Calculus AB Final Exam 2011
Name_________________________
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Teacher: Cordero Haupt King Verner
For both days of this exam, multiple choice problems will be worth 2.5 points each and free
response problems will be worth 9 points each.
Answer the following 7 multiple choice and 2 open response problems. No work need be shown for
multiple-choice problems. Clearly write the letters of your multiple-choice responses in the
appropriate spaces provided on this page. Open response problems will be graded in the same
manner as AP open response problems. The AP exam allows for 3 minutes per multiple-choice, 15
minutes per open response problem. You may want to allocate your time in the same manner.
1. ________
2. ________
3. ________
4. ________
5. ________
6. ________
7. ________
Calculus AB Final Exam 2011
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d2y
at the point (4, 3) ?
dx 2
1. If x 2 + y 2 = 25 , what is the value of
(A) −
25
27
(B) −
7
27
(C)
7
27
(D)
3
4
(E)
25
27
2. The function f given by f ( x ) = 2 x 3 − 3 x 2 − 12 x has a relative minimum at x =
(A) –1
(B) 0
(C) 2
(D)
3 − 105
4
(E)
3 + 105
4
3. A particle moves along the x-axis so that at any time t > 0, its velocity is given by v(t ) = 4 − 6t 2 .
If the particle is at position x = 7 at time t = 1, what is the position of the particle at time t = 2 ?
(A) –10
(B) –5
(C) –3
(D) 3
(E) 17
Calculus AB Final Exam 2011
4.
What is the slope of the line tangent to the graph of y =
(A) −
5.
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1
e
If f ′(x) =
(B) −
2
and f
x
(A) 2
3
4e
(C) −
(D)
1
4e
(E)
1
e
( e )= 5, then f (e) =
(B) ln(25)
(C) 5 +
6. At what point on the graph of y =
1 1
(A)  , − 
2 2
1
4e
e− x
at x = 1 ?
x +1
1 1
(B)  , 
 2 8
1
2
2 2
−
e e2
(D) 6
(E) 25
x 2 is the tangent line parallel to the line 2x – 4y = 3?
1

(C) 1, − 
4

 1
(D) 1, 
 2
(E) (2, 2)
Calculus AB Final Exam 2011
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 x 2 − 6 x + 9 for x ≤ 2
7. f ( x ) = 
for x > 2
 kx + b
The function f is defined above. For what values of k and b, if any, is f differentiable at x = 2?
(A) k = 2, b = 3
(B) k = 2, b = –3
(C) k = –2, b = –3
(D) k = –2, b = 5
(E) No value of k or b will make f differentiable at x = 2.
Calculus AB Final Exam 2011
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Free Response Question 1
dy −xy 2
=
. Let y = f ( x ) be the particular solution to this
dx
2
differential equations with the initial condition f (− 1) = 2 .
Consider the differential equation
y
(a) On the axes provided, sketch a slope field for
the given differential equation at the twelve
points indicated.
2
1
(b) Write an equation for the line tangent to the
graph of f at x = –1.
x
-1
O
1
2
(c) Find the solution y = f ( x ) to the given differential equation with the initial condition f (− 1) = 2 .
Calculus AB Final Exam 2011
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Free Response Question 2
The rate, in calories per minute, at which a person
using an exercise machine burns calories is
modeled by the function f. In the figure to the
1
3
right, f ( t ) = − t 3 + t 2 + 1 for 0 ≤ t ≤ 4 and f is
4
2
piecewise linear for 4 ≤ t ≤ 24.
(a) Find f ′(22) . Indicate units of measure.
QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.
(b) For the time interval 0 ≤ t ≤ 24, at what time t is f increasing at its greatest rate? Show the
reasoning that supports your answer.
(c) Find the total number of calories burned over the time interval 6 ≤ t ≤ 18 minutes.
(d) The setting on the machine is now changed so that the person burns f (t ) + c calories per minute.
For this setting, find c so that an average of 15 calories per minute is burned during the time
interval 6 ≤ t ≤ 18.