Interpreting the Behavior of Accumulation Functions Involving Area

#6.5 Interpreting the Behavior of Accumulation Functions Involving Area

Welcome back to AP Calculus! 🌶️

In this guide, we'll analyze the behavior of accumulation functions using the Fundamental Theorem of Calculus. We'll focus on graphical, numerical, analytical, and verbal representations to gain a comprehensive understanding of integrally-defined functions.

#👩‍🏫The Fundamental Theorem Of Calculus

The Fundamental Theorem of Calculus links accumulation functions, which are adapted from infinite Riemann sums, to antiderivatives, which “undo” a derivative. This theory states that if

$$F(n)=\int_{a}^{n}f(t)dt$$

Then,

$$F’(n) = f(n)$$

The first equation above states that $F(n)$ is the accumulation function of $f(t)$ which equals the area under the curve of $f(t)$ between the boundaries made by the lines $x =a$ and $x = n$. Here, a is some constant number while n is a variable. Visually, this looks like the following:

!Untitled

Drawing of the accumulation function with boundaries a and n and the area under the curve f(t) colored blue and labeled as F(x)

Image Courtesy of Julianna Fontanilla

The second equation says that the derivative of $F(n)$ is $f(n)$. This also means that the antiderivative of $f(n)$ is $F(n)$, and that $F'(n) = f(n)$.

To put these 2 pieces of information together, the area under a function is equal to the value of its antiderivative (calculated with the same bounds, of course).

#📒 Using the Fundamental Theorem Of Calculus

Just like previous functions, these integrally defined functions can be defined by their various characteristics listed below. Some AP multiple-choice or free-response questions will ask you to analyze these factors given a graph or a function with an integral.

#Quick Refresher Chart

| If F(x) is… | then F’(...