King Fahd University of Petroleum and Minerals
Department of Mathematics
Math 102
Final Exam
231
December 20, 2023
Net Time Allowed: 180 Minutes
MASTER VERSION
231, Math 102, Final Exam
Example 4/ Section 8.8
Page 1 of 14
MASTER
Z ∞
ex
1.
dx =
2x
−∞ 1 + e
π
2
π
(b)
4
(a)
(correct)
π
2
π
(d) −
4
(e) 0
(c) −
Question 15 / Section 8.8
Z 2
2.
0
1
dx
(x − 1)2
(a) diverges
(b) is equal to 1
(c) is equal to −2
(d) is equal to 2
(e) is equal to 0
(correct)
231, Math 102, Final Exam
Question 74 / Section 9.1
3. The sequence
√
n ln 1 + n1
(a) converges to 0
Page 2 of 14
MASTER
(correct)
(b) converges to 1
(c) converges to 2
(d) converges to e
(e) diverges
Question 36 / Section 9.2
4.
∞
X
(0.3)n =
n=0
10
7
11
(b)
7
12
(c)
7
9
(d)
7
8
(e)
7
(a)
(correct)
231, Math 102, Final Exam
Page 3 of 14
MASTER
Question 35 / Section
9.5
∞
X (−1)n+1
. The least number of terms required to approximate
3
n
n=1
the sum of the series with an error less than 0.001 is
(Hint: Use Alternating Series Remainder Theorem)
5. Consider the series
(a) 10
(correct)
(b) 8
(c) 12
(d) 14
(e) 6
Question 47 / Section 9.6
6. Using the Root test for the series
∞
X
n=1
1
3
2
(b)
3
(c) 1
4
(d)
3
1
(e)
2
(a)
an where an =
p
n
n
|an | =
,
we
have
lim
n
n→∞
3
(correct)
231, Math 102, Final Exam
Question 83 Z/ Section 5.4
sin x √
7. If F (x) =
t dt, then F
0
Page 4 of 14
0
π 6
MASTER
=
√
6
√4
(b) 6
√
6
(c)
√2
6
(d)
√3
2
(e)
2
(a)
(correct)
Question 91 / Section 5.5
8. The area of the region bounded by√the graphs of the equations
2
y = xe−x /4 , y = 0, x = 0 and x = 6 is
(a) 2 − 2e−3/2
(b) 1 − e−3/2
(c) 2 + 2e−3/2
(d) −2 − 2e−3/2
(e) e−3/2
(correct)
231, Math 102, Final Exam
Page 5 of 14
MASTER
Question
49 / Section 5.9
Z
cosh2 (x − 1) sinh(x − 1) dx =
9.
1
cosh3 (x − 1) + c
3
1
(b) sinh3 (x − 1) + c
3
(c) cosh3 (x − 1) + c
(a)
(correct)
(d) sinh3 (x − 1) + c
(e) tanh3 (x − 1) + c
Question 11 / Section 8.4
Z
10.
xp
4 + x2 dx =
2
1
(4 + x2 )3/2 + c
6
1
(b) (4 + x2 )3/2 + c
4
1
(c) (4 + x2 )1/2 + c
6
1
(d) (4 + x2 )1/2 + c
4
1
(e) (2 + x2 )3/2 + c
6
(a)
(correct)
231, Math 102, Final Exam
Page 6 of 14
MASTER
Example 3 / Section 8.5
2
2
Cx + D
2x3 − 4x − 8
= −
+ 2
, then C + D =
11. If 2
2
(x − x)(x + 4) x x − 1
x +4
(a) 6
(correct)
(b) 8
(c) 10
(d) 4
(e) 2
Question 83 / Section 9.1
12. Given that the sequence
its limit is equal to
(a) 2
√
(b) 2
√
(c) 3
(d) 3
(e) 2 +
√
2,
p
2+
√
q
2,
p
√
2 + 2 + 2, . . . is a convergent sequence,
(correct)
√
2
231, Math 102, Final Exam
Page 7 of 14
MASTER
Question 38 / Section 9.2
13.
∞
X
k=1
1
=
9k 2 + 3k − 2
1
6
1
(b)
5
1
(c)
7
1
(d)
4
1
(e)
3
(a)
Question 47 / Section 9.3
14. Using the integral test, and for positive values of p, the series
(correct)
∞
X
n
converges
2 )p
(1
+
n
n=1
if
(a) p > 1
(b) 0 < p < 1
(c) p ≥ 1
(d) 0 < p ≤ 1
(e) p > 0
(correct)
231, Math 102, Final Exam
Page 8 of 14
MASTER
Question 27 / Section 9.7
15. If P3 (x) = 2 + b(x − 1) + c(x − 1)2 + d(x − 1)3 is the 3rd Taylor polynomial for the
2
function f (x) = , centered at x = 1, then b + c + d =
x
(a) −2
(correct)
(b) −4
(c) 4
(d) 2
(e) 0
Example 4 / Section 5.2
16. The upper sum for the region bounded by the graph of f (x) = x2 and the x-axis
between x = 0 and x = 2 in terms of n (the number of subintervals) is
8 4
4
+ + 2
3 n 3n
5 2
2
(b) + + 2
3 n 3n
8 4
4
(c) − + 2
3 n 3n
5 2
2
(d) − + 2
3 n 3n
8
4
(e) + 2
3 3n
(a)
(correct)
231, Math 102, Final Exam
Page 9 of 14
MASTER
Question
29 / Section 5.7
Z
dx
√ =
1 + 2x
17.
(a)
√
√
2x − ln(1 +
√
√
2x) + c
(correct)
2x + ln(1 + 2x) + c
√
(c) ln(1 + 2x) + c
√
(d) 2x + c
√
√
(e) x − ln(1 + x) + c
(b)
Question 40 / Section 5.8
Z
18.
√
2 dx
=
4x − x2
x−2
+c
2
(b) arcsin x−2
+c
2
(c) arcsin x+2
+c
2
(d) 2 arcsin x+2
+c
2
(a) 2 arcsin
(e) 4 arcsin(x − 2) + c
(correct)
231, Math 102, Final Exam
Page 10 of 14
MASTER
Question 16 / Section 7.4
πi
19. The arc length of the graph of the function y = ln(cos x) over the interval 0,
is
3
h
(a) ln(2 +
(b) ln(1 +
(c) ln(2 +
√
√
√
3)
(correct)
3)
2)
√
(d) ln(2 − 3)
√
(e) ln(1 − 3)
Example 5 (b) / Section 9.2 and Questions 11, 26 /Section 9.4
20. Which of the following series is convergent?
∞
X
1
I.
n!
n=0
II.
∞
X
III.
sin
n=1
∞
X
1
n
n!
2(n!) + 1
n=0
(a) I only
(b) I and II only
(c) I and III only
(d) I, II and III
(e) II and III only
(correct)
231, Math 102, Final Exam
Page 11 of 14
MASTER
Questions 15, 21, 22 / Section 9.5
21. Which of the following series is convergent?
∞
X
(−1)n n
I.
ln(n + 1)
n=1
II.
∞
X
III.
sin
n=1
∞
X
(2n − 1)π
2
1
cos nπ
n
n=1
(a) III only
(correct)
(b) I and III only
(c) I and II only
(d) I only
(e) I, II and III
Questions 52, 54 / Section 9.5
22. Consider the following series
∞
X
(−1)n+1
I.
n4/3
n=1
∞
X
(−1)n
√
II.
.
n
+
4
n=0
Then
(a) I converges absolutely but II converges conditionally
(b) I and II converge absolutely
(c) I and II converge conditionally
(d) I converges absolutely but II diverges
(e) I diverges but II converges absolutely
(correct)
231, Math 102, Final Exam
Page 12 of 14
Question 25 / Section 9.8
23. The interval of convergence of the power series
∞
X
(−1)n+1 (x − 4)n
n=1
(a) (−5, 13]
MASTER
n9n
is
(correct)
(b) [−5, 13]
(c) [−5, 13)
(d) (−5, 13)
(e) (−∞, ∞)
Question 68 / Section 9.10
Z 1/2
24.
arctan(x2 ) dx =
0
(a)
(b)
∞
X
(−1)n
(4n + 3)(2n + 1)24n+3
n=0
∞
X
(−1)n
(4n − 3)(2n − 1)24n+3
n=0
∞
X
(−1)n
(c)
(4n + 3)(2n + 1)24n−3
n=0
∞
X
(−1)n+1
(d)
(4n + 3)(2n + 1)24n+3
n=0
∞
X
(−1)n
(e)
(4n + 3)(2n + 1)24n+3
n=1
(correct)
231, Math 102, Final Exam
Page 13 of 14
Question 39 / Section 9.9
25. The power series representation for the function f (x) =
(a)
(b)
(c)
(d)
(e)
∞
X
n=0
∞
X
n=0
∞
X
n=0
∞
X
n=0
∞
X
(2n + 1) xn , |x| < 1
MASTER
1+x
is
(1 − x)2
(correct)
(n + 1) xn , |x| < 1
(2n − 1) xn , |x| < 1
(2n + 1) xn+1 , |x| < 1
(n − 1) xn , |x| < 1
n=0
Question 23 / Section 7.3
26. The volume of the solid generated by revolving the region bounded by the graphs
of the equations y = 2x − x2 and y = 0 about the line x = 4 is
(a) 8 π
(b) 6 π
(c) 4 π
(d) 10 π
(e) 12 π
(correct)
231, Math 102, Final Exam
Page 14 of 14
MASTER
Question 23 / Section 8.2
xe2x
dx =
(2x + 1)2
Z
27.
e2x
(a)
+c
4(2x + 1)
e2x
(b)
+c
2(2x + 1)
e2x
(c)
+c
4(x + 1)
ex
(d)
+c
4(2x + 1)
ex
(e)
+c
2(2x + 1)
(correct)
Example 3 / Section 8.7
Z 2
28.
0
x
dx =
1 + e−x2
1
1 + e4
(a)
ln
2
2
1
1 + e−4
ln
(b)
2
2
1
(c) ln(1 + e4 )
2
1
(d) ln(1 + e−4 )
2 1 + e4
(e) ln
2
(correct)