Deck 10: Parametric Equations, Polar Coordinates, and Vectors

Question 1

​What is the total distance traveled by the particle in the time interval ? Round your answer to three decimal places.

Answer 1

5.885

Question 2

​The velocity vector of a particle in the xy-plane is for all real t​.If the position of the particle at is ,then the position vector of the particle is ​

A)​
B)​
C)​
D)​

Answer 2

​

Question 3

​The area enclosed inside the polar curve is ​

A)​14
B)​
C)​28
D)​

Answer 3

​14

Question 4

A particle moves in the xy​-plane for values of time t​ in the interval according to the parametric equations and ​ .For what value(s)of t in this interval is the line tangent to the path of the particle vertical?
​

A)​3 only
B)​1.5,2.5,and 3.5 only
C)​2 and 3 only
D)​1.5,2.5,3 and 3.5 only

Question 5

A polar curve is given by .The slope of the curve at is...(round your answer to three decimal places,if necessary) ​

A)​0
B)​
C)​0.5
D)​-0.5

Question 6

​A particle moves in the xy-plane such that its position for time is given by and .What is the slope of the tangent to the path of the particle when ? ​

A)​
B)​
C)​
D)​

Question 7

A particle moving in the xy-plane has acceleration vector for all .The particle is at rest at time .
Give the velocity vector of at time .

Question 8

​A plane curve has parametric equations and .An expression for the rate of the slope of tangent to the path of the curve is ​

A)​
B)​
C)​2
D)​

Question 9

​The path of a particle in the xy-plane is given by the parametric equations and for .An integral expression that represents the length of the path from to is ​

A)​
B)​
C)​
D)​

Question 10

Give the velocity vector of at time .Round your answer to three decimal places.

Question 11

​A particle moves in the xy-plane so that its velocity for time is given by the parametric equations and ​ .​ An expression for the distance traveled by the particle on the interval is ​

A)​
B)​
C)​
D)​

Question 12

​A particle moves in the xy-plane for so that and .At time ,the particle is moving ​

A)​to the right and up
B)​to the left and up
C)​to the left and down
D)​​to the right and down

Question 13

​A particle moves in the xy-plane so that its acceleration vector for time is .If the velocity vector at is ,then how fast is the particle moving when ? ​

A)​32.742
B)​17.437
C)​9.434
D)​33.720

Question 14

​The position vector of a particle moving in the xy-plane is for .For what value of y does the path of the particle have a horizontal tangent? ​

A)​-300
B)​
C)​25
D)​300

Question 15

How fast is the particle moving at time ? Round your answer to three decimal places.

Question 16

​A particle moves in the xy​-plane so that its velocity vector for time is .Which one of the following statements is true about the particle when ? ​

A)​The particle is speeding up.
B)​​The particle is slowing down.
C)​​The particle is at rest.
D)​​The speed of the particle is .

Question 17

The area inside the polar curve is ...(round your answer to three decimal places) ​ ​
​

A)​70.686
B)​32.987
C)​37.699
D)​127.235

Question 18

​A particle moving in the xy-plane has a position vector for .The acceleration vector of the particle is ​

A)​
B)​
C)​
D)​

Question 19

​The position vector of particle moving in the xy-plane is .What is the distance traveled by the particle from to ? Round your answer to three decimal places. ​

A)​9.870
B)​10.354
C)​12.335
D)​42.912

Question 20

The position vector describes the path of an object moving in the xy-plane.Sketch a graph of the path and sketch the velocity and acceleration vectors at the point . ​

A)​

B)​

C)​

D)​

E)​

Question 21

Find the domain of the vector-valued function given below. ​ ​

A)[ -∞,4)
B)[0,4]
C)[-4,4]
D)(-∞,∞)
E)[- 4,∞)

Question 22

The position vector describes the path of an object moving in space.Find the speed of the object. ​

A)
B) ​
C) ​
D) ​
E)

Question 23

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 above the horizontal.Use the model 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 24

Find given the functions below. ​ ​

A) ​
B) ​
C) ​
D) ​
E) ​

Question 25

The 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 with the horizontal.Find the speed of the football when it is 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
E)190.165 feet per second

Question 26

Find the domain of the vector-valued function given below. ​ ​
​
Where​ ​

A)(-∞,∞)
B)[-1,1]
C)[-4,∞)
D)[0,1]
E)[1,∞)

Question 27

The position vector describes the path of an object moving in space.Find the speed s(t)of the object. ​

A)21
B)
C)
D)149
E)

Question 28

Given the vector-valued function below,evaluate . ​ ​

A) ​
B) ​
C) ​
D) ​
E) ​

Question 29

The 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 with the horizontal.Find the time in seconds the receiver has 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 30

A baseball,hit 9 feet above the ground,leaves the bat at an angle of and is caught by an outfielder 9 feet above the ground and 500 feet from home plate.What is the initial speed of the ball? Round your answer to two decimal places. ​

A)31.62 ft/sec
B)126.49 ft/sec
C)189.74 ft/sec
D)158.11 ft/sec
E)63.25 ft/sec

Question 31

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 above the horizontal.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 32

The position vector describes the path of an object moving in space.Find the acceleration of the object. ​

A) ​
B) ​
C) ​
D) ​
E) ​

Question 33

Use the given acceleration function and initial conditions to find the position at time . ​ ​

A)
B)
C)
D)
E)

Question 34

Use the given acceleration function and initial conditions to find the position at time . ​ ​

A)
B)
C)
D)
E)

Question 35

A baseball,hit 5 feet above the ground,leaves the bat at an angle of and its caught by an outfielder 5 feet above the ground and 500 feet from home plate.How high does the ball rise? Round your answer to the nearest integer. ​

A)77 feet
B)10 feet
C)167 feet
D)118 feet
E)23 feet

Question 36

The 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 with the horizontal.Find the maximum height of the football.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 37

The position vector describes the path of an object moving in space.Find the velocity v(t)of the object. ​

A)
B)
C)
D)
E)

Question 38

Use the given acceleration function and initial conditions to find the position at time . ​ ​

A)
B)
C)
D)
E)

Question 39

A 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 40

The position vector describes the path of an object moving in space.Find the velocity v(t)of the object. ​

A) ​
B) ​
C) ​
D) ​
E) ​

Question 41

Find r' (t)given the following vector function. ​ ​

A) ​
B) ​
C) ​
D) ​
E) ​

Question 42

Find given the following vector function: ​ ​

A)
B)0
C)
D)
E)

Question 43

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

A)
B)
C)
D)
E)

Question 44

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

A) ​
B) ​
C) ​
D) ​
E) ​

Question 45

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. ​

A) ​
B) ​
C) ​
D) ​
E) ​

Question 46

Find given the following vector function. ​ ​

A)
B)
C)
D)
E)

Question 47

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

A) ​
B) ​
C) ​
D) ​
E) ​

Question 48

Find r' (t)given the following vector function. ​ ​

A) ​
B) ​
C) ​
D) ​
E) ​

Question 49

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

A)​

​

B)​

​

C)​

​

D)​

​

E)​

​

Question 50

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

A) ​
B) ​
C) ​
D) ​
E) ​

Question 51

Suppose the two particles travel along the space curves and .A collision will occur at the point of intersection P if both particles are at P at the same time.Find the point of collision. ​

A)(9,9,9)
B)(9,16,9)
C)(16,9,16)
D)(16,16,16)
E)(8,8,8)

Question 52

Find the vectors r(t)and r' (t)for the following vector function. ​ ​

A) ​
B) ​
C) ​
D) ​
E) ​

Question 53

Find the open interval on which the curve given by the vector-valued function is smooth. ​

A)
B)
C)
D)
E)

Question 54

Determine the interval on which the vector-valued function is continuous. ​

A)(9,∞)
B)(5,∞)
C)[5,∞)
D)[9,∞)
E)(-5,∞)

Question 55

Evaluate the limit given below. ​ ​

A) ​
B) ​
C) ​
D) ​
E)The limit does not exist.​

Question 56

Evaluate the limit given below. ​ ​

A) ​
B) ​
C) ​
D) ​
E)The limit does not exist.​

Question 57

Represent the following curve by a vector-valued function. ​ ​

A) ​
B) ​
C) ​
D) ​
E) ​

Question 58

Find given the following vector function. ​ ​

A)
B)
C)
D)
E)

Question 59

Find given the following vector function. ​ ​

A) ​
B) ​
C) ​
D) ​
E) ​

Question 60

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

A) ​
B) ​
C) ​
D) ​
E) ​

Question 61

Find given the following. ​ ​

A) ​
B) ​
C) ​
D) ​
E) ​

Question 62

Find the indefinite integral below. ​ ​
Do not include an arbitrary constant vector.
​

A) ​
B) ​
C) ​
D) ​
E) ​

Question 63

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

A) ​
B) ​
C) ​
D) ​
E) ​

Question 64

Match the equation with the graph shown in red below. ​ ​

A) ​
B) ​
C) ​
D) ​
E) ​

Question 65

Find the vector v whose initial and terminal points are given below. ​
(5)5,2.1), (9.5,6.1)
​

A)
B)
C)
D)
E)

Question 66

Given ,find . ​

A)
B)
C)
D)
E)

Question 67

A particle moves in the yz-plane along the curve represented by the vector-valued function .Find the maximum value of . ​

A)4
B)3
C)7
D)8
E)5

Question 68

Evaluate the definite integral below. ​
​ ​
​

A)
B)
C)
D)
E)

Question 69

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

A) ​
B) ​
C) ​
D) ​
E) ​

Question 70

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

A) ​
B) ​
C) ​
D) ​
E) ​

Question 71

Find vectors u and v whose initial and terminal points are given.Determine whether u and v are equivalent. ​
U: (6,5), (9,11)v: (3,-5), (6,1)
​

A)
B)
C)
D)
E)

Question 72

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

A)2 ​
B) ​
C)2t ​
D)5 ​
E)0 ​

Question 73

Given and ,find . ​

A)
B)
C)
D)
E)

Question 74

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

A) ​
B) ​
C) ​
D) ​
E) ​

Question 75

Find the vector v whose initial and terminal points are given below. ​
(5,5), (7,2)
​

A)
B)
C)
D)
E)

Question 76

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

A)
B)
C)
D)
E)

Question 77

Find the indefinite integral below. ​ ​
Do not include an arbitrary constant vector.
​

A)
B)
C)
D) ​
E)

Question 78

Given and ,find . ​

A)
B)
C)
D)
E)

Question 79

The vector v and its initial point is given.Find the terminal point. ​ ,initial point ​

A)
B)
C)
D)
E)

Question 80

A particle moves in the yz-plane along the curve represented by the vector-valued function .Find the minimum value of . ​

A)6
B)4
C)7
D)9
E)8