The Ultimate AP Calculus Quiz!

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Advanced Placement Calculus is a set of two distinct Advanced Placement calculus courses and exams offered by College Board. AP Calculus AB covers limits, derivatives, and integrals. AP Calculus BC covers all AP Calculus AB topics plus additional topics (including more integration techniques such as integration by parts, Taylor series, parametric equations, polar coordinate functions, and curve interpolations).
The test is not the actual AP. This, prepares you for the actual test.

• 1.

If a and b are integers such that 4a – 8 > 0 and 4b + 8 < 0, then which of the following must be true?

• A.

Ab is even

• B.

Ab is odd

• C.

Ab is negative

• D.

Ab is positive

C. Ab is negative
Explanation
If 4a - 8 > 0 and 4b + 8 < 0, it means that a is greater than 2 and b is less than -2. Therefore, when we multiply a positive number (a) with a negative number (b), the result will always be negative. Hence, ab is negative.

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• 2.

If z is a positive number not equal to 1, which of the following must also be positive?

• A.

B/b+1

• B.

B+6/b-3

• C.

1/2b-2

• D.

2-b

A. B/b+1
Explanation
The given expression b/b+1 represents the division of b by b+1. Since z is a positive number not equal to 1, b will also be positive. Dividing a positive number by a positive number will always result in a positive value. Therefore, the expression b/b+1 must also be positive.

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• 3.

For all x, let f(x) = (10 – x)^2. If y = f(6), which of the following is equal to 4y?

• A.

F(24)

• B.

F(18)

• C.

F(12)

• D.

F(8)

B. F(18)
Explanation
The function f(x) is defined as (10 - x)^2. We are given that y = f(6), which means y = (10 - 6)^2 = 16. We need to find 4y, which is equal to 4 * 16 = 64. To find the corresponding value of x for this result, we need to find the value of x that satisfies f(x) = 64. Evaluating the options, we find that f(18) = (10 - 18)^2 = 64, which matches the desired result. Therefore, the correct answer is f(18).

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• 4.

What is the distance between the b-intercept and the y-intercept of the line c = 2/3x – 6?

• A.

9

• B.

15

• C.

√89

• D.

√117

D. √117
Explanation
The distance between the b-intercept and the y-intercept of a line can be found by taking the absolute difference between their respective y-coordinates. In this case, the b-intercept is the y-intercept, which is the point where the line crosses the y-axis. To find this point, we set x = 0 in the equation c = 2/3x - 6. Solving for c gives us c = -6. Therefore, the y-intercept is (0, -6). The distance between this point and the b-intercept is the absolute value of -6, which is 6. However, the answer given is √117, which is not equal to 6. Therefore, the given answer is incorrect.

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• 5.

Which of the following is equal to 3a + 9b + 12?

• A.

3(a + 3b + 9)

• B.

3(a + 3b)+12

• C.

3(a + 3b)+4

• D.

3(a + 6b + 9)

B. 3(a + 3b)+12
Explanation
The given expression is 3a + 9b + 12. To find an equivalent expression, we can factor out the common factor of 3 from the first two terms, which gives us 3(a + 3b). Then, we add the constant term of 12 to get the final expression of 3(a + 3b) + 12.

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• 6.

If f(x) = √3x – 2, what is the smallest possible value of f(x)?

• A.

0

• B.

2/3

• C.

4

• D.

1

A. 0
Explanation
The function f(x) = √3x - 2 represents a square root function. Since the square root of any number is always non-negative, the smallest possible value of f(x) occurs when √3x - 2 is equal to 0. Solving for x, we get x = 2/3. Therefore, the smallest possible value of f(x) is 0.

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• 7.

If x​​​​​​2 – |5x| = –6, then what is one possible value of x?

• A.

6

• B.

2

• C.

0

• D.

5

B. 2
Explanation
The equation x^2 - |5x| = -6 can be rewritten as x^2 + 6 = |5x|. Since the absolute value of a number is always non-negative, the right side of the equation must be non-negative as well. This means that x^2 + 6 must be greater than or equal to 0. The only possible value of x that satisfies this condition is x = 2.

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• 8.

If 3(a – 2) = 24, then what does 5(a – 2) equal?

• A.

35

• B.

43

• C.

40

• D.

27

C. 40
Explanation
The given equation states that 3 multiplied by the quantity (a-2) is equal to 24. To find the value of 5 multiplied by the quantity (a-2), we can first solve the given equation. Dividing both sides of the equation by 3 gives us (a-2) = 8. Now, we can substitute this value into the expression 5(a-2). Multiplying 5 by 8 gives us 40, which is the answer.

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• 9.

If √a + 22 = 38, a =

• A.

234

• B.

256

• C.

125

• D.

250

B. 256
Explanation
The equation √a + 22 = 38 can be solved by subtracting 22 from both sides to isolate the square root term. This gives us √a = 16. Squaring both sides of the equation eliminates the square root, resulting in a = 256. Therefore, the correct answer is 256.

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• 10.

If b is an integer between 50 and 70 and can be expressed as 7z + 3 where z is an integer, what are the possible values of b?

• A.

50,59,66

• B.

52,59,66

• C.

52,58,66

• D.

52,59,65

B. 52,59,66
Explanation
The question states that b can be expressed as 7z + 3, where z is an integer. To find the possible values of b, we need to substitute different values of z and check if the resulting expression falls within the given range of 50 to 70.

For z = 7, 7z + 3 = 52, which falls within the range.
For z = 8, 7z + 3 = 59, which falls within the range.
For z = 9, 7z + 3 = 66, which falls within the range.

Therefore, the possible values of b are 52, 59, and 66.

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