Exam 13: Vector Functions

At what point on the curve $x = t ^ { 3 } , y = 9 t , z = t ^ { 4 }$ is the normal plane parallel to the plane $3 x + 9 y - 4 z = 4 ?$

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

Find the 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.

Find the integral $\int \left( 4 t \mathbf { i } + 6 t ^ { 2 } \mathbf { j } + 4 \mathbf { k } \right) d t$ . Select the correct answer.

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 $t$ . $\mathbf { r } ( t ) = ( 3 + 3 t ) \mathbf { i } + ( 4 + 9 t ) \mathbf { j } - ( 6 t ) \mathbf { k }$

Sketch the curve of the vector function $\mathbf { r } ( t ) = \left\langle t ^ { 2 } , t ^ { 3 } , t \right\rangle , t \geq 0$ , and indicate the orientation of the curve.

Find the length of the curve $\mathbf { r } ( t ) = 2 t \mathbf { i } + 4 \cos t \mathbf { j } + 4 \sin t \mathbf { k } , 0 \leq t \leq 2 \pi$ Select the correct answer.

Find the length of the curve $\mathbf { r } ( t ) = - 3 t \mathbf { i } + 2 t \mathbf { j } - t \mathbf { k } , - 2 \leq t \leq 1$ .

Find the unit tangent vector $T ( t )$ . $\mathbf { r } ( t ) = \langle 2 \sin t , 4 t , 2 \cos t \}$

Find the domain of the vector function $\mathbf { r } ( t ) = 7 t \mathbf { i } + \frac { 1 } { t - 9 } \mathbf { j }$ . Select the correct answer.

The curvature of the curve given by the vector function $r$ is $\mathrm { k } ( t ) = \frac { \left| \mathbf { r } ^ { t } ( t ) \times \mathbf { r } ^ { tt } ( t ) \right| } { \left| \mathbf { r } ^ { t } ( t ) \right| ^ { 3 } }$ Use the formula to find the curvature of $\mathbf { r } ( t ) = \left\langle \sqrt { 13 } t , e ^ { t } , e ^ { - t } \right)$ at the point $( 0,1,1 )$ . Select the correct answer.

Sketch the curve of the vector function $\mathbf { r } ( t ) = 5 \sin t \mathbf { i } + 6 \cos t \mathbf { j }$ , and indicate the orientation of the curve.

Find the speed of a particle with the given position function. $r ( t ) = t \mathbf { i } + 6 t ^ { 2 } \mathbf { j } + 5 t ^ { 6 } \mathbf { k }$

Sketch the curve of the vector function $\mathbf { r } ( t ) = 5 \sin t \mathbf { i } + 6 \cos t \mathbf { j }$ , and indicate the orientation of the curve.

Let $C$ be a smooth curve defined by $\mathbf { r } ( t ) = 2 \mathbf { i } + 3 t \mathbf { j } + 2 t ^ { 2 } \mathbf { k }$ , and let $\mathbf { T } ( t )$ and $\mathbf { N } ( t )$ 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 $t = 1$ .