Interpreting the Behavior of Accumulation Functions Involving Area - AP Calculus BC Study Guide

The Core Idea: Interpreting the Behavior of Accumulation Functions Involving Area

An accumulation function is a function defined by a definite integral with a variable upper limit, typically in the form $g(x)=\int [a,x]f(t)dt$. The fundamental concept of this topic is understanding that this new function, $g(x)$, represents the accumulated net area under the curve of $f(t)$ starting from a fixed point $t=a$ and ending at a variable point $t=x$.

The core task is to analyze the behavior of $g(x)$—where it increases, decreases, has maxima or minima, and how it curves (concavity)—by examining the properties of the function $f(t)$ being integrated. The relationship between these two functions is governed by the Fundamental Theorem of Calculus, which establishes a direct link between the derivative of the accumulation function $g(x)$ and the original function $f(x)$. By interpreting the graph or properties of $f(x)$, we can deduce the complete behavior of $g(x)$ without ever needing to find an explicit formula for $g(x)$.

Key Formulas/Rules/Theorems

The primary theorem governing the behavior of accumulation functions is the Second Fundamental Theorem of Calculus. This theorem provides the crucial link between the accumulation function and the function being integrated (the integrand).

The Second Fundamental Theorem of Calculus

If $g(x)$ is an accumulation function defined as:

$$g(x)={\int }_a^{x}f(t)dt$$

where $a$ is a constant, then the derivative of $g(x)$ with respect to $x$ is:

$$g^{\prime }(x)=\frac{d}{dx}\left[{\int }_a^{x}f(t)dt\right]=f(x)$$

Explanation:

This theorem states that the rate of change of the accumulated net area under $f$ at a point $x$ is simply the value of the function $f$ at that point $x$. In simpler terms, the derivative of an integral-defined function is the integrand itself, evaluated at the variable upper limit. This powerful result allows us to use all the tools of differential calculus to analyze $g(x)$ by simply looking at $f(x)$.

Understanding the Connection Between $g(x)$ and $f(x)$

The relationship $g^{\prime }(x)=f(x)$ is the key to interpreting the behavior of $g(x)$. By extending this, we also find that $g^{\prime \prime }(x)=f^{\prime }(x)$. This means that the graph of $f$ provides complete information about the first and second derivatives of $g$. The following table summarizes these critical connections:

Core Concepts & Rules

• Definition: An accumulation function, $g(x)=\int [a,x]f(t)dt$, calculates the net signed area between the graph of $f$ and the horizontal axis from a constant $a$ to a variable $x$.

• The Derivative Connection: The derivative of the accumulation function $g(x)$ is the integrand $f(x)$. That is, $g^{\prime }(x)=f(x)$.

• Increasing/Decreasing Behavior: The function $g(x)$ is increasing on intervals where its derivative, $f(x)$, is positive (i.e., the graph of $f$ is above the x-axis). $g(x)$ is decreasing where $f(x)$ is negative.

• Relative Extrema: The function $g(x)$ has a relative maximum where its derivative, $f(x)$, changes from positive to negative. It has a relative minimum where $f(x)$ changes from negative to positive.

• Concavity: The function $g(x)$ is concave up on intervals where its second derivative, $g^{\prime \prime }(x)=f^{\prime }(x)$, is positive. This occurs where the function $f(x)$ is increasing. $g(x)$ is concave down where $f(x)$ is decreasing.

• Points of Inflection: The function $g(x)$ has a point of inflection where its concavity changes. This occurs where $g^{\prime \prime }(x)=f^{\prime }(x)$ changes sign, which corresponds to points where $f(x)$ has a local maximum or minimum.

Step-by-Step Example 1: Basic Application

Let the function $f$ be the piecewise-linear function shown in the graph below. Let $g$ be the function defined by $g(x)=\int [-2,x]f(t)dt$.