Free Printable AP Calc MC Review 2023

1.

No Calc

a)

A

b)

C

c)

D

d)

E

2.

No Calc

a)

A

b)

B

c)

C

d)

D

3.

No Calc

a)

A

b)

B

c)

C

d)

E

4.

a)

A.

b)

B.

c)

C.

d)

D.

5.

What is the limit?

a)

DNE

b)

2/3

c)

1/4

d)

Infinity

6.

Based on the table, use LRAM and 4 sub-intervals to estimate the Area under the curve. (Choose the correct set-up.)

a)

5(3) + 1(4) + 2(5) + 1(7)

b)

5(4) + 1(5) + 2(7) + 1(6)

c)

5(3) + 6(4) + 8(5) + 9(7)

d)

0(3) + 5(4) + 6(5) + 8(7)

7.

a)

A

b)

B

c)

C

d)

D

8.

a)

A

b)

B

c)

C

d)

D

9.

a)

Volume using Washers

b)

Volume using Disks

c)

Volume using Shells

d)

Volume using Cross Sections

10.

a)

f '(x)

b)

0

c)

∞

d)

f(x)

11.

What is true about the following?

a)

A

b)

B

c)

C

d)

D

12.

f' is given, which could be f?

a)

A

b)

B

c)

C

13.

Find h'(3)

a)

-2

b)

0

c)

1

d)

3

14.

a)

A

b)

B

c)

C

d)

D

15.

a)

A

b)

B

c)

C

d)

D

16.

a)

A

b)

B

c)

C

d)

D

17.

a)

A

b)

B

c)

C

d)

D

18.

a)

A

b)

B

c)

C

d)

D

19.

a)

A

b)

B

c)

C

d)

D

20.

a)

A

b)

B

c)

C

d)

D

21.

a)

A

b)

B

c)

C

d)

D

22.

a)

A

b)

B

c)

C

d)

D

23.

a)

A

b)

B

c)

C

d)

D

24.

a)

A

b)

B

c)

C

d)

D

25.

a)

A

b)

B

c)

C

d)

D

26.

a)

A

b)

B

c)

C

d)

D

27.

a)

A

b)

B

c)

C

d)

D

28.

a)

A

b)

B

c)

C

d)

D

29.

a)

A

b)

B

c)

C

d)

D

30.

a)

A

b)

B

c)

C

d)

D

31.

a)

A

b)

B

c)

C

d)

D

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.

a)

A

b)

B

c)

C

d)

D

36.

a)

A

b)

B

c)

C

d)

D

37.

a)

A

b)

B

c)

C

d)

D

38.

a)

A

b)

B

c)

C

d)

D

39.

a)

A

b)

B

c)

C

d)

D

40.

Select all statements that are TRUE.

a)

b)

c)

d)

e)

41.

Find the Limit

a)

0

b)

-1/4

c)

-5/4

d)

1

e)

DNE

42.

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 concave up

d)

The function is concave down

43.

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

44.

If f''(x) > 0 over the interval (a, b), then what will be true about f'(x)?

a)

It's constant

b)

It's increasing

c)

It's decreasing

d)

Cannot be determined

45.

a)

Mean value theorem

b)

Instantaneous Rate of Change

c)

Intermediate Value Theorem

d)

Fundamental Theorem of Calculus

46.

a)

Average Rate of Change

b)

Instantaneous Rate of Change

c)

Intermediate Value Theorem

d)

Average Value of f

47.

What is this expression equivalent to?

a)

f '(g'(x))

b)

f '(x)g'(x)

c)

f '(g(x))g'(x)

d)

f '(g(x)

48.

a)

position

b)

acceleration

c)

total distance

d)

speed

49.

Find the derivative of f(x) = x²sinx

a)

f'(x) = 2xsinx - x²cosx

b)

f'(x) = 2xsinx + x²sinx

c)

f'(x) = 2xsinx + x²cosx

d)

f'(x) = 2xcosx

50.

Find the derivative:
y = 3ln(x²-3)

a)

6x/(x²-3)

b)

3/(x²-3)

c)

3x/(x²-3)

d)

9x/(x²-3)

51.

a)

A

b)

B

c)

C

d)

D

e)

E

52.

a)

A

b)

B

c)

C

d)

D

e)

E

53.

a)

A

b)

B

c)

C

d)

D

e)

E

54.

a)

A

b)

B

c)

C

d)

D

e)

E

55.

a)

A

b)

B

c)

C

d)

D

e)

E

56.

a)

A

b)

B

c)

C

d)

D

e)

E

57.

a)

A

b)

B

c)

C

d)

D

e)

E

58.

a)

A

b)

B

c)

C

d)

D

e)

E

59.

a)

A

b)

B

c)

C

d)

D

e)

E

60.

a)

A

b)

B

c)

C

d)

D

e)

E

61.

$\int_{ }^{ }\left(\cos\left(3x\right)dx\right)$

a)

$-3\sin\left(3x\right)+C$

b)

$-\frac{1}{3}\sin\left(3x\right)+C$

c)

$\frac{1}{3}\sin\left(3x\right)+C$

d)

$\sin\left(3x\right)+C$

62.

What is the slope of the line tangent to the graph of y = ln(2x) at the point where x = 4?

a)

1/8

b)

1/4

c)

3/4

d)

1/2

63.

$\frac{d}{dx}\left(\sin^3\left(x^2\right)\right)=?$

a)

$6x\sin^2\left(x^2\right)\cos\left(x^2\right)$

b)

$\cos^3\left(x^2\right)$

c)

$3\sin^2\left(x^2\right)$

d)

$3\sin^2\left(x^2\right)\cos\left(x^2\right)$

64.

$\lim_{x\rightarrow\infty}\left(\frac{x^3}{e^{3x}}\right)$

a)

0

b)

2/9

c)

2/3

d)

1

65.

$f'\left(x\right)=x\left(x-3\right)^2\left(x+1\right)$

The function, f, has first derivative given above. At what values of x does f have a relative maximum?

a)

-1 only

b)

0 only

c)

-1 and 0 only

d)

-1 and 3 only

66.

$t=\frac{\pi}{2}$

The velocity of a particle moving along the x-axis is given by v(t) = sin(2t) at time t. If the particle is at x = 4 when t = 0, what is the position of the particle at the t-value given above?

a)

5

b)

4

c)

6

d)

3

67.

For what values of x does the graph of y = 3x⁵+10x⁴ have a point of inflection?

a)

-8/3

b)

-2

c)

0

d)

0 and -8/3

68.

Functions w, x, and y are differentiable with respect to time and are related by the equation w = x²y. If x is decreasing at a rate of 1 unit per minute, and y is increasing at a rate of 4 units per minute, at what rate is w changing with respect to time when x = 6 and y = 20?

a)

-96

b)

-240

c)

-384

d)

276

69.

Given the function above, find $f'\left(-1\right)$

a)

nonexistent

b)

3

c)

2

d)

-2

70.

a)

1 only

b)

1 and 2 only

c)

2 only

d)

1, 2, and 3

71.

What is the area of the region enclosed by the graphs of f(x) = x - 2x² and g(x) = -5x?

a)

9

b)

36

c)

20/3

d)

16/3

72.

$\int_1^3f\left(x\right)dx$

If g(x) = x^2 - 3x + 4 and f(x) = g'(x), then find the value of the given integral.

a)

2

b)

4

c)

14/3

d)

-2

73.

$If\ x^2y-3x=y^3-3,\ find\ \frac{dy}{dx}\ at\ \left(-1,2\right)$

a)

-7/11

b)

-7/13

c)

-1/2

d)

-3/14

74.

$f\left(x\right)=\int_4^{2x}\sqrt{t^2-t}dt$

Given the function above, find f ' (2).

a)

$2\sqrt{12}$

b)

$\sqrt{12}$

c)

$0$

d)

$\sqrt{2}$

75.

$\int_2^x\left(3t^2-1\right)dt=$

a)

$x^3-x-6$

b)

$x^3-x$

c)

$3x^2-12$

d)

$3x^2-1$

76.

A vase has the shape obtained by revolving the curve y = 2 + sin(x) about the x-axis, where x and y are measured in inches. What is the volume, in cubic inches, of the vase?

a)

80.115

b)

71.113

c)

33.666

d)

25.501

77.

At time t = 0 years, a forest preserve has a population of 1500 deer. If the rate of growth of the population is modeled by R(t) = 2000e^0.23t deer per year, what is the total population at t = 3?

a)

3987

b)

10,141

c)

12,628

d)

5487

78.

Given the area of A is 14, the area of B is 16, and the area of C is 50, what is the average value of f on the interval [0,8]?

a)

6

b)

10

c)

40/3

d)

80/3

79.

$v\left(t\right)=\frac{t^2-1}{t^2+1}$

A particle moves along the x-axis so that it's velocity is given by the function above. What is the total distance traveled by the particle from t = 0 to t = 2?

a)

0.927

b)

1.600

c)

0.600

d)

0.214

80.

A differentiable function has a property that f'(x) < 3 on the x-interval [1,8] and f(5) = 6. Which of the following could be true?

1. f(2) = 0

2. f(6) = -2

3. f(7) = 13

a)

1 only

b)

1 and 2 only

c)

1, 2, and 3

d)

2 and 3 only

81.

$h\left(x\right)=\int_0^xf\left(t\right)dt$
The graph of f is shown above. Which of the following orders is correct?

a)

$h''\left(2\right)<h'\left(2\right)<h\left(2\right)$

b)

$h'\left(2\right)<h''\left(2\right)<h\left(2\right)$

c)

$h\left(2\right)<h'\left(2\right)<h''\left(2\right)$

d)

$h''\left(2\right)<h\left(2\right)<h'\left(2\right)$

82.

What is the area of the region enclosed by the graphs of y = e^x - 2, y = sin(x), and x = 0?

a)

0.745

b)

2.340

c)

3.472

d)

0.239

83.

A particle moves along a line so that its velocity is given by v(t) = -t³+2t²+2^-t for t > 0. For what values of t is the speed of the particle increasing?

a)

$\left(0.177,\ 1.256\right)\ and\ \left(2.057,\ \infty\right)$

b)

$\left(0.177,\ 1.256\right)\ only$

c)

$\left(0,\ 2.057\right)\ only$

d)

$\left(0,\ 0.177\right)\ and\ \left(1.256,\infty\right)$

84.

Given the table above, and given that g(x) = f ^-1(x), what is the value of g'(4)?

a)

1/4

b)

1/3

c)

-3/100

d)

-1/4

85.

A particle moves along the x-axis so that at any time t > 0 its velocity is given by v(t) = t²ln(t+2). What is the acceleration of the particle at t = 6?

a)

29.453

b)

20.453

c)

74.860

d)

1.500

86.

For t > 0 hours, H is a differentiable function of t that gives the temperature, in degrees C, at an Arctic weather station. Which of the following is the best interpretation of H ' (24) ?

a)

The rate at which the temperature is changing at t = 24 hours.

b)

The average rate at which the temperature changed during the 24th hour

c)

The change in the temperature during the first day

d)

The change in the temperature at t = 24 hours

87.

$r\left(t\right)=9\sin\left(\sqrt{t+1}\right)$

A spherical tank contains 81.637 gallons of water at t = 0 minutes. For the next 6 minutes, water flows out of the tank at a rate given above. How many gallons of water at in the tank at the end of 6 minutes?

a)

36.606

b)

45.031

c)

68.858

d)

126.668

88.

$f'\left(x\right)=x-4e^{-\sin\left(2x\right)}$

The first derivative of the function f is given above. How many points of inflection does the graph of f have on the interval 0 < x < 2pi?

a)

Three

b)

Four

c)

Five

d)

Six

89.

$f'\left(x\right)=\sin\left(x^3\right)$

Let f be the function with first derivative given above for x-values [0,2]. At what value of x does f attain its absolute maximum value on that interval?

a)

1.465

b)

1.845

c)

2

d)

1.162

90.

Find the derivative:
y = 3ln(x²-3)

a)

6x/(x²-3)

b)

3/(x²-3)

c)

3x/(x²-3)

d)

9x/(x²-3)

91.

Find the derivative: $y=3\ln\left(x^2-3\right)$

a)

$\frac{6x}{x^2-3}$

b)

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

c)

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

d)

$\frac{9x}{x^2-3}$

92.

Find $\frac{dy}{dx}$ for $y=e^{-5x^5}$ .

a)

$e^{-5x^5}$

b)

$-5e^{-5x^5}$

c)

$-25x^4e^{-5x^5}$

d)

$\ln\left(-5x^5\right)$

93.

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

94.

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

95.

What is the derivative of e^x?

a)

xe^x-1

b)

x^e

c)

e^x

d)

e^x-1

96.

What is the derivative of ln(x)?

a)

xln(x)

b)

$\frac{1}{x}$

c)

e^x

d)

ln(x+1)

97.

Find the derivative of $y=\ln3x$

a)

$\frac{1}{x}$

b)

$\frac{1}{3x}$

c)

$\ln3\ +\ \ln x$

d)

$\frac{1}{\ln3x}$

98.

Find the derivative of $y=e^{\tan x}$

a)

$e^{\tan x}$

b)

$e^{\sec^2x}$

c)

$\tan xe^{\sec x-1}$

d)

$\sec^2xe^{\tan x}$

99.

$\frac{d}{dx}\left[-3\left(5^{-4x}\right)\right]$

a)

$-12\cdot5^{-4x}\cdot\ln\left(4\right)$

b)

$60^{-4x}\cdot\ln\left(5\right)$

c)

$15^{-4x}\cdot\ln\left(15\right)$

d)

$12\cdot5^{-4x}\cdot\ln\left(5\right)$

100.

Find h'(3)

a)

-2

b)

0

c)

1

d)

3

101.

a)

1/2

b)

-1/2

c)

0

d)

3/2

102.

Given the table containing some values of differentiable functions and their derivatives, solve:
If $h\left(x\right)=\frac{f\left(x\right)}{g\left(x\right)}$ Find $h'\left(4\right)$

a)

$0$

b)

$2$

c)

$\frac{5}{6}$

d)

$-3$

103.

Part 1) Given h₁(x) = f (x) + g(x), find h₁'(1)

a)

3

b)

2

c)

0

d)

5

104.

Part 4) Given h₄(x) =f (x)g(x) , find h₄'(2)

a)

1/3

b)

10/3

c)

4/3

d)

7/3

105.

Part 5) Given h₅(x) = ( f (x))², find h₅'(1)

a)

-10

b)

-9

c)

-13

d)

-8

106.

Find the equation of the tangent line to the graph of $f\left(x\right)=\frac{1}{x}$ at the point (1,2).

a)

$\left(y-1\right)=\frac{1}{4}\left(x-2\right)$

b)

$\left(y-2\right)=\frac{1}{4}\left(x-1\right)$

c)

(y - 1) = (x - 2)

d)

(y - 2) = - (x - 1)

107.

Let f be the function defined by f(x)=4x³-10x+5. Find the equation of the tangent line to the graph of f at the point where x = 1

a)

y+1=2(x-1)

b)

y-1=2(x-1)

c)

y+1=2(x+1)

d)

y+1=-2(x-1)

108.

Find the slope of the line tangent to $f$ at $x=9$ given $f\left(x\right)=-x^2+5\sqrt{x}$

a)

$18+\frac{5}{6}$

b)

$-18+\frac{5}{6}$

c)

$-18-\frac{5}{6}$

d)

$18-\frac{5}{6}$

109.

f’(x) = 3x+4. Find the slope of the line tangent to f(x) at the point (0, 7)

a)

3

b)

4

c)

-7

d)

7

110.

If f '(x) = 0, what does that mean with respect to the tangent line?

a)

Tangent line is vertical

b)

Tangent line has a positive slope

c)

Tangent line is horizontal

d)

Tangent line has a negative slope

111.

What is the derivative of a constant, C?

a)

C

b)

1

c)

0

112.

Find the derivative of:

f\left(x\right)=\frac{2}{x^2}-\frac{9}{x}

a)

$f'\left(x\right)=-\frac{4}{x^3}+\frac{9}{x^2}$

b)

$f'\left(x\right)=-\frac{4}{x}+9$

c)

$f'\left(x\right)=\frac{2}{x^3}-\frac{9}{x^2}$

d)

$f'\left(x\right)=\frac{4}{x^3}-\frac{9}{x^2}$

113.

Find the derivative of:

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)

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

114.

Which option is using correctly the quotient rule?

a)

b)

c)

d)

115.

Find the derivative
f(t) = (t² + 2t)⁵

a)

f'(t) = 5(2t+2)⁴

b)

f'(t) = 5(t² + 2t)⁴

c)

f'(t) = 5(t² + 2t)⁴(2t)

d)

f'(t) = 5(t² + 2t)⁴(2t + 2)

116.

Use the product rule to find the derivative. $f\left(x\right)\ =\ -x^3\left(3x^4-2\right)$

a)

$-3x^2+12x^3$

b)

$-21x^3+6x$

c)

$-21x^6+6x^2$

d)

$-84x^6-6x^2$

117.

Solve f '(2) for
f(x) = x² + 2x

a)

2x + 2

b)

6

c)

12

d)

2x

118.

Find dy/dx if y = cos⁵x

a)

5cos⁴x

b)

-5sin⁴x

c)

5cos⁴xsinx

d)

-5cos⁴xsinx

119.

$\frac{\text{d}}{\text{d}x}\left[x^2\cdot\sin\left(x\right)\right]$

a)

$x^2\cos\left(x\right)-2x\left(\sin x\right)$

b)

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

c)

$x^2\sin\left(x\right)+2x\cos\left(x\right)$

d)

$x^2\cos\left(x\right)+2x\sin\left(x\right)$

120.

Find the slope of f(x) = -3x²-6x at x = 1.

a)

m = 0

b)

f'(x) = -6x - 6

c)

f'(x) = 6x

d)

m = -12

121.

a)

A

b)

B

c)

C

d)

D

AP Calc MC Review 2023

Total questions: 121

Worksheet time: 2hrs 28mins

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