Deck 13: Vector-Valued Functions

Question 1

Use a CAS to estimate the arc length of the curve.

A) 4.4
B) 4.3
C) 3.9
D) 5.7

Answer 1

4.4

Question 2

Use a CAS to estimate the arc length of the curve.

A) 17.6
B) 56.5
C) 23.5
D) 7.6

Answer 2

17.6

Question 3

Compute the arc length.

A)
B)
C)
D)

Answer 3

Question 4

Use graphing technology to sketch the curve traced out by the given vector-valued function.

A)
B)
C)
D)

Question 5

Match the vector-valued function with its graph.

A)
B)
C)
D)

Question 6

Determine all values of t at which the given vector-valued function is continuous.

A)
B)
C)
D) all real numbers

Question 7

Find parametric equations for the indicated curve.
The intersection of and z = 4

Question 8

Use graphing technology to sketch the curve traced out by the given vector-valued function. Use the viewpoint and (see figure below).

A)
B)
C)
D)

Question 9

Determine all values of t at which the given vector-valued function is continuous.

A)
B)
C)
D) all real numbers

Question 10

Find parametric equations for the indicated curve.
The intersection of and x + y = 2

Question 11

Show that the curve lies on the elliptical paraboloid . Show all your work.

Question 12

Match the vector-valued function with its graph.

A)
B)
C)
D)

Question 13

Sketch, by hand, the curve traced out by the given vector-valued function.

Question 14

Phillip claims that the trajectory for is identical to the trajectory for , where t represents time. Sharon suggests that Phillip's claim is not necessarily true. Who is correct and why?

Question 15

Sketch, by hand, the curve traced out by the given vector-valued function.

Question 16

Sketch, by hand, the curve traced out by the given vector-valued function.

Question 17

Find the limit if it exists.

A)
B)
C)
D) does not exist

Question 18

Use graphing technology to sketch the curve traced out by the given vector-valued function. Use the viewpoint and (see figure below).

A)
B)
C)
D)

Question 19

Match the vector-valued function with its graph.

A)
B)
C)
D)

Question 20

Find the limit if it exists.

A)
B)
C)
D) does not exist

Question 21

Find t such that and are perpendicular.

A)
B) t = 0
C)
D) The vectors and are not perpendicular for any value of t.

Question 22

Find the velocity function for the given position function.

A)
B)
C)
D)

Question 23

Show that there are no values of t such that and are parallel. Show your work.

Question 24

Find the acceleration function for the given position function.

A)
B)
C)
D)

Question 25

Find t such that and are perpendicular.

A) , k an integer
B) t = 0
C)
D) The vectors and are not perpendicular for any value of t.

Question 26

Explain your answer or give a counterexample. [Hint - it may be useful to think about circular motion about the origin.]

Question 27

Find t such that and are perpendicular.

A)
B) t = 0
C)
D) The vectors and are not perpendicular for any value of t.

Question 28

Find the derivative of the given vector-valued function.

A)
B)
C)
D)

Question 29

Find the velocity function for the given position function.

A)
B)
C)
D)

Question 30

Evaluate the given integral.

A)
B)
C)
D)

Question 31

Find the acceleration function for the given position function.

A)
B)
C)
D)

Question 32

Determine whether the following is true or false.

Question 33

Find the position function from the given velocity function.

A)
B)
C)
D)

Question 34

Evaluate the given integral.

A)
B)
C)
D)

Question 35

Find all values of t such that is parallel to the xy-plane.

A)
B)
C)
D)

Question 36

Find all values of t such that is parallel to the xy-plane.

A)
B)
C)
D) is not parallel to the xy-plane for any real value of t

Question 37

Find the derivative of the given vector-valued function.

A)
B)
C)
D)

Question 38

Evaluate the given integral.

A)
B)
C)
D)

Question 39

Evaluate the given integral.

A)
B)
C)
D)

Question 40

Find the position function from the given velocity function.

A)
B)
C)
D)

Question 41

Ted tells his incredulous nephews that he can turn a bucket of water upside down without the water pouring out! To illustrate, he puts a small amount of water in a bucket and swings it around in a circle so that the bucket travels from his knees, up over his head, and swings back down to his knees (see figure). If the water is 2.3 feet from the center of the circle of motion and is weightless at the top of the loop, what is the linear speed of the bucket? [Ignore the fact that the water in the bucket has depth; that is, pretend all the water is 2.3 feet from the loop's center.]

A) 8.6 ft/s
B) 3.7 ft/s
C) 4.3 ft/s
D) 17.2 ft/s

Question 42

A projectile is fired with initial speed feet per second from a height of h feet at an angle of above the horizontal. Assuming that the only force acting on the object is gravity, find the horizontal range. Round to the nearest tenth of a foot.

A) 928.8 ft
B) 2131.8 ft
C) 281.1 ft
D) 2127.5 ft

Question 43

On the highly-successful TV game show Wheel of Fortune, a contestant spins a large spinner wheel that determines the amount of money for which the contestant is playing. Typically, contestants spin the wheel at an initial rate of 3.1 rad/s, and the (constant) frictional force stops the wheel in about 2.5 seconds. If the frictional force produces a torque of 19.1 foot-pounds, estimate the moment of inertia of the wheel.

A) 15.4
B) 3.9
C) 0.0649
D) 7.1

Question 44

A projectile is fired with initial speed feet per second from a height of h feet at an angle of above the horizontal. Assuming that the only force acting on the object is gravity, find the maximum altitude. Round to the nearest tenth of a foot.

A) 760.8 ft
B) 2133.5 ft
C) 311.0 ft
D) 3013.1 ft

Question 45

Find the force acting on an object of mass 18 kg with the given position function (in meters and seconds).

A)
B)
C)
D)

Question 46

Find the magnitude of the net force on an object of mass 11 kg with the given position function (in units of meters and seconds).

A) 4235 Newtons
B) 2695 Newtons
C) 385 Newtons
D) 1925 Newtons

Question 47

Find the curvature at the given point.

A)
B)
C)
D)

Question 48

Which of the following is an arc length parameterization of the given two-dimensional curve? The line segment from to

A)
B)
C)
D)

Question 49

Find the force acting on an object of mass 9 kg with the given position function (in meters and seconds).

A)
B)
C)
D)

Question 50

Which of the following is an arc length parameterization of the given two-dimensional curve? The circle of radius 2 centered at the origin

A)
B)
C)
D)

Question 51

A golfer rotates a club with constant angular acceleration through an angle of radians. If the angular velocity increases from 0 to 6 rad/sec, find .

A) rad/sec²
B) rad/sec²
C) rad/sec²
D) rad/sec²

Question 52

Find the unit tangent vector to the curve at the indicated point.

A)
B)
C)
D)

Question 53

Find the centripetal force on an object of mass 13 kg with the given position function (in units of meters and seconds).

A)
B)
C)
D)

Question 54

A projectile is fired with initial speed feet per second from a height of h feet at an angle of above the horizontal. Assuming that the only force acting on the object is gravity, find the speed at impact. Round to the nearest tenth of a foot.

A) 333.3 ft/s
B) 933.6 ft/s
C) 205.8 ft/s
D) 1318.3 ft/s

Question 55

Find the unit tangent vector to the curve at the indicated point.

A)
B)
C)
D)

Question 56

A stationary merry-go-round of radius 8 feet is started in motion by a push consisting of a force of 7 pounds on the outside edge, tangent to the circular edge of the merry-go-round, for 4 seconds. The moment of inertia of the merry-go-round is . Find the resulting angular velocity of the merry-go-around.

A) 2 rad/sec
B) 8 rad/sec
C) 1 rad/sec
D) 7 rad/sec

Question 57

At Erik's sixth-year birthday party, the children are playing a game where each contestant tries to throw a tennis ball into a cardboard box located 19 feet away. The box is 24 inches tall and 15 inches wide (see figure). A child tosses a ball with an initial height of 14 inches and an angle of 60^o from the horizontal. If the initial speed of the ball is 25 feet per second, will the ball land in the box? Show all of your work. [Ignore the radius of the ball. Assume that the only force acting on the ball is gravity.Assume that the ball is "on-target" and that there is nothing behind the box to cause a rebound. Note that g = 32 ft/s² = 384 in/s².] blank
blank
24 inches
19 feet
15 inches
[The figure not necessarily drawn to scale.]

Question 58

Find the position function from the given acceleration function.

A)
B)
C)
D)

Question 59

Find the position function from the given acceleration function.

A)
B)
C)
D)

Question 60

Find the landing point for a projectile launched under the given conditions. Round distances to the nearest foot. Assign your coordinate system as indicated in the figure below. initial speed: initial height: h = 7 ft
Angle: launch direction: due east
Magnus force: northerly force of 19 pounds
Projectile mass: 3 slug

A)
B)
C)
D)

Question 61

Find the unit normal vector at the given point.

A)
B)
C)
D)

Question 62

Find the osculating circle at the given point.

A)
B)
C)
D)

Question 63

Use the graph of below to predict whether the curvature would be larger at or .

A)
B)
C)

Question 64

Find the unit normal vector at the given point.

A)
B)
C)
D)

Question 65

Find the curvature of the polar curve at the indicated point.

A)
B)
C)
D) undefined

Question 66

Find the curvature of the polar curve at the indicated point.

A)
B)
C)
D)

Question 67

Find the tangential and normal components of acceleration for the given position function at the indicated point.

A)
B)
C)
D)

Question 68

Find the osculating circle at the given point.

A)
B)
C)
D) There is no osculating circle at the given point.

Question 69

Graph the curvature function for and find the limit of the curvature as .

A)
B)
C)
D)

Question 70

Use the graph of below to predict whether the curvature would be larger at or .

A)
B)
C)

Question 71

Sketch the curve and find any points of maximum or minimum curvature.

A) maximum curvature: and ; minimum curvature:
B) no maximum curvature; minimum curvature: and
C) maximum curvature: and ; minimum curvature:
D) maximum curvature: and ; no minimum curvature

Question 72

Find the unit normal vector at the given point.

A)
B)
C)
D)

Question 73

Use the graph of below to predict whether the curvature would be larger at or .

A)
B)
C)

Question 74

Find the curvature at the given point.

A)
B)
C)
D)

Question 75

For the circle with radius c centered at , show that the curvature at any point is equal to . Show all of your work.

Question 76

Graph the curvature function for and find the limit of the curvature as .

A)
B)
C)
D)

Question 77

Sketch the curve and find any points of maximum or minimum curvature.

A) maximum curvature: ; minimum curvature:
B) maximum curvature: ; minimum curvature:
C) maximum curvature: and ; no minimum curvature
D) maximum curvature: ; no minimum curvature

Question 78

Sketch a plausible curve with domain that has the curvature function graphed below.

Question 79

Graph the curvature function for and find the limit of the curvature as .

A)
B)
C)
D)

Question 80

Consider the claim that there exists a curve for which . Do you agree with this claim? Explain or justify your answer. [You need not give a rigorous proof.]