Free Printable Mock Test_5-31_Multiple Integration

1.

Fill in the blank: The subscript n that appears in a_n is called an ______.

a)

term

b)

index

c)

sequence

d)

series

2.

What is a term in a sequence?

a)

A number in the sequence

b)

The sum of the sequence

c)

The product of the sequence

d)

The difference in the sequence

3.

Fill in the blank: Sequences are denoted in any of the following forms: {a_1, a_2, a_3, ..., a_n, ...}, {a_n}^∞_{n=1}, or ______.

a)

{a}

b)

{a_n}

c)

{n_a}

d)

{n}

4.

What is the value of a_4 in the sequence a_n = 1/2^n?

a)

1/16

b)

1/8

c)

1/4

d)

1/2

5.

Fill in the blank: The sequence {1, 4, 7, 10, ...} can be defined in two ways. One way is using a recurrence relation, and the other is using an __ formula.

a)

implicit

b)

explicit

c)

hidden

d)

complex

6.

In Example 3, what is the next term in the sequence {a_n} = {-2, 5, 12, 19, ...}?

a)

26

b)

27

c)

28

d)

29

7.

In Example 4, what is the limit of the sequence a. {(-1)^n / (n^2 + 1)}?

a)

1

b)

0

c)

-1

d)

Infinity

8.

Fill in the blank: The sequence c. {a_n} where a_(n+1) = -2a_n, a_1 = 1, has terms that ______ in magnitude.

a)

decrease

b)

increase

c)

remain constant

d)

oscillate

9.

What is the fourth term of the sequence c. {a_n} where a_(n+1) = -2a_n, a_1 = 1?

a)

4

b)

-4

c)

8

d)

-8

10.

What is the high point after the 20th bounce using the explicit formula?

a)

0.21 ft

b)

0.31 ft

c)

0.41 ft

d)

0.51 ft

11.

What is the reasonable conjecture for the value of the series 0.9 + 0.09 + 0.009 + \.?

a)

0.9

b)

0.99

c)

1

d)

0.999

12.

The notation for accumulation over an infinite interval in functions is ______.

a)

∫_a^b f(x) dx

b)

∑_(k=1)^n a_k

c)

∫ f(x) dx

d)

∑_(k=1)^∞ a_k

13.

In sequences/series, the dependent variable is denoted by ______.

a)

x

b)

n

c)

a_n

d)

f(x)

14.

What is the formula for the sequence $a_n = \frac{2^n + 1}{2^n + 1}$ ?

a)

$a_n = 1$

b)

$a_n = 2$

c)

$a_n = n$

d)

$a_n = 0$

15.

What is the explicit formula for the sequence $a_n = 5^n / 5^{n+1}$ ?

a)

$a_n = 1/5$

b)

$a_n = 5$

c)

$a_n = n$

d)

$a_n = 0$

16.

(a)

17.

Determine whether the sequence Converges or Diverges. If convergent, give the limit of the sequence.

(a)

18.

Determine whether the sequence is Bounded, Bounded above, Bounded below, or None of the above.

(a)

19.

(a)

20.

Note: The answer must be in fraction form. e.g. 1/3,4/9

(a)

21.

Determine whether the sequence Converges or Diverges.

(a)

22.

Answer the 1st blank in the statement:

In order to apply the Integral Test to a sequence {a_n}, the function a (n) = a_n must be c___________s, p___________e and d______________g.

(a)

23.

Answer the 2nd blank in the statement:

In order to apply the Integral Test to a sequence {a_n}, the function a (n) = a_n must be c___________s, p___________e and d______________g.

(a)

24.

Answer the 3rd blank in the statement:

In order to apply the Integral Test to a sequence {a_n}, the function a (n) = a_n must be c___________s, p___________e and d______________g.

(a)

25.

What tests in the Integral and Comparison Test topic do not work well with factorials?

(a)

26.

The Ratio Test is not effective when the terms of a sequence only contain functions.

(a)

27.

a)

Diverges; Limit Comparison Test

b)

Diverges; Limit Comparison Test with 1

c)

Converges; Integral Test

d)

Converges; Limit Comparison Test

28.

a)

Diverges; Limit Comparison Test

b)

Diverges; Limit Comparison Test with 1

c)

Converges; Integral Test

d)

Converges; Limit Comparison Test

29.

a)

Diverges; Limit Comparison Test

b)

Diverges; Limit Comparison Test with 1

c)

Converges; Integral Test

d)

Converges; Limit Comparison Test

30.

a)

Converges

b)

Converges (p-Series)

c)

Conditional

d)

Absolute

31.

A convergent alternating series is given along with its sum and a value of ε. Use Theorem 71 to find n such that the n^th partial sum of the series is within ε of the sum of the series.

(a)

32.

(a)

33.

(a)

34.

A power series is given. Find the radius of convergence.

(a)

35.

A power series is given. Find the radius of convergence.

(a)

36.

Which statement is true about the SEQUENCE $\left\{\arctan\left(\ln n\right)\right\}$ ?

a)

It diverges.

b)

It converges to 0.

c)

It converges to $\pi$ .

d)

It converges to $\frac{\pi}{2}$ .

37.

Which statement is true about the SERIES $\sum_{n=1}^{\infty}\frac{5}{\pi^n}$ ?

a)

It converges by the Geometric Series Test.

b)

It converges by the nth Term Test for Divergence.

c)

It diverges by the nth Term Test for Divergence.

d)

It diverges by the Geometric Series Test.

38.

Does the SERIES $\sum_{n=1}^{\infty}\frac{1}{4+e^{-x}}$ converge or diverge?

a)

Converge

b)

Diverge

39.

Find the sum: $\frac{2}{5}+\frac{4}{25}+\frac{8}{125}+\frac{16}{625}+\frac{32}{3125}+...$

a)

There is no sum. Diverges

b)

2/3

c)

5/3

d)

2/7

40.

Which statement is true about the SERIES $\sum_{n=1}^{\infty}\frac{6}{n\sqrt[]{n}}$ ?

a)

It is a convergent geometric series.

b)

It is a convergent P-series.

c)

It is a divergent P-series.

d)

It is a divergent geometric series.

41.

Which SERIES would best be compared to the harmonic series to perform a comparison test?

a)

$\sum_{n=1}^{\infty}\frac{n^3+3}{n+1}$

b)

$\sum_{n=1}^{\infty}\arctan\left(n\right)$

c)

$\sum_{n=1}^{\infty}\frac{n^2+3n+2}{\sqrt[]{n^6+5}}$

d)

$\sum_{n=1}^{\infty}\frac{4n+3}{n+4}$

42.

nth term test:

If the limit as n approaches infinity of a series is __________ zero, then the series will diverge.

a)

Not equal to

b)

equal to

43.

Geometric Series:

A geometric series will converge if __________.

a)

$\left|r\right|>1$

b)

$\left|r\right|<1$

44.

p-series:

A p-series will converge if __________ .

a)

$p<1$

b)

$p>1$

45.

Will the series converge or diverge?

\sum_{n=1}^{\infty}\frac{n!}{n\cdot3^n}

a)

converge

b)

diverge

46.

Will the series converge or diverge?

\sum_{n=1}^{\infty}\frac{\sqrt{3n-4}}{\sqrt{9n+2}}

a)

converge

b)

diverge

47.

Will the series converge or diverge?

\sum_{n=1}^{\infty}\frac{3^n}{8^{\left(n+1\right)}}

a)

converge

b)

diverge

48.

Will the series converge or diverge?

\sum_{n=1}^{\infty}\frac{n^{\frac{2}{3}}}{n^{\frac{13}{3}}}

a)

converge

b)

diverge

49.

Will the series converge or diverge?

\sum_{n=1}^{\infty}\frac{n+2}{\left(n+1\right)^3}

a)

converge

b)

diverge

50.

Which of the following tests would be the best choice to prove the series is divergent?

a)

nth term test

b)

integral test

c)

direct comparison test

d)

limit comparison test

51.

Which statement is true about the series?

a)

The sum is -4/3

b)

The sum is 2/3

c)

The sum is 9/2

d)

The series diverges

52.

Which of the following is true about the given series?

a)

it is a divergent p-series

b)

it is a convergent p-series

c)

it is a divergent geometric series

d)

it is a convergent geometric series

53.

Which test can NOT be used to prove that the series diverges?

a)

limit comparison test

b)

direct comparison test

c)

p-series

d)

integral test

54.

Select the correct comparison series you would use for the given series.

a)

$\Sigma\ \frac{1}{n}$

b)

$\Sigma\ \frac{1}{n^2}$

c)

$\Sigma\ \left(\frac{1}{2}\right)^n$

d)

$\Sigma\ \left(2\right)^n$

55.

What would be the test used, and what is the result of the test?

a)

alternating series, absolutely converge

b)

nth term test, diverge

c)

alternating series, diverge

d)

integral test, converge

56.

Which of the following is not true about ratio test?

a)

, then the series is convergent

b)

, then the series is divergent

c)

, then the series is inconclusive

d)

none of the choices

57.

Which of the following is not true about root test?

a)

, then the series is convergent

b)

, then the series is divergent

c)

, then the series is inconclusive

d)

all of the choices

58.

Which of the following is considered as partial sum?

a)

b)

$1+x+x^2+x^3+x^4$

c)

1! + 2! + 3! + 4!

d)

both b and c

59.

What is the center of convergence of the given power series?

a)

0

b)

1

c)

8

d)

-8

60.

What is the center of convergence of the given power series?

a)

0

b)

n

c)

x - c

d)

x = c

61.

What is the center of convergence of the given power series?

a)

0

b)

1/n

c)

x - c

d)

x = c

62.

What is the center of convergence of the given power series?

a)

0

b)

1/n

c)

-2

d)

2/3

63.

What is the center of convergence of the given power series?

a)

3n

b)

-4

c)

3

d)

4

64.

What is the center of convergence of the given power series?

a)

0

b)

-2

c)

2

d)

infinity

65.

What is the center of convergence of the given power series?

a)

-1

b)

1

c)

-1/2

d)

1/2

66.

Which of the following statement is not true?

a)

When the degree of numerator is less than degree of denominator the horizontal asymptote is at y = 0.

b)

When the degree of numerator is greater than degree of denominator by one, it has a slant asymptote.

c)

When the degree of numerator is equal to degree of denominator the horizontal asymptote is at the ratio of leading coefficients.

d)

none of the choices

67.

Which of the following statement is not true?

a)

Arithmetic series never converge and always diverge.

b)

Harmonic series always converge and never diverge.

c)

Power series converge at greater than 1 and diverge at greater than or equal to 1

d)

none of the choices

68.

Which is not true about divergence test?

a)

If the limit of a series exists, then the series diverge.

b)

If the limit of a series is not equal to zero, then the series diverge.

c)

If the limit of a series exists, then the series converge.

d)

If the limit of a series does not exist, then the series diverge.

69.

The series converges for which x?

a)

−1 < x < 5

b)

−1 ≤ x ≤ 5

c)

1 ≤ x ≤ 5

d)

−1 ≤ x < 5

70.

Find the partial sum S50 for the series

a)

b)

c)

d)

71.

Find the radius of convergence for the power series

a)

0

b)

$\infty$

c)

1

d)

2

72.

Which of the following functions grows fastest?

a)

x!

b)

$X^X$

c)

$X^{1,000,000,000,000}$

d)

$9,999,999,999,999,999^X$

73.

What is the radius of convergence for the power series?

a)

0

b)

1

c)

2

d)

no radius of convergence

74.

Which of these is another way to write the function 1 + $\frac{x}{5}$ + $\frac{x^2}{25}$ + $\frac{x^3}{125}$ + ….?

a)

$\sum_{n=0}^{^∞}\frac{x^n}{5n}$

b)

$\sum_{n=0}^{^∞}\frac{x^n}{5^n}$

c)

$\sum_{n=0}^{^∞}\left(5x\right)^n$

d)

$\sum_{n=0}^{^∞}5x^n$

75.

You're told that the interval of convergence for a given power series is (-7,5). What is the center of the power series?

a)

-7

b)

-1

c)

0

d)

5

76.

What is the interval of convergence for f(x)?

a)

2<x<4

b)

2<x≤4

c)

2≤x≤4

d)

2≤x<4

77.

You're told that the interval of convergence for a given power series is (1,5). What is the center of the power series?

a)

-6

b)

1

c)

3

d)

5

78.

You're told that the interval of convergence for a given power series is 2<x<4. What is the center of the power series?

a)

-6

b)

1

c)

3

d)

5

79.

What is the partial sum $\sum_{n=0}^35x^n$ ?

a)

1+ 25 + 125 + 625

b)

$5+5x+5x^2+5x^3$

c)

$0+5^n+10^n+15^n$

d)

inconclusive

80.

What is the partial sum $\sum_{n=1}^35^n$ ?

a)

1+ 25 + 125 + 625

b)

$5x+5x^2+5x^3$

c)

$5^n+10^n+15^n$

d)

5 + 25 + 125

81.

Which of the following has a center at x = -3?

a)

$\sum_{n=0}^∞(n-5)x^n$

b)

$\sum_{n=0}^∞n!(x-3)^n$

c)

$\sum_{n=0}^∞n+1(x+3)^n$

d)

none of the choices

82.

Which of the following has a center at x = 5?

a)

$\sum_{n=0}^∞(n-5)x^n$

b)

$\sum_{n=0}^∞(n-5)!(x-5)^n$

c)

$\sum_{n=0}^∞n+1(x+5)^n$

d)

none of the choices

83.

Which of the following has a center at x = - 5?

a)

$\sum_{n=0}^∞(n-5)x^n$

b)

$\sum_{n=0}^∞(n-5)!(x-5)^n$

c)

$\sum_{n=0}^∞n+1(x+5)^n$

d)

none of the choices

84.

Which of the following has a center at x = 5?

a)

$\sum_{n=0}^∞(n-5)x^n$

b)

$\sum_{n=0}^∞(n-5)!(x-5)^n$

c)

$\sum_{n=0}^∞n+1(x+5)^n$

d)

none of the choices

85.

Which of the following has a center at x = 0?

a)

$\sum_{n=0}^∞a_nx^n$

b)

$\sum_{n=0}^∞n!(x-3)^n$

c)

$\sum_{n=0}^∞n+1(x+3)^n$

d)

none of the choices

86.

Integrate $\int_{ }^{ }\left(\frac{1-\sin x}{\cos^2x}\right)dx$

a)

$\tan x-\sec x+c$

b)

$\tan x+\sec x+c$

c)

$\cot x-\operatorname{cosec}x+c$

d)

$\cot x+\operatorname{cosec}x+c$

87.

Integrate $\int_{ }^{ }\left(\frac{\cos2x}{\cos^2x}\right)dx$

a)

$2x-\tan x+c_{ }$

b)

$2x+\tan x+c$

c)

$x-\tan x+c$

d)

$x+\tan x+c$

88.

Integrate $\int_{ }^{ }\left(\frac{\cos^2x}{\sin^2x}\right)dx$

a)

$-\cot x-x+c$

b)

$\cot x+x+c$

c)

$-\cot x+x+c$

d)

$\cot x-x+c$

89.

Integrate $\int_{ }^{ }\left(\cos x-\sin x\right)^2dx$

a)

$x+\frac{1}{2}\cos2x+c$

b)

$x-\frac{1}{2}\cos2x+c$

c)

$x+\cos2x+c$

d)

$x-\cos2x+c$

90.

Integrate $\int_{ }^{ }\frac{\left(1+\cos x\right)^2}{\sin^2x}dx$

a)

$-2\cot x-x-2\operatorname{cosec}x+c$

b)

$2\cot x-x+2\operatorname{cosec}x+c$

c)

$-2\cot x-x+2\operatorname{cosec}x+c$

d)

$2\cot x-x-2\operatorname{cosec}x+c$

91.

Integrate $\int_{ }^{ }\left(\cot x-\tan x\right)^2dx$

a)

$-\cot x-4x+\tan x+c$

b)

$-\cot x-4x-\tan x+c$

c)

$\cot x-4x-\tan x+c$

d)

$\cot x-4x+\tan x+c$

92.

Integrate $\int_{ }^{ }\left(\cos x-\sec x\right)^2dx$

a)

$-\frac{3}{2}x+\frac{1}{4}\sin2x+\tan x+c$

b)

$-\frac{3}{2}x+\frac{1}{2}\sin2x+\tan x+c$

c)

$-\frac{3}{2}x-\frac{1}{4}\sin2x+\tan x+c$

d)

$-\frac{3}{2}x-\frac{1}{2}\sin2x+\tan x+c$

93.

Integrate $\int_{ }^{ }\left(\frac{1+\cos x}{\sin^2x}\right)dx$

a)

$-\cot x-\operatorname{cosec}x+c$

b)

$-\cot x+\operatorname{cosec}x+c$

c)

$\cot x+\operatorname{cosec}x+c$

d)

$\cot x-\operatorname{cosec}x+c$

94.

Integrate $\int_{ }^{ }\left(\frac{\cos2x}{1-\cos^22x}\right)dx$

a)

$-\frac{1}{2}\operatorname{cosec}2x+c$

b)

$\frac{1}{2}\operatorname{cosec}2x+c$

c)

$-\frac{1}{2}\sec2x+c$

d)

$\frac{1}{2}\sec2x+c$

95.

Evaluate the indefinite integral using integration by parts.
$\int3x\ e^{2x}\ dx\$

a)

$-\frac{xe^{2x}}{2}+\frac{e^{2x}}{4}+C$

b)

$\frac{3xe^{2x}}{2}-\frac{3e^{2x}}{4}+C$

c)

$xe^{-2x}+\frac{\left(1-x^2\right)^{ }}{2}+C$

d)

$-\frac{xe^{2x}}{2}+\frac{\ln e^{2x}}{4}+C$

96.

Given $\int_1^4f\left(x\right)dx=5$ , what is $\int_1^4\left(2f\left(x\right)-1\right)dx$ equal to?

a)

4

b)

10

c)

7

d)

0

97.

What is the value of the definite integral $\int_1^3\left(3x^2+\frac{2a}{x^2}\right)dx$ , where $a$ is a constant?

a)

$26-\frac{8a}{3}$

b)

$26-\frac{56a}{27}$

c)

$28-\frac{8a}{3}$

d)

$26+\frac{4a}{3}$

98.

Which one of the following statements is $not$ true?

a)

$\int_{-2}^{\ 2}x^3dx=0$

b)

$\int_{-2}^{\ 2}x^2dx=2\int_0^2x^2dx$

c)

$\int_0^32xdx=\int_2^32xdx+\int_0^22xdx$

d)

$\int_{-1}^32dx=4$

99.

What is the indefinite integral $\int_{ }^{ }\left(3x-3\right)^5\ dx$ equal to?

a)

$\left(3x-2\right)^6+C$

b)

$\frac{\left(3x-2\right)^6}{18}+C$

c)

$\frac{\left(3x-2\right)^6}{18}+C$

d)

$15\left(3x-2\right)^4+C$

100.

Given $\int_{-1}^2\left(kx+2\right)dx=4,$ what is the value of $k$ ?

a)

$-\frac{4}{3}$

b)

$-\frac{4}{5}$

c)

$\frac{8}{3}$

d)

$-\frac{2}{3}$

101.

The graph of $y=f\left(x\right)$ passes through the point $\left(2,\ -3\right),$ and $f'\left(x\right)=3x^2-2x+1.$ What is the equation of $f\left(x\right)$ ?

a)

$f\left(x\right)=6x-2$

b)

$f\left(x\right)=x^3-x^2+x-9$

c)

$f\left(x\right)=x^3-x^2+x+3$

d)

$f\left(x\right)=x^3-x^2-8$

102.

What is the area of the region enclosed by the curve $y=x^3$ and the $x$ -axis, between $x=-2$ and $x=3?$

a)

$16\frac{1}{4}$ square units

b)

65 square units

c)

$24\frac{1}{4}$ square units

d)

16 square units

103.

The line $y=x+6$ intersects the parabola $y=x^2$ at the points $\left(-2,\ 4\right)$ and $\left(3,\ 9\right).$ What is the area of the region contained between the line and the parabola?

a)

$18\frac{5}{6}$ square units

b)

$8\frac{1}{6}$ square units

c)

$20\frac{5}{6}$ square units

d)

$26\frac{5}{6}$ square units

104.

The graph of a function $y=f\left(x\right)$ is shown in the diagram. The graph consists of a horizontal interval $AB$ , a quarter-circle $BC$ , and an interval $CD.$

What is the value of the definite integral $\int_0^6f\left(x\right)dx?$

a)

$2+\pi$

b)

$6+\pi$

c)

$\pi-2$

d)

$\pi$

105.

By first differentiating $\cos x^3,$ what is the primitive of $6x^2\sin x^3?$

a)

$\sin x^3$

b)

$2\sin x^3$

c)

$-\cos x^3$

d)

$-2\cos x^3$

Mock Test_5-31_Multiple Integration

Total questions: 105

Worksheet time: 2hrs 10mins

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