AP Calculus BC Final Exam
January, 2009
Name: ______________________
Directions: Solve each of the following problems. Decide which of the choices given is the best and
fill in the corresponding oval on the answer sheet. You may NOT use your calculator on
this section.
1. What is the x-coordinate of the point of inflection on the graph y  13 x3  5 x 2  24 ?
(A) 5
2. lim
n 
(B) 0
(C) 
(B) 2
(C) 1
10
3
(D) -5
(E) -10
(D) 3
(E) nonexistent
3n3  5n
is
n 3  2n 2  1
(A) 5
3. If x  2 xy  y 2  2, then at the point 1,1 ,
(A) 
3
2
(B) 0
(C)
1
2
dy
dx

(D)
3
2
(E) nonexistent
4. A particle moves along the x-axis so that its position at time t is given by s  t   t 2  4t  4 .
What is the acceleration of the particle when t = 4?
(A) 0
(B) 2
(C) 4
(D) 8
(E) 12
5. A different particle moves on a plane curve so that at any time t  0 , its position can be represented
3
by: x  t   t 3  t and y  t    2t  1 . The acceleration vector of the particle at t  1 is
(A) (0, 1)
6. Evaluate: lim
x 4
(A) 
8
3
(B) (2, 3)
(C) (2, 6)
(D) (6, 12)
(C) 0
(D)
(E) (6, 24)
16  x 2
x 2  11x  28
(B) 
8
11
8
11
(E)
8
3
Use the figure below to answer #7 and #8
7. A bug begins to crawl up a vertical wire at time t = 0. The velocity v of the bug at time t, 0  t  8 ,
is given by the function whose graph is shown above. At what value of t does the bug change
direction?
(A) 2
(B) 4
(C) 6
(D) 7
(E) 8
8. What is the total distance the bug traveled from t = 0 to t = 8?
(A) 14
(B) 13
(C) 11
(D) 8
(E) 6
9. If f  x   sin 1 x, then f '  12  
(A)
2 3
3
(B)
4
5
(C) 
4
5
(D) 
2 3
3
(E)

2
10. An equation of the curve f(x) when f '  x   3x2  x at the point (2, 5) is
(A) f  x   6 x  6
(B) f  x   6 x  8
(D) f  x   x3  x  2
(E) f  x   x3 
x2
5
2
x2
(C) f  x   x   1
2
3
11.
d
dx
 3 
=

2 
 4 x 
(A)
(D)
6 x
4  x 
2
2
(B)
2
(E)
3
4  x 
2
3x
4  x 
2
(C)
2
6x
4  x 
2
2
3
2x
x
12. If F  x    t 3  1 dt , then F '  2  
0
(A) -3
(B) -2
(C) 2
(D) 3
(E) 18
13. If f  x   x 2 x  3, then f '  x  
(A)
3x  3
2x  3
(B)
x
2x  3
(D)
1
2x  3
(E)
5x  6
2x  3
(C)
x  3
2x  3
14. An equation of the line tangent to y  x3  3x 2  2 at its point of inflection is
(A) y   6 x  6
(D) y = 3x - 1
2
15. Integrate:
1
x
2
(B) y = -3x + 1
(E) y = 4x + 1
(C) y = 2x + 10
dx
1
(A) 
1
2
(B)
7
24
(C)
1
2
(D) 1
(E) 2 ln 2
16. The absolute maximum value of f  x   x3  3x2  12 on the closed interval [-2, 4] occurs at x =
(A) -2
(B) 0
(C) 1
(D) 2
(E) 4
17. If f  x   e x , then f '  ln 2  
(A)
18.
1
4
(B)
1
2
(C)
2
2
(D) 1
(E)
2
 x2 ,
x3
19. At x = 3, the function given by f  x   
is
6
x

9,
x

3

(A) undefined
(B) continuous but not differentiable
(C) differentiable but not continuous
(D) neither continuous nor differentiable
(E) both continuous and differentiable
20.
2
1
-2
-1
1
2
-1
-2
Shown above is a slope field for which of the following differential equations?
(A)
dy
 1 x
dx
(B)
dy
 x2
dx
(D)
dy x

dx y
(E)
dy
 ln y
dx
(C)
dy
 x y
dx
x
e2
21. Integrate:  dx
2
x
(A) e  x  C
(B) e 2  C
(D) 2e 2  C
(E) e x  C
x
(C) e 2  C
x
22. If x3  3xy  2 y3  17 , then in terms of x and y,
(A) 
x2  y
x  2 y2
(B) 
x2  y
(D) 
2 y2
x2  y
x  y2
dy

dx
(C) 
x2  y
x  2y
 x2
(E)
1 2 y2
23. What is the average value of y  x 2 x3  1 on the interval [0, 2]?
(A)
26
9
(B)
52
9
24. If x  e 2t and y  sin  2t  , then
(C)
26
3
52
3
(E) 24
dy

dx
(A) 4e2t cos  2t 
(B)
e 2t
cos  2t 
cos  2t 
2e 2t
(E)
cos  2t 
e2t
(D)
(D)
(C)
sin  2t 
2e2t
25. As shown in the figure below, a square with vertices (0, 0), (2, 0), (2, 2) and (0, 2) is divided
into two regions by the graph of y   x 2  2 x . If a point is picked at random from inside the
square, what is the probability that the point lies in the region above the parabola?
(A) 0
(B)
1
6
(C)
1
3
(D)
1
2
(E)
2
3
AP Calculus BC Final Exam
January, 2009
Name: ______________________
Free Response- Show your work. You may use your calculator. Write final answers on the answer
sheet provided.
26. Find lim
x 4
3x 2  3x  36
x2  6 x  8
27. If f  x   4 x 2 x 2  9, find f '  5 .
28. If
dy
x

and y = 5, when x = 4, find the equation of the curve.
dx
9  x2
29. The function f is continuous on the closed interval [0, 10] and has values that are given in the
10
table below. Using n = 5, what is the trapezoidal approximation of
 f  x  dx ?
0
x
f(x)
0
20
1
19.5
2
18
3
15.5
4
12
5
7.5
6
2
7
-4.5
8
-12
9
-20.5
10
-30
30. Find the slope of the curve 2 xy  y 2  21 at the point on the curve where y = 3.
31. Water is being pumped into a conical reservoir (vertex down) at the constant rate of 10 ft3/min.
If the reservoir has a radius of 4 ft and is 12 ft deep, how fast is the water rising when the water
is 6 ft deep?
32. A 20 foot ladder leans against a vertical wall. The foot of the ladder is sliding along the ground at
a rate of 2 ft/sec. How fast is the top of the ladder sliding down the wall at the instant that the foot
of the ladder is 16 feet from the base of the wall?
33. The function f  x   x3  ax2  bx  c has a relative maximum at (-3, 25) and a point of inflection
at x = -1. Find a, b, and c.
34. The volume of a cylindrical tin can with a top and bottom is to be 16 cubic inches. If a minimum
amount of tin is to be used to construct the can, what must be the dimensions (r and h), in inches of
the can?
35. State the set of values for which f  x    x  2  x  3 is BOTH increasing and concave up.
2
36. For what values of t does the curve given by the parametric equations
x t   t 3  t 2 1
y  t   t 4  2t 2  8t
have a
vertical tangent?
37. Find the maximum profit for the sale of widgets if the cost and revenue profits are as follows:
C  x   0.5 x  500
R  x 
50 x
x
38. If 0  k  2 and the area under the curve y  cos x from x  k to x  2 is 0.1, then find k.
39. Find the equation of the line tangent to the graph of f  x   x4  2x2 that has a slope of 1.
40. Integrate:
cos 2 x
dx
2
2x
 sin
AP Calculus BC Final Exam
January, 2009
Name: ______________________
Answer Sheet
26. ________________________________
36. _______________________________
27. ________________________________
37. _______________________________
28. ________________________________
38. _______________________________
29. ________________________________
39. _______________________________
30. ________________________________
40. _______________________________
31. ________________________________
32. ________________________________
33. ________________________________
34. ________________________________
35. ________________________________
Formulae
Area
Atriangle  12 bh
Aequilateral 
3s 2
4
Arect  w
Acircle   r 2
Atrapezoid  12  b1  b2  h
Ssphere  4 r 2
Scylinder  2 rh  1or 2 r 2
Volume
Vbox  wh
Vcylinder   r 2 h
Vsphere  43  r 3
Vcone  13  r 2 h
Miscellaneous
a 2  b2  c 2
d
 x2  x1    y2  y1 
2
P  2  2w
C  2 r
Profit = Revenue - Cost
2