \text { at } x = - 2$ . Round your answer to three decimal places.

Use the properties of the derivative to find $D _ { t } ( \mathbf { r } ( t ) \times \mathbf { u } ( t ) ) \text { given the following }$ vector-valued functions. (t)=44t+44t (t)=+44t+44t

$\text { The position vector } \mathbf { r } ( t ) = 8 t \mathbf { i } + + 4 t \mathbf { j } + 5 t \mathbf { k } \text { describes the path of an object moving in }$ space. Find the speed $s ( t )$ of the object.

The quarterback of a football team releases a pass at a height of 8 feet above the playing field, and the football is caught by a receiver 38 yards directly downfield at a height of 4 feet. The pass is released at an angle of $45 ^ { \circ } \text { with the horizontal. Find the speed of the football when it is }$ released. Round your answer to three decimal places.

Find the domain of the vector-valued function given below. $\mathbf { r } ( t ) = \sqrt { 16 - t ^ { 2 } } \mathbf { i } + \frac { 1 } { t + 6 } \mathbf { j } + ( t - 4 ) \mathbf { k }$

$\text { Find } a _ { \mathbf { T } } \text { at time } t = 2 \text { for the space curve } \mathbf { r } ( t ) = ( 6 t - 7 ) \mathbf { i } + t ^ { 2 } \mathbf { j } - 10 t \mathbf { k } \text {. Round your }$ answer to three decimal places.

$\text { Find } \mathbf { r } ^ { t } ( t ) \cdot \mathbf { r } ^ { tt } ( t ) \text { given the following vector function. }$ $\mathbf { r } ( t ) = 3 \cos t \mathbf { i } + 5 \sin t \mathbf { j }$

$\text { Find } a _ { \mathbb { N } } \text { at time } t = 2 \text { for the space curve } \mathbf { r } ( t ) = ( 5 t - 7 ) \mathbf { i } + t ^ { 2 } \mathbf { j } - 7 t \mathbf { k } \text {. Round your }$ answer to three decimal places.

Find the principle unit normal vector to the curve given below at the specified point. $\mathbf { r } ( t ) = 5 \cos t \mathbf { i } + 5 \sin t \mathbf { j } , \quad t = \frac { 5 \pi } { 3 }$

Given the vector-valued function below, evaluate $\mathbf { r } ( 11 + \Delta t ) - \mathbf { r } ( 11 )$ $\mathbf { r } ( t ) = \ln t \mathbf { i } + \frac { 8 } { t } \mathbf { j } + 4 t \mathbf { k }$

$\text { Find } \mathbf { r } ( t ) \text { given the following. }$ $\mathbf { r } ^ { t } ( t ) = 18 t ^ { 5 } \mathbf { j } + 6 t \mathbf { k } , \mathbf { r } ( 0 ) = 2 \mathbf { i } + 18 \mathbf { j }$

$\text { The position vector } \mathbf { r } ( t ) = 4 t \mathbf { i } + 2 t \mathbf { j } + \frac { 1 } { 6 } t ^ { 2 } \mathbf { k } \text { describes the path of an object moving in }$ space. Find the acceleration $\mathbf { a } ( t )$ of the object.