Exam 14: Vector-Valued Functions and Motion in Space

Provide an appropriate response. - $\text { Prove that } \int _ { a } ^ { b } \left( \mathbf { r } _ { 1 } ( t ) + r _ { 2 } ( t ) \right) d t = \int _ { a } ^ { b } r _ { 1 } ( t ) d t + \int _ { a } ^ { b } r _ { 2 } ( t ) d t$

Find the curvature of the curve r(t). -r(t) = (10 + 6 cos 8t) i - (4 + 6 sin 8t)j + 3k

Evaluate the integral. - $\left. \int _ { 0 } ^ { \pi / 4 } \left[ \left( 2 \sec ^ { 2 } t \right) \mathbf { i } - ( 3 + \sin t ) \mathbf { j } - ( 10 \sec t \tan t ) \mathbf { k } \right) \right] d t$

$\text { For the curve } r ( t ) \text {, write the acceleration in the form aTT } + \text { anN. }$ -r(t) = (cosh t)i + (sinh t)j + tk

Solve the initial value problem. -Differential Equation: $\frac { \mathrm { d } \mathbf { r } } { \mathrm { dt } } = \left( 9 \mathrm { t } ^ { 2 } - 5 \right) \mathbf { i } - \mathbf { j } + \frac { 1 } { \sqrt { 1 + \mathrm { t } } } \mathbf { k }$ Initial Condition: $\mathbf { r } ( 0 ) = - 9 \mathbf { i } + 5 \mathbf { j } + 11 \mathbf { k }$

Solve the initial value problem. -Differential Equation: $\frac { \mathrm { d } \mathbf { r } } { \mathrm { dt } } = - 9 \mathrm { ti } + 4 \mathrm { tj } + 7 \mathrm { tk }$ Initial Condition: $\mathbf { r } ( 0 ) = - 7 \mathbf { i } + 4 \mathbf { k }$

Find the curvature of the space curve. - $\mathbf { r } ( \mathrm { t } ) = \frac { 40 } { 9 } ( 1 + t ) ^ { 3 / 2 } \mathbf { i } + \frac { 40 } { 9 } ( 1 - t ) ^ { 3 / 2 } \mathbf { j } + \frac { 10 } { 3 } t \mathbf { k }$

Find T, N, and B for the given space curve. -r(t) = (8t sin t + 8cos t)i + (8t cos t - 8 sin t)j - 5k

Evaluate the integral. - $\int _ { 0 } ^ { 1 } \left[ 6 t i + 12 t ^ { 2 } j - \frac { 6 } { ( 1 + t ) ^ { 4 } } k \right] d t$

Solve the problem. Unless stated otherwise, assume that the projectile flight is ideal, that the launch angle is measured from the horizontal, and that the projectile is launched from the origin over a horizontal surface -A spring gun at ground level fires a tennis ball at an angle of $33 ^ { \circ }$ . The ball lands $12 \mathrm {~m}$ away. What was the ball's initial speed? Round your answer to the nearest tenth.

For the curve r(t), find an equation for the indicated plane at the given value of t. - $\mathbf { r } ( \mathrm { t } ) = ( 10 \mathrm { t } \sin t + 10 \cos \mathrm { t } ) \mathbf { i } + ( 10 \mathrm { t } \cos t - 10 \sin t ) \mathbf { j } + 2 \mathbf { k } ; \text { normal plane at } t = 5.5 \pi$

Solve the problem. -Find the values of $v _ { 0 }$ in the equation $e = \frac { r _ { 0 } v _ { 0 } ^ { 2 } } { G M } - 1$ that make the orbit described by $r = \frac { ( 1 + e ) r _ { 0 } } { 1 + e \cos \theta }$ parabolic.

Solve the problem. Unless stated otherwise, assume that the projectile flight is ideal, that the launch angle is measured from the horizontal, and that the projectile is launched from the origin over a horizontal surface -A fan in the bleachers at Wrigley Field throws an opposing player's home run baseball back onto the playing field. Assume that the fan is 30 feet above the field and that the ball is launched at an angle of $35 ^ { \circ }$ . When will the ball hit the ground if its initial speed is $25 \mathrm { ft } / \mathrm { sec }$ ? Round your answer to the nearest tenth.

Provide an appropriate response. -Show that if a particle's velocity vector is always orthogonal to its acceleration vector then the particle's speed is constant.

Provide an appropriate response. -Prove that if $\mathbf { u }$ is a differentiable function of $\mathrm { t }$ and $\mathrm { f }$ is a differentiable scalar function of $\mathrm { t }$ , then $\frac { d } { d t } ( f u ) = \frac { d f } { d t } \mathbf { u } + f \frac { d u } { d t } \text {. }$

Solve the initial value problem. -Differential Equation: $\frac { \mathrm { d } \mathbf { r } } { \mathrm { dt } } = \frac { 3 } { 2 } ( \mathrm { t } + 3 ) ^ { 1 / 2 } \mathbf { i } + \mathrm { e } ^ { \mathrm { t } } \mathbf { j }$ Initial Condition: $\mathbf { r } ( 0 ) = 0$

Find T, N, and B for the given space curve. - $r ( t ) = \frac { 4 } { 3 } ( 1 + t ) ^ { 3 / 2 } \mathbf { i } + \frac { 4 } { 3 } ( 1 - t ) ^ { 3 / 2 } \mathbf { j } + 1 \mathrm { tk }$

Find the length of the indicated portion of the curve. - $r ( t ) = \left( e ^ { t } \cos t \right) i + \left( e ^ { t } \sin t \right) j + 7 e ^ { t } k ^ { \prime } , - \ln 2 \leq t \leq 0$

Solve the problem. -The orbit of a satellite had a period of $\mathrm { T } = 93.11$ minutes. Calculate the semimajor axis of the orbit. (Earth's mass $= 5.975 \times 10 ^ { 24 } \mathrm {~kg}$ and $\mathrm { G } = 6.6720 \times 10 ^ { - 11 } \mathrm { Nm } ^ { 2 } \mathrm {~kg} ^ { - 2 }$ ).

Evaluate the integral. -