AP Calculus BC Flashcards: The Fundamental Theorem of Calculus and Accumulation Functions
Written by AP Content Team, Verified for 2026 AP Exams, Last updated: May 2026
Review key ideas with interactive flashcards. This set includes 10 cards to help you master important concepts.
What is the formal statement of the first part of the Fundamental Theorem of Calculus regarding derivatives of integrals?
If a function f is continuous on an interval containing a, then the derivative of its accumulation function is given by the formula d/dx ∫[a, x] f(t) dt = f(x).
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What is the formal statement of the first part of the Fundamental Theorem of Calculus regarding derivatives of integrals?
If a function f is continuous on an interval containing a, then the derivative of its accumulation function is given by the formula d/dx ∫[a, x] f(t) dt = f(x).
What is the relationship between differentiation and integration as shown by the Fundamental Theorem of Calculus?
The theorem d/dx ∫[a, x] f(t) dt = f(x) demonstrates that differentiation and integration are inverse operations; taking the derivative of an integral of a function returns the original function.
Write a definite integral to represent a function F(x) that accumulates the values of f(t) = t³ starting from t=0.
The accumulation function can be represented by the definite integral F(x) = ∫[0, x] t³ dt.
Define an accumulation function.
An accumulation function is a function defined by a definite integral, such as F(x) = ∫[a, x] f(t) dt, which measures the accumulated area under the curve of f(t) from a constant 'a' to a variable 'x'.
Find the derivative of the function h(x) = ∫[-1, x] √(t²+1) dt.
Using the Fundamental Theorem of Calculus, the derivative h'(x) is found by replacing the variable of integration 't' with 'x', resulting in h'(x) = √(x²+1).
What key condition must the function f satisfy for the rule d/dx ∫[a, x] f(t) dt = f(x) to be valid?
The function f must be continuous on the interval of integration that contains both the constant 'a' and the variable 'x'.
Given the accumulation function G(x) = ∫[3, x] sin(t) dt, what is G'(x)?
According to the Fundamental Theorem of Calculus, the derivative G'(x) is simply the integrand evaluated at x, so G'(x) = sin(x).
In the expression F(x) = ∫[a, x] f(t) dt, why is the variable of integration denoted by 't' instead of 'x'?
A different variable, 't', is used as a dummy variable for integration because 'x' is already used as the upper limit of the integral and the input for the function F(x).
What mathematical tool is used to represent an accumulation function?
An accumulation function is represented using a definite integral, which calculates the net accumulation of a quantity over an interval.
How can a definite integral be used to define a new function?
A new function, often called an accumulation function, can be defined by a definite integral where the upper limit of integration is a variable, such as F(x) = ∫[a, x] f(t) dt.