Math 1206
Common Final Exam
Fall 2010
Form B
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Mark your answers to the test questions in rows 1 - 15 of the op scan sheet. Your score on this test will be
the number of correct answers. You have one hour to complete this portion of the exam. Turn in the op
scan sheet with your answers and the question sheets, including this cover sheet, at the end of this part
of the final exam.
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Student ID #
1. Use the trapezoid rule with five evenly spaced partition points to approximate the integral
given the table of values
x
f (x)
A.
3
2
B. −
0
1
3
2
3
-1
6
-4
9
3
12
2
C. −3
D. −
1
2
2. The area of the region bounded above by y = x and y = 1, and below by y =
A.
5
6
B.
1
6
C.
8
3
D.
1
2
3
x2
is
4
R 12
0
f (x) dx,
3. The integral to find the moment with respect to the x-axis, Mx , for the region of density 3 bounded
by y = (x − 2)2 and y = 4 − x is
3
Z
2
x 4 − x − (x − 2)
A. Mx = 3
Z
3 3
B. Mx =
4 − x − (x − 2)2 dx
2 0
Z 3
D. Mx = 3
x (4 − x)2 − ((x − 2)2 )2 dx
dx
0
3
C. Mx =
2
3
Z
(4 − x)2 − ((x − 2)2 )2 dx
0
0
4. Find the volume of the solid produced by rotating the area between the curves y =
x = 2 about the y-axis.
A. 22π
B. 25π
5. Let g(x) =
g 0 (x).
R 2x
1
C. 8π
6
, y = 1, and
x
D. 16π
f (t) dt where the graph of f (t) is given below. Find g(1), g(0), and an expression for
f!t"
2.0
1.5
1.0
0.5
!0.5
!1.0
1
2
3
4
5
t
1
1
A. g(1) = − , g(0) = − , g 0 (x) = 2f (2x)
2
2
1
1
1
C. g(1) = − , g(0) = − , g 0 (x) = f (2x)
2
2
2
1
B. g(1) = 0, g(0) = , g 0 (x) = 2f (2x)
2
1
1
D. g(1) = 0, g(0) = , g 0 (x) = f (2x)
2
2
2
6. Find the average value of f (x) = sin(x) cos(x) over the interval [0, π2 ].
A.
1
2π
B.
1
π
C.
1
2
D. 1
√
7. Evaluate lim (e−x x).
x→∞
A. ∞
8. Evaluate
R
B. −∞
C. 0
x2 sin(4x) dx.
1
1
1
cos(4x) + C
B. − x2 cos(4x) + x sin(4x) +
4
8
32
1
1
1
D. x2 cos(4x) − x sin(4x) −
cos(4x) + C
4
8
32
A. 2x sin(4x) + 4x2 cos(4x) + C
C.
x3
sin(4x) + x2 cos(4x) + C
2
9. Which integral is obtained from
A.
C.
D. 1
R
x4
√
1
dx by an appropriate trig substitution?
2x2 − 3
√ !Z
2 2
1
dθ
9
sec3 θ
√ !Z
2 2
1
dθ
4
9
sec θ tan θ
4
√
9 3
Z
4
√
9 3
Z
B.
D.
3
1
dθ
sec3 θ
1
sec4 θ
tan θ
dθ
10. Evaluate
A.
R
x2
x
dx.
+x−2
1
2
ln |x + 1| + ln |x − 2| + C
3
3
C. −
1
2
ln |x − 1| − ln |x + 2| + C
3
3
11. After using the table formula
R √
x x4 − 4 dx becomes
R√
u2
−
a2 du
p
4 x2 p 4
4
x − 4 − ln x + x − 4 + C
A. 2
2
2
1 x2 p 4
4 2 p 4
C.
x − 4 − ln x + x − 4 + C
2 2
2
R0
−∞
1
ln |x2 + x − 2| + C
2
D.
1
2
ln |x − 1| + ln |x + 2| + C
3
3
√
u√ 2
a2 2
2
2
=
u −a −
ln u + u − a + C, the integral
2
2
p
x
4 2 p 4
4
B. 2
x − 4 − ln x + x − 4 + C
2
2
p
1 xp 4
4 4
D.
x − 4 − ln x + x − 4 + C
2 2
2
12. Evaluate
B.
ex
dx.
1 + (ex )2
A. ∞
B.
π
2
C. 0
D.
π
4
13. It requires a force of 3 pounds to hold a spring stretched two inches beyond its natural length. How
much work (in inch-pounds) is done in stretching the spring from its natural length to five inches
beyond its natural length.
A.
75
in-lb
4
B.
75
in-lb
2
25
in-lb
3
C.
4
D.
10
in-lb
3
14. Evaluate the integral
R√
x(1 − x2 ) dx.
A.
5 3/2
x − x7/2 + C
3
B.
2 5/2 2 9/2
x − x +C
3
9
C.
1 −1/2 5 3/2
x
− x +C
2
2
D.
2 3/2 2 7/2
x − x +C
3
7
15. Find f (x) if f 0 (x) = e2x + x2 sin(x3 ) and f (0) = 1.
A. f (x) = e2x +
1
1
cos(x3 ) −
3
3
1
1
1
B. f (x) = e2x + cos(x3 ) +
2
3
6
1
1
5
C. f (x) = e2x − cos(x3 ) +
2
3
6
D. f (x) = e2x −
5
1
1
cos(x3 ) +
3
3