Provide an appropriate response. -A baseball is hit when it is $3.2$ feet above the ground. It leaves the bat with an initial speed of $152 \mathrm { ft } / \mathrm { sec }$, making an angle of $18 ^ { \circ }$ with the horizontal. Assuming a drag coefficient $\mathrm { k } = 0.12$, how high does the ball go, and when dc it reach maximum height? For projectiles with linear drag: \[\begin{array} { l } x = x _ { 0 } + \frac { v _ { 0 } } { k } \left( 1 - e ^ { - k t } \right) \cos \alpha \\ y = y _ { 0 } + \frac { v _ { 0 } } { k } \left( 1 - e ^ { - k t } \right) \sin \alpha + \frac { g } { k ^ { 2 } } \left( 1 - k t - e ^ { - k t } \right) \end{array}\] where $\mathrm { k }$ is the drag coefficient, $\mathrm { v } _ { 0 }$ and $\alpha$ are the projectile's initial speed and launch angle, and $g$ is the acceleratic of gravity $\left( 32 \mathrm { ft } / \sec ^ { 2 } \right)$.

A) Maximum height = 3.48 feet; B) Maximum height = 0.21 feet; time to maximum height = 0.05 sec time to maximum height = -0.21 sec C) Maximum height = 3.51 feet; D) Maximum height = 8.71 feet; time to maximum height = 0.02 sec time to maximum height = 0.74 sec A) Maximum height = 3.48 feet; B) Maximum height = 0.21 feet; time to maximum height = 0.05 sec time to maximum height = -0.21 sec C) Maximum height = 3.51 feet; D) Maximum height = 8.71 feet; time to maximum height = 0.02 sec time to maximum height = 0.74 sec

Exam 14: Vector-Valued Functions and Motion in Space

Calculate the arc length of the indicated portion of the curve r(t). - $\mathbf { r } ( \mathrm { t } ) = \left( 8 + 2 \mathrm { t } ^ { 2 } \right) \mathbf { i } + \left( 2 \mathrm { t } ^ { 2 } - 7 \right) \mathbf { j } + \left( 7 - \mathrm { t } ^ { 2 } \right) \mathbf { k } , - 1 \leq t \leq 2$

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Question 1

Correct Answer:

Verified

A

Find the principal unit normal vector N for the curve r(t). - $\mathbf { r } ( t ) = ( 10 + t ) \mathbf { i } + ( 9 + \ln ( \cos t ) ) \mathbf { k } , - \pi / 2 < t < \pi / 2$

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Question 2

Correct Answer:

Verified

C

Provide an appropriate response. - $\text { Prove that } \int _ { a } ^ { b } k r ( t ) d t = k \int _ { a } ^ { b } r ( t ) d t \text { for any scalar constant } k$

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Question 3

Correct Answer:

Verified

$\begin{array}{l}\int _ { a } ^ { b } k r ( t ) d t = \int _ { a } ^ { b } k ( x ( t ) i + y ( t ) j ) d t = \int _ { a } ^ { b } k x ( t ) i + k y ( t ) j d t\\= k \int _ { a } ^ { b } x ( t ) i d t + k \int _ { a } ^ { b } y ( t ) j d t = k \left[ \int _ { a } ^ { b } x ( t ) i d t + \int _ { a } ^ { b } y ( t ) j d t \right]\\= k \left[ \int _ { a } ^ { b } ( x ( t ) i + y ( t ) j ) d t \right] = k \int _ { a } ^ { b } r ( t ) d t\end{array}$

For the curve r(t), find an equation for the indicated plane at the given value of t. -r(t) = (cosh t)i + (sinh t)j + tk; rectifying plane at t = 0.

(Multiple Choice)

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Question 4

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 -Find the muzzle speed of a gun whose maximum range is 18.8 km. Round your answer to the nearest tenth.

(Multiple Choice)

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Question 5

$\text { For the smooth curve } r ( t ) \text {, find the parametric equations for the line that is tangent to } r \text { at the given parameter value } t = t _ { 0 } \text {. }$ - $\mathbf { r } ( \mathrm { t } ) = \left( 4 \mathrm { t } ^ { 2 } - 3 \mathrm { t } \right) \mathbf { i } + ( \mathrm { t } + 7 ) \mathbf { j } + \mathbf { k } ; \mathrm { t } _ { 0 } = 2$

(Multiple Choice)

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Question 6

$\text { Find the arc length parameter along the curve from the point where } t = 0 \text { by evaluating } s = \int _ { 0 } ^ { t } | v ( \tau ) | d \tau \text {. }$ -r(t) = (1 + 3t)i + (1 + 6t)j + (4 - 4t)k

(Multiple Choice)

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Question 7

Evaluate the integral. - $\int _ { 1 } ^ { 2 } ( ( 2 - 7 t ) \mathbf { i } + 5 \sqrt { t } \mathbf { j } ) d t$

(Multiple Choice)

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Question 8

$\text { For the curve } r ( t ) \text {, write the acceleration in the form aTT } + \text { anN. }$ - $\mathbf { r } ( \mathrm { t } ) = \left( \mathrm { t } ^ { 2 } - 1 \right) \mathbf { i } + ( 2 \mathrm { t } - 3 ) \mathbf { j } + 6 \mathbf { k }$

(Multiple Choice)

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Question 9

Find the torsion of the space curve. - $\mathbf { r } ( \mathrm { t } ) = ( 2 \mathrm { t } - 7 ) \mathbf { i } + \left( \mathrm { t } ^ { 2 } - 2 \right) \mathbf { j } + 7 \mathbf { k }$

(Multiple Choice)

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Question 10

Find T, N, and B for the given space curve. - $\mathbf { r } ( t ) = ( 8 + t ) \mathbf { i } + ( 9 + \ln ( \sec t ) ) \mathbf { j } - 10 \mathbf { k } , - \pi / 2 < t < \pi / 2$

(Multiple Choice)

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Question 11

Provide an appropriate response. -Increasing the initial speed of a projectile by a factor of 6 increases its range by what factor? Assume the elevation is the same.

(Multiple Choice)

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Question 12

$\text { For the smooth curve } r ( t ) \text {, find the parametric equations for the line that is tangent to } r \text { at the given parameter value } t = t _ { 0 } \text {. }$ - $r ( t ) = ( 10 \sin t ) i - ( 3 \cos 3 t ) j + e ^ { - 8 t _ { k } } \mathbf { k } ; t _ { 0 } = 0$

(Multiple Choice)

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Question 13

Find the principal unit normal vector N for the curve r(t). - $r ( t ) = \left( t ^ { 2 } + 4 \right) \mathbf { j } + ( 2 t - 1 ) \mathbf { k }$

(Multiple Choice)

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Question 14

$\text { Find the acceleration vector in terms of } u _ { r } \text { and } u _ { \theta } \text {. }$ - $\mathrm { r } = 2 \cos 3 \mathrm { t }$ and $\theta = 4 \mathrm { t }$

(Multiple Choice)

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Question 15

The vector r(t) is the position vector of a particle at time t. Find the angle between the velocity and the acceleration vectors at time t = 0. - $r ( t ) = \left( 3 t ^ { 2 } + 3 \right) i + \left( 4 t ^ { 3 } - 2 t \right) k$

(Multiple Choice)

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Question 16

The vector r(t) is the position vector of a particle at time t. Find the angle between the velocity and the acceleration vectors at time t = 0. - $\mathbf { r } ( \mathrm { t } ) = \sqrt { 3 } \mathrm { ti } + \left( \mathrm { t } + \frac { \pi } { 3 } \mathrm { t } ^ { 2 } \right) \mathbf { k }$

(Multiple Choice)

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Question 17

Provide an appropriate response. -A baseball is hit when it is $3.2$ feet above the ground. It leaves the bat with an initial speed of $152 \mathrm { ft } / \mathrm { sec }$ , making an angle of $18 ^ { \circ }$ with the horizontal. Assuming a drag coefficient $\mathrm { k } = 0.12$ , how high does the ball go, and when dc it reach maximum height? For projectiles with linear drag: x=+ 1- \alpha y=+ 1- \alpha+ 1-kt- where $\mathrm { k }$ is the drag coefficient, $\mathrm { v } _ { 0 }$ and $\alpha$ are the projectile's initial speed and launch angle, and $g$ is the acceleratic of gravity $\left( 32 \mathrm { ft } / \sec ^ { 2 } \right)$ .

(Multiple Choice)

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Question 18

Find the torsion of the space curve. -r(t) = (cosh t)i + tj + (sinh t)k

(Multiple Choice)

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Question 19

Evaluate the integral. - $\int _ { 0 } ^ { 1 } \left[ \frac { 1 } { \sqrt { 2 t - t ^ { 2 } } } i + t e ^ { 4 } t j - \frac { 15 t ^ { 2 } } { \left( 4 + 5 t ^ { 3 } \right) ^ { 2 } } \mathbf { k } \right] d t$

(Multiple Choice)

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Question 20

Calculate the arc length of the indicated portion of the curve r(t). - $\mathbf { r } ( \mathrm { t } ) = \left( 8 + 2 \mathrm { t } ^ { 2 } \right) \mathbf { i } + \left( 2 \mathrm { t } ^ { 2 } - 7 \right) \mathbf { j } + \left( 7 - \mathrm { t } ^ { 2 } \right) \mathbf { k } , - 1 \leq t \leq 2$

Find the principal unit normal vector N for the curve r(t). - $\mathbf { r } ( t ) = ( 10 + t ) \mathbf { i } + ( 9 + \ln ( \cos t ) ) \mathbf { k } , - \pi / 2 < t < \pi / 2$

Provide an appropriate response. - $\text { Prove that } \int _ { a } ^ { b } k r ( t ) d t = k \int _ { a } ^ { b } r ( t ) d t \text { for any scalar constant } k$

For the curve r(t), find an equation for the indicated plane at the given value of t. -r(t) = (cosh t)i + (sinh t)j + tk; rectifying plane at t = 0.

$\text { Find the arc length parameter along the curve from the point where } t = 0 \text { by evaluating } s = \int _ { 0 } ^ { t } | v ( \tau ) | d \tau \text {. }$ -r(t) = (1 + 3t)i + (1 + 6t)j + (4 - 4t)k

Evaluate the integral. - $\int _ { 1 } ^ { 2 } ( ( 2 - 7 t ) \mathbf { i } + 5 \sqrt { t } \mathbf { j } ) d t$

$\text { For the curve } r ( t ) \text {, write the acceleration in the form aTT } + \text { anN. }$ - $\mathbf { r } ( \mathrm { t } ) = \left( \mathrm { t } ^ { 2 } - 1 \right) \mathbf { i } + ( 2 \mathrm { t } - 3 ) \mathbf { j } + 6 \mathbf { k }$

Find the torsion of the space curve. - $\mathbf { r } ( \mathrm { t } ) = ( 2 \mathrm { t } - 7 ) \mathbf { i } + \left( \mathrm { t } ^ { 2 } - 2 \right) \mathbf { j } + 7 \mathbf { k }$

Find T, N, and B for the given space curve. - $\mathbf { r } ( t ) = ( 8 + t ) \mathbf { i } + ( 9 + \ln ( \sec t ) ) \mathbf { j } - 10 \mathbf { k } , - \pi / 2 < t < \pi / 2$

Provide an appropriate response. -Increasing the initial speed of a projectile by a factor of 6 increases its range by what factor? Assume the elevation is the same.

$\text { For the smooth curve } r ( t ) \text {, find the parametric equations for the line that is tangent to } r \text { at the given parameter value } t = t _ { 0 } \text {. }$ - $r ( t ) = ( 10 \sin t ) i - ( 3 \cos 3 t ) j + e ^ { - 8 t _ { k } } \mathbf { k } ; t _ { 0 } = 0$

Find the principal unit normal vector N for the curve r(t). - $r ( t ) = \left( t ^ { 2 } + 4 \right) \mathbf { j } + ( 2 t - 1 ) \mathbf { k }$

$\text { Find the acceleration vector in terms of } u _ { r } \text { and } u _ { \theta } \text {. }$ - $\mathrm { r } = 2 \cos 3 \mathrm { t }$ and $\theta = 4 \mathrm { t }$

The vector r(t) is the position vector of a particle at time t. Find the angle between the velocity and the acceleration vectors at time t = 0. - $r ( t ) = \left( 3 t ^ { 2 } + 3 \right) i + \left( 4 t ^ { 3 } - 2 t \right) k$

The vector r(t) is the position vector of a particle at time t. Find the angle between the velocity and the acceleration vectors at time t = 0. - $\mathbf { r } ( \mathrm { t } ) = \sqrt { 3 } \mathrm { ti } + \left( \mathrm { t } + \frac { \pi } { 3 } \mathrm { t } ^ { 2 } \right) \mathbf { k }$

Find the torsion of the space curve. -r(t) = (cosh t)i + tj + (sinh t)k

Evaluate the integral. - $\int _ { 0 } ^ { 1 } \left[ \frac { 1 } { \sqrt { 2 t - t ^ { 2 } } } i + t e ^ { 4 } t j - \frac { 15 t ^ { 2 } } { \left( 4 + 5 t ^ { 3 } \right) ^ { 2 } } \mathbf { k } \right] d t$

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Exam 14: Vector-Valued Functions and Motion in Space

Find the principal unit normal vector N for the curve r(t). - r(t)=(10+t)i+(9+ln⁡(cos⁡t))k,−π/2<t<π/2\mathbf { r } ( t ) = ( 10 + t ) \mathbf { i } + ( 9 + \ln ( \cos t ) ) \mathbf { k } , - \pi / 2 < t < \pi / 2r(t)=(10+t)i+(9+ln(cost))k,−π/2<t<π/2

Provide an appropriate response. - Prove that ∫abkr(t)dt=k∫abr(t)dt for any scalar constant k\text { Prove that } \int _ { a } ^ { b } k r ( t ) d t = k \int _ { a } ^ { b } r ( t ) d t \text { for any scalar constant } k Prove that ∫ab​kr(t)dt=k∫ab​r(t)dt for any scalar constant k

For the curve r(t), find an equation for the indicated plane at the given value of t. -r(t) = (cosh t)i + (sinh t)j + tk; rectifying plane at t = 0.

Evaluate the integral. - ∫12((2−7t)i+5tj)dt\int _ { 1 } ^ { 2 } ( ( 2 - 7 t ) \mathbf { i } + 5 \sqrt { t } \mathbf { j } ) d t∫12​((2−7t)i+5t​j)dt

Find the torsion of the space curve. - r(t)=(2t−7)i+(t2−2)j+7k\mathbf { r } ( \mathrm { t } ) = ( 2 \mathrm { t } - 7 ) \mathbf { i } + \left( \mathrm { t } ^ { 2 } - 2 \right) \mathbf { j } + 7 \mathbf { k }r(t)=(2t−7)i+(t2−2)j+7k

Find T, N, and B for the given space curve. - r(t)=(8+t)i+(9+ln⁡(sec⁡t))j−10k,−π/2<t<π/2\mathbf { r } ( t ) = ( 8 + t ) \mathbf { i } + ( 9 + \ln ( \sec t ) ) \mathbf { j } - 10 \mathbf { k } , - \pi / 2 < t < \pi / 2r(t)=(8+t)i+(9+ln(sect))j−10k,−π/2<t<π/2

Provide an appropriate response. -Increasing the initial speed of a projectile by a factor of 6 increases its range by what factor? Assume the elevation is the same.

Find the principal unit normal vector N for the curve r(t). - r(t)=(t2+4)j+(2t−1)kr ( t ) = \left( t ^ { 2 } + 4 \right) \mathbf { j } + ( 2 t - 1 ) \mathbf { k }r(t)=(t2+4)j+(2t−1)k

The vector r(t) is the position vector of a particle at time t. Find the angle between the velocity and the acceleration vectors at time t = 0. - r(t)=(3t2+3)i+(4t3−2t)kr ( t ) = \left( 3 t ^ { 2 } + 3 \right) i + \left( 4 t ^ { 3 } - 2 t \right) kr(t)=(3t2+3)i+(4t3−2t)k

Find the torsion of the space curve. -r(t) = (cosh t)i + tj + (sinh t)k

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Find the principal unit normal vector N for the curve r(t). - $\mathbf { r } ( t ) = ( 10 + t ) \mathbf { i } + ( 9 + \ln ( \cos t ) ) \mathbf { k } , - \pi / 2 < t < \pi / 2$

Provide an appropriate response. - $\text { Prove that } \int _ { a } ^ { b } k r ( t ) d t = k \int _ { a } ^ { b } r ( t ) d t \text { for any scalar constant } k$

Evaluate the integral. - $\int _ { 1 } ^ { 2 } ( ( 2 - 7 t ) \mathbf { i } + 5 \sqrt { t } \mathbf { j } ) d t$

Find the torsion of the space curve. - $\mathbf { r } ( \mathrm { t } ) = ( 2 \mathrm { t } - 7 ) \mathbf { i } + \left( \mathrm { t } ^ { 2 } - 2 \right) \mathbf { j } + 7 \mathbf { k }$

Find T, N, and B for the given space curve. - $\mathbf { r } ( t ) = ( 8 + t ) \mathbf { i } + ( 9 + \ln ( \sec t ) ) \mathbf { j } - 10 \mathbf { k } , - \pi / 2 < t < \pi / 2$

Find the principal unit normal vector N for the curve r(t). - $r ( t ) = \left( t ^ { 2 } + 4 \right) \mathbf { j } + ( 2 t - 1 ) \mathbf { k }$

The vector r(t) is the position vector of a particle at time t. Find the angle between the velocity and the acceleration vectors at time t = 0. - $r ( t ) = \left( 3 t ^ { 2 } + 3 \right) i + \left( 4 t ^ { 3 } - 2 t \right) k$