Deck 12: Vector-Valued Functions

Question 1

Answer 1

E

Question 2

Given the vector-valued function below, evaluate

Answer 2

C

Question 3

Sketch the curve represented by the vector-valued function give the orientation of the curve.

Answer 3

C

Question 4

a vector-valued function, using the given parameter, to represent the intersection of the surfaces given below.

Question 5

Suppose the outer edge of a playground slide is in the shape of a helix of radius 10.5 meters. The slide has a height of 2 meters and makes one complete revolution from top to bottom.
Find a vector-valued function for the helix.

Question 6

Evaluate the limit given below.

Question 7

the domain of the vector-valued function given below.

Question 8

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

Question 9

Determine the interval on which the vector-valued function

Question 10

Question 11

Find a vector-valued function, using the given parameter, to represent the intersection of the surfaces given below. $\begin{array}{l}\text { Surfaces }&\text { Parameter }\\z=x^{2}+y^{2}, z=81&x=9 \cos 8 \pi t\end{array}$

A) $\mathbf { r } ( t ) = 9 \mathbf { i } + 9 \sin 8 \pi \mathbf { j } + 9 \cos 8 \pi \mathbf { k }$
B) $\mathbf { r } ( t ) = 9 \cos 8 \pi \mathbf { i } + 9 \sin \pi \mathbf { j } + 81 \mathbf { k }$
C) $\mathbf { r } ( t ) = 9 \cos 8 \pi \mathbf { i } + 9 \sin 8 \pi \mathbf { j } + 9 \mathbf { k }$
D) $\mathbf { r } ( t ) = 9 \cos 8 \pi t \mathbf { i } + 81 \mathbf { j } + 9 \sin 8 \pi \mathbf { k }$
E) $\mathbf { r } ( t ) = 9 \cos 8 \pi \mathbf { i } + 9 \sin 8 \pi \mathbf { j } + 81 \mathbf { k }$

Question 12

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) $\mathbf { r } ( t ) = 25 \sin 2 \pi \mathbf { i } + 25 \cos 2 \pi t \mathbf { j } + 5 \sin ^ { 2 } 2 \pi t \mathbf { k }$

Question 13

the domain of the vector-valued function given below.

Question 14

Question 15

Sketch the curve represented by the vector-valued function and give the orientation of the curve.

Question 16

Match the equation with the graph shown in red below.

Question 17

Evaluate the limit given below.

Question 18

Suppose the two particles travel along the space curves

Question 19

Represent the following curve by a vector-valued function.

Question 20

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

Question 21

Question 22

Question 23

the open interval on which the curve given by the vector-valued function

Question 24

the xy-plane. Sketch a graph of the path and sketch the velocity and acceleration vectors at the point

Question 25

Question 26

Question 27

Question 28

Evaluate the definite integral below.

Question 29

the indefinite integral below.

Question 30

Question 31

Use the properties of the derivative to find vector-valued functions.

Question 32

Use the properties of the derivative to find vector-valued functions.

Question 33

Use the properties of the derivative to find vector-valued functions.

Question 34

Question 35

the properties of the derivative to find vector-valued functions.

Question 36

particle moves in the yz-plane along the curve represented by the vector-valued

Question 37

vector-valued functions.

Question 38

Question 39

particle moves in the yz-plane along the curve represented by the vector-valued

Question 40

the indefinite integral below.

Question 41

baseball player at second base throws a ball 87 feet to the player at first base. The ball is thrown 5 feet above the ground with an initial velocity of 50 miles per hour and at an angle of
Above the horizontal. At which height does the player at first base catch the ball? Round your
Answer to three decimal places.

A)7.576 feet
B)6.676 feet
C)5.476 feet
D)6.076 feet
E)5.576 feet

Question 42

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 Use the model for projectile motion, assuming there is no air resistance. Round your answer to three
Decimal places.

A)14,289.757 feet
B)14,292.757 feet
C)7,146.378 feet
D)7,149.378 feet
E)14,298.757 feet

Question 43

quarterback of a football team releases a pass at a height of 6 feet above the playing field, and the football is caught by a receiver 42 yards directly downfield at a height of 3 feet.
The pass is released at an angle of Round your answer to one decimal place.

A)-295.8 feet
B)-194.1 feet
C)-278.9 feet
D)-224.3 feet
E)-197.7 feet

Question 44

three decimal places.

A)11.077
B)7.310
C)11.797
D)5.898
E)5.538

Question 45

Question 46

the principle unit normal vector to the curve given below at the specified point.

Question 47

Determine the range of a projectile fired at a height of 5 feet above the ground with an initial velocity of 1000 feet per second and at an angle of for projectile motion, assuming there is no air resistance. Round your answer to three decimal places.

A)15,627.500 feet
B)8,137.843 feet
C)31,252.187 feet
D)31,254.999 feet
E)16,268.138 feet

Question 48

the unit tangent vector to the curve given below at the specified point.

Question 49

the principle unit normal vector to the curve given below at the specified point.

Question 50

the given acceleration function and initial conditions to find the position at time t = 3.

Question 51

Question 52

Question 53

quarterback of a football team releases a pass at a height of 8 feet above the playing field, and the football is caught by a receiver 38 yards directly downfield at a height of 4 feet.
The pass is released at an angle of released. Round your answer to three decimal places.

A)33.169 feet per second
B)59.366 feet per second
C)19.462 feet per second
D)118.732 feet per second

Question 54

quarterback of a football team releases a pass at a height of 6 feet above the playing field, and the football is caught by a receiver 34 yards directly downfield at a height of 3 feet.
The pass is released at an angle of to reach the proper position after the quarterback releases the football. Round your answer to one
Decimal place.

A)0.8 second
B)4.6 seconds
C)7.8 seconds
D)1.3 seconds
E)2.6 seconds

Question 55

three decimal places.

A)6.581
B)4.818
C)3.407
D)2.154
E)4.962

Question 56

the principle unit normal vector to the curve given below at the specified point.

Question 57

the given acceleration function and initial conditions to find the position at time t

Question 58

the given acceleration function and initial conditions to find the position at time t = 1.

Question 59

the unit tangent vector

Question 60

the unit tangent vector to the curve given below at the specified point.

Question 61

three decimal places.

A)18.724
B)23.792
C)22.264
D)15.179
E)22.764

Question 62

the curvature K of the curve answer to three decimal places.

A)0.883
B)0.994
C)0.019
D)0.012
E)0.017

Question 63

three decimal places.

A)-40.625
B)-39.209
C)-111.667
D)-46.016
E)-43.988

Question 64

the radius of curvature of the plane curve your answer to three decimal places.

A)24.694
B)85.333
C)28.133
D)34.510
E)37.510

Question 65

answer to three decimal places.

A)0.062
B)0.071
C)0.659
D)0.000
E)0.753

Question 66

baseball is hit 5 feet above the ground at 140 feet per second and at an angle of with respect to the ground. Find the vector-valued function for the path of the baseball.

Question 67

the length of the plane curve given below.

Question 68

baseball is hit 3 feet above the ground at 80 feet per second and at an angle of with respect to the ground. Find the range. Round your answer to one decimal place.

A)203 feet
B)259.5 feet
C)275.7 feet
D)386.6 feet
E)215.6 feet

Question 69

answer to three decimal places.

A)0.970
B)0.465
C)0.930
D)1.814
E)1.060

Question 70

baseball is hit 3 feet above the ground at 80 feet per second and at an angle of with respect to the ground. Find the maximum height. Round your answer to the nearest integer.

A)66 feet
B)65 feet
C)53 feet
D)56 feet
E)61 feet

Question 71

baseball is hit 5 feet above the ground at 95 feet per second and at an angle of with respect to the ground. Find the arc length of the trajectory. Round your answer to one decimal place.

A)270.8 feet
B)298 feet
C)293.7 feet
D)239.1 feet
E)330.7 feet

Question 72

Because of a storm, ground controllers instruct the pilot of a plane flying at an altitude of 1.6 miles to make a of the plane during this maneuver is time in hours and is the distance in miles. Determine the speed of the plane. Round your answer to
The nearest integer.

A)50 mi/h
B)16 mi/h
C)54 mi/h
D)34 mi/h
E)79 mi/h

Question 73

the arc length for answer to two decimal places.

A)67.63
B)73.37
C)79.40
D)49.75
E)62.97

Question 74

A)16
B)0
C)5
D)25
E)4

Question 75

the length of the space curve given below.

Question 76

the curvature K of the curve given below.

Question 77

the curvature K of the curve Round your answer to three decimal places.

A)7.286
B)9.952
C)0.021
D)0.014
E)0.009

Question 78

the curvature of the plane curve three decimal places.

A)0.498
B)0.332
C)0.015
D)0.005
E)0.003

Question 79

answer to three decimal places.

A)0.781
B)0.724
C)0.343
D)0.686
E)0.649

Question 80

the point on the curve maximum.