Free Printable Math Major 3114: Calculus 3

1.

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

2.

What is the nature of the sequence?

a)

bounded

b)

bounded below

c)

bounded above

d)

neither bounded below nor above

3.

Find the second derivative of the function:
f (x) = 2x - 5x⁶

a)

f ''(x)= 2 - 30x

b)

f ''(x) = 2-30x⁵

c)

f ''(x) = -30x⁵

d)

f ''(x) = -150x⁴

4.

What is the recursive formula of the sequence below?

5, -3, -11, -19

a)

a₁=5, a_n=a_n-1+8

b)

a₁=5, a_n=a_n-1-8

c)

a₁=8, a_n=a_n-1+5

d)

a₁=-8, a_n=a_n-1-5

5.

For what values of r is {a_n} bounded?

a)

r>0

b)

r<0

c)

|r|>1

d)

|r|<1

e)

for no values of r

6.

Is the sequence convergent or divergent?

a)

Convergent

b)

Divergent

c)

Impossible to know

7.

The velocity of an object is given as v = 2t + 3t³. what is the acceleration of the object at t = 2 s?

a)

38 m/s/s

b)

27 m/s/s

c)

16 m/s/s

d)

49 m/s/s

8.

A sequence {a_n} is bounded and decreasing. What can you conclude about {a_n} ?

a)

{a_n} alternates between positive and negative values

b)

{a_n} is not monotonic

c)

{a_n} has a lower bound but no upper bound

d)

{a_n} converges to its lower bound

9.

If a sequence is convergent then its terms must have a finite limit as n aproaches iinfinity

a)

True

b)

False

10.

Wht is the 99^th term in the sequence {a_n} ?

a)

99

b)

1/99

c)

99/9802

d)

impossible to know

11.

Which of the following statementts is true?

a)

Every bounded sequence is monotonic

b)

Every bounded sequence is convergent

c)

Every bounded and monotonic sequence is convergent

d)

Every bounded and convergent sequence is monotonic

12.

Is the sequence {a_n} convergent or divergent?

a)

neither

b)

divergent

c)

convergent

d)

impossible to know

13.

By using Binomial Series, what is the formula for the 3rd terms of $\left(p\ +\ q\right)^8$ ?

a)

b)

c)

d)

14.

What is the coefficients of $\left(a\ +\ b\right)^3$ in Pascal Triangle?

a)

1

b)

1 1

c)

1 2 1

d)

1 3 3 1

15.

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

16.

What is the 3rd term in the sequence given by the rule below?

a)

6

b)

3

c)

16

d)

18

17.

Integrate $\int_{ }^{ }3\ dx$

a)

= 0

b)

$=\frac{3}{x}+c$

c)

$=\ 3x\ +c$

d)

$=4+c$

18.

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

19.

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

a)

b)

c)

d)

20.

What is an infinite sequence?

a)

a sequence that has a set end element

b)

a sequence that can have infinitely many terms

c)

a sequence that has values that add up to a value greater than infinity

d)

a sequence whose terms approach zero

21.

Find f(x) if f '(x) = 2x⁵

a)

10x⁴+ c

b)

5x⁶ + c

c)

⅓ x⁶ + c

d)

10x⁶+ c

22.

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)

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

23.

What is an antiderivative?

a)

The opposite of a derivative

b)

The same as a derivative

c)

A second derivative

d)

It always represents velocity.

24.

Find the enclosed area by the curve with equation $y=4-x^2$ ,the X-axis and the lines x = -1 and x = 2

a)

5

b)

7

c)

9

d)

11

25.

Using the areas of each region given
$\int_b^df\left(x\right)=$

a)

2

b)

22

c)

-2

d)

3

26.

Find the area between the graph of $y=\left(x-2\right)^5$ , the x-axis, and the vertical lines $x=2$ and $x=4$ .

a)

32/3

b)

4

c)

16/3

d)

16

27.

Given the limit at the left, what can we conclude about the sequence {a_n} ?

a)

{a_n} is monotonic

b)

{a_n} is decreasing

c)

{a_n} is convergent

d)

{a_n} is divergent

28.

Find $\int_2^9\int_5^8\ \left(\sin\ 2x\right)dxdy$

a)

$\approx0.792$

b)

$\approx0.518$

c)

$\approx-0.231$

d)

$\approx0.415$

29.

Given the function of the curve is $\mathrm{y=}x^2\mathrm{-4x\ +\ 3}$ , find the area of the shaded region.

a)

$-\frac{4}{3}$

b)

$\frac{2}{3}$

c)

$\frac{4}{3}$

d)

$-\frac{2}{3}$

30.

Find the area of the shaded region

a)

0

b)

$2\frac{2}{3}$

c)

$-\frac{4}{3}$

d)

$\frac{4}{3}$

31.

Find the Maclaurin polynomial with degree n = 2 of $f\left(x\right)=e^{3x}$ .

a)

$1+x+x^2$

b)

$1+3x+6x^2$

c)

$1+3x+\frac{9}{2}x^2$

d)

$1+3x+9x^2$

32.

Given the marginal cost function, $C'\left(x\right)=6x^2-3x+5$ and the fixed cost is 8. Find the cost function.

a)

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

$5x$

b)

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

$5x+8$

c)

$C\left(x\right)=12x-3$

d)

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

$5x+k$

33.

Another word for 'integral' is .. .

a)

Constant

b)

Derivative

c)

Antiderivative

d)

Theorem

34.

The area of the region bounded by the curve y = sinx between the ordinates x = 0, x = π/2 and the x-axis is

a)

2 sq.units

b)

4 sq . units

c)

3 sq. units

d)

1 sq. units

35.

The area of the region bounded by the curve y = sinx between the ordinates x = 0, x = π/2 and the x-axis is

a)

2 sq.units

b)

4 sq . units

c)

3 sq. units

d)

1 sq. units

36.

Which is the formula of Taylor series?

a)

$\sum_{n=0}^{\infty}\frac{f^{\left(n\right)}\left(0\right)x^n}{n!}$

b)

$\sum_{n=0}^{\infty}\frac{f^{\left(n\right)}\left(c\right)\left(x-c\right)^n}{n!}$

c)

$\sum_{n=1}^{\infty}\frac{f^{\left(n\right)}\left(0\right)x^n}{n!}$

d)

$\sum_{n=1}^{\infty}\frac{f^{\left(n\right)}\left(c\right)\left(x-c\right)^n}{n!}$

37.

Find the slope of f(x) = 2x²+3.

a)

4x

b)

4x²

c)

3

d)

2x²+3

38.

Taylor series has center at _________.

a)

x = 0

b)

x = 1

c)

x = e

d)

x = c

39.

What is the nature of the sequence?

a)

monotonically increasing

b)

monotonically decreasing

c)

not monotonic

d)

monotonic but neither increasing nor decreasing

40.

Which is the formula of Maclaurin series?

a)

$\sum_{n=0}^{\infty}\frac{f^{\left(n\right)}\left(0\right)x^n}{n!}$

b)

$\sum_{n=0}^{\infty}\frac{f^{\left(n\right)}\left(c\right)\left(x-c\right)^n}{n!}$

c)

$\sum_{n=1}^{\infty}\frac{f^{\left(n\right)}\left(0\right)x^n}{n!}$

d)

$\sum_{n=1}^{\infty}\frac{f^{\left(n\right)}\left(c\right)\left(x-c\right)^n}{n!}$

41.

Calculate the area for the shaded region on the function $y=x^3-3x^2+2x$

a)

1.4 units²

b)

0.5 units²

c)

0.56 units²

d)

2.67 units²

e)

0.33 units²

42.

Derivative means the same thing as

a)

slope of the tangent line

b)

slope of the normal line

c)

exponent

d)

potato

43.

Which of the following is the indefinite integral of $\frac{x^3}{2}+7$ ?

a)

$\frac{3x^2}{2}$

b)

$\frac{3x^2}{2}+7x+c$

c)

$\frac{x^4}{8}+7x+c$

d)

$\frac{x^4}{8}+c$

44.

Find the cubic approximate for f(x) = sin x at x = 3π/2 = a

a)

a.) Sin x = -1 - ½ (x - 3π/2)²

b)

b.) Sin x = -1 + ½ (x - 3π/2)²

c)

c.) Sin x = 1 - ½ (x + 3π/2)²

d)

d.) Sin x = -1 - ½ (x + 3π/2)²

45.

Estimating In (0.75) using a Taylor polynomial about x-1, what is the least degree of the polynomial that assures an error smaller than 0.001?

a)

a. 3

b)

b. 4

c)

c. 5

d)

d. 6

46.

Maclaurin series has center at __________.

a)

x = 0

b)

x = 1

c)

x = e

d)

x = c

47.

Use the second-degree polynomial to approximate the function 𝑓(𝑥)=2 + x + x² at the point a=1.

a)

a.) 4 + 3/2 (x-1) + (x-1)²

b)

b.) 4 + 3/2 (1-x) + 2/3 (1-x)²

c)

c.) 4 + 3 (x-1) + (x-1)²

d)

d.) 4 + 3 (x-1) + 2(x-1)²

48.

The area of the region bounded by the ellipse

\frac{x^2}{25}+\frac{y^2}{16}=1\ is

a)

20π sq units

b)

5π2 sq units

c)

16π sq units

d)

25π sq units

49.

If the position of a particle is represented by x(t) = -t² + 1, what is its instantaneous velocity at t = 1?

a)

v = 0

b)

v = 1

c)

v = -1

d)

v = -2

50.

Which is the basic Taylor series for $e^x$ ?

a)

$\sum_{n=0}^{\infty}x^n$

b)

$\sum_{n=0}^{\infty}\frac{\left(-1\right)^nx^{2n}}{\left(2n\right)!}$

c)

$\sum_{n=0}^{\infty}\frac{\left(-1\right)^nx^{2n+1}}{\left(2n+1\right)!}$

d)

$\sum_{n=0}^{\infty}\frac{x^n}{n!}$

Math Major 3114: Calculus 3

Total questions: 50

Worksheet time: 1hrs 20mins

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