Free Printable AP Calculus

1.

Name the discontinuity at x=2.

a)

Infinite Discontinuity

b)

Removable Discontinuity

c)

Jump Discontinuity

d)

Continuous at x=2

2.

Name the discontinuity at x=2.

a)

Infinite Discontinuity

b)

Removable Discontinuity

c)

Jump Discontinuity

d)

Continuous at x=2

3.

Name the discontinuity at x=2.

a)

Infinite Discontinuity

b)

Removable Discontinuity

c)

Jump Discontinuity

d)

Continuous at x=2

4.

What are the conditions that satisfy the mean value theorem, and what does it mean?

a)

Continuous on the open and differentiable on the closed, and then there is at least 2 numbers c and d in the interval (a,b) (that is a < c < b) such that

b)

Discontinuous on the closed and differentiable on the open, then there is at least one number c in the interval (a,b) (that is a < c < b) such that f'(c)=(f(b)- f(a))/ (b-a)

c)

Continuous on the closed and differentiable on the open,and then there is at least one number c in the interval (a,b) (that is a < c < b) such that f'(c)=(f(b)- f(a))/ (b-a)

d)

Continuous on the open and differentiable on the open, and then there is no number c in the interval (a,b) (that is a < c < b) such that f'(c)=(f(b)- f(a))/ (b-a)

5.

Describe an integral

a)

Area above a curve

b)

Area below a curve

c)

Area above a curve minus area below a curve

d)

All of the above

6.

What is the derivative of sin(x)?

a)

-sin(x)

b)

-cos(x)

c)

cos(x)

d)

sin(x)

7.

What is the derivative of cos(x)?

a)

sin(x)

b)

-sin(x)

c)

cos(x)

d)

-cos(x)

8.

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

9.

What is the derivative of tan(x)?

a)

-sec²(x)

b)

-csc²(x)

c)

sec²(x)

d)

csc²(x)

10.

What is the derivative of sec(x)?

a)

sec(x)tan(x)

b)

csc(x)cot(x)

c)

-sec(x)tan(x)

d)

-csc(x)cot(x)

11.

What is the derivative of cot(x)?

a)

sec²(x)

b)

-sec²(x)

c)

csc²(x)

d)

-csc²(x)

12.

What is the derivative of csc(x)?

a)

csc(x)cot(x)

b)

-csc(x)cot(x)

c)

-csc²(x)

d)

-cot²(x)

13.

Note: Not finding derivative, just finding what it's equal to...

a)

cos x

b)

csc x

c)

sec x

d)

cot x

14.

Note: Not finding derivative, just finding what it's equal to...

a)

csc x

b)

sec x

c)

cot x

d)

tan x

15.

Note: Not finding derivative, just finding what it's equal to...

a)

csc x

b)

sec x

c)

cot x

d)

tan x

16.

Equals

a)

_a∫^cf(x)dx

b)

_a∫^bf(x)dx

c)

_b∫^cf(x)dx

d)

_c∫^af(x)dx

17.

Equals

a)

∫ f(x)dx

b)

cx∫ f(x)dx

c)

c∫f(x)dx

d)

(1/c)∫f(x)dx

18.

equals

a)

f(x)

b)

f(x) + C

c)

f '(x)

d)

f '(x) +C

19.

a)

Intermediate Value Theorem

b)

Rolle's Theorem

c)

Average Rate of Change

d)

Average Value of f

20.

a)

Average Rate of Change

b)

Average Value of f

c)

Intermediate Value Theorem

d)

Rolle's Theorem

21.

Equals

a)

_d∫^cf(x)dx

b)

- _d∫^cf(x)dx

c)

f(d) - f(c)

d)

f(c) - f(d)

22.

a)

Volume using Washers

b)

Volume using Disks

c)

Volume using Shells

d)

Volume using Cross Sections

23.

a)

f(x)

b)

f(x) + C

c)

f '(x)

d)

f '(x) + C

24.

a)

Volume using Disks

b)

Volume using Washers

c)

Volume using Shells

d)

Volume using Cross-Sections

25.

a)

position

b)

acceleration

c)

total distance

d)

displacement

26.

a)

f '(x)

b)

0

c)

∞

d)

f(x)

27.

a)

f(a) - f(b)

b)

f(b) - f(a)

c)

f '(b) - f '(a)

d)

f '(a) - f '(b)

28.

a)

linear approximation

b)

integration by parts

c)

average rate of change

d)

product rule

29.

a)

Mean value theorem

b)

Rolle's theorem

c)

Intermediate Value Theorem

d)

Fundamental Theorem of Calculus

30.

a)

(1/n)⋅u^n-1(du/dx)

b)

n⋅uⁿ⁺¹(du/dx)

c)

n⋅u^n-1(du/dx)

d)

(1/n)⋅uⁿ⁺¹(du/dx)

31.

a)

(1/n) ⋅x^n-1

b)

(1/n) ⋅xⁿ⁺¹

c)

n ⋅x^n-1

d)

n ⋅xⁿ⁺¹

32.

a)

f '(x)⋅g(x) + f(x)⋅g'(x)

b)

f '(x)⋅g(x) - f(x)⋅g'(x)

c)

[f '(x)⋅g(x) + f(x)⋅g'(x)] / [g(x)]²

d)

[f '(x)⋅g(x) - f(x)⋅g'(x)] / [g(x)]²

33.

If y = f(x)/g(x), then dy/dx =

a)

f '(x)⋅g(x) + f(x)⋅g'(x)

b)

f '(x)⋅g(x) - f(x)⋅g'(x)

c)

[f '(x)⋅g(x) + f(x)⋅g'(x)] / [g(x)]²

d)

[f '(x)⋅g(x) - f(x)⋅g'(x)] / [g(x)]²

34.

a)

speed

b)

acceleration

c)

displacement

d)

total distance

35.

a)

Volume using Shells

b)

Volume using Disks

c)

Volume using Washers

d)

Volume using Cross Sections

36.

a)

cosx

b)

-cosx

c)

-sinx cosx

d)

sinx cosx

37.

a)

sinx

b)

-sinx

c)

-cscx cotx

d)

sec²x

38.

a)

cscx cotx

b)

secx tanx

c)

sec²x

d)

-cscx cotx

39.

a)

secx tanx

b)

sec²x

c)

cscx cotx

d)

-cscx cotx

40.

a)

secx tanx

b)

-secx tanx

c)

tan²x

d)

cscx cotx

41.

a)

-cscx cotx

b)

cscx cotx

c)

-csc²x

d)

csc²x

42.

The acceleration function is the first derivative of...

a)

position

b)

velocity

c)

calculus

d)

particle motion

43.

If f'(x) = 0 what does that imply about the x value?

a)

It is a critical point, it is a possible max, min, or point of inflection.

b)

That the limit does not exist.

44.

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

a)

It's positive

b)

It's negative

c)

It's zero

d)

Cannot be determined

45.

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

a)

It's postive

b)

It's negative

c)

It's zero

d)

Cannot be determined

46.

a)

position

b)

velocity

c)

acceleration

d)

total distance

47.

a)

position

b)

velocity

c)

acceleration

d)

total distance

48.

Suppose that $h\left(x\right)$ is continuous on the interval (2, 9) and that f(2) > 0 and f(9) < 0, then by the Intermediate Value Theorem

a)

$f\left(x\right)\$ is decreasing on the interval (2, 9)

b)

$f\left(x\right)\$ must have at least one x-intercept in the interval (2, 9)

c)

$f\left(x\right)$ has a positive Average Rate of Change on the interval (2, 9)

d)

$f\left(x\right)$ has at least one critical value in the interval (2, 9)

e)

$\lim_{\left(x\rightarrow2\right)}f\left(x\right)=9$

49.

What is velocity?

a)

the quickness of an object

b)

the location of an object

c)

acceleration

d)

speed in a specific direction

50.

v (t) > 0 means

a)

the particle is moving to the right

b)

the particle is speeding up

c)

the particle has positive position

d)

the particle is at rest

51.

The acceleration function is the first derivative of...

a)

position

b)

velocity

c)

calculus

d)

particle motion

52.

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

53.

What does a flat (horizontal) line on a velocity-time graph mean?

[Example section B on this graph]

a)

The object is not moving.

b)

The object is moving, but at a constant velocity.

c)

The object is accelerating, and its acceleration is positive.

d)

The object is accelerating, and its acceleration is negative.

54.

You can calculate the slope of a velocity-time graph.

If you calculate the slope of the graph between two points, what would that tell you?

a)

The velocity of the object.

b)

The distance travelled by the object.

c)

The acceleration of the object.

d)

The mass of the object

55.

Besides v(t) = 0, what else is needed to know an object changes directions at a specific time, t ?

a)

v(t) = 0 and a(t) = 0

b)

v(t)=0 and velocity changes sign

c)

v(t) = 0 and acceleration changes sign

56.

What is the derivative of a constant, C?

a)

C

b)

1

c)

0

57.

a)

b)

c)

d)

58.

a)

b)

c)

d)

59.

a)

b)

c)

d)

60.

Which option shows the same expression?

a)

b)

c)

d)

61.

How can you change this to power form:

f\left(x\right)=4\sqrt{x}

a)

$f\left(x\right)=\frac{2}{\sqrt{x}}$

b)

$f\left(x\right)=2\sqrt{x}$

c)

$f\left(x\right)=4x^{\frac{1}{2}}$

d)

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

62.

How do you rewrite this in power form:

f\left(t\right)=\frac{1}{t}-\frac{3}{t^2}+\frac{5}{t^3}

a)

$f\left(t\right)=t^{-1}-3t^{-2}+5t^{-3}$

b)

$f\left(t\right)=-t^{-2}+6t^{-3}-15t^{-4}$

c)

$f\left(t\right)=-t^{-2}+t^{-3}-t^{-4}$

d)

$f\left(t\right)=\ t^{-2}-6t^{-3}+15t^{-4}$

63.

$\frac{\text{d}}{\text{d}x}\left(\ln x\right)=$

a)

$\frac{1}{x}$

b)

$\frac{1}{x\ln x}$

c)

$\ln x$

d)

$e^x$

64.

$\frac{\text{d}}{\text{d}x}\left(\log_ax\right)=$

a)

$\frac{1}{x\ln x}$

b)

$\frac{1}{x\ln a}$

c)

$\frac{\ln a}{x}$

d)

$a^x$

65.

$\frac{\text{d}}{\text{d}x}\left(a^x\right)=$

a)

$a^x\ln a$

b)

$\frac{1}{x\ln a}$

c)

$a^x\ln x$

d)

$a^x$

66.

$\frac{\text{d}}{\text{d}x}\left(e^x\right)=$

a)

$e^x\ln x$

b)

$\frac{1}{x}$

c)

$\frac{1}{x\ln e}$

d)

$e^x$

67.

$\frac{\text{d}}{\text{d}x}\left(\arcsin x\right)=$

a)

$\frac{1}{\sqrt{1-x^2}}$

b)

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

c)

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

d)

$\frac{1}{1-x^2}$

68.

$\frac{\text{d}}{\text{d}x}\left(\arctan x\right)=$

a)

$\frac{1}{\sqrt{1-x^2}}$

b)

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

c)

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

d)

$\frac{1}{1-x^2}$

69.

$\frac{\text{d}}{\text{d}x}\left(\arccos x\right)=$

a)

$-\frac{1}{\sqrt{1-x^2}}$

b)

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

c)

$-\frac{1}{\sqrt{1+x^2}}$

d)

$-\frac{1}{1-x^2}$

70.

$\frac{\text{d}}{\text{d}x}\left(\operatorname{arccot}x\right)=$

a)

$-\frac{1}{\sqrt{1-x^2}}$

b)

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

c)

$-\frac{1}{\sqrt{1+x^2}}$

d)

$-\frac{1}{1-x^2}$

71.

$\frac{\text{d}}{\text{d}x}\left(\operatorname{arcsec}x\right)=$

a)

$\frac{1}{x\sqrt{x^2-1}}$

b)

$\frac{1}{x\sqrt{1-x^2}}$

c)

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

d)

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

72.

$\frac{\text{d}}{\text{d}x}\left(\operatorname{arccsc}x\right)=$

a)

$-\frac{1}{x\sqrt{x^2-1}}$

b)

$-\frac{1}{x\sqrt{1-x^2}}$

c)

$-\frac{1}{x\sqrt{x^2+1}}$

d)

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

73.

Find the derivative.

a)

1/x

b)

x

c)

2/x

d)

2x

74.

The three situations where derivatives fail to exist are at corners or cusps, at a vertical tangent, and...

a)

horizontial tangent

b)

discontinuity

c)

curve

d)

intercepts

75.

The derivative of which function is itself?

a)

e^x

b)

x

c)

sinx

d)

tanx

76.

f'(g(x)) ⋅ g'(x) is what derivative rule?

a)

Product

b)

Quotient

c)

Chain

d)

It is the definition of a derivative.

77.

Which of the following must be true for a function to be continuous at a point, a? (select all that apply)

a)

f(a) must exist

b)

the limit as x approaches a must exist

c)

the function has to go to infinity

d)

f(a) and the limit as x approaches a must be equal

e)

you won't need to do any factoring/simplifying when finding the limit

78.

For a limit to exist, the left and the right hand limits must be equal.

a)

true

b)

false

79.

What is a horizontal tangent?

a)

Has a positive slope.

b)

Has a negative slope.

c)

Has a slope of zero.

d)

Has an undefined slope.

80.

What is a vertical tangent?

a)

Has a positive slope.

b)

Has a negative slope.

c)

Has a slope of zero.

d)

Has an undefined slope.

81.

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 function is constant

d)

Neither increasing nor decreasing

82.

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

a)

The function is increasing

b)

The function is decreasing

c)

The function is concave up

d)

The function is concave down

83.

How many extrema are in the picture?

a)

2

b)

3

c)

4

d)

5

84.

If f"(x) < 0, what is true about f(x)?

a)

it is concave up there

b)

it has an inflection point

c)

It's zero

d)

it is concave down there

85.

The integral is calculating

a)

"the slope of a tangent line"

b)

"the area under a curve"

c)

"the steepness of a space"

d)

"the length of a circle arc"

86.

What's the resulting function of

$\int dx$ ?

a)

d

b)

x

c)

x+c

d)

1

87.

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.

88.

What does C represent in an antiderivative?

a)

A variable

b)

A constant

c)

None of these

d)

Unknown

89.

A slope field is a pictorial representation of all of the possible solutions to a given differential equation.

a)

True

b)

False

c)

I don't know

90.

Find the antiderivative of
x²

a)

(1/3)x³+C

b)

x³

c)

(1/3)x³

d)

2x

91.

∫ 4 dx

a)

0

b)

4t + C

c)

4x + C

d)

2x² + C

92.

∫secxtanx dx =

a)

secx + C

b)

-secx + C

c)

tanx + C

d)

-tanx + C

93.

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

94.

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

95.

When looking for critical points we did....

a)

took the limit of the function. 2. Graphed the critical points.

b)

Found f'(x). 2. Set f'(x) = 0 and solved for x. 3. Created a sign diagram. 4. Took the limit

c)

Found f'(x). 2. Set f'(x) = 0 and solved for x. 3. Created a sign diagram. 4. Checked out intervals.

96.

When applying calculus. The second derivative helps find...

a)

the distance traveled by an object.

b)

The velocity of a particle at any given point

c)

acceleration of an object at any given time

97.

Integration is the inverse of differentiation but it applications it can be used to....

a)

find the area under a curve

b)

calculate the force of an object

c)

Find the altitude of an objects perimeter

98.

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

99.

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

a)

negative

b)

positive

c)

decreasing

d)

increasing

100.

To determine if a piecewise function is continuous at one of the breaks in the domain...

a)

set each piece equal and plug in the value of x you are examining

b)

set each derivative equal and plug in the value of x you are examining

101.

To determine if a piecewise function is differentiable at one of the breaks in the domain...

a)

set each piece equal and plug in the value of x you are examining

b)

set each derivative equal and plug in the value of x you are examining

102.

A function is even if it is symmetric over the

a)

x-axis

b)

y-axis

c)

origin

103.

A function is odd if it is symmetric over the

a)

x-axis

b)

y-axis

c)

origin

104.

If f(-x)=f(x), the function is

a)

even

b)

odd

c)

neither

105.

If f(x)=-f(x), the function is

a)

even

b)

odd

c)

neither

106.

The derivative of an odd function is

a)

odd

b)

even

c)

neither

107.

The derivative of an even function is

a)

odd

b)

even

c)

neither

108.

sin (π/2)

a)

√3/2

b)

√2/2

c)

1/2

d)

1

109.

cos (π/4)

a)

√3/2

b)

√2/2

c)

1/2

d)

1

110.

cos (π/6)

a)

√3/2

b)

√2/2

c)

1/2

d)

1

111.

tan 2π =

a)

0

b)

1

c)

-1

d)

undefined

112.

cos(π)

a)

√3/2

b)

-√2/2

c)

-1

d)

1

113.

sin 0 =

a)

0

b)

¹/₂

c)

^√2/₂

d)

^√3/₂

e)

1

114.

cos 0 =

a)

0

b)

¹/₂

c)

^√2/₂

d)

^√3/₂

e)

1

115.

If evaluating a limit produces 0/0, then...

a)

the limit is 0

b)

the limit does not exist

c)

rewrite the expression and evaluate again

116.

What is lne?

a)

0

b)

1

c)

e

d)

DNE

117.

What is ln1?

a)

0

b)

1

c)

e

d)

DNE

118.

When evaluating a limit approaching infinity, if the degree of the numerator > the degree of the denominator...

a)

the limit DNE (goes to +/- infinity)

b)

take the ratio of the leading coefficients

c)

y=0

119.

When evaluating a limit approaching infinity, if the degree of the numerator = the degree of the denominator...

a)

the limit DNE (goes to +/- infinity)

b)

take the ratio of the leading coefficients

c)

y=0

120.

When evaluating a limit approaching infinity, if the degree of the numerator < the degree of the denominator...

a)

the limit DNE (goes to +/- infinity)

b)

take the ratio of the leading coefficients

c)

y=0

121.

To find the equation of a horizontal asymptote of a rational function...

a)

compare the degrees of the numerator and denominator

b)

set the numerator = 0

c)

set the denominator = 0

122.

To find the equation of a vertical asymptote of a rational function...

a)

compare the degrees of the numerator and denominator

b)

set the numerator = 0

c)

set the denominator = 0

123.

To find critical numbers...

a)

set the derivative =0

b)

see where the function is undefined

c)

set the derivative = 0 and see where the function is undefined

124.

When you see "average rate of change", think...

a)

slope formula

b)

take a derivative

125.

When you see "instantaneous rate of change", think...

a)

slope formula

b)

take a derivative

126.

Which of the following is true?

a)

If a function is continuous at a point, it must be differentiable at that point.

b)

If a function is differentiable at a point, it must be continuous at that point.

127.

Sine or Cosine?

a)

y=sinx

b)

y=cosx

128.

Sine or Cosine?

a)

y=sinx

b)

y=cosx

129.

Simplify ln e²

a)

e²

b)

2 ln e

c)

1

d)

2

130.

What does this picture represent?

a)

Left Riemann Sum

b)

Right Riemann Sum

c)

Middle Riemann Sum

d)

Trapezoidal Sum

131.

What does this picture represent?

a)

Left Riemann Sum

b)

Right Riemann Sum

c)

Middle Riemann Sum

d)

Trapezoidal Sum

132.

What does picture represent?

a)

Left Riemann Sum

b)

Right Riemann Sum

c)

Middle Riemann Sum

d)

Trapezoidal Sum

133.

For a function that is strictly decreasing, a right hand Riemann Sum is which of the following:

a)

Overestimate

b)

Underestimate

c)

Exact Solution

d)

Unable to Determine

134.

For a function that is strictly increasing, a right hand Riemann Sum is which of the following:

a)

Overestimate

b)

Underestimate

c)

Unable to Determine

d)

Exact Solution

135.

a)

Average Value

b)

Net Area

c)

Total Area

d)

Average Rate of Change

136.

a)

Distance

b)

Displacement

c)

Velocity

d)

Acceleration

137.

∫ - sinx dx

a)

-cos x + C

b)

cos x + C

c)

tan x + C

d)

1/√1- x²

138.

∫cosx dx =

a)

-secx + C

b)

secx + C

c)

sinx + C

d)

-sinx + C

139.

∫ 1/x dx

a)

ln x

b)

ln x + C

c)

-1/x²

d)

-1/x² + C

140.

a)

cos x + C

b)

tan x + C

c)

-cos x + C

d)

-tan x + C

141.

$If\int_2^5\ f\left(x\right)dx=5\ and\ \ \int_4^5\ f\left(x\right)dx=\pi,\ find\ \int_5^4\ f\left(x\right)dx.$

a)

$0$

b)

$-1$

c)

$-\pi$

d)

$\pi$

142.

$\int_2^5\ f\left(x\right)dx=5\ and\ \ \int_4^5\ f\left(x\right)dx=\pi,\ find\ \int_2^4\ f\left(x\right)dx.$ If

a)

$\pi-5$

b)

$2$

c)

$5-\pi$

d)

$-\left(5-\pi\right)$

143.

Derivative means the same thing as

a)

slope of the tangent line

b)

slope of the normal line

c)

exponent

d)

area under a curve

144.

The derivative of a function is its

a)

Sign

b)

Maximum/Minimum

c)

Instantaneous rate of change

d)

Common Denominator

145.

When do you use the chain rule?

a)

anytime you want

b)

when there is a function in a function

c)

when two functions are being multiplied

d)

when two functions are being divided

146.

How do the slope of tangent lines and normal lines compare?

a)

Their slopes are the same.

b)

Their slopes are negative reciprocals.

AP Calculus

Total questions: 146

Worksheet time: 3hrs 32mins

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