78
Sample Examination III
81
3
8.
x dx
=
x2 + 1
(A) ln 5
(B) ln 10
(C) 2 ln 5
(D) 1 ln 5
2
(E) ln b 5 l
2
Answer
9.
d ln
1
=
dx d x 2 - 1 n
(A)
1
x -1
(B)
2x
1 - x2
(C)
2x
x -1
2
2
(D) 2x 3 - 2x
(E) 2x - 2x 3
Answer
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Sample Examination III
79
10. For 0 # x 1 r , an antiderivative of 2 tan x is
2
(A) ln ^sec 2xh
(B) 2 sec 2 x
(C) ln ^sec 2 xh
(D) 2 ln ^cos xh
(E) ln ^2 sec xh
Answer
11. A function f is continuous on the closed interval 6 4, 6@ and twice differentiable on the open
interval ^4, 6h. If f l ]5g =-3, and f is concave downwards on the given interval, which of the
following could be a table of values for f ?
(A)
(D)
x
f (x)
4
5
6
8
4
0
x
f (x)
4
5
6
6
4
2
(B)
(E)
x
f (x)
4
5
6
8
6
2
x
f (x)
4
5
6
8
3
2
(C)
x
f (x)
4
5
6
8
6
5
Answer
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80
Sample Examination III
12. The equation of the line tangent to the curve y = kx + 8 at x =-2 is y = x + 4.
k+x
What is the value of k ?
(A) –3
(B) –1
(C) 1
(D) 3
(E) 4
Answer
x
5
6
9
11
12
f (x)
10
7
11
12
8
13. A function f is continuous on the closed interval 6 5, 12 @ and differentiable on the open interval
^5, 12h and f has the values given in the table above. Using the subintervals 6 5, 6 @, 66, 9@, 69, 11@,
and 611, 12 @, what is the right Riemann sum approximation to
(A) 64
(B) 65
(C) 66
(D) 68.5
75
12
f ^ xh dx ?
(E) 72
Answer
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Sample Examination III
81
14. A function f ^ xh has a vertical asymptote at x = 2. The derivative of f ^ xh is positive for
all x =
Y 2. Which of the following statements are true?
I. lim f ^ xh =+ 3
x"2
II. lim f ^ xh =+ 3
x " 2+
III. lim f ^ xh =+ 3
x " 2-
(A) I only
(B) II only
(C) III only
(D) I and II only
(E) I, II, and III
Answer
15. The following statements concerning the location of an extreme value of a twice-differentiable
funtion, f, are all true. Which statement also includes the correct justification?
(A) The function has a maximum at x = 5 because f l ^5h = 0.
(B) The function has a maximum at x = 5 because f l ^ xh 1 0 for x 1 5 and f l ^ xh 2 0 for
x 2 5.
(C) The function has a minimum at x = 3 because the tangent line at x = 3 is horizontal.
(D) The function has a minimum at x = 3 because f l ^ xh 1 0 for x 1 3 and f l ^ xh 2 0 for
x > 3.
(E) The function has a minimum at x = 3 because f ll ^3h 1 0.
Answer
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82
Sample Examination III
4
3
2
1
1
2
3
4
5
6
7
Graph of f
16. The graph of a differentiable function f is shown above. The graph has a relative minimum at
x = 1 and a relative maximum at x = 5. Let g be the function defined by g ^ xh =
70
For what value of x does the graph of g change from concave up to concave down?
(A) 0
(B) 1
(C) 2
(D) 5
x
f ]t g dt.
(E) 7
Answer
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