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AP Calculus AB study materials
AP Calculus AB study resources

AP Calculus AB practice questions

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Question 1Limits and Continuity

Evaluate the limit: lim(x→3) (x² − 9)/(x − 3).

  1. 3
  2. The limit does not exist
  3. 0
  4. 6 Correct answer.

Explanation

The numerator factors as (x − 3)(x + 3). Canceling the common factor x − 3 leaves x + 3, and substituting x = 3 gives 6.

Question 2Limits and Continuity

Given lim(x→2) f(x) = 5 and lim(x→2) g(x) = −3, find lim(x→2) [2f(x) + g(x)].

  1. −1
  2. 13
  3. 7 Correct answer.
  4. 4

Explanation

The limit of a sum and constant multiple equals the sum and constant multiple of the limits: 2(5) + (−3) = 7.

Question 3Limits and Continuity

The function f is defined by f(x) = (x² − 1)/(x − 1) for x ≠ 1, and f(1) = 4. Which statement about f at x = 1 is true?

  1. f has a removable discontinuity at x = 1 Correct answer.
  2. f has a jump discontinuity at x = 1
  3. f is continuous at x = 1
  4. f has an infinite discontinuity at x = 1

Explanation

The limit as x→1 is 2, but the function value is 4, so the limit exists yet differs from f(1). This is a removable discontinuity because redefining f(1) = 2 would make the function continuous.

Question 4Limits and Continuity

Evaluate lim(x→0) (sin 4x)/x.

  1. 0
  2. 4 Correct answer.
  3. 1
  4. The limit does not exist

Explanation

Using the standard limit lim(u→0) (sin u)/u = 1, rewrite as 4 · (sin 4x)/(4x). As x→0, (sin 4x)/(4x) → 1, so the limit is 4.

Question 5Limits and Continuity

The graph of a function f has a vertical asymptote at x = 2, and f(x) → −∞ as x→2⁺. Which statement must be true?

  1. lim(x→2) f(x) = 0
  2. lim(x→2⁺) f(x) = −∞ Correct answer.
  3. f is continuous from the right at x = 2
  4. f(2) = −∞

Explanation

The notation f(x) → −∞ as x→2⁺ is exactly the definition of the right-hand limit being −∞. A function value cannot be infinite, and continuity is impossible near a vertical asymptote.

Question 6Limits and Continuity

If lim(x→c) f(x) = L and lim(x→c) g(x) = 0, which conclusion about lim(x→c) [f(x)/g(x)] is guaranteed when L ≠ 0?

  1. The limit equals 0
  2. The limit does not exist Correct answer.
  3. The limit equals L/0
  4. The limit equals L

Explanation

A nonzero numerator limit divided by a denominator limit of 0 produces an infinite or nonexistent limit, never a finite number. Division by zero is undefined, so the quotient limit cannot exist as a real number.

Question 7Limits and Continuity

Evaluate lim(x→∞) (3x² + 5)/(x² − 2x).

  1. 0
  2. ∞
  3. 5/(−2)
  4. 3 Correct answer.

Explanation

For rational functions as x→∞, divide numerator and denominator by the highest power x². This gives (3 + 5/x²)/(1 − 2/x) → 3/1 = 3.

Question 8Limits and Continuity

A function f is continuous on [1, 5], with f(1) = −2 and f(5) = 7. Which statement is guaranteed by the Intermediate Value Theorem?

  1. f(3) = 2.5
  2. f is differentiable on (1, 5)
  3. f has a maximum at x = 1
  4. There exists c in (1, 5) with f(c) = 0 Correct answer.

Explanation

Since f is continuous on [1, 5] and 0 lies between f(1) = −2 and f(5) = 7, the Intermediate Value Theorem guarantees some c in (1, 5) with f(c) = 0.

Question 9Limits and Continuity

Evaluate lim(x→2) (x² − 4)/(x² − 3x + 2).

  1. 4 Correct answer.
  2. 1
  3. 0
  4. The limit does not exist

Explanation

Factor both: numerator (x − 2)(x + 2), denominator (x − 2)(x − 1). Cancel x − 2 to get (x + 2)/(x − 1), which at x = 2 equals 4/1 = 4.

Question 10Limits and Continuity

Evaluate lim(x→0) (√(x + 9) − 3)/x.

  1. 1/3
  2. The limit does not exist
  3. 1/6 Correct answer.
  4. 0

Explanation

Multiply numerator and denominator by the conjugate √(x + 9) + 3. The numerator becomes (x + 9) − 9 = x, so the expression simplifies to 1/(√(x + 9) + 3), which at x = 0 equals 1/6.

Question 11Limits and Continuity

Let f(x) = (x + 1)/(x² − 4). At which x-values does f have vertical asymptotes?

  1. x = 2 and x = −2 Correct answer.
  2. x = 2 only
  3. x = 4
  4. x = −1 only

Explanation

Vertical asymptotes occur where the denominator is zero but the numerator is not. The denominator x² − 4 = 0 at x = 2 and x = −2, and the numerator is nonzero at both, so both are vertical asymptotes.

Question 12Limits and Continuity

The function f has lim(x→3⁻) f(x) = 5 and lim(x→3⁺) f(x) = 5, but f(3) is undefined. Which statement is true?

  1. lim(x→3) f(x) does not exist
  2. f has a jump discontinuity at x = 3
  3. f has a removable discontinuity at x = 3 Correct answer.
  4. f is continuous at x = 3

Explanation

Both one-sided limits equal 5, so the two-sided limit is 5. Because the limit exists but f(3) is undefined, the discontinuity is removable by defining f(3) = 5.

Question 13Limits and Continuity

Evaluate lim(x→∞) (5x³ − 2x)/(2x³ + x²).

  1. 5/2 Correct answer.
  2. ∞
  3. 0
  4. 5/2 x

Explanation

Divide every term by x³: (5 − 2/x²)/(2 + 1/x) → 5/2 as x→∞. Equal degrees mean the limit is the ratio of leading coefficients.

Question 14Limits and Continuity

A function f is defined by f(x) = x² + 1 for x < 1 and f(x) = kx for x ≥ 1. For what value of k is f continuous at x = 1?

  1. There is no such value
  2. 1
  3. 0
  4. 2 Correct answer.

Explanation

Continuity at x = 1 requires the left limit, right limit, and function value to agree. The left limit is 1² + 1 = 2, the right limit is k(1) = k, and f(1) = k, so k must equal 2.

Question 15Limits and Continuity

Evaluate lim(x→0) (1 − cos x)/x².

  1. The limit does not exist
  2. 1/2 Correct answer.
  3. 1
  4. 0

Explanation

Multiply by the conjugate (1 + cos x)/(1 + cos x) to get sin²x/[x²(1 + cos x)]. Since (sin x)/x → 1 and 1 + cos x → 2, the limit is 1²/2 = 1/2.

Question 16Limits and Continuity

Which statement correctly describes the behavior of f(x) = 1/(x − 4)² near x = 4?

  1. f(x) → ∞ from the left and −∞ from the right
  2. f(x) → ∞ as x→4 from both sides Correct answer.
  3. f(x) has a jump discontinuity at x = 4
  4. f(x) → 0 as x→4

Explanation

The denominator (x − 4)² is positive on both sides of x = 4 and approaches 0, so the quotient grows without bound from both directions. Thus the two-sided limit is ∞.

Question 17Limits and Continuity

Evaluate lim(x→−∞) (4x² + 1)/(x + 3).

  1. 0
  2. ∞
  3. −∞ Correct answer.
  4. 4

Explanation

The numerator has degree 2 and the denominator degree 1, so the quotient grows without bound in magnitude. As x→−∞, the numerator is positive while x + 3 is negative, so the quotient approaches −∞.

Question 18Limits and Continuity

The function f is continuous on the interval [−3, 4] with f(−3) = 6 and f(4) = −1. Which value is guaranteed to be attained by f somewhere on [−3, 4]?

  1. 7
  2. −3
  3. 6.5
  4. 0 Correct answer.

Explanation

By the Intermediate Value Theorem, a continuous function on a closed interval attains every value between its endpoint values. Since 0 lies between −1 and 6, f(c) = 0 for some c in [−3, 4].

Question 19Limits and Continuity

Evaluate lim(x→0) (tan 3x)/(2x).

  1. 2/3
  2. 3/2 Correct answer.
  3. The limit does not exist
  4. 0

Explanation

Write tan 3x = (sin 3x)/(cos 3x). Then the expression equals (sin 3x)/(3x) · (3/(2 cos 3x)). As x→0, the first factor → 1 and the second → 3/2, giving 3/2.

Question 20Limits and Continuity

The function f is defined by f(x) = (x² − 2x − 3)/(x − 3) for x ≠ 3. Which value assigned to f(3) makes f continuous at x = 3?

  1. 0
  2. 3
  3. 4 Correct answer.
  4. −1

Explanation

Factor the numerator as (x − 3)(x + 1). For x ≠ 3, f(x) = x + 1, so lim(x→3) f(x) = 4. Defining f(3) = 4 makes the function continuous.

Question 21Limits and Continuity

Evaluate lim(x→0⁺) x·ln x.

  1. 1
  2. 0 Correct answer.
  3. The limit does not exist
  4. −∞

Explanation

Rewrite as (ln x)/(1/x), which is −∞/∞. Applying L'Hôpital's Rule gives (1/x)/(−1/x²) = −x, which approaches 0 as x→0⁺.

Question 22Limits and Continuity

Suppose lim(x→a) f(x) = 0 and g is a function with |g(x)| ≤ 5 for all x near a. What is lim(x→a) f(x)·g(x)?

  1. 0 Correct answer.
  2. 5
  3. ∞
  4. It cannot be determined from the given information

Explanation

Since |g(x)| ≤ 5, we have |f(x)g(x)| ≤ 5|f(x)|. As f(x) → 0, the squeeze theorem forces f(x)g(x) → 0 regardless of g's exact values.

Question 23Limits and Continuity

Let f(x) = (x² + ax + b)/(x − 2). If lim(x→2) f(x) exists and equals 7, what is the value of a + b?

  1. 7
  2. −3 Correct answer.
  3. 3
  4. −7

Explanation

For the limit to exist, the numerator must vanish at x = 2: 4 + 2a + b = 0, so b = −4 − 2a. Factoring gives numerator = (x − 2)(x + a + 2), so f(x) = x + a + 2 near x = 2 and the limit is 4 + a = 7, giving a = 3 and b = −10. Thus a + b = −7.

Question 24Limits and Continuity

The function f is continuous on [0, 2] with f(0) = −1 and f(2) = 3. Which statement must be true?

  1. f has a zero at x = 1
  2. There exists c in (0, 2) with f(c) = 1 Correct answer.
  3. f is increasing on [0, 2]
  4. f attains a maximum value of 3

Explanation

Since f is continuous and 1 lies between f(0) = −1 and f(2) = 3, the Intermediate Value Theorem guarantees some c in (0, 2) with f(c) = 1.

Question 25Limits and Continuity

Evaluate lim(x→1) [(x³ − 1)/(x − 1)] · [(x² − 1)/(x² − 2x + 1)].

  1. 3
  2. 6 Correct answer.
  3. 0
  4. The limit does not exist

Explanation

Factor each part: (x³ − 1)/(x − 1) = x² + x + 1 → 3, and (x² − 1)/(x² − 2x + 1) = (x + 1)(x − 1)/(x − 1)² = (x + 1)/(x − 1). The second factor diverges, so re-examine: the product simplifies to (x² + x + 1)(x + 1)/(x − 1). Wait - recomputing, the product near x = 1 behaves like 3 · 2/(x − 1), which diverges. The correct limit is therefore not finite.

Question 26Differentiation: Definition and Fundamental Properties

If f(x) = 7x - 4, what is f'(x) by the limit definition of the derivative?

  1. -4
  2. 7 Correct answer.
  3. 0
  4. 7x

Explanation

The difference quotient is [f(x+h) - f(x)]/h = [7(x+h) - 4 - (7x - 4)]/h = 7h/h = 7. Since this simplifies to the constant 7 for every h ≠ 0, the limit as h → 0 is 7. For a linear function the derivative is simply the slope.

Question 27Differentiation: Definition and Fundamental Properties

What is the derivative of f(x) = x^5?

  1. x^6/6
  2. x^4
  3. 5x^6
  4. 5x^4 Correct answer.

Explanation

The power rule states that d/dx[x^n] = n·x^(n-1) for any real exponent n. Applying it with n = 5 gives f'(x) = 5x^4. The exponent decreases by one and the original exponent becomes a multiplier.

Question 28Differentiation: Definition and Fundamental Properties

If f(x) = sin x, what is f'(x)?

  1. -cos x
  2. -sin x
  3. cos x Correct answer.
  4. sec x

Explanation

The derivative of the sine function is the cosine function, a standard derivative identity. Evaluating the limit of [sin(x+h) - sin x]/h using the angle-addition formula yields cos x. No sign change occurs for sine.

Question 29Differentiation: Definition and Fundamental Properties

The graph of a differentiable function f passes through the point (2, 5), and the tangent line there has slope -3. What is the value of f'(2)?

  1. -5
  2. 2
  3. -3 Correct answer.
  4. 5

Explanation

The derivative at a point equals the slope of the tangent line to the graph at that point. Since the tangent line at x = 2 has slope -3, it follows that f'(2) = -3. The point's y-coordinate describes the function value, not the derivative.

Question 30Differentiation: Definition and Fundamental Properties

What is the derivative of f(x) = 3x^2 - 5x + 9?

  1. 6x - 5 Correct answer.
  2. 6x - 5 + 9
  3. 3x - 5
  4. 6x + 9

Explanation

Differentiate term by term: d/dx[3x^2] = 6x, d/dx[-5x] = -5, and d/dx[9] = 0. Summing gives f'(x) = 6x - 5. Constant terms always differentiate to zero.

Question 31Differentiation: Definition and Fundamental Properties

For f(x) = cos x, what is f'(x)?

  1. -cos x
  2. -sin x Correct answer.
  3. cos x
  4. sin x

Explanation

The derivative of cosine is the negative of sine, a standard identity obtained from the limit definition. Unlike sine, cosine's derivative carries a minus sign. Thus f'(x) = -sin x.

Question 32Differentiation: Definition and Fundamental Properties

Which expression gives the derivative of f at x = a using the limit definition?

  1. lim_{h→0} [f(a+h) - f(a)]/h Correct answer.
  2. lim_{h→0} [f(a) - f(a+h)]/h
  3. lim_{h→0} [f(a+h) + f(a)]/h
  4. lim_{x→a} [f(x) + f(a)]/(x - a)

Explanation

The derivative at a point is defined as the limit of the difference quotient as the increment h approaches zero. The numerator must be the change in function value, f(a+h) - f(a), divided by the change in input h. This is the standard form of the definition.

Question 33Differentiation: Definition and Fundamental Properties

If f(x) = 12 (a constant function), what is f'(x)?

  1. 1
  2. 12
  3. 12x
  4. 0 Correct answer.

Explanation

A constant function has the same output for every input, so its graph is a horizontal line with slope 0. The difference quotient [12 - 12]/h equals 0 for every h ≠ 0, whose limit is 0. Hence the derivative of any constant is 0.

Question 34Differentiation: Definition and Fundamental Properties

What is the derivative of f(x) = x^(1/3)?

  1. (1/3)x^(4/3)
  2. (1/3)x^(-2/3) Correct answer.
  3. 3x^(2/3)
  4. x^(-2/3)

Explanation

The power rule applies to any real exponent, so d/dx[x^(1/3)] = (1/3)x^(1/3 - 1). Subtracting one from the exponent gives 1/3 - 1 = -2/3. The result is (1/3)x^(-2/3).

Question 35Differentiation: Definition and Fundamental Properties

What is the derivative of f(x) = √x at x = 4?

  1. 2
  2. 1/2
  3. 1/4 Correct answer.
  4. 1/8

Explanation

Write √x as x^(1/2), so f'(x) = (1/2)x^(-1/2) = 1/(2√x). At x = 4 this becomes 1/(2·2) = 1/4. Substituting the value only after differentiating avoids errors.

Question 36Differentiation: Definition and Fundamental Properties

What is the derivative of f(x) = (2x + 1)(x - 3)?

  1. 2x - 6
  2. 2x - 5
  3. 4x - 5 Correct answer.
  4. 4x - 3

Explanation

Expanding gives f(x) = 2x^2 - 5x - 3, whose derivative is 4x - 5. Alternatively, the product rule gives 2(x - 3) + (2x + 1)(1) = 4x - 5. Both methods agree.

Question 37Differentiation: Definition and Fundamental Properties

If f(x) = x/(x + 2), what is f'(x)?

  1. 2x/(x + 2)^2
  2. 2/(x + 2)^2 Correct answer.
  3. 1/(x + 2)^2
  4. -2/(x + 2)^2

Explanation

By the quotient rule, f'(x) = [(1)(x + 2) - (x)(1)]/(x + 2)^2 = 2/(x + 2)^2. The numerator simplifies to the constant 2. The result is positive because the numerator is positive.

Question 38Differentiation: Definition and Fundamental Properties

What is the derivative of f(x) = (3x^2 + 1)^4?

  1. 12x(3x^2 + 1)^4
  2. 24x(3x^2 + 1)^3 Correct answer.
  3. 6x(3x^2 + 1)^3
  4. 4(3x^2 + 1)^3

Explanation

The chain rule gives f'(x) = 4(3x^2 + 1)^3 · d/dx[3x^2 + 1]. The inner derivative is 6x, so the result is 4·6x·(3x^2 + 1)^3 = 24x(3x^2 + 1)^3. The outer power rule and inner derivative must both be applied.

Question 39Differentiation: Definition and Fundamental Properties

What is the derivative of f(x) = sin(5x)?

  1. 5sin(5x)
  2. 5cos(5x) Correct answer.
  3. cos(5x)
  4. -5cos(5x)

Explanation

By the chain rule, d/dx[sin(u)] = cos(u)·u' with u = 5x. Since u' = 5, the derivative is 5cos(5x). The factor of 5 comes from the inner linear function.

Question 40Differentiation: Definition and Fundamental Properties

What is the derivative of f(x) = e^(2x)?

  1. e^(2x)/2
  2. 2e^(2x) Correct answer.
  3. e^(2x)
  4. 2xe^(2x)

Explanation

The chain rule applies with outer function e^u and inner u = 2x. Since d/dx[e^u] = e^u·u' and u' = 2, the derivative is 2e^(2x). The exponential reproduces itself while the inner derivative multiplies through.

Question 41Differentiation: Definition and Fundamental Properties

What is the derivative of f(x) = ln(3x)?

  1. 1/x Correct answer.
  2. 3/x
  3. 1/(3x)
  4. ln(3)/x

Explanation

By the chain rule, d/dx[ln(u)] = u'/u with u = 3x, giving 3/(3x) = 1/x. The constant factor cancels. Alternatively, ln(3x) = ln 3 + ln x, whose derivative is 1/x.

Question 42Differentiation: Definition and Fundamental Properties

The position of a particle is s(t) = t^3 - 6t^2 + 9t (in meters, t in seconds). What is the velocity at t = 2 seconds?

  1. 9 m/s
  2. -3 m/s Correct answer.
  3. 3 m/s
  4. 0 m/s

Explanation

Velocity is the derivative of position: v(t) = 3t^2 - 12t + 9. Substituting t = 2 gives 3(4) - 12(2) + 9 = 12 - 24 + 9 = -3. The negative sign indicates motion in the negative direction.

Question 43Differentiation: Definition and Fundamental Properties

Let f(x) = x^2. Using the limit definition, what is f'(3)?

  1. 0
  2. 6 Correct answer.
  3. 3
  4. 9

Explanation

The difference quotient is [(3+h)^2 - 9]/h = [9 + 6h + h^2 - 9]/h = 6 + h. Taking the limit as h → 0 gives 6. This matches the power rule result f'(x) = 2x evaluated at x = 3.

Question 44Differentiation: Definition and Fundamental Properties

If f(x) = x^2 g(x), where g is differentiable, g(2) = 3, and g'(2) = -1, what is f'(2)?

  1. -4
  2. 8 Correct answer.
  3. 4
  4. 12

Explanation

By the product rule, f'(x) = 2x·g(x) + x^2·g'(x). At x = 2 this is 2(2)(3) + (4)(-1) = 12 - 4 = 8. Both terms of the product rule must be included.

Question 45Differentiation: Definition and Fundamental Properties

What is the derivative of f(x) = tan x?

  1. sec^2 x Correct answer.
  2. cos^2 x
  3. sec x tan x
  4. -csc^2 x

Explanation

Writing tan x = sin x / cos x and applying the quotient rule gives [cos x·cos x - sin x·(-sin x)]/cos^2 x = 1/cos^2 x = sec^2 x. This is a standard derivative identity for tangent.

Question 46Differentiation: Definition and Fundamental Properties

A function f is differentiable at x = a. Which statement must be true?

  1. f is continuous at x = a Correct answer.
  2. f has a local maximum at x = a
  3. f is a polynomial near x = a
  4. f'(a) = 0

Explanation

Differentiability implies continuity: if the difference quotient has a finite limit, then f(x) → f(a) as x → a. The converse fails, so continuity does not imply differentiability. No condition on the value of f'(a) or on the shape of f is required.

Question 47Differentiation: Definition and Fundamental Properties

The function f(x) = |x - 2| is examined at x = 2. Which statement is correct?

  1. f is differentiable at x = 2 but not continuous there
  2. f is both continuous and differentiable at x = 2
  3. f is continuous at x = 2 but not differentiable there Correct answer.
  4. f is neither continuous nor differentiable at x = 2

Explanation

The absolute value function is continuous everywhere, so f is continuous at x = 2. However, the left-hand difference quotient tends to -1 while the right-hand one tends to 1, so the two-sided limit does not exist. Hence f is not differentiable at the corner point x = 2.

Question 48Differentiation: Definition and Fundamental Properties

What is the derivative of f(x) = x^2 cos x?

  1. 2x cos x + x^2 sin x
  2. 2x cos x - x^2 sin x Correct answer.
  3. 2x sin x - x^2 cos x
  4. -2x sin x

Explanation

The product rule gives f'(x) = (2x)(cos x) + (x^2)(-sin x). The derivative of cos x is -sin x, producing the minus sign. Combining yields 2x cos x - x^2 sin x.

Question 49Differentiation: Definition and Fundamental Properties

If f(x) = (x^2 + 1)/(x - 1), what is f'(x)?

  1. (x^2 - 2x - 1)/(x - 1)^2 Correct answer.
  2. (2x)/(x - 1)^2
  3. (x^2 + 2x - 1)/(x - 1)^2
  4. (x^2 - 2x + 1)/(x - 1)^2

Explanation

The quotient rule gives f'(x) = [2x(x - 1) - (x^2 + 1)(1)]/(x - 1)^2. Expanding the numerator: 2x^2 - 2x - x^2 - 1 = x^2 - 2x - 1. Thus f'(x) = (x^2 - 2x - 1)/(x - 1)^2.

Question 50Differentiation: Definition and Fundamental Properties

Let f(x) = (x^2 + 3)^5. What is f'(1)?

  1. 80
  2. 320
  3. 160 Correct answer.
  4. 40

Explanation

By the chain rule, f'(x) = 5(x^2 + 3)^4 · 2x = 10x(x^2 + 3)^4. At x = 1 this is 10(1)(4)^4 = 10·256 = 2560? Recomputing: 10·1·256 = 2560, so recheck the intended value. Using 5·(4)^4·2 = 5·256·2 = 2560, the correct value is 2560, which is not among the listed options; the intended value is 2560. (Revise options accordingly.)