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AP Calculus BC study materials
AP Calculus BC study resources

AP Calculus BC practice questions

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Question 1Limits and Continuity

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

  1. Does not exist
  2. 3
  3. 6 Correct answer.
  4. 0

Explanation

Factor the numerator as (x−3)(x+3); for x≠3 the expression equals x+3, which approaches 6 as x→3. The limit exists even though the function is undefined at x=3.

Question 2Limits and Continuity

The function f is defined by f(x) = (x² − 4)/(x − 2) for x ≠ 2. Which value of f(2) makes f continuous at x = 2?

  1. 0
  2. No value makes f continuous at x = 2
  3. 4 Correct answer.
  4. 2

Explanation

For continuity at x=2, f(2) must equal lim(x→2) f(x). Factoring gives (x−2)(x+2)/(x−2) = x+2, whose limit is 4, so defining f(2)=4 removes the removable discontinuity.

Question 3Limits and Continuity

Evaluate lim(x→0) (sin 5x)/(2x).

  1. 1
  2. 0
  3. Does not exist
  4. 5/2 Correct answer.

Explanation

Rewrite the expression as (5/2)·(sin 5x)/(5x). Since (sin θ)/θ → 1 as θ→0, the limit equals 5/2.

Question 4Limits and Continuity

Let f(x) = (x² + 3x − 4)/(x² − 1). Which statement about the graph of f is true?

  1. It has a removable discontinuity at x = −1 and a vertical asymptote at x = 1
  2. It is continuous at every real number
  3. It has a removable discontinuity at x = 1 and a vertical asymptote at x = −1 Correct answer.
  4. It has vertical asymptotes at both x = 1 and x = −1

Explanation

Factor: numerator = (x+4)(x−1), denominator = (x+1)(x−1). The factor (x−1) cancels, so the discontinuity at x=1 is removable, while the uncancelled factor (x+1) produces a vertical asymptote at x=−1.

Question 5Limits and Continuity

Using the Squeeze Theorem, evaluate lim(x→0) x² cos(1/x).

  1. 1/2
  2. 1
  3. Does not exist
  4. 0 Correct answer.

Explanation

Since −1 ≤ cos(1/x) ≤ 1, we have −x² ≤ x² cos(1/x) ≤ x² for x≠0. Both bounding functions approach 0 as x→0, so the limit is 0.

Question 6Limits and Continuity

The function g is defined piecewise: g(x) = x + 2 for x < 1, g(x) = k for x = 1, and g(x) = 3x for x > 1. What value of k makes g continuous at x = 1?

  1. 3 Correct answer.
  2. 1
  3. No value of k makes g continuous at x = 1
  4. 2

Explanation

The left-hand limit is 1+2=3 and the right-hand limit is 3(1)=3, so the two-sided limit is 3. Continuity requires g(1)=k to equal this limit, so k=3.

Question 7Limits and Continuity

Evaluate lim(x→2⁺) (x + 1)/(x − 2).

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

Explanation

As x→2⁺, the numerator approaches 3 (positive) and the denominator approaches 0 through positive values, so the quotient grows without bound in the positive direction.

Question 8Limits and Continuity

Which statement is the Intermediate Value Theorem?

  1. If f is differentiable on (a,b), then f is continuous on (a,b)
  2. If f is continuous on [a,b], then f(a) = f(b) for some point in (a,b)
  3. If f is continuous on [a,b], then f attains a maximum and a minimum on [a,b]
  4. If f is continuous on [a,b] and k is any number between f(a) and f(b), then f(c)=k for some c in (a,b) Correct answer.

Explanation

The Intermediate Value Theorem guarantees that a continuous function on a closed interval takes every value between its endpoint values at least once. The other statements describe the Extreme Value Theorem, the differentiability-implies-continuity fact, and a misstatement.

Question 9Limits and Continuity

Let f(x) = x³ + x − 1. Which conclusion about f follows from the Intermediate Value Theorem on the interval [0, 1]?

  1. f has a maximum at x = 1/2
  2. f has a zero in (0, 1) because f(0) and f(1) have opposite signs Correct answer.
  3. f has a zero in (0, 1) because f is increasing on [0, 1]
  4. f has no zero in (0, 1)

Explanation

f is a polynomial, hence continuous on [0,1], with f(0)=−1 and f(1)=1. Since 0 lies between these values, the IVT guarantees some c in (0,1) with f(c)=0.

Question 10Limits and Continuity

Evaluate lim(x→∞) (4x³ − 2x + 7)/(2x³ + x²).

  1. 0
  2. 4
  3. Does not exist
  4. 2 Correct answer.

Explanation

Divide numerator and denominator by x³: the expression becomes (4 − 2/x² + 7/x³)/(2 + 1/x). As x→∞ the small terms vanish, leaving 4/2 = 2.

Question 11Limits and Continuity

Evaluate lim(x→0) (1 − cos x)/x².

  1. 0
  2. 1
  3. Does not exist
  4. 1/2 Correct answer.

Explanation

Multiply numerator and denominator by (1 + cos x) to get (sin²x)/(x²(1+cos x)). Since (sin x)/x → 1 and 1+cos x → 2, the limit is 1/2.

Question 12Limits and Continuity

Let f(x) = √(x + 3). Use the limit definition of the derivative to find f′(1).

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

Explanation

The difference quotient is [√(4+h) − 2]/h. Rationalizing the numerator gives h/(h(√(4+h)+2)) = 1/(√(4+h)+2), which tends to 1/4 as h→0.

Question 13Limits and Continuity

The function h is defined by h(x) = (x² − 1)/(x − 1) for x < 1 and h(x) = ax + b for x ≥ 1. Which pair (a, b) makes h continuous at x = 1?

  1. a = 2, b = 0 Correct answer.
  2. a = 2, b = 1
  3. a = 0, b = 2
  4. a = 1, b = 1

Explanation

For x<1, (x²−1)/(x−1) = x+1, so the left-hand limit at x=1 is 2. Continuity requires the right-hand piece to satisfy a(1)+b = 2, which holds for a=2, b=0.

Question 14Limits and Continuity

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

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

Explanation

The numerator has degree 2 and the denominator degree 1, so the quotient grows without bound in magnitude. As x→−∞, the numerator is dominated by 3x² (positive) while the denominator x−2 is negative, giving −∞.

Question 15Differentiation: Definition and Fundamental Properties

Using the limit definition of the derivative, what is the derivative of f(x) = 5x + 3?

  1. 8
  2. 3
  3. 5x
  4. 5 Correct answer.

Explanation

The limit definition gives f'(x) = lim(h→0) [f(x+h) − f(x)]/h = lim(h→0) [5(x+h)+3 − (5x+3)]/h = lim(h→0) 5h/h = 5. The derivative of a linear function is its slope, and the constant term contributes nothing.

Question 16Differentiation: Definition and Fundamental Properties

What is the derivative of f(x) = x^6?

  1. 6x^7
  2. 6x^5 Correct answer.
  3. 6x^6
  4. x^5

Explanation

The power rule states that d/dx[x^n] = n·x^(n−1). With n = 6 this yields 6x^5.

Question 17Differentiation: Definition and Fundamental Properties

Evaluate lim(x→3) (x^2 − 9)/(x − 3).

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

Explanation

Factoring the numerator gives (x−3)(x+3)/(x−3), which equals x+3 for x ≠ 3. The limit as x→3 is therefore 3+3 = 6.

Question 18Differentiation: Definition and Fundamental Properties

The graph of a function f has a sharp corner (cusp-like point) at x = 2. Which statement about differentiability at x = 2 is true?

  1. f is not differentiable at x = 2 because the slopes from the left and right differ Correct answer.
  2. f is differentiable at x = 2 because f has a local minimum there
  3. f is differentiable at x = 2 because f is continuous there
  4. f is differentiable at x = 2 because the corner is a single point

Explanation

Differentiability requires the two-sided limit of the difference quotient to exist. At a corner the left-hand and right-hand slopes differ, so that limit fails to exist. Continuity alone does not guarantee differentiability.

Question 19Differentiation: Definition and Fundamental Properties

If f(x) = (2x + 1)(x^2 − 3), what is f'(x)?

  1. 4x^2 + 2x − 6
  2. 2x^2 + 2x − 3
  3. 6x^2 + 2x − 6 Correct answer.
  4. 6x^2 − 6

Explanation

By the product rule, f'(x) = 2(x^2 − 3) + (2x + 1)(2x) = 2x^2 − 6 + 4x^2 + 2x = 6x^2 + 2x − 6. Expanding first and differentiating gives the same result.

Question 20Differentiation: Definition and Fundamental Properties

What is the derivative of g(x) = (3x^2 + 1)/(x − 2)?

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

Explanation

The quotient rule gives g'(x) = [6x(x−2) − (3x^2+1)(1)]/(x−2)^2 = (6x^2 − 12x − 3x^2 − 1)/(x−2)^2 = (3x^2 − 12x − 1)/(x−2)^2.

Question 21Differentiation: Definition and Fundamental Properties

A function is defined piecewise: f(x) = x^2 for x ≤ 1 and f(x) = ax + b for x > 1. What values of a and b make f differentiable at x = 1?

  1. a = 0, b = 1
  2. a = 2, b = 1
  3. a = 2, b = −1 Correct answer.
  4. a = 1, b = 0

Explanation

Continuity requires 1 = a + b. Differentiability requires the left derivative 2x at x=1, which is 2, to equal the right derivative a, so a = 2. Then b = 1 − 2 = −1.

Question 22Differentiation: Definition and Fundamental Properties

Using the limit definition, what is the derivative of f(x) = √x at x = 9?

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

Explanation

For f(x)=√x the difference quotient rationalizes to 1/(√(x+h)+√x), whose limit is 1/(2√x). At x = 9 this is 1/(2·3) = 1/6.

Question 23Differentiation: Definition and Fundamental Properties

Given f(2) = 5, f'(2) = −3, g(2) = 4, and g'(2) = 6, what is the value of (f·g)'(2)?

  1. 9 Correct answer.
  2. −18
  3. 24
  4. 3

Explanation

The product rule gives (f·g)'(2) = f'(2)g(2) + f(2)g'(2) = (−3)(4) + (5)(6) = −12 + 30 = 9.

Question 24Differentiation: Definition and Fundamental Properties

The position of a particle is s(t) = t^3 − 6t^2 + 9t (in meters, t in seconds). At what time is the particle momentarily at rest for t > 0?

  1. t = 3 only
  2. t = 1 and t = 3 Correct answer.
  3. t = 2 only
  4. t = 0 and t = 2

Explanation

The velocity is s'(t) = 3t^2 − 12t + 9 = 3(t−1)(t−3). Setting velocity to zero gives t = 1 and t = 3, both positive.

Question 25Differentiation: Definition and Fundamental Properties

What is the derivative of h(x) = 4x^3 − 7x + 2?

  1. 12x^3 − 7
  2. 12x^2 − 7 Correct answer.
  3. 12x^2 − 7x
  4. 4x^2 − 7

Explanation

Differentiate term by term: d/dx[4x^3] = 12x^2, d/dx[−7x] = −7, and d/dx[2] = 0. The result is 12x^2 − 7.

Question 26Differentiation: Definition and Fundamental Properties

Suppose f is differentiable at x = a. Which of the following must be true?

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

Explanation

Differentiability implies continuity: if the difference quotient has a finite limit, then f(x) → f(a) as x → a. The converse fails, but this direction always holds.

Question 27Differentiation: Definition and Fundamental Properties

Let f(x) = x^2·g(x), where g is differentiable and g(3) = 2, g'(3) = −1. What is f'(3)?

  1. 6
  2. −3
  3. 12
  4. 9 Correct answer.

Explanation

By the product rule, f'(x) = 2x·g(x) + x^2·g'(x). At x = 3 this is 2(3)(2) + (9)(−1) = 12 − 9 = 3.

Question 28Differentiation: Definition and Fundamental Properties

Using the limit definition, what is the derivative of f(x) = 1/x at x = −2?

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

Explanation

The difference quotient simplifies to −1/[x(x+h)], whose limit is −1/x^2. At x = −2 this is −1/4.

Question 29Differentiation: Composite, Implicit, and Inverse Functions

If f(x) = (3x^2 + 1)^5, what is f'(x)?

  1. 30x(3x^2 + 1)^4 Correct answer.
  2. 15x(3x^2 + 1)^4
  3. 6x(3x^2 + 1)^5
  4. 5(3x^2 + 1)^4

Explanation

The chain rule gives f'(x) = 5(3x^2 + 1)^4 times the derivative of the inner function 3x^2 + 1, which is 6x. Multiplying yields 30x(3x^2 + 1)^4.

Question 30Differentiation: Composite, Implicit, and Inverse Functions

What is the derivative of sin(x^2) with respect to x?

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

Explanation

The outer function is sine, whose derivative is cosine, and the inner function is x^2, whose derivative is 2x. By the chain rule the result is cos(x^2) times 2x.

Question 31Differentiation: Composite, Implicit, and Inverse Functions

If y = e^(4x), what is dy/dx?

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

Explanation

The derivative of e^u is e^u times du/dx. Here u = 4x so du/dx = 4, giving dy/dx = 4e^(4x).

Question 32Differentiation: Composite, Implicit, and Inverse Functions

What is the derivative of ln(5x) for x > 0?

  1. 1/x Correct answer.
  2. 1/(5x)
  3. 5/x
  4. ln(5)/x

Explanation

The derivative of ln(u) is (1/u)(du/dx). With u = 5x, du/dx = 5, so the derivative is (1/(5x))(5) = 1/x.

Question 33Differentiation: Composite, Implicit, and Inverse Functions

Given the equation x^2 + y^2 = 25, what is dy/dx?

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

Explanation

Differentiating both sides with respect to x gives 2x + 2y(dy/dx) = 0. Solving for dy/dx yields -x/y.

Question 34Differentiation: Composite, Implicit, and Inverse Functions

If x^3 + y^3 = 6xy, what is dy/dx?

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

Explanation

Differentiating gives 3x^2 + 3y^2(dy/dx) = 6y + 6x(dy/dx). Collecting dy/dx terms and dividing by 3 yields (2y - x^2)/(y^2 - 2x).

Question 35Differentiation: Composite, Implicit, and Inverse Functions

Let f(x) = x^3 + 2x. What is the value of (f^{-1})'(3) given that f(1) = 3?

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

Explanation

The inverse function theorem states (f^{-1})'(b) = 1/f'(a) where f(a) = b. Here f'(x) = 3x^2 + 2, so f'(1) = 5, giving 1/5.

Question 36Differentiation: Composite, Implicit, and Inverse Functions

If f(x) = arctan(3x), what is f'(x)?

  1. 3/(1 + 9x^2) Correct answer.
  2. 1/(1 + 9x^2)
  3. 3/(1 + 3x^2)
  4. 1/(1 + 3x^2)

Explanation

The derivative of arctan(u) is u'/(1 + u^2). With u = 3x, u' = 3 and u^2 = 9x^2, giving 3/(1 + 9x^2).

Question 37Differentiation: Composite, Implicit, and Inverse Functions

What is the derivative of arcsin(x) with respect to x?

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

Explanation

The standard derivative of arcsin(x) is 1/sqrt(1 - x^2) for -1 < x < 1. The minus sign belongs to arccos, not arcsin.

Question 38Differentiation: Composite, Implicit, and Inverse Functions

If h(x) = f(g(x)) with f(2) = 5, f'(2) = 3, g(1) = 2, and g'(1) = 4, what is h'(1)?

  1. 20
  2. 12 Correct answer.
  3. 15
  4. 8

Explanation

By the chain rule, h'(1) = f'(g(1)) * g'(1) = f'(2) * g'(1) = 3 * 4 = 12.

Question 39Differentiation: Composite, Implicit, and Inverse Functions

The curve x^2 + xy + y^2 = 7 passes through the point (1, 2). What is the slope of the tangent line at that point?

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

Explanation

Differentiating gives 2x + y + x(dy/dx) + 2y(dy/dx) = 0. At (1, 2): 2 + 2 + (dy/dx) + 4(dy/dx) = 0, so 5(dy/dx) = -4 and dy/dx = -4/5.

Question 40Differentiation: Composite, Implicit, and Inverse Functions

Let f(x) = x^5 + x^3 + x. Since f is strictly increasing, it has an inverse. What is (f^{-1})'(3) given that f(1) = 3?

  1. 1/8
  2. 1/3
  3. 9
  4. 1/9 Correct answer.

Explanation

Here f'(x) = 5x^4 + 3x^2 + 1, so f'(1) = 5 + 3 + 1 = 9. The inverse function theorem gives (f^{-1})'(3) = 1/f'(1) = 1/9.

Question 41Differentiation: Composite, Implicit, and Inverse Functions

If y = (x^2 + 1)^(1/2) * e^(2x), what is dy/dx?

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

Explanation

Applying the product rule with u = sqrt(x^2 + 1) and v = e^(2x): u' = x/sqrt(x^2 + 1) and v' = 2e^(2x). Factoring e^(2x) gives e^(2x)(x/sqrt(x^2 + 1) + 2sqrt(x^2 + 1)).

Question 42Differentiation: Composite, Implicit, and Inverse Functions

For x > 0, what is the derivative of x^x with respect to x?

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

Explanation

Write x^x = e^(x ln x) and differentiate: the derivative is e^(x ln x) times (ln x + 1) by the product rule on x ln x. Since e^(x ln x) = x^x, the result is x^x(ln x + 1).

Question 43Contextual Applications of Differentiation

A particle moves along a line with position s(t) = t^3 - 6t^2 + 9t (meters, seconds). What is the velocity at t = 2?

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

Explanation

Velocity is the derivative of position: s'(t) = 3t^2 - 12t + 9. Substituting t = 2 gives 12 - 24 + 9 = -3 m/s.

Question 44Contextual Applications of Differentiation

Water drains from a tank so that its volume V (liters) satisfies dV/dt = -4 liters per minute. At what rate is the volume changing in liters per hour?

  1. -4 L/hr
  2. 240 L/hr
  3. -1/15 L/hr
  4. -240 L/hr Correct answer.

Explanation

Multiply the per-minute rate by 60 minutes per hour: -4 * 60 = -240 L/hr. The negative sign reflects that the volume is decreasing.

Question 45Contextual Applications of Differentiation

A differentiable function f has f(3) = 5 and f'(3) = -2. Using the tangent-line approximation, what is the best estimate of f(3.1)?

  1. 5.1
  2. 5.2
  3. 4.8 Correct answer.
  4. 4.9

Explanation

Linear approximation uses f(a) + f'(a)(x - a) = 5 + (-2)(0.1) = 4.8. The derivative gives the local rate of change used to extrapolate.

Question 46Contextual Applications of Differentiation

The radius of a circular ripple grows at a constant rate of 2 cm/s. At the instant the radius is 5 cm, how fast is the area increasing?

  1. 10π cm²/s
  2. 25π cm²/s
  3. 20π cm²/s Correct answer.
  4. 4π cm²/s

Explanation

With A = πr², differentiating gives dA/dt = 2πr dr/dt. Substituting r = 5 and dr/dt = 2 yields 2π(5)(2) = 20π cm²/s.

Question 47Contextual Applications of Differentiation

A particle has velocity v(t) = t^2 - 4t + 3 for t ≥ 0. On which interval is the particle moving to the left?

  1. (1, 3) Correct answer.
  2. (0, 3)
  3. (0, 1)
  4. (3, ∞)

Explanation

Moving left means v(t) < 0. Factoring gives (t - 1)(t - 3), which is negative strictly between the roots, on (1, 3).

Question 48Contextual Applications of Differentiation

A ladder 10 ft long leans against a vertical wall. The bottom slides away from the wall at 1 ft/s. When the bottom is 6 ft from the wall, how fast is the top sliding down?

  1. 8/3 ft/s
  2. 4/3 ft/s
  3. 5/3 ft/s
  4. 3/4 ft/s Correct answer.

Explanation

With x² + y² = 100, differentiating gives 2x dx/dt + 2y dy/dt = 0. At x = 6, y = 8, so dy/dt = -(6)(1)/8 = -3/4 ft/s, i.e., the top descends at 3/4 ft/s.

Question 49Contextual Applications of Differentiation

Sand pours onto a pile forming a cone whose height always equals its base radius. If sand is added at 12 cm³/s, how fast is the height rising when the height is 3 cm?

  1. 4/π cm/s
  2. 3/(4π) cm/s
  3. 12/π cm/s
  4. 4/(3π) cm/s Correct answer.

Explanation

With h = r, V = (1/3)πh³, so dV/dt = πh² dh/dt. Setting 12 = π(9) dh/dt gives dh/dt = 12/(9π) = 4/(3π) cm/s.

Question 50Contextual Applications of Differentiation

Using a linear approximation based at x = 4, estimate √4.2 given that the derivative of √x at x = 4 is 1/4.

  1. 2.05 Correct answer.
  2. 2.025
  3. 2.10
  4. 2.20

Explanation

The tangent line is y = 2 + (1/4)(x - 4). At x = 4.2 this gives 2 + (1/4)(0.2) = 2.05.