Free Printable AP Calculus Exam Review

1.

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

2.

a)

Intermediate Value Theorem

b)

Rolle's Theorem

c)

Average Rate of Change

d)

Average Value of f

3.

a)

Average Rate of Change

b)

Average Value of f

c)

Intermediate Value Theorem

d)

Rolle's Theorem

4.

a)

f(a) - f(b)

b)

f(b) - f(a)

c)

f '(b) - f '(a)

d)

f '(a) - f '(b)

5.

a)

Mean value theorem

b)

Rolle's theorem

c)

Intermediate Value Theorem

d)

Fundamental Theorem of Calculus

6.

a)

speed

b)

acceleration

c)

displacement

d)

total distance

7.

a)

Volume using Shells

b)

Volume using Disks

c)

Volume using Washers

d)

Volume using Cross Sections

8.

a)

A

b)

B

c)

C

d)

E

9.

a)

2/3

b)

1

c)

4/3

d)

2

10.

a)

position

b)

acceleration

c)

total distance

d)

speed

11.

a)

position

b)

velocity

c)

acceleration

d)

total distance

12.

a)

position

b)

velocity

c)

acceleration

d)

total distance

13.

a)

A

b)

B

c)

C

d)

D

14.

If g is the inverse function of F and f(2)=3, find the value of g'(3) for F(x)=6x²+2x-1

a)

1/24

b)

1/25

c)

1/26

d)

26

15.

a)

A

b)

B

c)

C

d)

D

e)

E

16.

a)

A

b)

B

c)

C

d)

D

e)

E

17.

a)

A

b)

B

c)

C

d)

D

e)

E

18.

a)

A

b)

B

c)

C

d)

D

e)

E

19.

Find the limit of the function as x approaches 2.

a)

1

b)

-1

c)

5

d)

DNE

20.

INTEGRATE

a)

A

b)

B

c)

C

d)

D

21.

Find g(2)

a)

-1/2

b)

-1

c)

1

d)

1/2

22.

Find g'(2)

a)

-1/2

b)

-1

c)

1

d)

DNE

23.

Find g(-3)

a)

-2π

b)

-π

c)

π

d)

2π

24.

Where is g decreasing?

a)

[-1,2]

b)

[1,3]

c)

[-3,1] ∪ [3,4]

d)

[1,4]

25.

Where is g an absolute maximum? (Dont forget how to justify...)

a)

1

b)

-3

c)

2

d)

4

26.

INTEGRATE

a)

A

b)

B

c)

C

d)

D

27.

Use LRAM and 4 subintervals to estimate Area under the curve

a)

A

b)

B

c)

C

d)

D

28.

Find F'

a)

A

b)

B

c)

C

d)

D

29.

In drawing the slope field for the differential equation , I would place short slope lines of _________ at the points (0,1), (1,-1), and (2,-2)
$\frac{dy}{dx}=2x-3y$

Select all that apply

a)

-3

b)

-1

c)

2

d)

5

e)

10

30.

Which choice below represents this slope field?

a)

dy/dx = 2x

b)

dy/dx = -x

c)

dy/dx = yx

d)

dy/dx = x²

31.

Which slope field is represented by dy/dx = x²?

a)

b)

c)

d)

32.

A puppy gains weight, w, at a rate approximately inversely proportional to its age, t, in months.

a)

A

b)

B

c)

C

d)

D

33.

dy/dx = 4x/y. Suppose y(0)=1
The particular solution is

a)

B

b)

C

c)

D

d)

E

34.

Consider the differential equation dy/dx = x + 2y for which g(x) is the solution. Which of the following statements is true if the particular solution contains (0,-1)

a)

g(x) is increasing and concave up

b)

g(x) is increasing and concave down

c)

g(x) is decreasing and concave up

d)

g(x) is decreasing and concave down

35.

∫dx

a)

x+c

b)

2x+c

c)

3x+c

d)

4x+c

36.

Find the Particular Solution

a)

$y=2+e^{\left(\frac{x^2}{2}+x\right)}$

b)

$y=2e^{\left(\frac{x^2}{2}+x\right)}$

c)

$y=\ln\left|\frac{x^2}{2}+x+1\right|+2$

d)

$y=\ln\left|\frac{x^2}{2}+x+e^2\right|$

37.

Of the following, which is a solution to the differential equation, select all that apply:
$y''-6y'+8y=0$

a)

$y=2\sin\left(4x\right)$

b)

$y=3e^{2x}$

c)

$y=Ce^{4x},\$ where C is a constant

38.

a)

A

b)

B

c)

C

d)

D

39.

a)

A

b)

B

c)

C

d)

D

40.

Find $f'\left(4\right)$

a)

-4

b)

$-\frac{1}{4}$

c)

$\frac{1}{4}$

d)

4

41.

Find $h'\left(-4\right)$

a)

-4

b)

-1/4

c)

1/4

d)

4

42.

Find dy/dx
xy+y²=2

a)

-y/(x+2y)

b)

y/(x+2y)

c)

-3y/x

d)

-3x/y

43.

Find dy/dx at a given point.

a)

5/4

b)

4/5

c)

1

d)

-5

44.

Find dy/dx at the given point

x³ +2xy -y²=11 at (2,3)

a)

-4/7

b)

12

c)

-9

d)

9

45.

Find f(4)

a)

1

b)

5

c)

0

d)

DNE

46.

Select all the values where f(x) is NOT continuous

a)

1

b)

4

c)

5

d)

9

47.

What is the limit?

a)

DNE

b)

Infinity

c)

6

d)

12

48.

a)

1/2

b)

0

c)

Positive Infinity

d)

Negative Infinity

49.

a)

I only

b)

I and II Only

c)

I, II, and III

d)

II Only

50.

Find the limit as x approaches -3

a)

0

b)

1

c)

-6

d)

DNE

51.

Find the Limit

a)

0

b)

3

c)

1

d)

DNE

52.

Is this function continuous?

a)

Yes

b)

No, The limit does not exist

c)

No, the value does not exist

d)

No, the value does not equal the limit.

53.

List the horizontal and vertical asymptotes of the function.

a)

x = 3

b)

x = -3

c)

y = -2

d)

y = 2

54.

a)

A

b)

B

c)

C

d)

D

55.

a)

16/3

b)

-16/3

c)

4/3

d)

-4/3

56.

a)

5

b)

infinity

c)

-1

d)

1

57.

a)

0

b)

1

c)

-1

d)

DNE

58.

a)

A

b)

B

c)

C

d)

D

59.

The derivative of

a)

y'=2x-x^-2

b)

y'=x^-1+8x

c)

y'=x^-2+8x

d)

y=8x-x^-2

60.

$f\left(x\right)=\frac{9}{x^{\left(\frac{1}{3}\right)}}$

a)

9x^-1/3

b)

-3x^-4/3

c)

-9x^2/3

d)

-3x^2/3

61.

Which of the following are synonyms for find the derivative?

a)

f'(x)

b)

dy/dx

c)

y'

d)

slope of the tangent line

e)

instantaneous rate of change

62.

Given u(x) = f(x)g(x), f(2) = 3, g(2) = -3, f '(2) = 5, and g'(2) = 1,

then u'(2) = ?

a)

-12

b)

4

c)

5

d)

12

63.

The derivative of the function f is given by $f'\left(x\right)=-x^3\cdot\sin\left(x^2\right)$ , How many points of inflection does the graph of $f$ have on the open interval $\left(-2.5,2.5\right)?$

a)

1

b)

2

c)

3

d)

4

e)

5

64.

The graph of $f'$ , the derivative of the function f, is shown above. Which of the following could be the graph of $f$

a)

b)

c)

d)

65.

A Particle moves on the x-axis. Given that $v\left(t\right)=-t^2+2t+5$ is the particle SPEEDING UP or SLOWING DOWN at t=4?

a)

Speeding Up

b)

Slowing Down

c)

Neither

66.

Which integral would find the volume of the bounded area revolved around the x-axis

a)

$\pi\int_0^1\left(1-x^2\right)^2\ \ dx$

b)

$\pi\int_0^1\left(\sqrt{1-y}\right)^2\ \ dy$

c)

$\int_0^1\left(1-y^2\right)\ \ dy$

d)

$\pi\int_0^1\left(\sqrt{1-x}\right)^2\ \ dx$

67.

Which integral would find the volume of the bounded area revolved around the y-axis

a)

$\pi\int_0^1\left(1-x^2\right)^2\ \ dx$

b)

$\pi\int_0^1\left(\sqrt{1-y}\right)^2\ \ dy$

c)

$\int_0^1\left(1-y^2\right)\ \ dy$

d)

$\pi\int_0^1\left(\sqrt{1-x}\right)^2\ \ dx$

68.

Find the volume of the shaded region revolved around the y axis.

a)

$\int_0^2\left(\left(y^2+2\right)^2-1\right)dy$

b)

$\int_0^2\left(\left(y^2+1\right)^2\right)dy$

c)

$\int_0^4\left(\left(y^2+2\right)^2-1\right)dy$

d)

$\int_0^4\left(\left(y^2+1\right)^2\right)dy$

69.

Find the volume of the shape whose cross sections are squares whose sides lie in the shaded region and perpendicular to the x axis.

a)

$\int_{-\pi}^0\left(\left(f\left(x\right)-g\left(x\right)\right)^2\right)dx$

b)

$\int_0^{\pi}\left(\left(f\left(x\right)-g\left(x\right)\right)^2\right)dy$

c)

$\int_{-\pi}^0\left(\left(g\left(x\right)-f\left(x\right)\right)^2\right)dx$

d)

$\int_0^{\pi}\left(\left(g\left(x\right)-f\left(x\right)\right)^2\right)dy$

70.

find y' for y= sin(2x²)

a)

4sin(2x²)

b)

4xcos(2x²)

c)

2xcos(2x²)

d)

xcos(4x²)

71.

Find $\frac{dy}{dx}\ if\ y=\frac{x^3}{x-1}$

a)

$\frac{2x^3-3x^2}{\left(x-1\right)^2}$

b)

$\frac{2x^3+3x^2}{\left(x-1\right)^2}$

c)

$\frac{-2x^3-3x^2}{\left(x-1\right)^2}$

d)

$\frac{-2x^3+3x^2}{\left(x-1\right)^2}$

72.

Find H'(1)

(a)

73.

Find the derivative: $\sin^2\left(3x+2\right)$

a)

$6\sin\left(3x+2\right)\cos\left(3x+2\right)$

b)

$-6\sin\left(3x+2\right)\cos\left(3x+2\right)$

c)

$6\cos\left(3x+2\right)$

d)

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

74.

Find the Derivative

a)

A

b)

B

c)

C

d)

D

e)

E

75.

d/dx(e^sinx) =

a)

e^sinx

b)

-e^sinx(cosx)

c)

e^sinx(cosx)

d)

e^cosx

76.

d/dx(2^3x+4)

a)

3(ln2)(2^3x+4)

b)

3(2^3x+4)

c)

3(2^3x+4/(ln2))

d)

2^3x+4

77.

d/dx ln(x³)

a)

3/x

b)

1/x³

c)

ln(3x²)

d)

1 / 3x²

78.

Find the Derivative

a)

A

b)

B

c)

C

d)

D

e)

E

79.

If s(t) represents the velocity graph, when is the object moving to the left?

a)

(0, 3) U (8, 12)

b)

(0, 7) ∪ (10, 12)

c)

(0, 6) ∪ (10, 12)

d)

(3, 8)

80.

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

a)

2

b)

4

c)

6

d)

8

81.

Find a value for d'(8)

a)

3.3 feet per second

b)

-10 feet per second

c)

-3.3 feet per second

d)

10 feet per second

82.

a)

A

b)

B

c)

C

d)

D

83.

∫5 e^5x dx

a)

25 e^5x + C

b)

5 e^{5x - 1} + C

c)

e^5x + C

d)

(1/5) e^5x+ C

84.

What should "u" equal in this integral?

a)

x

b)

(lnx)/x

c)

lnx

d)

sin(lnx)

85.

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

a)

10

b)

20

c)

0

d)

Not enough information

86.

$\int_{0\ }^{a\ }f\left(x\right)dx\ =?$

a)

-2

b)

2

c)

6

d)

-6

87.

$\int_{a\ }^{b\ }f\left(x\right)dx\ =?$

a)

-2

b)

2

c)

10

d)

-6

88.

$\int_0^1f\left(x\right)dx\$ is

a)

Positive

b)

Negative

c)

0

d)

Cannot be determined

89.

$\int_3^1f\left(x\right)\ dx$ is

a)

Positive

b)

Negative

c)

0

d)

Cannot be determined

90.

Calculate $\int_2^4f\left(x\right)\ dx$

a)

1

b)

2

c)

4

d)

-4

91.

Calculate $\int_{-4}^6\ f\left(x\right)\ dx$

a)

$6-2\pi$

b)

$3-\pi$

c)

$5+2\pi$

d)

$3-2\pi$

92.

R(t) represents the rate at which people enter a line, measured in people per minute. What does R'(3) = 4 represent?

a)

At t = 3 minutes, people are entering the line at a rate of 4 people per minute.

b)

At t = 3 minutes, the rate at which people are entering the line is increasing at a rate of 4 people per minute per minute.

c)

At t = 3 minutes, there are 4 people in line.

d)

At t = 4 minutes, 3 people are in line.

93.

B(t) represents the temperature of the biscuits after t minutes. What does the following notation represent?

a)

The temperature of the biscuits at 10 minutes

b)

The average temperature of the biscuits on the time interval 3 to 10 minutes.

c)

The average rate of temperature on the interval 3 to 10 minutes.

d)

The change in temperature of the biscuits on the time interval 3 to 10 minutes.

94.

B(t) represents the temperature of the biscuits after t minutes. What does the following notation represent?

a)

The temperature of the biscuits after 10 minutes.

b)

The average rate of the temperature of the biscuits on the time interval 3 to 10 minutes.

c)

The average temperature of the biscuits on the time interval 3 to 10 minutes.

d)

The temperature of the biscuits on the time interval 3 to 10 minutes.

95.

A particle travels along the x-axis. The position of the particle at t = 4 is 10. Which of the following would give you the position of the particle at t = 7?

a)

b)

c)

v(7)

d)

v(4) + v(7)

96.

A particle has a positive velocity and a negative acceleration. Which of the following is true?

a)

The speed is increasing.

b)

The speed is decreasing.

c)

The speed is neither increasing or decreasing.

d)

The particle is not moving.

97.

The rate at which the temperature outside is changing is represented by the function O(t). What does the following represent?

a)

The temperature after 12 minutes is 15 degrees.

b)

The temperature has increased 15 degrees on the time interval 5 to 12 minutes.

c)

The rate of change of temperature on the interval 5 to 12 minutes is 15.

d)

The temperature at 5 minutes is 15 degrees.

98.

H(t) represents the number of hats purchased at a store after t hours. What does the above notation represent ?

a)

The number of hats sold on the time interval 4 to 8 hours was 32.

b)

The number of hats sold at t = 8 hours was 32.

c)

The rate at which hats were sold on the interval 4 to 8 hours was 32.

d)

The number of hats sold at t = 4 hours was 32.

99.

What is the derivative of the function f(x)=∫₁^2x(6e^x)dx

a)

6e^2x

b)

12e^2x

c)

6e^2x-6e^x

d)

6e^2x

100.

Over what interval(s) is f(x) increasing? (Be careful this is a graph of f'!)

a)

(-∞, -3) ∪ (1, ∞)

b)

(-3, 1)

c)

(-5, 0) ∪ (2, ∞)

d)

(-5, ∞)

101.

At what x-value(s) does f(x) have a minimum? (Be careful this is a graph of f'!)

a)

x = -3

b)

x = 1

c)

x = 5

d)

x = -3 and x = 5

102.

Given f'(x), what x values could be critical numbers? (select all that apply)

f'\left(x\right)=\frac{3x}{x-2}

a)

0

b)

1

c)

2

d)

3

e)

4

103.

Lef f be a function and let $f'\left(x\right)=x^2\left(x-2\right)\left(x+3\right)$ At how many points does the graph of f have a relative maxmimum OR minimum?

a)

0

b)

1

c)

2

d)

3

104.

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

Where does the function have a local minimum?

a)

0

b)

1

c)

2

d)

3

105.

Find the Limit

a)

4x²-3

b)

8x

c)

4x+4h-3

d)

4x

106.

A climber is on a hike. After 2 hours he is at an altitude of 400 feet. After 6 hours, he is at an altitude of 700 feet. What is the average rate of change?

a)

200

b)

150

c)

300

d)

75

107.

Which could be f?

a)

A

b)

B

c)

C

108.

Given a graph of f'', which could be a Point of Inflection?

a)

A

b)

B

c)

C

d)

D

109.

a)

[-2,1]

b)

[-2,3]

c)

[3,5]

d)

[0,1.5] and [3,5]

110.

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)

12

d)

44

111.

What is x-value at which the function below has the same instantaneous rate of change as the average rate of change over the indicated interval?

f(x) = x² - 3x - 28, [-4,7]

a)

1.5

b)

-1.5

c)

0.5

d)

-0.5

112.

The value of c guaranteed to exist by the MVT for $f\left(x\right)=x^2$ on the interval [0,3] is:

a)

1

b)

2

c)

3/2

d)

1/2

113.

Find

a)

0

b)

1/ln(5)

c)

1

d)

ln(5)

114.

Suppose f (x) = 3x² is graphed in the x-y plane. Will any linear approximations to f overestimate or underestimate the actual function values?

a)

overestimate

b)

underestimate

115.

Suppose f(x) = x³ – x.

Use a linear approximation at x = 2 to estimate f(2.5).

a)

10.5

b)

11

c)

11.5

d)

12

116.

The following is a description of what theorem?

If a function is differentiable, then the derivative must be equal to the Average Rate of Change somewhere on that interval.

a)

Intermediate Value Theorem

b)

Mean Value Theorem

c)

Exteme Value Theorem

d)

Differentiable Value Theorem

117.

What is needed to be shown in order to get full credit on a Free Response L'Hopital's Rule Question?

a)

You MUST STATE THE FUNCTION IS CONTINUOUS

b)

You MUST show the limit of the top is equal to 0

c)

You MUST show the limit of the bottom is equal to 0

d)

You MUST use Limit Notation

e)

You MUST state by L’Hopital’s Rule

118.

If asked to find the absolute maximum of a function and then JUSTIFY, what is the best way to justify on a free response question?

a)

A table including end points and critical points, and the values you get when you plug into the function

b)

A table including just critical points and the values you get when you plug into the function

c)

A sign chart including just critical points to determine all of the maxes.

d)

A sign chart including critical points and end points to determine all of the maxes.

119.

Find all the critical points on the interval from 0 to 2pi:
$f\left(x\right)=\sin x\cdot\cos x$

a)

$\frac{\pi}{4}$

b)

$\frac{3\pi}{4}$

c)

$\frac{5\pi}{4}$

d)

$\frac{7\pi}{4}$

e)

All of the above.

120.

Given the following derivative, Find all the critical points of the following. Select all that apply: $f'\left(x\right)=x\ln x-4\ln x$

a)

-2

b)

0

c)

1

d)

4

121.

a)

A

b)

B

c)

D

d)

E

122.

a)

1/3

b)

8/3

c)

16/3

d)

24/3

123.

Chris is sitting on the edge of a dock tossing rocks into the water. As each rock hits the water, small circles appear traveling outward from the point of impact. The radius of the circle is changing at a rate of 5 in/sec. How fast is the area changing when the circumference is 4 in?

a)

40 in/sec

b)

20 in/sec

c)

20pi in/sec

d)

40pi in/sec

124.

Oil spilled from a tanker spreads in a circle whose circumference increases at a rate of 40 ft/sec. How fast is the area of the spill increasing when the circumference of the circle is 100π ft?

What rate is given in the problem?

a)

$\frac{dC}{dt}=40$

b)

$\frac{dr}{dt}=40$

c)

$\frac{dA}{dt}=40$

d)

$\frac{dπ}{dt}=40$

125.

Oil spilled from a tanker spreads in a circle whose circumference increases at a rate of 40 ft/sec. How fast is the area of the spill increasing when the circumference of the circle is 100π ft?

What rate are we looking for and when?

a)

$\frac{dA}{dt}\ when\ C=100π$

b)

$\frac{dC}{dt}\ when\ A\ =\ 100π$

c)

$\frac{dA}{dt}\ when\ A\ =\ 100π$

d)

$\frac{dC}{dt}\ when\ r\ =\ 100π$

126.

Oil spilled from a tanker spreads in a circle whose circumference increases at a rate of 40 ft/sec. How fast is the area of the spill increasing when the circumference of the circle is 100π ft?

At what rate is the radius changing when the circumference is 100π ft?

a)

$\frac{dr}{dt}=\frac{20}{π}$

b)

$\frac{dr}{dt}=\frac{50}{π}$

c)

$\frac{dr}{dt}=50$

d)

$\frac{dr}{dt}=20$

127.

Oil spilled from a tanker spreads in a circle whose circumference increases at a rate of 40 ft/sec. How fast is the area of the spill increasing when the circumference of the circle is 100π ft?

a)

$\frac{dA}{dt}=2000\ \frac{ft^2}{\sec}$

b)

$\frac{dA}{dt}=40\ \frac{ft^2}{\sec}$

c)

$\frac{dA}{dt}=100\ \frac{ft^2}{\sec}$

d)

$\frac{dA}{dt}=200\ \frac{ft^2}{\sec}$

AP Calculus Exam Review

Total questions: 127

Worksheet time: 4hrs 28mins

Similar Resources on Wayground