AP Calculus BC
Review Unit 10
Name: _________________________
SHOW ALL WORK! 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.
Topics to study:
☺ Integration by partial fraction decomposition
☺ Integration by parts
☺ L’Hopital’s Rule
Evaluate each expression completely:
1.

ln( 5 x)dx
6.
 x 5
dx
 8x  15
9.
x
cos(9 x)  1
x0
x2
5.
x

1
x
1
1
13.
3.
 3  25 x
2
dx
8.
6
x
2
1
1 3 5 x
 3 x e dx
11. lim
3
14.
2x
dx
x3
x

dx
2.

10.
1
 4 x sin( 2 x)dx
4. lim
7.
☺ Integration of Inverse Trig & Exponential Base a
☺ Choosing the proper integration technique
☺ Improper Integrals
2
2
6 x 3
x3
12.
1
dx
 8 x  25
15.
1
xe4 x dx
2
x2
dx
x  39
dx
 x  12
1
x
2
0
x
dx
 6x  8
lim x 2e x
x
16.
 9 x cos(3x  1)dx 
9 x cos(3x  1)  sin( 3x  1)  C
(D) 3x sin( 3x  1)  cos(3x  1)  C
3x 2 sin( 3x  1)  C
(C) 3x sin( 3x  1)  cos(3x  1)  C
(A)
(B)
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17. Let
f (x) be a differentiable function with the properties that lim f ( x 2  3x)  0 and
x0
lim f ' ( x 2  3x)  4 . Find lim
x0
x0
f ( x  3x)
sin x
2
(A) 0
(B) 3
(C) 4
(D) 12
(E) nonexistent
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18. Let
f ( x) 
k
, where x  0 and k is some finite positive constant, as shown in the graph. Let
x
1

0
1
L   f ( x)dx and M   f ( x)dx . Which of the following statements is true?
(A) L  M
(B) L  M
(C) L  M
(D) The relative values of L and M depend on the value of k.
(E) No conclusion can be made about the relative values of L and M.
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19. The area of the region bounded by the graphs of
(A)
2 ln( 2)  1
(B)
y  xex , x  0 , x  ln 2 and the x-axis is
2 ln( 2)
(C) ln( 2) 
(D) 2 ln( 2)  1
(E) ln( 2)   1
_____________________________________________________________________________________________
2
2
20. Let f(x) be a differentiable function with the properties that f(1) = 5 and lim f ( x)  8 .
x 
(A) -13
(B) -8
(C) 0
(D) 5

 f ' ( x)dx =
1
(E) 
21.
cos x  e x

x0 ln( 1  x)
lim
(A) 0
(B) -1
(C) 1
(D) e
(E) 
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22. Which of the following is an improper integral? Explain your answer
0
(A)

4
4
(C)
2
dx
x4
(B)
x
2
2
xdx
0 1  x 2
(D)

1
dx
9
2
dx
x
(E) none of these
_____________________________________________________________________________________________
x
23. Let
F ( x)   5tet dt for t  0 and x  0 . The graph of f (t )  5tet is shown.
0
(A) Find an expression for
F (x) , in terms of x only, that does not involve an integral.
(B) Using your answer in part (A), find lim F ( x ) . Justify your answer.
x
(C) Using your answer in part (B), explain what is meant by the expression lim F ( x ) .
x
_____________________________________________________________________________________________
2x
23. Calculator allowed. Let g be the function defined for x  1 and g is defined by g ( x) 
Write the equation of the tangent line to g at x  2 .
 te
t
dt .
1
_____________________________________________________________________________________________
dy
 2 y ( x  1) . Let y  f (x) be the particular solution to this
dx
f ( x)  1
differential equation with the initial condition f (0)  1 . Find lim
. (Hint: First show that the limit is
x0 3 sin( 2 x)
24. Consider the differential equation
indeterminate)