Free Printable Calculus AP BC fun quiz

1.

FInd the derivative:

f(x) = (-2/3)x³ - (1/2)x² + 9x

a)

-2x² - x + 9

b)

-2x² + x + 9

c)

2x² - x + 9

d)

2x² - x - 9

2.

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

3.

Set up the derivative of y=(3x⁴- 7)*(5x²+ 1)

a)

(12x³)*(10x)

b)

(3x⁴ - 7)*(10x) + (12x³)*(5x² + 1)

c)

(3x⁴ - 7)*(10x) - (12x³)*(5x² + 1)

d)

(3x⁴ - 7)/(10x) + (12x³)/(5x² + 1)

4.

Find the derivative of y = 2x² (3x - 4)

a)

6x³ - 8x²

b)

18x² - 16x

c)

15x²- 6x

d)

6x - 4

5.

Differentiate f(x) =(2/ x⁵) - 5.

a)

x⁵-3

b)

-10x⁶

c)

x⁵-5

d)

-10x^-6

6.

Find the derivative of the given equation
f(x) = x⁴ + 4x³- 2x²

a)

x³ + x²- x

b)

4x³ + 12x²+ 4x

c)

4x + 12x - 4x

d)

4x³ + 12x²- 4x

7.

Find the derivative.

a)

x⁴cosx - 4x³sinx

b)

x⁴cosx + 4x³sinx

c)

4x³cosx

d)

-4x³cosx

8.

Find the derivative f(x) = xe^x

a)

f'(x) = e^x

b)

f'(x) = xe^x + xe^x

c)

f'(x) = e^x - xe^x

d)

f'(x) = e^x + xe^x

9.

Which of the following is true?

a)

the derivative is a way to show rate of change, that is - the amount by which a function is changing at one given point

b)

dx/dy is the derivative

10.

a)

cosx

b)

-cosx

c)

-sinx cosx

d)

sinx cosx

11.

a)

sinx

b)

-sinx

c)

-cscx cotx

d)

sec²x

12.

Find f(2).

a)

1

b)

-1

c)

5

d)

DNE

13.

Find the limit of the function as x approaches 2.

a)

1

b)

-1

c)

5

d)

DNE

14.

If a function has a derivative that is negative, what does that tell you?

a)

The function is increasing

b)

The function is decreasing

c)

The concavity of the function is up

d)

The concavity of the function is down

15.

a)

b)

c)

d)

16.

a)

b)

c)

d)

17.

a)

b)

c)

d)

18.

What is the derivative of cot(x)?

a)

sec²(x)

b)

-sec²(x)

c)

csc²(x)

d)

-csc²(x)

19.

Find the limits

\lim_{x\rightarrow2}2x

a)

4

b)

-4

c)

0

20.

find the limits

\lim_{x\rightarrow2}\left(x^5-2x^3+x^2-8\right)

a)

11

b)

15

c)

12

21.

What is the limit?

a)

DNE

b)

Infinity

c)

6

d)

12

22.

It is the foundation of knowledge in Calculus?

a)

Basic Operations

b)

Integers

c)

Function

d)

Limit

23.

What is the answer in the given expression?

\lim_{x\rightarrow50}25

a)

50

b)

25

c)

2

d)

125

24.

As x gets close to 1, then $\frac{x^2-1}{x-1}$ gets close to

a)

1.5

b)

1.9

c)

1.99

d)

2

25.

Evaluate the limit of the given expression.

a)

1

b)

2

c)

3

d)

4

26.

Evaluate the limit.

a)

5

b)

6

c)

7

d)

8

27.

Evaluate the limit using the given graph

a)

-2

b)

4

c)

1/2

d)

6

28.

Evaluate the limit.

a)

1

b)

-1

c)

2

d)

4

29.

It is the distance from the center to any point on the circle.

a)

Radius

b)

Diameter

c)

Secant

d)

Tangent

30.

Find the limit as x approaches 3 from the left

a)

4

b)

3

c)

2

d)

DNE

31.

What is the limit of the function as x approaches -4 from the left?

a)

2

b)

-4

c)

DNE

d)

-2

32.

a)

A

b)

B

c)

C

d)

D

33.

a)

A

b)

B

c)

C

d)

D

34.

a)

A

b)

B

c)

C

d)

D

35.

The diagram shows the curve y = (x − 2)² and the line y + 2x = 7, which intersect at points A and B. Find the area of the shaded region.

a)

$10\ \frac{5}{7}$

b)

$10\ \frac{3}{4\ }$

c)

$10\ \frac{1}{3\ }$

d)

$10\ \frac{2}{3}$

36.

$\int_{ }^{ }\left(x^{3\ }+\frac{1}{x^3}\right)\ dx$

a)

$\frac{x^{4\ }}{4}-\frac{x^{-2}}{2\ }+c$

b)

$\frac{x^{4\ }}{4}+\frac{x^{-2}}{2}+c$

c)

$\frac{x^4}{4}-\frac{x^2}{2\ }+c$

d)

$\frac{-x^4}{4}+\frac{x^{2\ }}{2\ }+c$

37.

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

a)

Position = 0

b)

Position = 1

c)

Position = 2

d)

Position = -1

38.

If the position function for a particle is s(t) = -t² - t, what is the instantaneous velocity function for the particle?

a)

v(t) = -2

b)

v(t) -2t - 1

c)

v(t) = t³

d)

v(t) = -t

39.

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

a)

Velocity = 0

b)

Velocity = 1

c)

Velocity = -1

d)

Velocity = -2

40.

The acceleration function is the first derivative of...

a)

position

b)

velocity

c)

calculus

d)

particle motion

41.

A bug begins to crawl up a vertical wire at time t = 0. The velocity v of the bug at time t, 0 < t < 8, is given by the function whose graph is shown behind this text. At what value of t does the bug change direction?

a)

2

b)

4

c)

6.5

d)

7

42.

The position function x(t)=7t²-18t-7 is given. What is the velocity function?

a)

v(t)=14t-18

b)

v(t)=14t+18

c)

v(t)=18t-14

d)

v(t)=18t+14

43.

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

a)

38 m/s/s

b)

27 m/s/s

c)

16 m/s/s

d)

49 m/s/s

44.

Given x(t)=t³-4t²+7, what is the initial position?

a)

11

b)

-10

c)

7

d)

0

45.

A particle's speed is decreasing if

a)

it's acceleration is positive

b)

it's velocity is positive

c)

it's velocity and acceleration have the same sign

d)

it's velocity and acceleration have different signs

46.

A particle has positive velocity if it's position graph is

a)

negative

b)

positive

c)

decreasing

d)

increasing

47.

A particle has negative velocity if it's position graph is

a)

negative

b)

positive

c)

decreasing

d)

increasing

48.

v(t)>0 means

a)

the particle is moving to the right

b)

the particle is moving to the left

c)

the particle has positive position

d)

the particle is at rest

49.

Which of the following can be used to determine when a particle is at rest?

a)

x(t)=0

b)

v(t)=0

c)

a(t)=0

50.

s(t) = t² - 20
Find the average velocity from t = 3 to t = 5.

a)

2

b)

4

c)

6

d)

8

51.

An objects distance from its starting point at time t is given by the equation
s(t) = t³ - 6t² - 4.
What is the speed of the object when its acceleration is 0?

a)

2

b)

-24

c)

22

d)

44

52.

What is the total distance of the object represented in this motion graph?

a)

0 meters

b)

300 meters

c)

450 meters

d)

600 meters

53.

What is the total displacement of the object represented in this motion graph?

a)

0 meters

b)

300 meters

c)

450 meters

d)

600 meters

54.

What is the total distance of the object represented in this motion graph?

a)

0 meters

b)

100 meters

c)

140 meters

d)

200 meters

55.

What is the total displacement of the object represented in this motion graph?

a)

0 meters

b)

100 meters

c)

140 meters

d)

200 meters

56.

When we "take the derivative" of a function what are we finding?

a)

What's a derivative?

b)

The rate at which our struggles in Calculus are increasing.

c)

The slope of the secant line

d)

The slope of the tangent line

57.

Find the sum of the first 4 terms of the series. If $a_1=1$ and $r\ =\ 2.8$

(a)

58.

what is the differentiation of log y

a)

1/x

b)

2/y

c)

1/y

d)

2/x

59.

differentiation is denoted by

a)

dy/dx

b)

d/dx

c)

d²/dx²

d)

d²y/dx²

60.

what are the major concepts in calculus

a)

integral,derivatives,limits

b)

limits,integral

c)

integral,derivatives

61.

opposite for integral

a)

derivative

b)

summation

c)

multiplication

d)

subtraction

62.

_{Evaluate the integral}

a)

30/3

b)

31/3

c)

29/3

d)

32/3

63.

a)

12/3

b)

6.5

c)

5.5

d)

11/3

64.

A particle moves in the xy-plane so that its position at any time is given by $x\left(t\right)=4\sin\left(\pi t\right)$ and $y\left(t\right)=\left(3t-1\right)^2$ . What is the speed of the particle at $t=2$ ?

a)

13.708

b)

30.209

c)

32.526

d)

42.566

65.

Determine whether the series is absolutely convergent, conditionally convergent, or divergent.

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

a)

Absolutely Convergent

b)

Conditionally Convergent

c)

Divergent

66.

$\int\left(\frac{4}{5}x^3-\ 4x^2+\frac{4}{x^2}-7\right)dx\ =\ ...$

a)

$\frac{1}{5}x^4-\frac{4}{3}x^3-4x^{-1}-7x\ +\ c$

b)

$\frac{1}{5}x^4-\frac{4}{3}x^3+4x^{-1}-7x+c$

c)

$\frac{1}{5}x^4-\frac{4}{3}x^3-4x^{-1}-7+c$

d)

$\frac{1}{5}x^4-8x^3-\frac{4}{3}x^2-7x+c$

e)

$\frac{12}{5}x^4-8x^3-4x^{-1}-7x+c$

67.

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

68.

$1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+...$

Does this series Converge or Diverge?

a)

Converge

b)

Diverge

69.

Find the derivative for $f\left(x\right)=e^{\sin x}$

a)

$e^{\cos x}$

b)

$\cos x\ \left(e^{\sin x}\right)$

c)

$\frac{1}{\cos\ x}e^{\sin x}$

d)

$-e^{\cos x}$

70.

Find $F'\left(x\right)$ . $F\left(x\right)=\int_0^{2x}\sin\left(t^2\right)dt$

a)

$2\sin\left(4x^2\right)$

b)

$2\cdot\sin\left(2x\right)$

c)

$\frac{\cos\left(2x^2\right)}{2}$

d)

$x^{2\ }\cdot\sin\left(x^{12}\right)$

71.

List all of the x-locations of the minimum points of the function whose derivative is graphed.

a)

{-2, 1}

b)

{-1, 1}

c)

{-1}

d)

Cannot be determined}

72.

$if\ f\left(0\right)=0,\ f'\left(0\right)=1,\ f''\left(0\right)=0,\ \&\ f'''\left(0\right)=2,$ Find the 3rd order Taylor Polynomial generated by f(x) at x=0

a)

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

b)

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

c)

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

d)

$2x^3+x$

73.

What method would you use here?

a)

antiderivatives

b)

u-substitution

c)

integration by parts

d)

this is elementary integral

74.

What method would you use here?

a)

antiderivatives

b)

u-substitution

c)

integration by parts

d)

this is elementary integral

75.

What method would you use here?

a)

antiderivatives

b)

u-substitution

c)

integration by parts

d)

this is elementary integral

76.

What method would you use here?

a)

antiderivatives

b)

u-substitution

c)

integration by parts

d)

this is elementary integral

77.

What would you choose for your u here if you used integration by parts?

a)

x

b)

sinx

c)

cosx

d)

e^x

78.

What would you choose for your u here if you used integration by parts?

a)

x

b)

x⁴

c)

e^-x

d)

e^x

79.

How many times would you need integration by parts here

a)

0 (trick question)

b)

1

c)

2

d)

∞

80.

How many times would you need integration by parts here

a)

0 (trick question)

b)

1

c)

2

d)

4

81.

Solve via integration by parts

a)

3t(e^2t)/2 - (e^2t)/4 + C

b)

3t(e^2t)⋅2 - 3(e^2t)⋅4 + C

c)

3(e^2t) - (e^2t)/4 + C

d)

3t(e^2t)/2 - 3(e^2t)/4 + C

82.

$\int_{-8}^5f\left(x\right)dx=$

(a)

83.

$\int_0^{-4}f\left(x\right)dx=$

(a)

84.

What is an appropriate calculus-based justification for the fact that g has an inflection point at x = c. Choose 1.

a)

f has an inflection point at x = c

b)

f has a relative minimum at x = c

c)

f is positive

85.

The graph of the function g is shown. Let

f\left(x\right)=\int_{-3}^xg\left(t\right)dt.\

What is f(1)?

(a)

86.

$F'\left(x\right)=?$

(a)

87.

The graph of h(x) is shown. What is

\int_{-17}^{-9}h\left(x\right)dx?

(a)

88.

Find

\int_{-3}^5f\left(x\right)dx

Give an exact answer as a multiple of pi.

(a)

89.

The graph of f is shown. Let

g\left(x\right)=\int_0^xf\left(t\right)dt

What is an appropriate calculus-based justification for the fact that g is increasing?

a)

f is positive

b)

f is increasing

c)

f is concave up

90.

a)

B

b)

C

c)

D

d)

E

91.

a)

B

b)

C

c)

D

d)

E

92.

a)

A

b)

B

c)

C

d)

E

93.

List all of the critical values (x-locations of the critical points) of the graphed function.

a)

{-2. 0}

b)

{-2, 1}

c)

{-4, -2, 0}

d)

Cannot be determined

94.

List all of the x-values of the points of inflection of the graphed function.

a)

{-3, 1}

b)

{-4, -3, -1}

c)

{-4, -2, 0}

d)

Cannot be determined.

95.

List all of the critical values (x-locations of the critical points) of the function whose derivative is graphed.

a)

{-3, -1, 1}

b)

{-3, -1}

c)

{-4, -2, 0}

d)

Cannot be determined

96.

List all of the x-locations of the points of inflection of the function whose derivative is graphed.

a)

{-4, -2, 0}

b)

{-2, 0}

c)

{-2, 0, 1}

d)

Cannot be determined

97.

List all of the x-locations of the minimum points of the function whose derivative is graphed.

a)

{-2, 1}

b)

{-1, 1}

c)

{-1}

d)

Cannot be determined}

98.

List all of the x-values of the critical points of the function whose second derivative is graphed.

a)

{-4, -2}

b)

{-3, -1}

c)

{-3, -1, 1}

d)

Cannot be determined

99.

List all of the x-values of the extremes of the function whose second derivative is graphed.

a)

{-4, -2}

b)

{-3, -1}

c)

{-3, -1, 1}

d)

Cannot be determined

100.

List all of the x-values of the points of inflection of the function whose second derivative is graphed.

a)

{-3, 1}

b)

{-3, -1, 1}

c)

{-4, 2}

d)

Cannot be determined

101.

An absolute maximum must occur at a critical point or at an endpoint.

a)

True

b)

False

102.

A critical number, c, or a function f is a number in the domain of f such that:

a)

f'(c) = 0

b)

f'(c) is undefined

c)

f'(c) = o or f'(c) is undefined

d)

None of the above

103.

If c is a critical number then f(c) is either a local maximum or a local minimum

a)

True

b)

False

104.

Find the critical points of f(x) = 2x⁴- 4x² + 1

a)

x= 0

b)

x = -1, 1

c)

x = -1, 0, 1

d)

no critical points

105.

Find the point(s) of inflection for f(x) = 2(x)^1/5 + 3

a)

None

b)

x = 0

c)

x = 0, 2

d)

x = -2, 0, 2

106.

If given f'(x) how would you find if f(x) is concave down?

a)

f'(x) = 0

b)

f'(x) (increasing) or f"(x) > 0

c)

f'(x) <(decreasing) of f"(x) < 0

d)

Can't be determined

107.

If given f'(x) how would you find the intervals which the graph f(x) has a positive slope?

a)

f'(x) < 0 (negative)

b)

f'(x) > 0 (positive)

c)

f'(x) is increasing

d)

Can't be determined

108.

The slope of a function is described by its ____________.

a)

first derivative

b)

second derivative

c)

third derivative

d)

expression

109.

The concavity of a function is described by its _______________.

a)

first derivative

b)

second derivative

c)

third derivative

d)

expression

110.

What is a point of inflection?

a)

When a function goes from increasing to decreasing.

b)

When the concavity changes.

c)

When the derivative changes.

d)

When the function crosses the x-axis.

111.

How do you find the critical numbers of a function?

a)

Critical numbers only occur when f'(x) = 0.

b)

Critical numbers only occur when f'(x) is undefined.

c)

Critical numbers occur when f'(x) = 0 or where f'(x) is undefined.

d)

Critical numbers occur as x-intercepts on the graph.

112.

If f '(3) = 0 and f"(3) < 0, then which of the following must be true?

a)

There is a local max at x=3

b)

There is a local min at x = 3

c)

There is an inflection point at x = 3

d)

There is an x-intercept at x = 3

113.

For a function f(x), f'(-3) = 5 indicates f(x) is ___________ at x=-3.

a)

increasing

b)

decreasing

c)

concave up

d)

concave down

114.

On what interval(s) is the function y=x³ + 6x²

concave down?

a)

(-∞,-4)

b)

(-∞,-2)

c)

(-2,∞)

d)

(0,∞)

115.

f' is given, which could be f?

a)

A

b)

B

c)

C

116.

If (a,b) is a local maximum, then what will be true about f''(a)?

a)

It's positive

b)

It's negative

c)

It's zero

d)

Cannot be determined

117.

Use the sign chart for f'(x). There is(are) ...

a)

a local maximum at x = -2.

b)

a local minimum at
x = -2 and a local maximum at x = 4.

c)

a local maximum at
x = -2 and a local minimum at x = 4.

d)

no extrema.

118.

The area under a curve is calculated using which mathematical concept?

a)

antiderivative

b)

indefinite integral

c)

definite integral

d)

derivative

119.

The result of $\int_1^3\left(x-2\right)\ dx$ is:

a)

positive

b)

negative

c)

zero

120.

The result of $\int_0^2\left(x^2-2x+1\right)\ dx$ is:

a)

positive

b)

negative

c)

zero

121.

The result of $\int_0^2\left(x^3-2x^2-x+1\right)\ dx$ is:

a)

positive

b)

negative

c)

zero

122.

The shaded area in the figure is represented by which of the following integral expressions?

a)

$\int_{-1}^2f\left(x\right)\ dx$

b)

$-\int_2^{-1}f\left(x\right)\ dx$

c)

$-\int_{-1}^2f\left(x\right)\ dx$

d)

$\int_{-2}^1f\left(x\right)\ dx$

123.

Which of the following is incorrect?

a)

$\int_b^af\left(x\right)\ dx=-\int_a^bf\left(x\right)\ dx$

b)

$\int_a^af\left(x\right)\ dx=0$

c)

$\int_a^bf\left(x\right)\ dx+\int_b^cf\left(x\right)\ dx$
$=\int_a^cf\left(x\right)\ dx$

d)

$\int_a^bf\left(x\right)\ dx-\int_b^cg\left(x\right)\ dx$
$=\int_a^c\left(f\left(x\right)-g\left(x\right)\right)\ dx$

e)

$\int_a^bf\left(x\right)\ dx-\int_a^bg\left(x\right)\ dx$
$=\int_a^b\left(f\left(x\right)-g\left(x\right)\right)\ dx$

124.

a)

2

b)

4

c)

1

d)

Not possible

125.

a)

-8

b)

8

c)

-2

d)

2

126.

If $\int_2^{10}f\left(x\right)dx=-6$
Find the value of $\int_{10}^2f\left(x\right)dx$

a)

- 6

b)

6

c)

2

d)

10

127.

If $\int_2^{10}f\left(x\right)dx=-6$
Find the value of $\int_2^{10}-2f\left(x\right)dx$

a)

- 12

b)

4

c)

- 8

d)

12

128.

If $\int_2^{10}f\left(x\right)dx=-6$
Find the value of $\int_2^{10}\ \frac{1}{3}f\left(x\right)dx$

a)

2

b)

18

c)

- 2

d)

- 18

129.

If $\int_0^2f\left(x\right)\ dx=5$ , $\int_2^4f\left(x\right)dx=3$ , and $\int_2^6\ f\left(x\right)dx\ =12$ , then $\int_0^6\ f\left(x\right)\ dx=$

a)

5

b)

-5

c)

17

d)

-17

130.

If $\int_1^5f\left(x\right)dx=8$ , then $\int_2^2f\left(x\right)dx=$

a)

8

b)

-8

c)

0

d)

4

131.

$\int_{-a}^af\left(x\right)dx=$

a)

$0,\ if\ f\left(-x\right)=f\left(x\right)$

b)

$2\int_0^af\left(x\right),\ if\ f\left(-x\right)=f\left(x\right)$

c)

$2\int_0^af\left(x\right),\ if\ f\left(-x\right)=-f\left(x\right)$

d)

none of these

132.

$\int_{-3}^3\left|x+1\right|dx=$

a)

10

b)

0

c)

8

d)

none of these

133.

The graph of $y=3x^2-x^3$ has a relative max at

a)

(0, 0) only

b)

(1, 2) only

c)

(2, 4) only

d)

(4, -16) only

134.

What is the second derivative of f(x) = cos 4x?

a)

16 cos 4x

b)

-16 cos 4x

c)

16 sin 4x

d)

-16 sin 4x

135.

Which of the following is the value of x given $7e^{3x-5}+1=50$ ?

a)

$x=\frac{5+\ln7}{3}$

b)

$x=\ln4$

c)

$x=\frac{\frac{7}{e}+5}{3}$

d)

No Solution

136.

Integrate $\int_{ }\ \frac{x^2+3x-2}{x}\ \ dx$

a)

$\frac{x^2}{2}+3x-2\ln\left|x\right|+c$

b)

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

c)

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

d)

$2x+3$

137.

If f(x) = (x² – 3)⁴, then f'(1) =

a)

–64

b)

–32

c)

–16

d)

32

Calculus AP BC fun quiz

Total questions: 137

Worksheet time: 6hrs 59mins

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