Find a vector-valued function, using the given parameter, to represent the intersection of the surfaces given below. $\begin{array}{ll} \text { Surfaces } & \text { Parameter } \\ z=\frac{x^{2}}{25}+\frac{y^{2}}{64}, y+8 x=0 & x=t \end{array}$

A) $\mathbf { r } ( t ) = t \mathbf { i } - 8 t \mathbf { j } + \frac { 25 } { 26 } t ^ { 2 } \mathbf { k }$ B) $\mathbf { r } ( t ) = t \mathbf { i } + \frac { 26 } { 25 } t ^ { 2 } \mathbf { j } - 8 t \mathbf { k }$ C) $\mathbf { r } ( t ) = t \mathbf { i } + 8 t \mathbf { j } + \frac { 26 } { 25 } t ^ { 2 } \mathbf { k }$ D) $\mathbf { r } ( t ) = t \mathbf { i } + \frac { 25 } { 26 } t ^ { 2 } \mathbf { j } - 8 t \mathbf { k }$ E) $\mathbf { r } ( t ) = t \mathbf { i } - 8 t \mathbf { j } + \frac { 26 } { 25 } t ^ { 2 } \mathbf { k }$ A) $\mathbf { r } ( t ) = t \mathbf { i } - 8 t \mathbf { j } + \frac { 25 } { 26 } t ^ { 2 } \mathbf { k }$ B) $\mathbf { r } ( t ) = t \mathbf { i } + \frac { 26 } { 25 } t ^ { 2 } \mathbf { j } - 8 t \mathbf { k }$ C) $\mathbf { r } ( t ) = t \mathbf { i } + 8 t \mathbf { j } + \frac { 26 } { 25 } t ^ { 2 } \mathbf { k }$ D) $\mathbf { r } ( t ) = t \mathbf { i } + \frac { 25 } { 26 } t ^ { 2 } \mathbf { j } - 8 t \mathbf { k }$ E) $\mathbf { r } ( t ) = t \mathbf { i } - 8 t \mathbf { j } + \frac { 26 } { 25 } t ^ { 2 } \mathbf { k }$

Find a vector-valued function, using the given parameter, to represent the intersection of the surfaces given below. $\begin{array}{ll} \text { Surfaces } & \text { Parameter } \\ x^{2}+y^{2}=25, z=x^{2} & x=5 \sin 2 \pi t \end{array}$

A) $\mathbf { r } ( t ) = 25 \sin 2 \pi t \mathbf { i } + 25 \cos 2 \pi t \mathbf { j } - 5 \sin ^ { 2 } 2 \pi t \mathbf { k }$ B) $\mathbf { r } ( t ) = 5 \sin 2 \pi t \mathbf { j } + 5 \cos 2 \pi t \mathbf { i } + 25 \sin ^ { 2 } 2 \pi \mathbf { k }$ C) $\mathbf { r } ( t ) = 5 \sin 2 \pi t \mathbf { i } + 5 \cos 2 \pi t \mathbf { j } - 25 \sin ^ { 2 } 2 \pi \mathbf { k }$ D) $\mathbf { r } ( t ) = 5 \sin 2 \pi t \mathbf { i } + 5 \cos 2 \pi t \mathbf { j } + 25 \sin ^ { 2 } 2 \pi t \mathbf { k }$ E) $\mathbf { r } ( t ) = 25 \sin 2 \pi \mathbf { i } + 25 \cos 2 \pi t \mathbf { j } + 5 \sin ^ { 2 } 2 \pi t \mathbf { k }$ A) $\mathbf { r } ( t ) = 25 \sin 2 \pi t \mathbf { i } + 25 \cos 2 \pi t \mathbf { j } - 5 \sin ^ { 2 } 2 \pi t \mathbf { k }$ B) $\mathbf { r } ( t ) = 5 \sin 2 \pi t \mathbf { j } + 5 \cos 2 \pi t \mathbf { i } + 25 \sin ^ { 2 } 2 \pi \mathbf { k }$ C) $\mathbf { r } ( t ) = 5 \sin 2 \pi t \mathbf { i } + 5 \cos 2 \pi t \mathbf { j } - 25 \sin ^ { 2 } 2 \pi \mathbf { k }$ D) $\mathbf { r } ( t ) = 5 \sin 2 \pi t \mathbf { i } + 5 \cos 2 \pi t \mathbf { j } + 25 \sin ^ { 2 } 2 \pi t \mathbf { k }$ E) $\mathbf { r } ( t ) = 25 \sin 2 \pi \mathbf { i } + 25 \cos 2 \pi t \mathbf { j } + 5 \sin ^ { 2 } 2 \pi t \mathbf { k }$

$D _ { t } [ \mathbf { r } ( t ) \times \mathbf { u } ( t ) ] \text { given the following }$ Use the properties of the derivative to find vector-valued functions. \[\begin{array} { l } \mathbf { r } ( t ) = 2 t \mathbf { i } - 3 t ^ { 3 } \mathbf { j } + 3 t ^ { 2 } \mathbf { k } \\ \mathbf { u } ( t ) = 3 \mathbf { i } + 5 t ^ { 2 } \mathbf { j } + 4 t ^ { 3 } \mathbf { k } \end{array}\]

A) $\left( - 60 t ^ { 3 } - 72 t ^ { 5 } \right) \mathbf { i } + \left( 18 t ^ { 2 } - 32 t ^ { 3 } \right) \mathbf { j } + 57 t ^ { 2 } \mathbf { k }$ B) $\left( - 60 t ^ { 3 } - 72 t ^ { 5 } \right) \mathbf { i } + \left( 32 t + 18 t ^ { 3 } \right) \mathbf { j } + 57 t ^ { 2 } \mathbf { k }$ C) $\left( - 60 t ^ { 3 } - 72 t ^ { 5 } \right) \mathbf { i } + \left( 18 t - 32 t ^ { 3 } \right) \mathbf { j } - 3 t ^ { 2 } \mathbf { k }$ D) $\left( - 60 t ^ { 3 } - 72 t ^ { 5 } \right) \mathbf { i } + \left( 18 t - 32 t ^ { 3 } \right) \mathbf { j } + 57 t ^ { 2 } \mathbf { k }$ E) $\left( - 60 t ^ { 3 } - 72 t ^ { 5 } \right) \mathbf { i } + \left( 18 t + 32 t ^ { 3 } \right) \mathbf { j } + 57 t ^ { 3 } \mathbf { k }$ A) $\left( - 60 t ^ { 3 } - 72 t ^ { 5 } \right) \mathbf { i } + \left( 18 t ^ { 2 } - 32 t ^ { 3 } \right) \mathbf { j } + 57 t ^ { 2 } \mathbf { k }$ B) $\left( - 60 t ^ { 3 } - 72 t ^ { 5 } \right) \mathbf { i } + \left( 32 t + 18 t ^ { 3 } \right) \mathbf { j } + 57 t ^ { 2 } \mathbf { k }$ C) $\left( - 60 t ^ { 3 } - 72 t ^ { 5 } \right) \mathbf { i } + \left( 18 t - 32 t ^ { 3 } \right) \mathbf { j } - 3 t ^ { 2 } \mathbf { k }$ D) $\left( - 60 t ^ { 3 } - 72 t ^ { 5 } \right) \mathbf { i } + \left( 18 t - 32 t ^ { 3 } \right) \mathbf { j } + 57 t ^ { 2 } \mathbf { k }$ E) $\left( - 60 t ^ { 3 } - 72 t ^ { 5 } \right) \mathbf { i } + \left( 18 t + 32 t ^ { 3 } \right) \mathbf { j } + 57 t ^ { 3 } \mathbf { k }$

Find the domain of the vector-valued function given below. $\mathbf { r } ( t ) = \mathbf { F } ( t ) \times \mathbf { G } ( t )$ where $\begin{array} { l } \mathbf { F } ( t ) = t ^ { 3 } \mathbf { i } + \frac { 1 } { t + 4 } \mathbf { j } + ( t - 4 ) \mathbf { k } \\ \mathbf { G } ( t ) = \sqrt { 1 - t ^ { 2 } } \mathbf { i } + t \mathbf { j } + ( t - 1 ) \mathbf { k } \end{array}$

A) $( - \infty , \infty )$ B) $[ - 1,1 ]$ C) $[ - 4 , \infty )$ D) $[ 0,1 ]$ E) $[ 1 , \infty )$ A) $( - \infty , \infty )$ B) $[ - 1,1 ]$ C) $[ - 4 , \infty )$ D) $[ 0,1 ]$ E) $[ 1 , \infty )$

$\text { Use the properties of the derivative to find } D _ { t } [ \mathbf { r } ( t ) \cdot \mathbf { u } ( t ) ] \text { given the following }$ vector-valued functions. $\begin{array} { l } \mathbf { r } ( t ) = 3 t \mathbf { i } + 6 t ^ { 3 } \mathbf { j } + 3 t ^ { 3 } \mathbf { j } \\ \mathbf { u } ( t ) = 2 \mathbf { i } + 2 t ^ { 2 } \mathbf { j } - 3 t ^ { 3 } \mathbf { k } \end{array}$

A) $6 t + 60 t ^ { 4 } - 54 t ^ { 5 }$ B) $6 + 60 t ^ { 4 } - 54 t ^ { 5 }$ C) $- 6 + 60 t ^ { 4 } + 54 t ^ { 5 }$ D) $6 + 60 t ^ { 4 } + 54 t ^ { 5 }$ E) $- 6 + 60 t ^ { 4 } - 16 t ^ { 5 }$ A) $6 t + 60 t ^ { 4 } - 54 t ^ { 5 }$ B) $6 + 60 t ^ { 4 } - 54 t ^ { 5 }$ C) $- 6 + 60 t ^ { 4 } + 54 t ^ { 5 }$ D) $6 + 60 t ^ { 4 } + 54 t ^ { 5 }$ E) $- 6 + 60 t ^ { 4 } - 16 t ^ { 5 }$

Exam 12: Vector-Valued Functions

$\text { Find } \mathbf { r } ^ { t } ( t ) \cdot \mathbf { r } ^ {tt } ( t ) \text { given the following vector function. }$ $\mathbf { r } ( t ) = \left( 2 t ^ { 2 } + 2 t \right) \mathbf { i } + \left( 3 t ^ { 2 } + 4 t \right) \mathbf { j }$

(Multiple Choice)

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Question 21

$\text { Find } a _ { \mathbf { T } } \text { at time } t = \frac { \pi } { 3 } \text { for the space curve } \mathbf { r } ( t ) = 7 \cos t \mathbf { i } + 7 \sin t \mathbf { j } + 8 t \mathbf { k } \text {. Round your }$ answer to three decimal places.

(Multiple Choice)

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Question 22

Find a vector-valued function, using the given parameter, to represent the intersection of the surfaces given below. Surfaces Parameter z=+,y+8x=0 x=t

(Multiple Choice)

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Question 23

$\text { Find } a _ { \mathbf { N } } \text { at time } t = \frac { \pi } { 3 } \text { for the space curve } \mathbf { r } ( t ) = 4 \cos t \mathbf { i } + 4 \sin t \mathbf { j } + 5 t \mathbf { k }$

(Multiple Choice)

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Question 24

Find the curvature K of the curve $\mathbf { r } ( t ) = t \mathbf { i } + 5 t ^ { 2 } \mathbf { j } + \frac { 7 t ^ { 3 } } { 4 } \mathbf { k } \text { at the point } P ( 2,20,14 )$ Round your answer to three decimal places.

(Multiple Choice)

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Question 25

Find a vector-valued function, using the given parameter, to represent the intersection of the surfaces given below. Surfaces Parameter +=25,z= x=52\pit

(Multiple Choice)

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Question 26

A baseball is hit 5 feet above the ground at 95 feet per second and at an angle of $\text { of } 45 ^ { \circ }$ with respect to the ground. Find the arc length of the trajectory. Round your answer to one decimal place.

(Multiple Choice)

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Question 27

$\text { Find } \mathbf { r } ( t ) \cdot \mathbf { u } ( t ) \text { given the functions below. }$ $\mathbf { r } ( t ) = 3 t \mathbf { i } + \frac { t } { 5 } \mathbf { j } + 2 t ^ { 2 } \mathbf { k } , \mathbf { u } ( t ) = 6 t \mathbf { i } + 6 \mathbf { j } - 5 t \mathbf { k }$

(Multiple Choice)

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Question 28

$D _ { t } [ \mathbf { r } ( t ) \times \mathbf { u } ( t ) ] \text { given the following }$ Use the properties of the derivative to find vector-valued functions. (t)=2t-3+3 (t)=3+5+4

(Multiple Choice)

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Question 29

A 5700-pound vehicle is driven at a speed of 25 miles per hour on a circular interchange of radius 90 feet. To keep the vehicle from skidding off course, what frictional force must The road surface exert on the tires? Round your answer to one decimal place.

(Multiple Choice)

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Question 30

Determine the interval on which the vector-valued function $r ( t ) = 9 e ^ { - t } \mathbf { i } + e ^ { - t } \mathbf { j } + \ln ( t - 5 ) \mathbf { k }$ is continuous.

(Multiple Choice)

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Question 31

Find the domain of the vector-valued function given below. $\mathbf { r } ( t ) = \mathbf { F } ( t ) \times \mathbf { G } ( t )$ where (t)=++(t-4) (t)=+t+(t-1)

(Multiple Choice)

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Question 32

$\text { Find } a _ { \mathbf { N } } \text { at time } t = 0 \text { for the plane curve } r ( t ) = e ^ { 2 t } \mathbf { i } + e ^ { - 6 t } \mathbf { j } \text {. Round your answer to }$ three decimal places.

(Multiple Choice)

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Question 33

$\text { Find the vectors } \mathbf { r } ( t ) \text { and } \mathbf { r } ^ { t } ( t ) \text { for the following vector function. }$ $\mathbf { r } ( t ) = ( 1 + 2 t ) \mathbf { i } + \left( 2 + 5 t ^ { 2 } \right) \mathbf { j } + 2 \mathbf { k }$

(Multiple Choice)

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Question 34

$\text { The position vector } \mathbf { r } ( t ) = \left\langle 2 \cos t , 4 \sin t , t ^ { 2 } \right\rangle \text { describes the path of an object moving }$ in space. Find the velocity $\mathbf { v } ( t )$ of the object.

(Multiple Choice)

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Question 35

$\text { Use the properties of the derivative to find } D _ { t } [ \mathbf { r } ( t ) \cdot \mathbf { u } ( t ) ] \text { given the following }$ vector-valued functions. (t)=3t+6+3 (t)=2+2-3

(Multiple Choice)

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Question 36

Find the point on the curve $y = ( x - 8 ) ^ { 2 } + 9 \text { at which the curvature } K \text { is a }$ maximum.

(Multiple Choice)

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Question 37

Determine the maximum height of a projectile fired at a height of 6 feet above the ground with an initial velocity of 700 feet per second and at an angle of $75 ^ { \circ } \text { above the horizontal. }$ Use the model for projectile motion, assuming there is no air resistance. Round your answer to three Decimal places.

(Multiple Choice)

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Question 38

Use the given acceleration function and initial conditions to find the position at time t $= 1$ , $\mathbf { a } ( t ) = 6 \mathbf { i } + 10 \mathbf { j } + 8 \mathbf { k } , \quad \mathbf { v } ( 0 ) = 4 \mathbf { k } , \quad \mathbf { r } ( 0 ) = \mathbf { 0 }$

(Multiple Choice)

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Question 39

The graph below is most likely the graph of which of the following equations?

(Multiple Choice)

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Question 40

$\text { Find } \mathbf { r } ^ { t } ( t ) \cdot \mathbf { r } ^ {tt } ( t ) \text { given the following vector function. }$ $\mathbf { r } ( t ) = \left( 2 t ^ { 2 } + 2 t \right) \mathbf { i } + \left( 3 t ^ { 2 } + 4 t \right) \mathbf { j }$

$\text { Find } a _ { \mathbf { T } } \text { at time } t = \frac { \pi } { 3 } \text { for the space curve } \mathbf { r } ( t ) = 7 \cos t \mathbf { i } + 7 \sin t \mathbf { j } + 8 t \mathbf { k } \text {. Round your }$ answer to three decimal places.

Find a vector-valued function, using the given parameter, to represent the intersection of the surfaces given below. Surfaces Parameter z=+,y+8x=0 x=t

$\text { Find } a _ { \mathbf { N } } \text { at time } t = \frac { \pi } { 3 } \text { for the space curve } \mathbf { r } ( t ) = 4 \cos t \mathbf { i } + 4 \sin t \mathbf { j } + 5 t \mathbf { k }$

Find the curvature K of the curve $\mathbf { r } ( t ) = t \mathbf { i } + 5 t ^ { 2 } \mathbf { j } + \frac { 7 t ^ { 3 } } { 4 } \mathbf { k } \text { at the point } P ( 2,20,14 )$ Round your answer to three decimal places.

Find a vector-valued function, using the given parameter, to represent the intersection of the surfaces given below. Surfaces Parameter +=25,z= x=52\pit

A baseball is hit 5 feet above the ground at 95 feet per second and at an angle of $\text { of } 45 ^ { \circ }$ with respect to the ground. Find the arc length of the trajectory. Round your answer to one decimal place.

$\text { Find } \mathbf { r } ( t ) \cdot \mathbf { u } ( t ) \text { given the functions below. }$ $\mathbf { r } ( t ) = 3 t \mathbf { i } + \frac { t } { 5 } \mathbf { j } + 2 t ^ { 2 } \mathbf { k } , \mathbf { u } ( t ) = 6 t \mathbf { i } + 6 \mathbf { j } - 5 t \mathbf { k }$

$D _ { t } [ \mathbf { r } ( t ) \times \mathbf { u } ( t ) ] \text { given the following }$ Use the properties of the derivative to find vector-valued functions. (t)=2t-3+3 (t)=3+5+4

A 5700-pound vehicle is driven at a speed of 25 miles per hour on a circular interchange of radius 90 feet. To keep the vehicle from skidding off course, what frictional force must The road surface exert on the tires? Round your answer to one decimal place.

Determine the interval on which the vector-valued function $r ( t ) = 9 e ^ { - t } \mathbf { i } + e ^ { - t } \mathbf { j } + \ln ( t - 5 ) \mathbf { k }$ is continuous.

Find the domain of the vector-valued function given below. $\mathbf { r } ( t ) = \mathbf { F } ( t ) \times \mathbf { G } ( t )$ where (t)=++(t-4) (t)=+t+(t-1)

$\text { Find } a _ { \mathbf { N } } \text { at time } t = 0 \text { for the plane curve } r ( t ) = e ^ { 2 t } \mathbf { i } + e ^ { - 6 t } \mathbf { j } \text {. Round your answer to }$ three decimal places.

$\text { Find the vectors } \mathbf { r } ( t ) \text { and } \mathbf { r } ^ { t } ( t ) \text { for the following vector function. }$ $\mathbf { r } ( t ) = ( 1 + 2 t ) \mathbf { i } + \left( 2 + 5 t ^ { 2 } \right) \mathbf { j } + 2 \mathbf { k }$

$\text { The position vector } \mathbf { r } ( t ) = \left\langle 2 \cos t , 4 \sin t , t ^ { 2 } \right\rangle \text { describes the path of an object moving }$ in space. Find the velocity $\mathbf { v } ( t )$ of the object.

$\text { Use the properties of the derivative to find } D _ { t } [ \mathbf { r } ( t ) \cdot \mathbf { u } ( t ) ] \text { given the following }$ vector-valued functions. (t)=3t+6+3 (t)=2+2-3

Find the point on the curve $y = ( x - 8 ) ^ { 2 } + 9 \text { at which the curvature } K \text { is a }$ maximum.

Use the given acceleration function and initial conditions to find the position at time t $= 1$ , $\mathbf { a } ( t ) = 6 \mathbf { i } + 10 \mathbf { j } + 8 \mathbf { k } , \quad \mathbf { v } ( 0 ) = 4 \mathbf { k } , \quad \mathbf { r } ( 0 ) = \mathbf { 0 }$

The graph below is most likely the graph of which of the following equations?

Graphs and Models

A Preview of Calculus

The Derivative and the Tangent Line Problem

Extrema on an Interval

Antiderivatives and Indefinite Integration

Slope Fields and Eulers Method

Area of a Region Between Two Curves

Basic Integration Rules

Sequences

Conics and Calculus

Vectors in the Plane

Introduction to Functions of Several Variables

Iterated Integrals and Area in the Plane

Vector Fields

Exact First-Order Equations

Filters

Exam 12: Vector-Valued Functions

Find a vector-valued function, using the given parameter, to represent the intersection of the surfaces given below. Surfaces Parameter z=+,y+8x=0 x=t

Find a vector-valued function, using the given parameter, to represent the intersection of the surfaces given below. Surfaces Parameter +=25,z= x=52\pit

A baseball is hit 5 feet above the ground at 95 feet per second and at an angle of of 45∘\text { of } 45 ^ { \circ } of 45∘ with respect to the ground. Find the arc length of the trajectory. Round your answer to one decimal place.

Dt[r(t)×u(t)] given the following D _ { t } [ \mathbf { r } ( t ) \times \mathbf { u } ( t ) ] \text { given the following }Dt​[r(t)×u(t)] given the following Use the properties of the derivative to find vector-valued functions. (t)=2t-3+3 (t)=3+5+4

A 5700-pound vehicle is driven at a speed of 25 miles per hour on a circular interchange of radius 90 feet. To keep the vehicle from skidding off course, what frictional force must The road surface exert on the tires? Round your answer to one decimal place.

Determine the interval on which the vector-valued function r(t)=9e−ti+e−tj+ln⁡(t−5)kr ( t ) = 9 e ^ { - t } \mathbf { i } + e ^ { - t } \mathbf { j } + \ln ( t - 5 ) \mathbf { k }r(t)=9e−ti+e−tj+ln(t−5)k is continuous.

Find the domain of the vector-valued function given below. r(t)=F(t)×G(t)\mathbf { r } ( t ) = \mathbf { F } ( t ) \times \mathbf { G } ( t )r(t)=F(t)×G(t) where (t)=++(t-4) (t)=+t+(t-1)

Find the point on the curve y=(x−8)2+9 at which the curvature K is a y = ( x - 8 ) ^ { 2 } + 9 \text { at which the curvature } K \text { is a }y=(x−8)2+9 at which the curvature K is a maximum.

The graph below is most likely the graph of which of the following equations?

Graphs and Models

A Preview of Calculus

The Derivative and the Tangent Line Problem

Extrema on an Interval

Antiderivatives and Indefinite Integration

Slope Fields and Eulers Method

Area of a Region Between Two Curves

Basic Integration Rules

Sequences

Conics and Calculus

Vectors in the Plane

Introduction to Functions of Several Variables

Iterated Integrals and Area in the Plane

Vector Fields

Exact First-Order Equations

Filters

A baseball is hit 5 feet above the ground at 95 feet per second and at an angle of $\text { of } 45 ^ { \circ }$ with respect to the ground. Find the arc length of the trajectory. Round your answer to one decimal place.

$D _ { t } [ \mathbf { r } ( t ) \times \mathbf { u } ( t ) ] \text { given the following }$ Use the properties of the derivative to find vector-valued functions. (t)=2t-3+3 (t)=3+5+4

Determine the interval on which the vector-valued function $r ( t ) = 9 e ^ { - t } \mathbf { i } + e ^ { - t } \mathbf { j } + \ln ( t - 5 ) \mathbf { k }$ is continuous.

Find the domain of the vector-valued function given below. $\mathbf { r } ( t ) = \mathbf { F } ( t ) \times \mathbf { G } ( t )$ where (t)=++(t-4) (t)=+t+(t-1)

Find the point on the curve $y = ( x - 8 ) ^ { 2 } + 9 \text { at which the curvature } K \text { is a }$ maximum.