MATH 252 Final Exam
Spring 2013
Version 01
Name
ID#
Instructor
Section
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On the separate answer sheet, fill in your name and identification number and code the appropriate spaces with a #2 pencil. Use the spaces marked “Year” under Birth Date to code the version
of the exam you are taking.
There are 35 multiple choice questions on this exam. Select the one best answer for each problem.
Mark all answers on the separate answer sheet with a #2 pencil. Make your marks heavy and black.
Mark only one answer for each question. If you make a mistake or wish to change an answer, be
sure to erase your first choice completely. Answer all questions. There is no penalty for guessing.
In the test booklet, do all of the work on the back of the preceding page and circle the letter of
the answer for each problem.
When you finish the exam, place your answer sheet inside the front cover of the test booklet and
turn it in to your instructor.
Z
1. What is
arctan(x) dx?
(a) sec2 x + C
(b) ln (sec x) + C
1
(c)
+C
1 + x2
(d) arctan(x) + ln x2 + 1 + C
1
(e) x arctan(x) − ln x2 + 1 + C
2
2. What is
(a)
(b)
(c)
(d)
(e)
R
sin2 (x) dx?
x 1
+ sin 2x + C
2 2
x 1
− sin 2x + C
2 4
x 1
+ cos 2x + C
2 4
1 3
sin x + C
3
1 − cos2 x + C
3. In order to solve
(a) x = 2 sin θ
Z √
4 + x2 dx, what substitution should be made?
(b) x = 2 cos θ
(c) x = 2 csc θ
(d) x = 2 tan θ
(e) x = 2 sec θ
Z
4. What is
(a)
(b)
(c)
(d)
(e)
tan5 x sec3 x dx?
2
1
1
sec7 x − sec5 x + sec3 x + C
7
5
3
1
1
1
sec6 x − sec6 x + sec2 x + C
6
3
2
1
1
1
tan8 x − tan6 x + tan4 x + C
8
3
4
1
1
tan6 x + sec5 x + C
6
5
1
tan6 x sec4 x + C
24
5. What is the template for the partial fraction expansion of
(a)
(b)
(c)
(d)
(e)
x2
?
(x2 + 1)(x + 1)2
Cx + D
E
F
A B
+ 2+ 2
+
+
x x
x +1
x + 1 (x + 1)2
A
B
+
x2 + 1 (x + 1)2
A
B
C
+
+
x2 + 1 x + 1 (x + 1)2
C
Ax + B
+
2
x +1
(x + 1)2
Ax + B
C
D
+
+
2
x +1
x + 1 (x + 1)2
Z
6. Approximate
1
sin(24x) dx using Simpson’s rule with n = 4. Round your answer to the
0
nearest ten thousandth.
(a) − .5084
(b) .0240
(c) − .5049
(d) 0.5100
(e) − .3917
Z
7. Evaluate the integral
0
1
1
x99/100
(a) The integral doesn’t exist
dx
(b) 1
(c) 10
(d) 100
(e) 1000
8. What is exact arc length of 1 + 6x3/2 for 0 ≤ x ≤ 1?
(a) 6.10322
2
823/2 − 1
(b)
243
12
(c) 1 +
5
(d) 9
7 1/2
(7 − 1)
(e)
172
9. What is the exact surface area of the surface obtained by rotating the graph of f (x) =
for 0 ≤ x ≤ 1 around the x-axis?
√
4 − x2
(a) 16π
(b)
1
6
√
3 3 + 2π
π
3
(d) 4π
(c)
(e) 4π(2 −
√
3)
10. What is the y-coordinate of the center of mass of the region under f (x) = sin(x) for 0 ≤ x ≤ π?
(a) π/8
(b) 0
(c) π
(d) π/2
(e) π/10
11. Suppose 3m by 5m rectangular plate is immersed in water, of density 1000kg/m3 , with the 3m
edge just touching the surface of the water. If the accelartion due to gravity is g = 9.81m/s2 ,
which one of the following integrals computes the pressure on the plate?
1
(a) 1000 · 9.81 ·
2
Z 3
1000 · 9.81 · 5(3 − x) dx
(b)
0
Z 3
(c)
1000 · 9.81 · 3(5 − x) dx
0
Z 5
(d)
1000 · 9.81 · 3(5 − x) dx
0
Z 5
1000 · 9.81 · 5(3 − x) dx
(e)
0
(
Ce−x
12. If f (x) =
0
(a) 0
(b) 1
x≥1
then what value of C makes f (x) a probability distribution function?
x<1
(c) e
(d)
1
e
(e) No C will work
13. If the amount in capital that a company has at time t is√f (t), then its derivative f (t) is the
net investment flow. Suppose the net investment flow is t million dollars per year (where t
is measured in years). Find the increase in capital (in millions) from the fourth year to the
eighth year.
(a) $9.75
(b) $24.0
(c) $0.83
(d) $4.00
(e) $7.50
14. Let f (x) = xe−x if x ≥ 0 and f (x) = 0 otherwise. Find P (1 ≤ X).
(a) 0
(b) 1
(c) e
2
(d)
e
1
(e)
2
15. Which of the following is a solution of the differential equation y 00 + y = sin x?
(a) y = sin x
(b) y = cos x
1
(c) y = sin x
2
1
(d) y = − cos x
2
1
(e) y = (sin x − cos x)
2
16. If f (x) is the solution to the initial value problem y 0 = xy with y(0) = 2, what is y(2)?
(a) 0
1
(b) e2
2
(c) e2
(d) 2e2
(e) 4e2
17. What integrating factor, I(x), is used to solve x2 y 0 + x4 y = sin(ex )?
(a) ex
2 /2
(b) ee
x
sin ex
(c) ex
5 /5
(d) ex
3 /3
(e) ex
18. What is the Cartesian equation of the parametric curve x(t) = t2 , y(t) =
q
√
√
2
(a) y = x +1
(b) y = x+1
(c) y =
x+1
(d) y = (x+1)2
√
t + 1?
(e) y = x2
19. What is the derivative of the parametric curve x(t) = t2 , y(t) = t3 at the point determined by
t = 1?
(a) 0
(b) 1
3
(c)
2
2
(d)
3
(e) undefined
20. What is the derivative of the polar curve f (θ) = θ2 ?
2θ sin θ + θ2 cos θ
θ3
(d)
(a) undefined
(b) 2θ
(c)
3
2θ cos θ − θ2 sin θ
2θ cos θ − θ2 sin θ
(e)
2θ sin θ + θ2 cos θ
21. What is the arc length of the polar curve f (θ) = 2 cos θ for 0 ≤ θ ≤ π?
(a) π
(b) 2π
(c) 3π
(d) 4π
(e) 5π
22. Find the equation of the parabola with vertex (2, 3) and directrix y = −1.
(a) y = (x − 2)2 + 3
1
(b) y = (x − 2)2 + 3
2
1
(c) y = (x − 2)2 + 3
4
1
(d) y = (x − 2)2 + 3
16
(e) y = 2(x − 2)2 + 3
23. What kind of conic section is given by the polar equation r =
(a) circle
(b) parabola
(c) ellipse
24. What is the nth term of the sequence
(d) hyperbola
4
?
5 − 4 sin θ
(e) not a conic section
2 3 4
5 6
7
,− , ,− ,
,−
, ... , if the first term is a1 ?
3 9 27 81 243 729
n
3n
n
(−1)n+1 · 3
n
n
+
1
(−1)n · n
3
n+1
(−1)n−1 · n
3
n
+2
(−1)n+1 ·
3n
(a) (−1)n+1 ·
(b)
(c)
(d)
(e)
25. What is the value of the series
∞
X
n=1
1
?
n(n + 2)
(a) The series diverges
(b) 5
3
(c)
2
(d) 1
3
(e)
4
26. What is the value of the series 2 +
(a) The series diverges
(b) 23
(c) 1
(d) 2
(e) 3
2 2
2
2
2
+ +
+
+
+ . . .?
3 9 27 81 243
27. The interval of convergence for
∞
X
n=1
(a) [−1, 1]
(b) [−1, 1)
28. Consider the two series
X
xn
is
n2 + 1
(c) (−1, 1]
(−1)n
(d) (−1, 1)
(e) None of these
X 1
1
and
n
n2
(a) They both converge, but to different numbers
(b) They both converge to the same number
(c) The first converges and the second diverges
(d) The first diverges and the second converges
(e) They both diverge
29. The series
X n5n
n!
(a) converges absolutely by the ratio test
(b) converges conditionally by the ratio test
(c) diverges by the integral test
(d) converges by the integral test
(e) diverges by the test for divergence
30. What is the Taylor series for f (x) = x2 + 3x + 6 centered at a = 2?
(a) f (x) = 6 + (x − 2)2
(b) f (x) = 16 + 7(x − 2) + 2(x − 2)2
(c) f (x) = 16 + 7(x − 2) + (x − 2)2
(d) f (x) = 2 + 7x + (x − 2)2
(e) f (x) = 4(x + 2) + (x + 2)2
31. What is the Maclaurin series for ln(1 + x2 )?
(a)
(b)
(c)
(d)
(e)
∞
X
xn
n=1
∞
X
n=1
∞
X
n=1
∞
X
n=1
∞
X
2n
(−1)n
x2n
n
(−1)n+1
(−1)n
xn
n
n+1 x
(−1)
n=1
32. The series
xn
n
X
2n
n
1
√
n2 + 1
X1
(a) converges by direct comparison with
n
X1
(b) diverges by direct comparison with
n
X1
(c) converges by limit comparison with
n
X1
(d) diverges by limit comparison with
n
(e) diverges by the test for divergence
33. The series
∞
X
n=2
1
n(ln n)2
(a) converges by the integral test
(b) diverges by the integral test
(c) converges by the alternating series test
(d) diverges by the test for divergence
(e) converges by the test for divergence
34. Suppose an is a sequence with limit 7. What can you say about
X
an ?
(a) The sum is also 7
(b) The series converges absolutely
(c) The series converges, but may not converge absolutely
(d) The series diverges
(e) The series could either converge or diverge
Z
35. Estimate
2
sin(x2 ) dx using a MacLaurin series with three nonzero terms.
0
(a) 0.8048
(b) 1.1706
(c) 1.1892
(d) 1.4222
(e) 1.8922
√
36. What is limx→∞
1 + x − 1 − 12 x
?
x2
(a) The limit doesn’t exist
(b) 0
1
(c)
8
(d) −
(e) 1
1
8
37. The radius of convergence of
∞
X
x2n
n=0
4n
is
(a) 0
(b) 1
(c) 2
(d) 4
(e) ∞
38. Find the coefficient of x2 in the McLaurin series of ln(cos x).
(a) 0
(b) 1/2
(c) −1/2
(d) 1
(e) −1
1. E
11. D
21. A
31. E
2. B
12. C
22. D
32. D
3. D
13. A
23. C
33. A
4. A
14. D
24. D
5. E
15. E
25. E
6. A
16. D
26. E
7. D
17. D
27. A
8. B
18. C
28. A
9. D
19. C
29. A
10. A
20. D
30. C
34. D
35. B
36. D
37. C
38. C