Exam 10: Vector Functions

Find the unit tangent and the unit normal to the graph of the vector function $\mathbf { r } ( t ) = \left\langle t ^ { 2 } - 2,2 t - t ^ { 3 } \right\rangle \text { at } t = 1$

A particle is traveling along a helix whose vector equation is given by $\mathbf { r } ( t ) = \left\langle R _ { 1 } \cos \alpha t , R _ { 2 } \sin \alpha t , \beta t \right\rangle$ , where $R _ { 1 } \geq R _ { 2 }$ . Find its maximum and minimum speeds.

Let $\mathbf { u } ( t ) = 2 t \mathbf { i } + \sin t \mathbf { j } - \cos t \mathbf { k } \text { and } \mathbf { v } ( t ) = \mathbf { i } + t ^ { 2 } \mathbf { j } - t \mathbf { k }$ Find $\frac { d } { d t } [ \mathbf { u } ( t ) \times \mathbf { v } ( t ) ]$

Find $\lim _ { t \rightarrow 0 } \left\langle 2 t , \cos t , e ^ { t } \right\rangle$

Find a space curve which parametrizes the intersection of the paraboloid z = -x² - y² and the plane y = x.

Find the derivative of the vector function r (t) = $\left\langle t , 1 / t , e ^ { t } \right\rangle$ when t = 1.

Show that the curve with vector equation r (t) = 2 $\cos ^ { 2 } t$ i + sin (2t) j + 2 sin t k is the curve of intersection of the surfaces $( x - 1 ) ^ { 2 } + y ^ { 2 } = 1$ and $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 4$ Use this fact to sketch the curve.

Find the intersection point of curves $\mathbf { r } _ { 1 } = \left\langle t , t ^ { 2 } , t ^ { 3 } \right\} \text { and } \mathbf { r } _ { 2 } = \left\langle 4 t + 6,4 t ^ { 2 } , 7 - t \right\rangle$

Find the center of the osculating circle of the curve described by $x = 4 \sin t , y = 3 t , z = 4 \cos t \text { at } ( 0,0,4 )$

Show that if $\mathbf { r } ^ { \prime } ( t )$ and $\mathbf { r } ^ { \prime \prime } ( t )$ are parallel at some point on the curve described by $\mathbf { r } ( t )$ , then the curvature at that point is 0. Give an example of a curve $\mathbf { r } ( t )$ for which $\mathbf { r } ^ { \prime } ( t )$ and $\mathbf { r } ^ { \prime \prime } ( t )$ are always parallel.

Identify the surface with the vector equation $\mathbf { r } ( u , v ) = ( 1 + 2 u + 3 v ) \mathbf { i } + ( 5 - u + 4 v ) \mathbf { j } + ( 3 + 5 u - 7 v ) \mathbf { k }$

Let $\mathbf { r } ( t ) = \langle 5 t , \sin 3 t , \cos 3 t \rangle$ . Show that the velocity vector is perpendicular to the acceleration vector.

Find the equation of the osculating circle of the ellipse whose equation is given by $\left( \operatorname { acos } t , \frac { a } { 2 } \sin t , 0 \right) \text { at } t = 0$

Find parametric equations of the tangent line to the curve $\mathbf { r } ( t ) = e ^ { 2 t } \mathbf { i } + 2 \sin 3 t \mathbf { j }$ at t = 0.

Find the curvature of the ellipse whose equation is given by $( a \cos t , b \sin t , 0 )$

Find the unit normal vector N(t) to the curve r (t) = $\langle \sin t , t , \cos t \rangle$ when t = 0.

Find a parametric representation for the surface $z = x ^ { 2 } + y ^ { 2 }$

Identify the geometric object that is represented by parametric equations $\mathbf { r } ( t , s ) = \langle s + t , 3 t + 1,3 s - 5 t \rangle$ .

A paper carrier is traveling 60 miles per hour down a straight road in the direction of the vector i when he throws a paper out the car window with a velocity (relative to the car) in the direction of j and of magnitude 10 miles per hour.(a) Find the velocity of the paper relative to the ground when the paper carrier releases it.(b) Find the speed of the paper at that time.

Identify the geometric object that is represented by parametric equations $\mathbf { r } ( t , s ) = \langle 3 \cos s , 3 \sin s , t \rangle$ .