MATH 172.503
Examination 2
March 11, 2010
NAME
SIGNATURE
This exam consists of 9 problems, numbered 1–9. For partial credit you must present your
work clearly and understandably and justify your answers.
The use of calculators is not permitted on this exam.
The point value for each question is shown next to each question.
CHECK THIS EXAMINATION BOOKLET BEFORE
YOU START. THERE SHOULD BE 9 PROBLEMS ON
6 PAGES (INCLUDING THIS ONE).
Do not mark in the box below.
1–5
6
7
8
9
Total
Points
Possible Credit
30
15
25
15
15
100
NAME
MATH 172.503
Examination 2
Page 2
Multiple Choice: [6 points each] In each of Problems 1–5, circle the best answer.
1.
After trigonometric substitution, the integral
Z
x2
√
dx
9x2 + 4
becomes
Z
tan3 θ
2
dθ
(A)
9
sec θ
Z
1
tan2 θ
(B)
dθ
3
sec θ
Z
4
(C)
tan3 θ dθ
9
Z
2
tan3 θ sec θ dθ
(D)
3
Z
4
tan2 θ sec θ dθ
(E)
27
Z
2.
Evaluate
1
(A)
1
2e2
(B)
1
e2
(C)
1
e
∞
1
dx.
e2x
(D) e
(E) The integral diverges.
March 11, 2010
NAME
3.
MATH 172.503
Examination 2
Page 3
The partial fraction decomposition of
1
x4 − 1
is
4.
(A)
A
(x − 1)4
(B)
A
B
Cx + D
+
+ 2
x−1 x+1
x +1
(C)
Ax + B Cx + D
+ 2
x2 − 1
x +1
(D)
A
B
D
C
+
+
+
2
x − 1 (x − 1)
(x + 1) (x + 1)2
(E)
A
B
C
+
+ 2
x−1 x+1 x +1
If we use the trapezoidal rule to approximate
Z 3
√
x dx
1
with 6 equal subintervals, we obtain which expression below?
r
r
r
r
√
1
4
5
7
8 √
(A)
1+2
+2
+2 2+2
+2
+ 3
n
3
3
3
3
r
r
r
√
3
1
3
5 √
(B)
2
+2+2
+2 2+2
+ 3
2n
2
2
2
r
r
r
r
2
4
5 √
7
8 √
(C)
+
+ 2+
+
+ 3
1+
n
3
3
3
3
r
r
r
3
1
3 √
5 √
(D)
+1+
+ 2+
+ 3
n
2
2
2
√
2
(E) 2 3 −
3
March 11, 2010
NAME
5.
MATH 172.503
Consider the following improper integrals:
Z ∞
Z ∞
1
I.
sin x dx
dx
II.
x
1
−∞
Examination 2
Z
III.
0
1
Page 4
1
√ dx
x
Which of these integrals converges?
(A) I only
(B) II only
(C) III only
(D) I and II only
(E) II and III only
6.
[15 points] Find the arc length of the curve y = 2x3/2 between x = 0 and x = 2.
March 11, 2010
NAME
7.
MATH 172.503
Examination 2
Page 5
[25 points] Evaluate the following indefinite integrals.
Z
x4
dx
x2 + 4
Z
6x − 4
dx
x3 − x
(a)
(b)
March 11, 2010
NAME
8.
MATH 172.503
Examination 2
Page 6
[15 points] Solve the differential equation
dy
2y
+√
= 0.
dx
1 − x2
9.
[15 points] Solve the initial value problem
x2
dy
+ 2xy = sin(3x),
dx
y(π/6) = 2.
March 11, 2010