AP Calculus BC Exam Review

Parametric, Vector, and Polar Functions

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Eliminate the parameter from the given parametric equations to find an equation that relates x and y directly.

1) x = cos t , y = 4 sin t

A) 16x2 - y2 = 16 B) 16x2 + y2 = 16 C) 4x2 + y2 = 4 D) x2 + 16y2 = 16

Graph the pair of parametric equations in the rectangular coordinate system.

2) x = 9t - 1 , y = 4t + 3, for t in the interval [0, 3]

x-30 -20 -10 10 20 30

y30

20

10

-10

-20

-30

x-30 -20 -10 10 20 30

y30

20

10

-10

-20

-30

A)

x-30 -20 -10 10 20 30

y30

20

10

-10

-20

-30

x-30 -20 -10 10 20 30

y30

20

10

-10

-20

-30B)

x-30 -20 -10 10 20 30

y30

20

10

-10

-20

-30

x-30 -20 -10 10 20 30

y30

20

10

-10

-20

-30

1

C)

x-30 -20 -10 10 20 30

y30

20

10

-10

-20

-30

x-30 -20 -10 10 20 30

y30

20

10

-10

-20

-30D)

x-30 -20 -10 10 20 30

y30

20

10

-10

-20

-30

x-30 -20 -10 10 20 30

y30

20

10

-10

-20

-30

Find dy/dx in terms of t.

3) x = ln(5t), y = ln(7t)9

A)5

7t B)

9

tC) 9 D)

5

7

Find d2y/dx2 in terms of t.

4) x = 5 sin t, y = 4 cos t

A) - 4

5 sec3t B)

4

5 cot t C) -

4

25 sec t D) -

4

25 sec3t

Determine analytically at the given value of t whether the parametric curve is increasing, decreasing, or neither.

5) x = 5 sin t, y = 8 sin t, t = 3π/4

A) Neither B) Increasing C) Decreasing

Determine analytically at the given value of t whether the parametric curve is concave up, concave down, or neither.

6) x = t2

2 + 9t, y =

t2

2 - 2t, t = 2

A) Concave up B) Concave down C) Neither

Solve the problem.

7) Find the points at which the tangent to the curve x = -5 + sin t, y = 4 + cos t is horizontal.

A) (-5, 5) and (-5, 3) B) (0, 5) and (π, 3) C) (0, 1) and (0, -1) D) (-5, 4)

Find the length of the curve.

8) x = 7 cos t + 7 t sin t, y = 7 sin t - 7t cos t, 0 ≤ t ≤ π/6

A)7

72π B)

1

7π2 C)

7

72π2 D)

49

72π2

2

Find the component form of the specified vector.

9) the vector OP , where O is the origin and P is the midpoint of the segment RS connecting R = (0, -2) and S = (-6,

7).

A) - 3, 5

2B) 6, -9 C) 3, -

9

2D) -6, 5

Find the magnitude of the vector and the direction angle θ it forms with the positive x-axis (0 ≤ θ ≤ 360°).

10) -7 2, 7 2

A) 14, 135° B) 14 2, 135° C) 49, 315° D) 14 2, 315°

Find the magnitude of the given vector.

11) 11, 5

A) 5 + 11 B) ≈ 11.225 C) 36 D) 6

Find the component form of the vector with the given magnitude that forms the given directional angle with the positive

x-axis.

12) 11, π/2 radians

A) 0, -11 B) 11 2, 11 2 C) 11, 0 D) 0, 11

Find the component form of the vector with the given magnitude that forms the given directional angle with the positive

x-axis. Round each component to the nearest hundredth.

13) 46, 251°

A) -0.33, -0.95 B) -43.49, -14.98 C) -15.84, -46 D) -14.98, -43.49

Find the indicated vector in component form.

14) Let u = -9, 2 . Find -5u.

A) 45, 10 B) -45, -10 C) -45, 10 D) 45, -10

Solve the problem.

15) An airplane flying due east at 509 mph in still air, encounters a 74-mph tail wind acting in the direction 60°

north of east. The airplane holds its compass heading but because of the wind acquires a new ground speed and

direction. What is its new speed?

A) 574.3 mph B) 476.3 mph C) 610.1 mph D) 549.7 mph

A particle travels in the plane with position vector r(t). Find the velocity vector v(t).

16) r(t) = sin 10t, cos 8t

A) v(t) = -10 cos 10t, 8 sin 8t B) v(t) = cos 10t, - sin 8t

C) v(t) = 10 cos 10t, -8 sin 8t D) v(t) = -10 sin 10t, 8 cos 8t

3

A particle travels in the plane with position vector r(t). Find the acceleration vector a(t).

17) r(t) = 4t2 - 4, 1

18t3

A) a(t) = 8t, 1

3t2 B) a(t) = 8,

1

6t C) a(t) = 8,

1

3t D) a(t) = 8t,

1

6

Use the given information to draw the path of the particle moving in the plane.

18) A particle moves in the plane with position vector 2 tan πt, 5 sec πt . Draw the path of the particle for t in [0,

2].

x

y

x

y

A)

x-8 -4 4 8

y8

4

-4

-8

x-8 -4 4 8

y8

4

-4

-8

B)

x-8 -4 4 8

y8

4

-4

-8

x-8 -4 4 8

y8

4

-4

-8

C)

x-8 -4 4 8

y8

4

-4

-8

x-8 -4 4 8

y8

4

-4

-8

D)

x-12 -8 -4 4 8 12

y12

8

4

-4

-8

-12

x-12 -8 -4 4 8 12

y12

8

4

-4

-8

-12

4

A particle moves in the plane with the given position vector. Find the velocity or acceleration vector, as indicated, at the

specified time.

19) r(t) = cos(5t), 6 ln(t - 3) . Find the velocity vector at t = 0.

A) -5, - 2 B) 0, - 2 C) 5, 6 D) 0, 2

If r(t) is the position vector of a particle in the plane at time t, find the particle's speed at the given value of t.

20) r(t) = et sin t, et cos t , t = 4

A) e4 B) 2 e4sin 4 cos 4 C) 2e4 D) 2e4

Solve the problem.

21) A particle moves in the plane so that its position at any time t ≥ 0 is given by

x = 5 sec πt , y = 2 tan πt

Eliminate the parameter and find an equation in terms of x and y for the path of the particle.

A) x2

5 -

y2

2 = 1 B)

y2

25 -

x2

4 = 1 C)

x2

25 -

y2

4 = 1 D)

x2

25 +

y2

4 = 1

The velocity v(t) of a particle moving in the plane is given, along with the position of the particle at time t = 0. Find the

position of the particle at time t = t1.

22) v(t) = (t + 1)-1, (t + 1)-2 , initial position = 4, 3 , t1 = 3

A) ln 4 , 15

4B) ln

1

4 + 4,

15

4C) ln

1

4 ,

15

4D) ln 4 + 4,

15

4

Solve the problem.

23) The velocity v(t) of a particle moving in the plane is given by the vector e3t - 2t, e3t + 2t . Find the distance

traveled by the particle from t = 0 to t = 2.

A) 212.7 B) 93.78 C) 165.2 D) 189.9

24) The position of a particle in the plane at time t is given by r(t) = 4t + cos t, 6t + sin t . Find an expression that

represents the distance the particle travels from time t = 0 to t = 5.

A)5

0

53 dt∫ B)5

0

1 + 52t2 + 2t(4 cos t + 6 sin t) dt∫

C)5

0

10 - sin t + cos t dt∫ D)5

0

53 + 12 cos t - 8 sin t dt∫

5

Plot the point whose polar coordinates are given.

25) (4, 60°)

A)

r-5 -4 -3 -2 -1 1 2 3 4 5

5

4

3

2

1

-1

-2

-3

-4

-5

r-5 -4 -3 -2 -1 1 2 3 4 5

5

4

3

2

1

-1

-2

-3

-4

-5B)

r-5 -4 -3 -2 -1 1 2 3 4 5

5

4

3

2

1

-1

-2

-3

-4

-5

r-5 -4 -3 -2 -1 1 2 3 4 5

5

4

3

2

1

-1

-2

-3

-4

-5

C)

r-5 -4 -3 -2 -1 1 2 3 4 5

5

4

3

2

1

-1

-2

-3

-4

-5

r-5 -4 -3 -2 -1 1 2 3 4 5

5

4

3

2

1

-1

-2

-3

-4

-5D)

r-5 -4 -3 -2 -1 1 2 3 4 5

5

4

3

2

1

-1

-2

-3

-4

-5

r-5 -4 -3 -2 -1 1 2 3 4 5

5

4

3

2

1

-1

-2

-3

-4

-5

Change the given polar coordinates (r, θ) to rectangular coordinates (x, y).

26) (3, π)

A) (0, 3) B) (0, -3) C) (-3, 0) D) (3, 0)

The rectangular coordinates of a point are given. Express the point in polar coordinates with r ≥ 0 and 0 ≤ θ < 2π.

27) (2, 2)

A) (2 2, 3π/4) B) (2, π/4) C) (2 2, π/4) D) (2, π/2)

Find two sets of polar coordinates for the point with the given rectangular coordinates.

28) (-7, 7 3)

Find two pairs of polar coordinates each with -π < θ ≤ π

A) 7, 2π

3 and 7, -

4π

3B) 14,

5π

6 and -14, -

π

6

C) 7, 5π

6 and 7, -

7π

6D) 14,

2π

3 and -14, -

π

3

Graph the set of points whose polar coordinates satisfy the given equation or inequality.

6

29) r = 6

r-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

6

5

4

3

2

1

-1

-2

-3

-4

-5

-6

r-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

6

5

4

3

2

1

-1

-2

-3

-4

-5

-6

A)

r-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

6

5

4

3

2

1

-1

-2

-3

-4

-5

-6

r-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

6

5

4

3

2

1

-1

-2

-3

-4

-5

-6

B)

r-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

6

5

4

3

2

1

-1

-2

-3

-4

-5

-6

r-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

6

5

4

3

2

1

-1

-2

-3

-4

-5

-6

C)

r-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

6

5

4

3

2

1

-1

-2

-3

-4

-5

-6

r-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

6

5

4

3

2

1

-1

-2

-3

-4

-5

-6

D)

r-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

6

5

4

3

2

1

-1

-2

-3

-4

-5

-6

r-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

6

5

4

3

2

1

-1

-2

-3

-4

-5

-6

For the given polar equation, write an equivalent rectangular equation.

30) 2r cos θ + 9r sin θ = 1

A) 2y + 9x = 1 B) 2x + 9y = 1 C)x

2 +

y

9 = 1 D) 2x + 9y = x2 + y2

7

Describe the graph of the polar equation.

31) r = -6 csc θ

A) Vertical line through (-6, 0) B) Horizontal line through (0, -6)

C) Line through origin with slope -6 D) Circle of radius 6 centered at origin

Identify the type of polar graph.

32) r = 2

A) rose with 2 petals B) straight line C) circle D) limacon

Graph the polar equation for 0 ≤ θ < 2π.

33) r = 4 + 4 sin θ

r-10 -5 5 10

10

5

-5

-10

r-10 -5 5 10

10

5

-5

-10

A)

r-10 -5 5 10

10

5

-5

-10

r-10 -5 5 10

10

5

-5

-10

B)

r-10 -5 5 10

10

5

-5

-10

r-10 -5 5 10

10

5

-5

-10

C)

r-10 -5 5 10

10

5

-5

-10

r-10 -5 5 10

10

5

-5

-10

D)

r-10 -5 5 10

10

5

-5

-10

r-10 -5 5 10

10

5

-5

-10

8

34) r = 5 sin 3θ

r-10 -5 5 10

10

5

-5

-10

r-10 -5 5 10

10

5

-5

-10

A)

r-10 -5 5 10

10

5

-5

-10

r-10 -5 5 10

10

5

-5

-10

B)

r-10 -5 5 10

10

5

-5

-10

r-10 -5 5 10

10

5

-5

-10

C)

r-10 -5 5 10

10

5

-5

-10

r-10 -5 5 10

10

5

-5

-10

D)

r-10 -5 5 10

10

5

-5

-10

r-10 -5 5 10

10

5

-5

-10

Graph the polar curve for 0 ≤ θ < 2π and identify the type of curve by name.

9

35) r = 9

3 + 8 cos θ

rr

A)

r-7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7

7

6

5

4

3

2

1

-1

-2

-3

-4

-5

-6

-7

r-7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7

7

6

5

4

3

2

1

-1

-2

-3

-4

-5

-6

-7

Ellipse

B)

r-10 -8 -6 -4 -2 2 4 6 8 10

10

8

6

4

2

-2

-4

-6

-8

-10

r-10 -8 -6 -4 -2 2 4 6 8 10

10

8

6

4

2

-2

-4

-6

-8

-10

Hyperbola

C)

r-10 -8 -6 -4 -2 2 4 6 8 10

10

8

6

4

2

-2

-4

-6

-8

-10

r-10 -8 -6 -4 -2 2 4 6 8 10

10

8

6

4

2

-2

-4

-6

-8

-10

Hyperbola

D)

r-7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7

7

6

5

4

3

2

1

-1

-2

-3

-4

-5

-6

-7

r-7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7

7

6

5

4

3

2

1

-1

-2

-3

-4

-5

-6

-7

Ellipse

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

36) r = 3 + 6 cos θ, θ = π

2

A) - 2 B)1

2C) 2 D) -

1

2

10

Find the area of the specified region.

37) inside the three-leaved rose r = 6 cos 3θ

A)9

2π B) 9π C) 18π D) 3π

38) shared by the circles r = 8 cos θ and r = 8 sin θ

A)8

3(4π - 3 3) B) 4(2 - 2) C) 8(π - 2) D) 16π

11

Answer KeyTestname: PARAMETRIC, VECTOR, AND POLAR FUNCTIONS

1) B

2) D

3) C

4) D

5) B

6) A

7) A

8) C

9) A

10) A

11) D

12) D

13) D

14) D

15) D

16) C

17) C

18) A

19) B

20) D

21) C

22) D

23) D

24) D

25) A

26) C

27) C

28) D

29) B

30) B

31) B

32) C

33) D

34) D

35) C

36) C

37) B

38) C

12