AP Calculus AB Questions and FRQ

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Choose only one answer for Multiple Choice.

Determine the domain of the function.

f as a function of x is equal to the square root of one minus x.

·	· All real numbers
·	· x > 1
·	· x ≤ 1
·	· All real numbers except 1

Determine the domain of the function.

f as a function of x is equal to the square root of x plus three divided by x plus eight times x minus two.

	· All real numbers except -8, -3, and 2
	· x ≥ 0
	· All real numbers
	· x ≥ -3, x ≠ 2

f(x) = 3x + 2; g(x) = 3x - 5

Find f/g.

	· (f/g)(x) = ; domain {x\|x ≠ - }
	· (f/g)(x) = ; domain {x\|x ≠ }
	· (f/g)(x) = ; domain {x\|x ≠ }
	· (f/g)(x) = ; domain {x\|x ≠ - }

Use your graphing calculator to graph f(x) = |x + 1| and determine where the function is increasing and decreasing.

	· Increasing x > -1; Decreasing x < -1
	· Increasing x < 1; Decreasing x > 1
	· Increasing x < -1, Decreasing x > -1
	· Increasing x > 1; Decreasing x < 1

Select true or false:

The function -3(x + 2)(x - 5)3 > 0, when x < -2 or x > 5.

·	· True
·	· False

f(x) = 2x + 6, g(x) = 4x2

Find (f + g)(x).

	· 8x3 + 24x
	·
	· 4x2 + 2x + 6
	· -4x2 + 2x + 6

f(x) = Square root of quantity x plus eight. ; g(x) = 8x - 12

Find f(g(x)).

	· f(g(x)) = 2
	· f(g(x)) = 8 - 12
	· f(g(x)) = 2
	· f(g(x)) = 8

Describe how the graph of y = x2 can be transformed to the graph of the given equation:

y = x2 - 20

·	· Shift the graph of y = x2 left 20 units.
·	· Shift the graph of y = x2 up 20 units.
·	· Shift the graph of y = x2 down 20 units.
·	· Shift the graph of y = x2 right 20 units.

Is the function of f(x) = |4x| + 3 over x even, odd, or neither?

	· Even
	· Odd
	· Neither

10.

State the domain of the rational function.

f(x) = thirteen divided by quantity ten minus x.

	· All real numbers except -10 and 10
	· All real numbers except 13
	· All real numbers except 10
	· All real numbers except -13 and 13

11.

If limit as x approaches a of f of x equals four and limit as x approaches a of g of x equals two , then limit as x approaches a of the quantity three times f of x squared minus four times g of x equals forty .

	· True
	· False

12.

Find the limit of the function algebraically.

limit as x approaches zero of quantity negative seven plus x divided by x squared.

·	· Does not exist
·	· 7
·	· 0
·	· -7

13.

Find limit as x approaches 3 for f of x equals the quantity 3x plus 1 for x less than or equal to 32 and f of x equals 2 times x squared for x greater than 3 .

	· 10
	· 18
	· Does not exist
	· 10 or 18

14.

Evaluate limit as x goes to 0 of 1 minus the quotient of the square of cosine of x, and x .

·	· 0
·	· −∞
·	· ∞
·	· Does not exist

15.

Evaluate limit as x goes to 0 of the quotient of the quantity x raised to the 4th power minus 1 and x minus 1 .

·	· 1
·	· 0
·	·
·	· Does not exist

16.

Evaluate limit as x goes to infinity of the quotient of 4 times x cubed plus 2 times x squared plus 3 times x and negative 9 times x squared plus 5 times x plus 5

·	· ∞
·	· -∞
·	· 0
·	·

17.

Find the equation of the horizontal asymptote for the function, f of x equals the quotient of the quantity x minus 100 and x raised to the 2nd power minus 100 .

	· There is no horizontal asymptote.
	· y = 0
	· y = 1
	· y = x

18.

Which of the following is false for f of x equals the quotient of 5 times x cubed minus 5 times x squared minus 10 times x and the quantity 2 times x raised to the fifth power minus 2 times x ?

	· The x-axis is an asymptote of f(x).
	· x = -1 is not an asymptote of f(x).
	· x = 1 is an asymptote of f(x).
	· The y-axis is an asymptote of f(x).

19.

To two decimal places, find the value of k that will make the function f(x) continuous everywhere.

f of x equals the quantity 3 times x plus k for x less than or equal to negative 4 and is equal to k times x squared minus 5 for x greater than negative 4

	· 11.00
	· -2.47
	· -0.47
	· None of these

20.

Where is f of x equals the quotient of x minus 1 and the quantity x squared minus 5 times x plus 4 discontinuous?

	· f(x) is continuous everywhere
	· 1
	· 1, 4
	· 4

21.

Is the function f of x equals the quantity 2 times x plus 1 for x less than or equal to 4 and is equal to x squared plus 12 for x greater than 4 continuous?

	· Yes
	· No

22.

List the discontinuities for the function f(x) = cot( 5x over 3 ).

	· There are no discontinuities.
	· n( ), where n is an integer
	· n( ), where n is an integer
	· n( ), where n is an integer

23.

Which of the following is true for f of x equals the quotient of the quantity x squared minus 9 and the quantity x minus 3 ?

	· There is a removable discontinuity at x = 3.
	· There is a non-removable discontinuity at x = 3.
	· The function is continuous for all real numbers.

24.

What is the instantaneous slope of y = negative 5 over x at x = 5?

·	·
·	· 1
·	· -1
·	·

25.

What is the average rate of change of y with respect to x over the interval [-2, 6] for the function y = 5x + 2?

·	· 5
·	· 2
·	·
·	· 10

26.

What is the slope for the function y = -3x2 + 2 at the point x = 2?

	· -4
	· -10
	· -12
	· The slope cannot be determined.

27.

The surface area, S, of a sphere of radius r feet is S = S(r) = 4πr2. Find the instantaneous rate of change of the surface area with respect to the radius r at r = 4.

·	· 32π
·	· 16π
·	· 64π
·	· 4π

28.

A ball is thrown vertically upward from the top of a 100 foot tower, with an initial velocity of 10 ft/sec. Its position function is s(t) = -16t2 + 10t + 100. What is its velocity in ft/sec when t = 2 seconds?

·	· -32
·	· -38
·	· -54
·	· 80

29.

Using the graph of f(x) below, find limit as x approaches 3 of f of x .

·	· −5
·	· −∞
·	· 0
·	· 1.7

30.

Find limit as x approaches zero of the quotient of sine of negative 2x and the sine of negative 5x. .

	· Does not exist
	· 0
	·
	·

31.

What is limits as x approaches negative 4 from the right of the quotient of x and the quantity x plus 4. ?

·	· ∞
·	· 0
·	· −4
·	· −∞

32.

What is limit as x approaches 9 of the quotient of the quotient of the quantity 9 minus x and the square root of x minus 2. ?

·	· −6
·	· 0
·	· 1
·	· Does not exist

33.

Find the limit of the function by using direct substitution.

limit as x approaches zero of quantity x squared minus five.

	· Does not exist
	· 0
	· 5
	· -5

34.

Use your graphing calculator to evaluate limit as x goes to infinity of the quantity 1 plus x raised to the power of 3 divided by x .

	· 0
	· π
	· e3
	· 1

35.

Use your calculator to select the best answer below:

limit as x goes to infinity of the quotient of the quantity the natural log of the absolute value of x minus 1 and the quantity cosine x raised to the x power

·	· does not exist
·	· 1
·	· -1
·	· 0

36.

limit as x approaches a of the quotient of the quantity x minus a and the quantity the square root of x minus the square root of a equals

·	·
·	·
·	·
·	· 2a

37.

Find the limit as x goes to 0 of the quotient of the quantity x plus 1 minus the square of cosine x and 3 times the sine of x .

	· does not exist
	· 3
	· 0
	·

38.

If limit as x approaches zero of f of x equals two and , then find .

·	· 64
·	· -4
·	· 16
·	· 28

39.

Evaluate limit as x approaches 0 of the quotient of the absolute value of x and x .

·	· 0
·	· does not exist
·	· 1
·	· -1

40.

Evaluate limit as x goes to 3 of the quotient of the quantity 1 divided by x minus 1 third and the quantity x minus .

·	·
·	·
·	·
·	·

41.

Evaluate limit as x goes to 0 of the quotient of the sine of 5 times x and 6x .

·	· 1
·	·
·	·
·	· does not exist

42.

If f is a continuous function with even symmetry and limit as x approaches infinity of f of x equals 10 , which of the following statements must be true?

I. the limit as x goes to negative infinity of f of x equals 10 II.There are no vertical asymptotes. III.The lines y = 10 and y = -10 are horizontal asymptotes.

·	· I only
·	· II only
·	· I and II only
·	· All statements are true.

43.

What are the horizontal asymptotes of the function f of x equals the quotient of the square root of the quantity x squared plus 1 and x ?

·	· y = 1 only
·	· y = -1 only
·	· y = 0
·	· y = -1 and y = 1

44.

Which one or ones of the following statements is/are true?

I. If the line y = 2 is a horizontal asymptote of y = f(x), then f is not defined at y = 2.

II. If f(5) > 0 and f(6) < 0, then there exists a number c between 5 and 6 such that f(c) = 0.

III. If f is continuous at 2 and f(2)=8 and f(4)=3, then the limit as x approaches 2 of f of the quantity 4 times x squared minus 14 equals 8 .

·	· All statements are true.
·	· I only
·	· II only
·	· III only

45.

Find limit as x goes to infinity of the quotient of 7 times x cubed minus 3 times x squared plus 3 times x and negative 8 times x squared plus 4 times x plus 3

·	·
·	· 0
·	· -∞
·	· ∞

46.

Evaluate limit as x goes to 2 from the left of the quotient of x and the quantity the square root of the quantity x squared minus 4 .

·	· 1
·	· 0
·	· 3
·	· does not exist

47.

Which of the following are the equations of all horizontal and vertical asymptotes for the graph of f of x equals x divided by the quantity x times the quantity x squared minus 16 ?

	· y = 0, x = -4, x = 4
	· y = 1, x = -4, x = 4
	· y = 0, x = -4, x = 0, x = 4
	· y = 1, x = -4, x = 0, x = 4

48.

Evaluate limit as x approaches 1 at f of x for f of x equals the quantity 5 times x minus 11 for x less than 1, equals 1 for x equals 1 and equals negative 3 times x plus 6 for x greater than 1 .

	· 3
	· -6
	· 1
	· does not exist

49.

Where is f of x equals the quotient of x plus 3 and x squared plus 2 times x minus 3 discontinuous?

	· f is continuous everywhere
	· x = 1
	· x = -3
	· x = -3 and x = 1

50.

Which of the following are continuous for all real values of x?

I. f of x equals the quotient of the quantity x squared plus 5 and the quantity x squared minus 1

II. g of x equals the quotient of 3 and the quantity x squared plus 1

III. h of x equals the absolute value of the quantity x minus 1

	· II and III only
	· I and II only
	· I only
	· II only

51.

Which of the following must be true for the graph of the function f of x equals the quotient of the quantity x squared minus 9 and the quantity 3 times x minus 9 ?

There is:

I. a vertical asymptote at x = 3

II. a removable discontinuity at x = 3

III. an infinite discontinuity at x = 3

	· I only
	· II only
	· III only
	· I, II, and III

52.

What is the average rate of change of y with respect to x over the interval [-3, 5] for the function y = 2x + 2?

·	· 16
·	·
·	· 2
·	·

53.

What is the instantaneous slope of y = negative two over x at x = 3?

·	·
·	·
·	·
·	·

54.

The height, s, of a ball thrown straight down with initial speed 32 ft/sec from a cliff 48 feet high is s(t) = -16t2 - 32t + 48, where t is the time elapsed that the ball is in the air. What is the instantaneous velocity of the ball when it hits the ground?

·	· 64 ft/sec
·	· 0 ft/sec
·	· 256 ft/sec
·	· -64 ft/sec

55.

The surface area of a right circular cylinder of height 5 feet and radius r feet is given by S(r)=2πrh+2πr2. Find the instantaneous rate of change of the surface area with respect to the radius, r, when r = 6.

·	· 24π
·	· 34π
·	· 64π
·	· 20π

56.

Use your graphing calculator to evaluate limit as x goes to infinity of the quantity x raised to the power of 1 divided by 2 times x .

·	· 1
·	·
·	· π
·	· 0

57.

Describe the discontinuity for the function f of x equals the quotient of the quantity x squared plus 9 and the quantity x minus 3 .

·	· There is a hole at x = -9.
·	· There is a vertical asymptote at x = 3.
·	· There is a removable discontinuity at x = 3.
·	· There is no discontinuity at x = 3.

58.

Find limit as x goes to 0 of the quotient of the sine of negative 4 times x and the sine of negative 2 times x .

	·
	· does not exist
	· 2
	· 0

59.

Evaluate limit as x goes to negative 8 from the right of the quotient of x and the quantity x plus 8 .

·	· ∞
·	· -∞
·	· 0
·	· -8

60.

Evaluate limit as x goes to infinity of the quotient of the negative 4 times x cubed plus 3 times x squared plus 6 times x and the quantity 6 times x squared minus 4 times x minus 5 .

	· -2
	·
	· 0
	· -∞

61.

Which of the following is the graph of which function has y = -1 as an asymptote?

	·
	· y = ln(x + 1)
	·
	·

62.

If f of x equals the quotient of the quantity x squared minus 16 and the quantity x plus 4 is continuous at x = -4, find f(-4).

·	· 4
·	· -4
·	· 8
·	· -8

63.

Where is f of x equals the quotient of the quantity x plus 4 and x squared plus 6 times x plus 8 discontinuous?

	· f(x) is continuous everywhere
	· x = -4
	· x = -2
	· x = -4 and x = -2

64.

If f(x) is discontinuous, determine the reason.

f of x equals the quantity x squared plus 4 for x less than or equal to 1 and equals x plus 4 for x greater than 1

	· f(x) is continuous for all real numbers
	· The limit as x approaches 1 does not exist
	· f(1) does not equal the limit as x approaches 1
	· f(1) is not defined

65.

If f(x) is a continuous function defined for all real numbers, f(-1) = 1, f(-5) = -10, and f(x) = 0 for one and only one value of x, then which of the following could be that x value?

·	· -6
·	· -5
·	· -4
·	· 0

66. Use the graph below to list the x value(s) where the limits as x approaches from the left and right of those integer values(s) are not equal.

_______________________________

67.

Find limit as x approaches 4 of the quotient of the quantity the square root of the quantity 3 times x plus 4 minus the square root of 4 times x and the quantity x squared minus 4 times x . You must show your work or explain your work in words.

_______________________________

68.

Find limit as x approaches 4 from the left of the quotient of the absolute value of the quantity x minus 4, and the quantity x minus 4 . You must show your work or explain your work in words.

_______________________________

69.

A ball's position, in meters, as it travels every second is represented by the position function s(t) = 4.9t2 + 350.

What is the velocity of the ball after 2 seconds?

Include units in your answer.

_______________________________

70.

The cost in dollars of producing x units of a particular telephone is C(x) = x2 - 2500. (10 points)

1. Find the average rate of change of C with respect to x when the production level is changed from x = 100 to x = 103. Include units in your answer.

2. Find the instantaneous rate of change of C with respect to x when x = 100. Include units in your answer.

_______________________________

71.

State the domain and range for the function f(x) = 2x2 - 7.

_____________________________

72. Show that the function f(x) = x3 + five over x is even, odd, or neither.

_____________________________

73. Determine the equation of a line, in slope-intercept form, that passes through the points (3, 6) and (6, 8).

____________________________

74. Write the equation of the function g(x) if g(x) = f(x - 2) +4 and f(x) = x3 + 2.

_____________________________

75. Identify the maximum and minimum values of the function y = 10 cos x in the interval [-2π, 2π]. Use your understanding of transformations, not your graphing calculator.

_____________________________