AP Calculus BC Practice Quiz: The Fundamental Theorem of Calculus and Accumulation Functions

Written by AP Content Team, Verified for 2026 AP Exams, Last updated: July 2026

Test your understanding with short quizzes. This quiz has 9 questions to check your progress.

Question 1 of 9

A function F(x) is defined as the net accumulation of a quantity whose rate of change is given by the continuous function f(t) from time t=a to time t=x. How can F(x) be represented?

A)F(x) = ∫ₐˣ f(t) dtB)F(x) = f(x) - f(a)C)F(x) = d/dx f(x)D)F(x) = ∫ₐᵇ f(x) dx

All Questions (9)

A function F(x) is defined as the net accumulation of a quantity whose rate of change is given by the continuous function f(t) from time t=a to time t=x. How can F(x) be represented?

A) F(x) = ∫ₐˣ f(t) dt

B) F(x) = f(x) - f(a)

C) F(x) = d/dx f(x)

D) F(x) = ∫ₐᵇ f(x) dx

Correct Answer: A

An accumulation function that measures the net change of a quantity from a fixed starting point 'a' to a variable endpoint 'x', based on a rate function f(t), is defined by the definite integral ∫ₐˣ f(t) dt. This aligns with the definition of new functions using definite integrals. [cite: 2641, 2642]

Let G(x) = ∫₃ˣ (t² + 5t) dt. What is G'(x)?

A) x³/3 + 5x²/2

B) x² + 5x

C) 2t + 5

D) x² + 5x - (3² + 5(3))

Correct Answer: B

According to the Fundamental Theorem of Calculus, if G(x) = ∫ₐˣ f(t) dt, then G'(x) = f(x). In this case, f(t) = t² + 5t, so G'(x) = f(x) = x² + 5x. [cite: 2643]

Find the derivative of the function H(x) = ∫₋₁ˣ e^(cos(t)) dt.

A) e^(cos(x))

B) e^(cos(x)) - e^(cos(-1))

C) -sin(x) e^(cos(x))

D) e^(cos(t))

Correct Answer: A

By the Fundamental Theorem of Calculus, d/dx ∫ₐˣ f(t) dt = f(x). Here, the integrand is f(t) = e^(cos(t)). Therefore, the derivative of H(x) is H'(x) = f(x) = e^(cos(x)). [cite: 2643]

Let F(x) be a function defined by F(x) = ∫₀ˣ √(t³ + 1) dt. What is the value of F'(2)?

A) 2

B) 3

C) √17

D) 0

Correct Answer: B

First, find the derivative F'(x) using the Fundamental Theorem of Calculus. If F(x) = ∫₀ˣ √(t³ + 1) dt, then F'(x) = √(x³ + 1). To find F'(2), substitute x=2 into the derivative: F'(2) = √(2³ + 1) = √(8 + 1) = √9 = 3. [cite: 2643]

What is d/dx ∫ₓ⁵ ln(t² + 2) dt?

A) ln(x² + 2)

B) -ln(x² + 2)

C) ln(27) - ln(x² + 2)

D) 2x / (x² + 2)

Correct Answer: B

First, use the property of definite integrals that ∫ₐᵇ f(t) dt = -∫ₐᵇ f(t) dt. So, ∫ₓ⁵ ln(t² + 2) dt = -∫₅ˣ ln(t² + 2) dt. Now, applying the Fundamental Theorem of Calculus: d/dx (-∫₅ˣ ln(t² + 2) dt) = - (d/dx ∫₅ˣ ln(t² + 2) dt) = -ln(x² + 2). [cite: 2643]

A function g is continuous, and a new function G is defined by G(x) = ∫₁ˣ g(t) dt. If g(x) < 0 for all x on the interval [1, 5], which of the following must be true about the function G on the interval (1, 5)?

A) G is increasing on (1, 5).

B) G is decreasing on (1, 5).

C) G is concave up on (1, 5).

D) G has a local maximum on (1, 5).

Correct Answer: B

According to the Fundamental Theorem of Calculus, G'(x) = g(x). The problem states that g(x) < 0 on the interval [1, 5]. Therefore, G'(x) < 0 on (1, 5). A function whose derivative is negative on an interval is decreasing on that interval. [cite: 2643]

The graph of a continuous function f is shown above. Let H(x) be the function defined by H(x) = ∫₋₂ˣ f(t) dt. What is the value of H'(1)?

A) -1

B) 0

C) 1

D) 2

Correct Answer: A

By the Fundamental Theorem of Calculus, H'(x) = d/dx ∫₋₂ˣ f(t) dt = f(x). Therefore, H'(1) = f(1). Looking at the provided graph of f, the value of the function at x=1 is f(1) = -1. [cite: 2643]

The rate at which a pollutant is entering a lake is given by the continuous function P(t), measured in gallons per hour, where t is the number of hours since midnight. Which expression represents the total amount of pollutant, in gallons, that has entered the lake between 4 AM and 10 AM?

A) P(10) - P(4)

B) (P(10) - P(4)) / (10 - 4)

C) ∫₄¹⁰ P(t) dt

D) d/dt P(10)

Correct Answer: C

The total accumulation of a quantity over an interval is found by integrating the rate of change of that quantity over the interval. The rate is P(t), and the interval is from t=4 to t=10. Therefore, the total amount of pollutant is represented by the definite integral ∫₄¹⁰ P(t) dt. [cite: 2641, 2647]

Let f be a continuous function and let F(x) = ∫₂ˣ f(t) dt. If f(2) = 5, what is the equation of the line tangent to the graph of F at x=2?

A) y = 5x - 10

B) y = 5x

C) y = 5x + 2

D) y = 2

Correct Answer: A

To find the equation of the tangent line, we need a point and a slope. The point is (2, F(2)). The value of the function at x=2 is F(2) = ∫₂² f(t) dt = 0. So the point is (2, 0). The slope of the tangent line is given by the derivative F'(x) at x=2. Using the Fundamental Theorem of Calculus, F'(x) = f(x). Therefore, the slope at x=2 is F'(2) = f(2) = 5. Using the point-slope form y - y₁ = m(x - x₁), we get y - 0 = 5(x - 2), which simplifies to y = 5x - 10. [cite: 2641, 2643]