Deck 10: Parametric Equations and Polar Coordinates

Question 1

Find parametric equations describing the given curve.
The line segment from to

Answer 1

Answers may vary. One possible answer follows.

Question 2

Sketch the plane curve defined by the given parametric equations.

A)
B)
C)
D)

Answer 2

B

Question 3

Sketch the plane curve defined by the given parametric equations.

A)
B)
C)
D)

Answer 3

B

Question 4

Use a CAS or graphing calculator to sketch the plane curve defined by the given parametric equations. Use a window size of [-5, 5] by [-5, 5].

A)
B)
C)
D)

Question 5

Which graph below corresponds to the given parametric equations?

A)
B)
C)
D)

Question 6

Find an x-y equation for the given parametric equations.

A)
B)
C)
D)

Question 7

Use a CAS or graphing calculator to sketch the plane curve defined by the given parametric equations. Use a window size of [-5, 5] by [-5, 5].

A)
B)
C)
D)

Question 8

Use a CAS or graphing calculator to sketch the plane curve defined by the given parametric equations. Use a window size of [-5, 5] by [-5, 5].

A)
B)
C)
D)

Question 9

For the given parametric equations, find a corresponding x-y equation for the curve.

A)
B)
C)
D)

Question 10

Sketch the plane curve defined by the given parametric equations.

A)
B)
C)
D)

Question 11

Use a CAS or graphing calculator to sketch the plane curve defined by the given parametric equations. Use a window size of [-5, 5] by [-5, 5].

A)
B) 17
C)
D)

Question 12

Find parametric equations describing the given curve.
The portion of the parabola from to

Question 13

Sketch the plane curve defined by the given parametric equations.

A)
B)
C)
D)

Question 14

For the given parametric equations, find a corresponding x-y equation for the curve.

A)
B)
C)
D)

Question 15

Find parametric equations for the path of a projectile launched from height ft with initial speed ft/s at angle from the horizontal.

A)
B)
C)
D)

Question 16

For the given parametric equations, find a corresponding x-y equation for the curve.

A)
B)
C)
D)

Question 17

Find parametric equations describing the given curve.
The circle of radius 4 centered at , drawn counterclockwise

Question 18

Find parametric equations for the path of a projectile launched from height ft with initial speed ft/s at angle from the horizontal.

A)
B)
C)
D)

Question 19

Find parametric equations describing the given curve.
The line segment from to

Question 20

For the given parametric equations, find a corresponding x-y equation for the curve.

A)
B)
C)
D)

Question 21

Identify all points at which the curve has a vertical tangent.

A)
B) (1,0)
C)
D)

Question 22

Find the area enclosed by the given curve.

A)
B) 0
C)
D)

Question 23

Find all points of intersection of the two curves.

A)
B)
C)
D) The curves have no point of intersection.

Question 24

Find all points of intersection of the two curves.

A)
B)
C)
D)

Question 25

Parametric equations for the position of an object are given. Find the object's velocity and speed at the given time.

A) horizontal component of velocity: -34
Vertical component of velocity: -17
Speed:
B) horizontal component of velocity: 34
Vertical component of velocity: 47
Speed:
C) horizontal component of velocity: 34
Vertical component of velocity: -47
Speed:
D) horizontal component of velocity: -34
Vertical component of velocity: 17
Speed:

Question 26

Identify all points at which the curve has a vertical tangent.

A)
B)
C)
D)

Question 27

Find the slope of the tangent line to the given curve at the indicated point.

A)
B)
C)
D)

Question 28

Find the speed of the object each time it crosses the x-axis. Show all your work.

Question 29

Find all points of intersection of the two curves.

A)
B)
C)
D) The curves have no points of intersection.

Question 30

Find the slope of the tangent line to the given curve at the indicated point.

A) -1
B)
C) undefined
D) -5

Question 31

Find the area enclosed by the given curve.

A)
B)
C)
D)

Question 32

Parametric equations for the position of an object are given. Find the object's velocity and speed at the given time.

A) horizontal component of velocity: 0
Vertical component of velocity: 7
Speed: 7
B) horizontal component of velocity: -3
Vertical component of velocity: 0
Speed: 3
C) horizontal component of velocity: 3
Vertical component of velocity: 0
Speed: -3
D) horizontal component of velocity:
Vertical component of velocity:
Speed:

Question 33

Which graph below corresponds to the given parametric equations?

A)
B)
C)
D)

Question 34

At time t = 0 seconds, a hockey puck is headed towards the "board" (the wall around the perimeter of the area of play) as shown in the figure (the view is from above the playing surface). At that same instant, a hockey player at a different position on the ice is headed straight up the ice in order to intercept the puck (see figure). The trajectories of the puck and the player are described by the following set of parametric equations: Find the location(s) of intersection of the trajectories of the player and the puck. Distances are in feet and time is in seconds. [figure is not necessarily drawn to scale]

A) and
B)
C) and
D)

Question 35

Find the area enclosed by the given curve.

Question 36

Which graph below corresponds to the given parametric equations?

A)
B)
C)
D)

Question 37

Which graph below corresponds to the given parametric equations?

A)
B)
C)
D)

Question 38

Find all slopes of the curve at the given point. [It may be helpful to graph the curve.]

A)
B) -3 and 0
C) -3 and 3
D) 0, undefined

Question 39

Identify all points at which the curve has a horizontal tangent.

A)
B)
C) (1, 1), (1,-1)
D) no points of intersection

Question 40

Find the slope of the tangent line to the given curve at the indicated point.

A)
B)
C)
D)

Question 41

Compute the surface area of the surface obtained by revolving the given curve about the indicated axis.

A) 989.5894
B) 184.000
C) 690.6874
D) 549.4286

Question 42

Compute the surface area of the surface obtained by revolving the given curve about the indicated axis.

A) 237.13
B) 470.92
C) 840.14
D) 34.55

Question 43

Plot the given polar point.

A)
B)
C)
D)

Question 44

Plot the given polar point.

A)
B)
C)
D)

Question 45

Compute the surface area of the surface obtained by revolving the given curve about the indicated axis.

A) 563.25
B) 156.87
C) 840.14
D) 76.91

Question 46

Plot the given polar point.

A)
B)
C)
D)

Question 47

Find the arc length of the curve. Approximate numerically, if needed.

A) 7.667
B) 6.787
C) 3.005
D) 8.445

Question 48

Find all polar coordinate representations of the given rectangular point.

A)
B)
C)
D)

Question 49

Find all polar coordinate representations of the given rectangular point.

A)
B)
C)
D)

Question 50

The tires on an automobile have a radius of 0.34 meters. If the tires make 12 revolutions per second, find the speed of the car in kilometers per hour (kph). Round to the nearest tenth. [1 km = 1000 m and 1 hr = 3600 s]

A) 92.3 kph
B) 46.1 kph
C) 14.7 kph
D) 29.4 kph

Question 51

Find all polar coordinate representations of the given rectangular point.

A)
B)
C)
D)

Question 52

The time required for a skier, initially at rest, to ski from point A to point B on an inclined plane (where B is not directly downhill from
A) 6.6855k

A) is given by: . In the equation above, a is the value of the parameter u at point A, b is the value of u at point B, and k is a constant. The expressions x(u) and y(u) represent parametrically the trajectory taken by the skier. The "straight downhill" direction is assumed to be in the direction of increasing values of y.
Find the time required to complete the trip if the skier chooses the following trajectory: .
B) 2.5546k
C) 11.7447k
D) 7.8653k

Question 53

Find the arc length of the curve. Approximate numerically, if needed.

A) 12
B) 6
C) 12 $\pi$
D) 2 $\pi$

Question 54

Plot the given polar point.

A)
B)
C)
D)

Question 55

The time required for a skier, initially at rest, to ski from point A to point B on an inclined plane (where B is not directly downhill from
A) 5.4802k

A) is given by: . In the equation above, a is the value of the parameter u at point A, b is the value of u at point B, and k is a constant. The expressions x(u) and y(u) represent parametrically the trajectory taken by the skier. The "straight downhill" direction is assumed to be in the direction of increasing values of y.
Find the time required to complete the trip if the skier chooses the following trajectory: .
B) 5.0956k
C) 91.310k
D) 90.2388k

Question 56

Find rectangular coordinates for the given polar point.

A)
B)
C)
D)

Question 57

Find the arc length of the curve. Approximate numerically, if needed.

A) 28.274
B) 14.181
C) 28.362
D) 14.137

Question 58

Find the arc length of the curve exactly.

A)
B)
C)
D)

Question 59

Find the arc length of the curve exactly.

Question 60

Find all polar coordinate representations of the given rectangular point.

A)
B)
C)
D)

Question 61

Find a corresponding x-y equation.

A)
B)
C)
D)

Question 62

Find rectangular coordinates for the given polar point.

A)
B)
C)
D)

Question 63

Identify all values of where r = 0.

A)
B)
C)
D)

Question 64

Find a corresponding x-y equation. r = 4

A)
B) x² + y² = 16
C)
D) x² + y² = 4

Question 65

Sketch the graph of the polar equation. (Find a corresponding x-y equation if it helps.) r = 2

A)
B)
C)
D)

Question 66

Find the rectangular representation of the given polar point.

A)
B)
C)
D)

Question 67

Find rectangular coordinates for the given polar point.

A)
B)
C)
D)

Question 68

Sketch the graph of the polar equation. (Find a corresponding x-y equation if it helps.)

A)
B)
C)
D)

Question 69

Identify a range of values of that produces one copy of the graph.

A)
B)
C)
D)

Question 70

Find the rectangular representation of the given polar point.

A)
B)
C)
D)

Question 71

Find a corresponding x-y equation.

A)
B)
C) x = 5
D) x² + y² = 5

Question 72

Sketch the graph of the polar equation. (Find a corresponding x-y equation if it helps.)

A)
B)
C)
D)

Question 73

Sketch the graph of the polar equation. (Find a corresponding x-y equation if it helps.)

A)
B)
C)
D)

Question 74

Identify a range of values of that produces one copy of the graph.

A)
B)
C)
D)

Question 75

Identify all values of where r = 0.

A)
B)
C)
D)

Question 76

Sketch the graph of the polar equation. (Find a corresponding x-y equation if it helps.)

A)
B)
C)
D)

Question 77

Find the rectangular representation of the given polar point.

A)
B)
C)
D)

Question 78

Sketch the graph of the polar equation. (Find a corresponding x-y equation if it helps.)

A)
B)
C)
D)

Question 79

Identify a range of values of that produces one copy of the graph.

A)
B)
C)
D)

Question 80

Find the rectangular representation of the given polar point.

A)
B)
C)
D)