Exam 13: Vector Functions

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

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

Find the domain of the vector function $\mathbf { r } ( t ) = \left\langle 8 \sqrt { t } , \frac { 1 } { t - 9 } , \ln t \right\rangle$

Let $\mathbf { r } ( t ) = \left\{ \sqrt { 6 - t } , \frac { \left( e ^ { t } - 8 \right) } { t } , \ln ( t + 1 ) \right\}$ Find the domain of $r$ .

The following table gives coordinates of a particle moving through space along a smooth curve. x y z 0.5 5.8 9.1 4.3 1 12.6 14.9 16.8 1.5 25.6 21.2 29.4 2 39.2 39.5 37.9 2.5 42.4 42.4 43 Find the average velocity over the time interval $[ 0.5,1.5 ]$ . Select the correct answer.

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

A particle moves with position function $r ( t ) = 4 \sqrt { 2 } t \mathbf { i } + e ^ { 4 t } \mathbf { j } + e ^ { - 4 t } \mathbf { k }$ . Find the acceleration of the particle.

Find a vector function that represents the curve of intersection of the two surfaces: the top half of the ellipsoid $x ^ { 2 } + 5 y ^ { 2 } + 5 z ^ { 2 } = 25$ and the parabolic cylinder $y = x ^ { 2 }$ .

The helix $\mathbf { r } _ { 1 } ( t ) = 4 \cos t \mathbf { i } + \sin t \mathbf { j } + t \mathbf { k }$ intersects the curve $\mathbf { r } _ { 2 } ( t ) = ( 4 + t ) \mathbf { i } + 6 t ^ { 2 } \mathbf { j } + 5 t ^ { 3 } \mathbf { k }$ at the point $( 4,0,0 )$ . Find the angle of intersection.

Find parametric equations for the tangent line to the curve with the given parametric equations at the specified point. $x = \cos t , y = 4 e ^ { 6 t } , z = 4 e ^ { - 6 t } ; ( 1,4,4 )$

Reparametrize the curve with respect to arc length measured from the point where $t = 0$ in the direction of increasing $t$ . $\mathbf { r } ( t ) = ( 7 + 3 t ) \mathbf { i } + ( 10 + 9 t ) \mathbf { j } - ( 6 t ) \mathbf { k }$

Find the acceleration of a particle with the given position function. $\mathbf { r } ( t ) = 9 \sin t \mathbf { i } + 10 t \mathbf { j } - 8 \cos t \mathbf { k }$

A projectile is fired with an initial speed of $200 \mathrm {~m} / \mathrm { s }$ and angle of elevation $60 ^ { \circ }$ . Find the range of the projectile. Select the correct answer.

If $\mathbf { r } ( t ) = \left\langle t , t ^ { 5 } , t ^ { 2 } \right\rangle$ , find $\mathbf { r } ^ { \prime \prime } ( t )$

If $\mathbf { r } ( t ) = \left( t , t ^ { 9 } , t ^ { 11 } \right)$ , find $\mathbf { r } ^ { tt } ( t )$ Select the correct answer.

Find the limit. $\lim _ { t \rightarrow 0 ^ { + } } ( 4 \cos t , 12 \sin t , 7 t \ln t )$

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

The torsion of a curve defined by $\mathbf { r } ( t )$ is given by $\tau = \frac { \left( \mathbf { r } ^ { t } \times \mathbf { r } ^ { tt } \right) \cdot \mathbf { r } ^ { ttt } } { \left| \mathbf { r } ^ { t } \times \mathbf { r } ^ { t t } \right| ^ { 2 } }$ Find the torsion of the curve defined by $\mathbf { r } ( t ) = \cos 2 t \mathbf { i } + \sin 2 t \mathbf { j } + 5 t \mathbf { k }$ .