MATH 155/GRACEY
PRACTICE FINAL
Name___________________________________
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Provide an appropriate response.
1) Which of the following statements is false?
A) The integral test does not apply to divergent sequences.
B) If a n and f(n) satisfy the requirements of the Integral Test, and if
1)
∫
∞
f(x)dx converges, then
1
∞
∑
n=1
∞
C)
∑
n=2
∞
D)
∑
n=1
an =
∫
∞
f(x) dx.
1
1
n(ln n)p
converges if p > 1.
1
converges if p >1 and diverges if p ≤ 1.
np
Choose the equation that matches the graph.
2)
A)
x2 y2
+
=1
9
25
B)
2)
x2 y2
=1
9
25
C)
x2 y2
=1
25
9
D)
y2 x2
=1
9
25
3)
3)
A)
x2 y2
+
=1
25
9
B)
y2 x2
=1
25
9
C)
1
y2 x2
+
=1
25
9
D)
y2 x2
+
=1
5
3
4)
4)
y
20
10
-20
-10
10
20
x
-10
-20
A) y2 = 6x
B) 6y2 = x
C) y2 = -6x
D) x2 = 6y
5)
5)
A)
x2 y2
+
=1
5
3
B)
x2 y2
+
=1
9
25
C)
x2 y2
=1
25
9
D)
x2 y2
+
=1
25
9
6)
6)
y
20
10
-20
-10
10
20
x
-10
-20
A) 4x2 = y
B) y2 = 4x
C) x2 = 4y
2
D) x2 = -4y
7)
7)
A)
y2 x2
+
=1
9
25
B)
y2 x2
=1
25
9
C)
x2 y2
=1
9
25
D)
y2 x2
=1
9
25
8)
8)
y
20
10
-4
-2
2
4
x
-10
-20
A) 4y2 = x
B) 4x2 = y
C) 4x2 = -y
D) x2 = 4y
9)
9)
y
20
10
-20
-10
10
20
x
-10
-20
A) y2 = -4x
B) y2 = 4x
C) -4y2 = x
3
D) x2 = -4y
10)
10)
y
20
10
-20
-10
10
20
x
-10
-20
A) y2 = -8x
B) -8x2 = y
C) x2 = 8y
D) x2 = -8y
11)
11)
A) 9x2 - 16y2 = 144
C) 16x2 + 9y2 = 144
B) 16x2 - 9y2 = 144
D) 9x2 + 16y2 = 144
12)
12)
A) 4y2 - 25x2 = 100
C) 25x2 - 4y2 = 100
B) 25x2 + 4y2 = 100
D) 4x2 - 25y2 = 100
Evaluate the integral.
13)
∫
19x sin x dx
13)
A) 19 sin x - 19x cos x + C
C) 19 sin x - x cos x + C
B) 19 sin x - 19 cos x + C
D) 19 sin x + 19x cos x + C
4
Evaluate the integral by making a substitution and then using a table of integrals.
ex
14)
dx
e2x - 4
∫
A)
1
2-x
ln
+C
4
x+2
B)
1
ex + 2
ln
+C
4
ex - 2
C)
1
2 - ex
ln
+C
4
ex + 2
D)
1
2 - e2x
ln
+C
4
e2x + 2
Solve the problem.
15) Find the volume of the solid generated by revolving the region in the first quadrant bounded by
the x-axis and the curve y = x cos x, 0 ≤ x ≤ π/2 about the y-axis.
π3
π3
π2
π3
A)
B)
C)
D)
+ 2π2 - 4π
- 8π
- 4π
- 4π
2
2
2
2
Solve the differential equation.
16) 4y ′ = ex/4 + y
14)
15)
16)
xex/4 + C
A) y =
4
-xex/4 + Cex/4
B) y =
4
C) y = xex/4 + Cex/4
D) y =
xex/4 + Cex/4
4
Find the volume.
17) Find the volume of the solid generated by revolving the region under the curve y = 8e-2x in the
first quadrant about the y-axis.
A) 4π
B) 8π
C) 8π3
D) 64π
Express the integrand as a sum of partial fractions and evaluate the integral.
160 dx
18)
x3 - 16x
∫
A) -10 ln x +
1
x
tan -1 + C
4
4
17)
18)
B) 10 ln x - 5ln x - 4 - 5ln x + 4 + C
C) -10 ln x + 5ln x - 4 + 5ln x + 4 + C
D) -
10
+ 5ln x - 4 + 5ln x + 4 + C
x
Evaluate the integral.
dx
19)
(4 - x2 )2
∫
19)
A)
1
x+2
ln
+C
4
x-2
B)
C)
1
x
1
x+2
- ln
+C
8 4 - x2 4
x-2
D)
5
x
8(4 - x2 )
+C
1
x
1
x+2
+ ln
+C
8 4 - x2 4
x-2
20)
∫
π
1 - cos2 x dx
20)
0
A) 2
2
2
B)
C)
2
D) 0
Integrate the function.
1
8
21)
dt
1 + 64t2
-1
∫
21)
A) 2tan -1 8
B) 2tan -1
1
8
C)
π
2
D) 2sin -1 8
Evaluate the integral by making a substitution and then using a table of integrals.
e2x
22)
dx
5ex + 6
∫
A)
6
+ ln 5ex + 6
5ex + 6
C)
x
6
ln 5x + 6
5 25
+C
+C
B)
ex
6
ln 5ex + 6
5
25
D)
ex
6
sin -1 5ex + 6
+
5
25
Express the integrand as a sum of partial fractions and evaluate the integral.
7x3 + 34x2 + 57x + 24
23)
dx
(x + 4)(x + 1)3
∫
A) ln (x + 4)4 (x + 1)3 -
4
1
+
+C
(x + 1) (x + 1)2
B) ln (x + 4)4 (x + 1)5 -
1
4
+
+C
(x + 1) (x + 1)2
C) ln (x + 4)4 (x + 1)3 D) ln (x + 4)4 (x + 1)3 +
5
(x + 1)2
3
(x + 1)2
22)
+C
+C
23)
+C
+C
Evaluate the integral.
(x + 8)2 tan -1 x + (5x - 25) (x + 8)
24)
dx
(x2 + 1) (x + 8)2
∫
24)
(tan -1 x)2
1
- 3 tan -1 x + ln(x2 + 1) + C
2
2
(tan -1 x)2
1
B)
- ln x + 8 - 3 tan -1 x + ln(x2 + 1) + C
2
2
(tan -1 x)2
C)
- ln x + 8 - 5 tan -1 x + ln(x2 + 1) + C
2
(tan -1 x)2
1
D)
- ln x + 8 + ln(x2 + 1) + C
2
2
A)
6
Evaluate the improper integral or state that it is divergent.
∞
25)
15xe2x dx
0
A) Divergent
B) 2.6667
∫
26)
∫
∞
19
8x(x + 1)2
1
25)
C) 1.3333
D) 1.6667
dx
A) -1.569
26)
B) 0.458
C) Divergent
D) 1.569
Evaluate the integral by using a substitution prior to integration by parts.
27)
∫
ln 3x 2 dx
27)
A) x ln 3x 2 - x ln 3x + x + C
C) x ln 3x 2 + 2x ln 3x - 2x + C
B) x ln 3x 2 - 2x ln 3x + C
D) x ln 3x 2 - 2x ln 3x + 2x + C
Evaluate the integral.
28)
∫ 3 sin 6x sin 3x dx
28)
A) 3
cos 3x cos 9x
+C
6
18
B) 3 sin 3x - sin 9x + C
C) 3
sin 3x sin 9x
+
+C
6
18
D) 3
Find a formula for the nth term of the sequence.
29) -9, -8, -7, -6, -5 (integers beginning with -9)
A) a n = n - 10
B) a n = n + 9
sin 3x sin 9x
+C
6
18
29)
C) a n = n - 9
D) a n = n - 8
Determine convergence or divergence of the alternating series.
∞
(-6)n
30) ∑
3n 4 + 7 n
n=1
A) Diverges
B) Converges
30)
Use the root test to determine if the series converges or diverges.
∞
n
31) ∑
1/n - 1 n
n=1 7n
A) Converges
B) Diverges
31)
∞
32)
∑
ln n n
2n - 9
32)
n=1
A) Diverges
B) Converges
7
Determine if the series converges or diverges. If the series converges, find its sum.
∞
1
1
33) ∑
1/(n+1)
1/n
2
2
n=1
1
A) diverges
B) converges; C) converges;1
2
33)
D) converges;
1
2
Find the limit of the sequence or determine that the limit does not exist.
6 n
34) an = 1 +
n
A) does not exist
B) 1
C) e
34)
D) e6
Use the root test to determine if the series converges or diverges.
∞
(n!)3n
35) ∑
(3n)! n
n=1
A) Converges
B) Diverges
35)
Use the ratio test to determine if the series converges or diverges.
∞
8n!
36) ∑
nn
n=1
A) Diverges
B) Converges
36)
Use the integral test to determine whether the series converges.
∞
1
37) ∑
5n
n=1
A) converges
B) diverges
37)
Determine if the geometric series converges or diverges. If it converges, find its sum.
2
2 2 2 3
2 4
38) 1 + +
+
+
+...
5
5
5
5
A) converges, 2
B) converges,
5
3
C) diverges
Determine if the series converges or diverges. If the series converges, find its sum.
∞
1
1
39) ∑
n+1
n+2
n=1
1
1
A) converges;
B) converges;
C) diverges
3
2
8
38)
D) converges,
7
3
39)
D) converges;
1
6
Write the first four elements of the sequence.
ln (n + 1)
40)
n3
40)
A) 0,
ln 2 ln 3 ln 4
,
,
8
27 64
B) ln 2,
C) 0,
ln 2 ln 3 ln 4
,
,
27 64 81
D)
ln 3 ln 4 ln 5
,
,
8
27 64
ln 2 ln 3 ln 4 ln 5
,
,
,
8
27 64 81
Find the limit of the sequence or determine that the limit does not exist.
41) an = ln(8n + 6) - ln(3n + 6)
A) does not exist
B) ln 5
C) ln
3
8
Determine if the series converges absolutely, converges, or diverges.
∞
(n!)2 3 n
42) ∑ (-1)n
(2n + 1)!
n=1
A) converges conditionally
B) diverges
41)
D) ln
8
3
42)
C) converges absolutely
A recursion formula and the initial term(s) of a sequence are given. Write out the first five terms of the sequence.
43) a1 = 1, an+ 1 = a n 6
43)
A) 1, 1, 1, 1, 1
C) 1, 6, 36, 216, 1296
B) 1, 6, 36, 216, 1296, 7776
D) 1, 7, 13, 19, 25
Use the Comparison Test to determine if the series converges or diverges.
∞
1
44) ∑
4 n-1 + 1
n=1
A) converges
B) diverges
44)
Find the Taylor polynomial of order 3 generated by f at a.
45) f(x) = x2 + x + 1, a = 4
45)
A) P3 (x) = 21 + 9(x - 4) + (x - 4)2
B) P3 (x) = 1 + 3(x - 4) + 3(x - 4)2 + (x - 4)3
C) P3 (x) = 21 + 9(x - 4) + 9(x - 4)2 + 21(x - 4)3
D) P3 (x) = 5 + 9(x - 4) + 13(x - 4)2
Find the function represented by the power series.
∞
46) ∑ (x - 10)n
46)
n=1
A)
x - 10
x - 11
B)
x - 10
x-9
C) -
9
x - 10
x - 11
D) -
x - 10
x-9
Find the interval of convergence of the series.
∞
47) ∑ (x - 8)n
n=0
A) 7 < x < 9
B) -9 < x < 9
47)
C) x < 9
D) 7 ≤ x < 9
Find the Taylor series generated by f at x = a.
48) f(x) = e2x, a = 8
∞
A)
∑
48)
∞
e16 2 n (x - 8)n
n!
B)
n=0
∞ 16 n+1
e 2
(x - 8)n
C) ∑
n!
n=0
49) f(x) =
∑
n=0
∞
C)
e16 2 n+1 (x - 8)n
(n + 1)!
n=0
∞ 16 n
e 2 (x - 8)n
D) ∑
(n + 1)!
n=0
1
,a=9
x
∞
A)
∑
∑
n=0
49)
∞
(x - 9)n
9n
B)
(-1)n (x - 9)n
9 n+1
D)
∑
n=0
∞
∑
n=0
(x - 9)n
9n+1
(-1)n (x - 9)n
9n
Solve the problem.
50) Use a Taylor series to estimate the integral's value to within an error of magnitude less than 10-3 .
0.2
1 + x3 dx
0
A) 0.2105
B) 0.2002
C) .6845
D) 0.1105
50)
∫
Find the Taylor polynomial of order 3 generated by f at a.
51) f(x) = x3 , a = 9
51)
A) P3 (x) = 2916 + 243(x - 9) + 18(x - 9)2 + (x - 9)3
B) P3 (x) = 729 + 243(x - 9) + 27(x - 9)2 + (x - 9)3
C) P3 (x) = 6 + 3(x - 9) + (x - 9)2 + (x - 9)3
D) P3 (x) = 729 + 81(x - 9) + 81(x - 9)2 + (x - 9)3
Find the Cartesian coordinates of the given point.
π
52) 18, 4
52)
A) (9 2, -9 2)
C) (-18 2, -18 2)
B) (9 3, -9)
D) (-9 2, -9 2)
10
Solve the problem.
53) Find the foci and asymptotes of the following hyperbola:
16x2 - y2 = 16
53)
A) Foci: ( 17, 0), (- 17, 0); Asymptotes: y = 4x, y = -4x
B) Foci: (0, 17), (0, - 17); Asymptotes: y = 4x, y = -4x
1
1
C) Foci: ( 17, 0), (- 17, 0); Asymptotes: y = x, y = - x
4
4
D) Foci: (4, 0), (-4, 0); Asymptotes: y =
1
1
x, y = - x
4
4
Find the standard-form equation of the ellipse centered at the origin and satisfying the given conditions.
54) An ellipse with length of major axis 14 and y-intercepts (0, ±2)
x2
y2
x2
y2
x2
y2
x2
y2
A)
B)
C)
D)
+
=1
+
=1
+
=1
+
=1
2
7
7
2
49
4
4
49
Replace the polar equation with an equivalent Cartesian equation.
55) r = 8 cot θ csc θ
B) y2 = 8x
A) y = 8x
Find the area of the specified region.
56) Inside one leaf of the four-leaved rose r = 7 sin 2θ
49π
49π
A)
B)
2
8
55)
8
x
C) y = 8x2
D) y =
7π
C)
2
49π 49
D)
8
16
56)
Find the standard-form equation of the ellipse centered at the origin and satisfying the given conditions.
57) An ellipse with intercepts (±4, 0) and (0, ±5)
x2
y2
x2
y2
x2
y2
x2
y2
A)
B)
C)
D)
+
=1
+
=1
+
=1
+
=1
16
25
5
4
25
16
4
5
Plot the point whose polar coordinates are given.
58) (-4, 0)
57)
58)
5
-5
54)
5
-5
11
A)
B)
5
-5
5
5
-5
-5
5
-5
C)
D)
5
-5
5
5
-5
-5
5
-5
Solve the problem.
59) Find the vertices and foci of the following hyperbola:
x2 y2
=1
25 96
59)
A) Vertices: (0, 11), (0, -11); Foci: (0, 5), (0, -5)
B) Vertices: (25, 0), (-25, 0); Foci: (11, 0), (-11, 0)
C) Vertices: (0, 5), (0, -5); Foci: (0, 11), (0, -11)
D) Vertices: (5, 0), (-5, 0); Foci: (11, 0), (-11, 0)
Determine if the given polar coordinates represent the same point.
60) (8, π/3), (-8, -2π/3)
A) Yes
B) No
12
60)
Solve the problem by integration.
61) The general expression for the slope of a curve is
4x + 6
. Find the equation of the curve if it
x2 + 6x
passes through (1, 0).
x(x - 6)3
A) y = ln
343
C) y = ln
B) y = ln x(x + 6)3
x( x + 6)3
7
D) y = ln
x(x + 6)3
343
Evaluate the improper integral or state that it is divergent.
0
62)
4 ex sin x dx
∫
-∞
A) - 2
62)
B) -4
C) 2
D) 0
Give the appropriate form of the partial fraction decomposition.
x2 - 4x + 30
63)
x2 - 10x + 24
A) 1 +
C)
64)
21
-15
+
x-6 x-4
63)
B) 1 +
x + 21 -x - 15
+
x-6
x-4
D)
21
15
+
x-6 x-4
21
-15
+
x-6 x-4
t4 + t2 - 9t - 12
t4 + 3t2
64)
A) 1 +
3t + 2
-3 -4
+
+
2
t
t +3
t2
B) 1 +
3t + 2 -4
+
t2 + 3 t2
C) 1 +
3t + 2
-3 -4
+
+
t
t2 + 3
t2
D) 1 +
2
-3 -4
+
+
t
t2 + 3
t2
Write the first four elements of the sequence.
n+1
65)
3n - 1
A) 0,
1 1 3
, ,
3 2 4
61)
B)
65)
1 1 3 2
, , ,
3 2 4 3
C) -1, 1,
3 1
,
5 2
Determine convergence or divergence of the alternating series.
∞
n+ n
66) ∑ (-1)n+1
n2 + 1
n=1
A) Diverges
B) Converges
13
D) 1,
3 1 5
, ,
5 2 11
66)
Determine if the series converges or diverges. If the series converges, find its sum.
∞
9
67) ∑
n(n + 1)(n + 2)
n=1
27
9
A) converges;
B) diverges
C) converges;
4
4
∞
68)
∑
n=1
67)
D) converges; 6
7n
(2n - 1)2 (2n + 1)2
A) converges;
49
6
68)
B) converges;
21
8
C) converges;
35
6
D) converges;
7
8
Use the integral test to determine whether the series converges.
∞
1
69) ∑
ln 6 n
n=1
A) diverges
B) converges
69)
Find the series' radius of convergence.
∞
(x - 4)n
70) ∑
ln (n + 4)
n=1
A) 2
B) 1
70)
D) ∞, for all x
C) 0
Find the first four terms of the binomial series for the given function.
x 1/3
71) 1 7
A) 1 -
1
1 2
5 3
x+
x x
21
441
9261
B) 1 -
1
1 2
5
x+
x x3
21
441
27,783
C) 1 -
1
1 2
5
xx x3
21
441
27,783
D) 1 -
1
1 2
5 3
xx x
21
147
9261
72) (1 + 4x2 )3
71)
72)
A) 1 + 12x2 + 12x4 + x6
C) 1 + 12x2 + 48x4 + 64x6
B) 1 + 12x2 + 36x4 + 108x6
D) 1 + 12x2 + 80x4 + 448x6
14
Graph.
73) 36x2 + 12y2 = 192
73)
y
10
5
-10
-5
5
10
x
-5
-10
A)
B)
y
-10
y
10
10
5
5
-5
5
10
x
-10
-5
-5
-5
-10
-10
C)
5
10
x
5
10
x
D)
y
-10
y
10
10
5
5
-5
5
10
x
-10
-5
-5
-5
-10
-10
Find the vertices and foci of the ellipse.
74) 16x2 + 121y2 = 1936
74)
A) Vertices: (0, ±4); Foci (0, ± 105)
C) Vertices: (±11, 0); Foci: (± 105, 0)
B) Vertices: (±4, 0); Foci: (± 105, 0)
D) Vertices: (0, ±11); Foci (0, ± 105)
15
Evaluate the integral.
75)
∫
-6x cos 9x dx
75)
A) -
2
cos 9x - 6x sin 9x + C
3
B) -
2
2
cos 9x - x sin 9x + C
27
3
C) -
2
2
cos 9x - sin 9x + C
27
3
D) -
2
2
cos 9x - x sin 6x + C
27
3
Use reduction formulas to evaluate the integral.
76)
∫ sin5 4x dx
76)
A) -
1
4
2
sin 4 4x cos 4x sin 2 4x cos 4x - cos 4x + C
5
15
3
B) -
1
1
2
sin 4 4x cos 4x sin 2 4x cos 4x cos 4x + C
20
15
15
C) -
1
2
1
cos4 4x sin 4x cos2 4x sin 4x cos 4x + C
20
15
30
D) -
1
1
1
x
sin 4 4x cos 4x sin 2 4x cos 4x - sin 8x + + C
20
15
6
2
Find the sum of the series.
∞
1
(-1)n
77) ∑
n
6
6n
n=0
6
A)
37
77)
B)
12
37
C)
6
35
D)
12
35
Use the root test to determine if the series converges or diverges.
∞ 9n 1/n 1 n
78) ∑
1/n
2n
-1
n=1
A) Diverges
B) Converges
Find the first four nonzero terms in the Maclaurin series for the function.
79) f(x) = x sin(3x)
9
81 6 243 8
9
27 5 729 7
A) 3x2 - x4 +
x x +...
B) 3x - x3 +
x x +...
2
40
560
2
8
80
C) 3x2 +
9 4 81 6 243 8
x +
x +
x +...
2
40
560
D) 3x -
78)
79)
9 4 243 6 729 8
x +
x x +...
2
40
560
Find the vertices and foci of the ellipse.
80) 9x2 + 25y2 = 225
80)
A) Vertices: (±9, 0); Foci: (±4, 0)
C) Vertices: (0, ±5); Foci: (0, ±4)
B) Vertices: (0, ±9); Foci: (0, ±5)
D) Vertices: (±5, 0); Foci: (±4, 0)
16
Find the limit of the sequence or determine that the limit does not exist.
(ln n)7
81) an =
n
A) e7
B) does not exist
81)
C) 0
Find the function represented by the power series.
∞ x 9 n
+
82) ∑
7
n=0
7
7
A) B)
x-2
x-2
D) ln 7
82)
C) -
7
x+2
D)
7
x+2
Parametric equations and a parameter interval for the motion of a particle in the xy-plane are given. Identify the
particle's path by finding a Cartesian equation for it. Graph the Cartesian equation. Indicate the portion of the graph
traced by the particle and the direction of motion.
83) x = 2t + 4, y = 6t + 6, -∞ ≤ t ≤ ∞
83)
y
5
4
3
2
1
-5
-4
-3
-2
-1
1
2
3
5 x
4
-1
-2
-3
-4
-5
A) y = 3x + 6
B) y = 3x - 6
y
y
-5
-4
-3
-2
5
5
4
4
3
3
2
2
1
1
-1
1
2
3
4
5 x
-5
-4
-3
-2
-1
1
-1
-1
-2
-2
-3
-3
-4
-4
-5
-5
2
Entire line, from left to right
Entire line, from left to right
17
3
4
5 x
C) y = 3x + 6
D) y = 3x - 6
y
y
-5
-4
-3
-2
5
5
4
4
3
3
2
2
1
1
-1
1
2
3
4
5 x
-5
-4
-3
-2
-1
1
-1
-1
-2
-2
-3
-3
-4
-4
-5
-5
2
Entire line, from right to left
Entire line, from right to left
18
3
4
5 x
Answer Key
Testname: M155_PRACTICE FINAL
1)
2)
3)
4)
5)
6)
7)
8)
9)
10)
11)
12)
13)
14)
15)
16)
17)
18)
19)
20)
21)
22)
23)
24)
25)
26)
27)
28)
29)
30)
31)
32)
33)
34)
35)
36)
37)
38)
39)
40)
41)
42)
43)
44)
45)
46)
47)
48)
49)
50)
B
B
C
A
D
C
D
B
A
D
C
C
A
C
D
D
B
C
D
A
A
B
A
B
A
B
D
D
A
B
A
B
D
D
A
B
A
B
B
B
D
B
A
A
A
C
A
A
C
B
19
Answer Key
Testname: M155_PRACTICE FINAL
51)
52)
53)
54)
55)
56)
57)
58)
59)
60)
61)
62)
63)
64)
65)
66)
67)
68)
69)
70)
71)
72)
73)
74)
75)
76)
77)
78)
79)
80)
81)
82)
83)
B
A
A
C
B
B
A
B
D
A
D
A
A
A
D
B
C
D
B
B
C
C
A
C
B
B
D
A
A
D
C
C
B
20