Free Printable Math Major 3114: Calculus 3

1.

a)

Divergent

b)

Convergent

2.

*hint - you will need to use an adaption of the definition of e when you take your limit.

a)

Convergent

b)

Divergent

3.

$\sum_{n=1}^{\infty}\frac{n}{\sqrt{n^5}}$

a)

Converges

b)

Diverges

4.

$\sum_{n=1}^{\infty}\frac{n+10}{10n+1}$

a)

Converges

b)

Diverges

5.

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

a)

Converges

b)

Diverges

6.

Let f be a positive, continuous, decreasing function such that

a_n=f\left(n\right)

. If

\sum_{n=1}^{\infty}a_n

converges to k, which of the following must be true?

a)

$\lim_{n\rightarrow\infty\ }a_n=k$

b)

$\int_1^nf\left(x\right)dx=k$

c)

$\int_1^{\infty}f\left(x\right)dx$ diverges

d)

$\int_1^{\infty}f\left(x\right)dx\$ converges

e)

$\int_1^{\infty}f\left(x\right)dx=k$

7.

Which of the following series diverge? (check all that apply)

a)

$\sum_{n=0}^{\infty}\left(\frac{\sin2}{\pi}\right)^n$

b)

$\sum_{n=1}^{\infty}\frac{1}{\sqrt{n}}$

c)

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

8.

What is the sum of the converging geometric series below?

$\sum_{n=1}^{\infty}\frac{7^{n+1}}{10^n}$

(a)

9.

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.

10.

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

11.

Geometric Series:

A geometric series will converge if __________.

a)

$\left|r\right|>1$

b)

$\left|r\right|<1$

12.

p-series:

A p-series will converge if __________ .

a)

$p<1$

b)

$p>1$

13.

Will the series converge or diverge?

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

a)

converge

b)

diverge

14.

Will the series converge or diverge?

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

a)

converge

b)

diverge

15.

Will the series converge or diverge?

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

a)

converge

b)

diverge

16.

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

17.

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

18.

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

19.

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)

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

20.

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

21.

Which test could be used to show that the series $\sum_{n=2}^{\infty}\frac{1}{n\ln n}$ diverges?

a)

Geometric Series Test

b)

p-Series Test

c)

Integral Test

d)

nth-Term Test for Divergence

22.

$\sum_{n=1}^{\infty}\frac{n}{\sqrt{n^5}}$

a)

Converges

b)

Diverges

23.

$\sum_{n=1}^{\infty}\frac{n+10}{10n+1}$

a)

Converges

b)

Diverges

24.

$\sum_{n=1}^{\infty}\frac{6}{4^{3n}}$

a)

Converges

b)

Diverges

25.

$\sum_{n=1}^{\infty}ne^{-n}$

a)

Converges

b)

Diverges

26.

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

a)

Converges

b)

Diverges

27.

$\sum_{n=1}^{\infty}\frac{1}{n^{1.06}}$

a)

Converges

b)

Diverges

28.

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

The series above is a

a)

geometric series

b)

p-series

29.

Let f be a positive, continuous, decreasing function such that

a_n=f\left(n\right)

. If

\sum_{n=1}^{\infty}a_n

converges to k, which of the following must be true?

a)

$\lim_{n\rightarrow\infty\ }a_n=k$

b)

$\int_1^nf\left(x\right)dx=k$

c)

$\int_1^{\infty}f\left(x\right)dx$ diverges

d)

$\int_1^{\infty}f\left(x\right)dx\$ converges

e)

$\int_1^{\infty}f\left(x\right)dx=k$

30.

Which of the following series diverge? (check all that apply)

a)

$\sum_{n=0}^{\infty}\left(\frac{\sin2}{\pi}\right)^n$

b)

$\sum_{n=1}^{\infty}\frac{1}{\sqrt{n}}$

c)

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

31.

What is the sum of the converging geometric series below?

$\sum_{n=1}^{\infty}\frac{7^{n+1}}{10^n}$

(a)

32.

Which of the following series should you use the ratio test to determine convergence or divergence?

a)

$\sum_{n=1}^{\infty}\frac{n^2+n}{3n^2+5}$

b)

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

c)

$\sum_{n=1}^{\infty}\frac{\ln\ n}{n}$

d)

$\sum_{n=1}^{\infty}\frac{1}{2^n}$

33.

Which series diverges by the nth term test?

a)

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

b)

$\sum_{n=1}^{\infty}\frac{3^n}{5^n}$

c)

$\sum_{n=1}^{\infty}\frac{5n^2}{3n^3+n^2}$

d)

$\sum_{n=1}^{\infty}\frac{1}{\tan^{-1}n}$

34.

Which series can the integral test be used to determine convergence or divergence?

a)

$\sum_{n=1}^{\infty}\frac{n}{e^{n^2}+n}$

b)

$\sum_{n=1}^{\infty}\frac{\ln n}{n}$

c)

$\sum_{n=1}^{\infty}\frac{\sin^{-1}n}{\sqrt{1-n^2}}$

d)

all of the above

35.

Find the second derivative of
f(x) = x²+ e^x - cosx

a)

f"(x) = 2 + e^x + cosx

b)

f"(x) = 2x + e^x + cosx

c)

f"(x) = 2x + xe^x - cosx

d)

f"(x) = 2x + e^x + sinx

36.

What does this calculate?

a)

Average Rate of Change of f over [a,b]

b)

Average Value of f

c)

Intermediate Value Theorem

d)

L'Hopital's Rule for limits

37.

Find the slope of the line that passes through the points (2, 4) and (6, 12)

a)

1/2

b)

-1/2

c)

2

d)

-2

38.

f(x) = 2x² + 12x + 16
What is the average rate of change of f(x) on the interval [-3, -2].

a)

2

b)

1/2

c)

-2

d)

-1/2

39.

Average Rate of Change is also known as...

a)

y-intercept

b)

x-intercept

c)

linear

d)

slope

40.

What is Mrs. Gordon's rate from 0 to 2 seconds?

a)

2.5 feet per second

b)

.4 feet per second

c)

5 feet per second

d)

.2 feet per second

41.

Calculate the average rate of change from 20 to 40 minutes.

a)

-2 minutes per gallon

b)

-20 minutes per gallon

c)

-2 gallons per minute

d)

-20 gallons per minute

42.

The graph of the function, f(x) is shown above. What is $\lim_{x\rightarrow3}f\left(x\right)?$

a)

1

b)

2

c)

3

d)

dne

43.

The graph of the function, f(x) is shown above. What is $\lim_{x\rightarrow2+}f\left(x\right)?$

a)

1

b)

2

c)

3

d)

dne

44.

The graph of the function, f(x) is shown above. What is $\lim_{x\rightarrow0+}f\left(x\right)?$

a)

1

b)

2

c)

3

d)

dne

45.

The graph of the function, f(x) is shown above. What is $\lim_{x\rightarrow2}f\left(x\right)?$

a)

1

b)

2

c)

3

d)

dne

46.

Is the sequence {a_n} convergent or divergent?

a)

neither

b)

divergent

c)

convergent

d)

impossible to know

47.

Given the sequence:

25, 21, 17, 13,...

What is the explicit formula that models the sequence?

a)

a_n+1 = a_n - 4

b)

a_n = -4n + 25

c)

a_n= -4n + 29

d)

a_n+1 = a_n + 25

48.

Given $f\left(x\right)=\left(1\ +3x\right)^4$ . What is the value of $f^{\left(2\right)}\left(0\right)$ ?

a)

1

b)

12

c)

108

d)

648

49.

Given $f\left(x\right)=e^{2x}$ . What is the 3rd terms in Maclaurin's Series?

a)

b)

c)

d)

50.

If y = $ax^n$ , then $\int_{ }^{ }y\ dx\ =$

a)

$\frac{ax^{n+1}}{n+1}$

b)

$\frac{ax^{n-1}}{n-1}+\ c$

c)

$\frac{ax^{n+1}}{n}+\ c$

d)

$\frac{ax^{n+1}}{n+1}+\ c$

51.

What does this calculate?

a)

Average Rate of Change of f over [a,b]

b)

Average Value of f

c)

Intermediate Value Theorem

d)

L'Hopital's Rule for limits

52.

Find the slope of the line that passes through the points (2, 4) and (6, 12)

a)

1/2

b)

-1/2

c)

2

d)

-2

53.

f(x) = 2x² + 12x + 16
What is the average rate of change of f(x) on the interval [-3, -2].

a)

2

b)

1/2

c)

-2

d)

-1/2

54.

Average Rate of Change is also known as...

a)

y-intercept

b)

x-intercept

c)

linear

d)

slope

55.

What is Mrs. Gordon's rate from 0 to 2 seconds?

a)

2.5 feet per second

b)

.4 feet per second

c)

5 feet per second

d)

.2 feet per second

56.

Calculate the average rate of change from 20 to 40 minutes.

a)

-2 minutes per gallon

b)

-20 minutes per gallon

c)

-2 gallons per minute

d)

-20 gallons per minute

57.

The graph of the function, f(x) is shown above. What is $\lim_{x\rightarrow3}f\left(x\right)?$

a)

1

b)

2

c)

3

d)

dne

58.

The graph of the function, f(x) is shown above. What is $\lim_{x\rightarrow2+}f\left(x\right)?$

a)

1

b)

2

c)

3

d)

dne

59.

The graph of the function, f(x) is shown above. What is $\lim_{x\rightarrow0+}f\left(x\right)?$

a)

1

b)

2

c)

3

d)

dne

60.

The graph of the function, f(x) is shown above. What is $\lim_{x\rightarrow2}f\left(x\right)?$

a)

1

b)

2

c)

3

d)

dne

Math Major 3114: Calculus 3

Total questions: 60

Worksheet time: 1hrs 12mins

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