AP Calculus AB 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.
If $F(x) = \int_{-5}^{x} (t^3 - 2t) dt$, what is $F'(x)$?
Using the Fundamental Theorem of Calculus, $F'(x) = x^3 - 2x$.
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If $F(x) = \int_{-5}^{x} (t^3 - 2t) dt$, what is $F'(x)$?
Using the Fundamental Theorem of Calculus, $F'(x) = x^3 - 2x$.
What is the formula for the derivative of an accumulation function, as given by the Fundamental Theorem of Calculus?
The formula is $\\frac{d}{dx} \int_{a}^{x} f(t) dt = f(x)$.
What is the name for a function defined by a definite integral with a variable upper limit, such as $g(x) = \int_{a}^{x} f(t) dt$?
This type of function is called an accumulation function.
How does the Fundamental Theorem of Calculus describe the relationship between differentiation and integration?
It establishes that differentiation and integration are inverse operations; taking the derivative of an integral of a function returns the original function.
Evaluate $\\frac{d}{dx} \int_{0}^{x} e^{t^2} dt$.
By applying the Fundamental Theorem of Calculus, the result is $e^{x^2}$.
State the part of the Fundamental Theorem of Calculus that relates the derivative of an integral.
If $f$ is a continuous function on an interval containing $a$, then the derivative of the accumulation function $\int_{a}^{x} f(t) dt$ with respect to $x$ is equal to $f(x)$.
Find the derivative of the function $g(x) = \int_{2}^{x} \\cos(t) dt$.
According to the Fundamental Theorem of Calculus, the derivative is $\\cos(x)$.
How can an accumulation function be represented?
An accumulation function can be represented using a definite integral where the upper limit of integration is a variable, such as $F(x) = \int_{a}^{x} f(t) dt$.
What condition must the function $f$ satisfy to apply the theorem $\\frac{d}{dx} \int_{a}^{x} f(t) dt = f(x)$?
The function $f$ must be continuous on the interval of integration that contains both $a$ and $x$.
What is the primary way that definite integrals can be used to create new functions?
Definite integrals can define new functions, known as accumulation functions, by measuring the accumulated area under a curve from a fixed point to a variable point.