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AP Calculus BC
Review Unit 11 Parametric, Vector, and Polar Equations
Name: _________________________
NO calculator allowed unless otherwise indicated. On the test, you will be required to show all of your work. The test will be divided into
calculator and no calculator parts. This review is not comprehensive. Please look back over your notes, your homework, and your quizzes to
help you study for the test. This review will count as part of your test grade!
Topics to study:
 Parametric equations
 Vectors
 Polar Equations and area
No Calculator is allowed for problems 1-2, 6- 7, 9, 12-14. You may use a calculator on problems 3 - 5, 8, 10-11. Show all setups.
1. The position of a particle at any time t  0 is given by


2
x t  t 2  2, y t  t 3.
3
(A) Find the magnitude of the velocity vector at t = 2.
(B) Set up an integral expression to find the total distance traveled by the particle from t = 0 to t = 4.
(C) Find
dy
as a function of x.
dx
(D) At what time t is the particle on the y-axis? Find the acceleration vector at this time.
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2. An object moving along a curve in the xy-plane has position
x t , y t at time t with the velocity vector
 1

v t 
, 2t  . At time t = 1, the object is at (ln 2, 4).
 t 1 

(A) Find the position vector.
(B) Write an equation for the line tangent to the curve when t = 1.
(C) Find the magnitude of the velocity vector when t = 1.
(D) At what time t > 0 does the line tangent to the particle at
x t , y t have a slope of 12?
3. An object moving along a curve in the xy-plane has position
x t , y t at time t with the parametric
equations x  2  cos t and y  1 sin t . Calculator allowed.
(A) Find an expression for the slope of the tangent to the path of the curve.
(B) Find an expression for the rate of change of the slope of the tangent to the path of the curve.
(C) For what value of x does the path of the object have a vertical tangent?
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4. A particle moving along a curve in the xy-plane has position


x t , y t , with
x t  2t  3sin t and y t  t 2  2cost, where 0  t  10. Find the velocity vector at
the time when the particle’s vertical position is y = 7. Calculator allowed.
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5. A particle moving along a curve in the xy-plane has position
The derivative
x t , y t at time t with dxdt  1 sin t .
3
    has a slope
dy
is not explicitly given. For any t  0, the line tangent to the curve at x t , y t
dt
of t + 3. Find the acceleration vector of the object at time t = 2. Calculator allowed.
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6. For what values of t does the curve given by the parametric equations
x(t )  t 4  4t 3  8t 2 and
y  t 3  3t 2  9t have a horizontal tangent?
(A) t= 0 only
(B) t = -1 and 4
(C) t = 6 and -2 (D) t = 1 and -3 (E) no value of t
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7. A section of the path of a racetrack follows the curve defined by the parametric expressions
x(t )  (ln t ) 2 and
1
y (t )  (3t  1)3 for 1  t  2 . Which of the following represents the distance for this section of the track?
3
2
(A)

1
2
 (3t  1) 2 
 dt
1  
 2 ln t 
(B)
(2 ln t ) 2  (3t  1) 4 dt
(D)
2
1
2
2
(C)

1


1
2 ln t
 3(3t  1) 2 dt
t
2
1
4
 2   (3t  1) dt
t 
2
(E)

1
2
 2 ln t 
4

  9(3t  1) dt
 t 
8. An object moving along a curve in the xy-plane has position
x t , y t at time t with
dx
 sin( 3t ) and
dt
dy
 4 cos t for 0  t  2 . At time t = 1, the object is at the point (3, 2). Calculator allowed.
dt
(A) Find the equation of the tangent line to the curve at the point where t = 1.
(B) Find the speed of the object at t = 1.
(C) Find the total distance traveled by the object over the time interval 0  t  2.
(D) Find the position of the object at time t = 4.
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9. Given the equation r  4 sin  . Find
dy
for    .
dx
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10. What is the slope of the line tangent to the polar curve
r  4  3 cos 
at


6
? Calculator allowed.
(A) 3.284
(B) 0.524
(C) 1.001
(D) 0.027
(E) -4.837
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11. Calculator allowed. The area enclosed by the lemniscate given by the polar equation
shown in the figure is
(A) 2
(B) 4
(C)
2
(D) 8
(E)
8
r 2  4 sin( 2 )
12. Sketch a graph and setup and an expression to find the area inside the cardioid r  3 3sin  but do not
evaluate.
y
x
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13. Sketch a graph, and set up an expression to find the area of the common interior of the cardioid
r  3 3sin  and the circle r  9 sin  but do not evaluate.
y
x
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14. Which of the following expressions does not give the area of one loop (or one petal) of the graph of the polar
equation r  4 cos( 2 ) ? Sketch a graph.
y
3
(A)
8  cos 2 (2 )d
`

4
(B)
4
8  cos 2 (2 )d
x
`0
4
3
1 4
4 cos(2 )2 d
(C)

2 `
4

(D)
5
4
16  cos (2 )d
2
`0
(E)
4
8  cos 2 ( 2 )d
`3
4