Topic 6.5 Notes – Interpreting the Behavior of Accumulation Functions Involving Area

What an Accumulation Function Is

An accumulation function has the form

$$g(x) = \int_a^x f(t)\,dt$$

Here’s what that means:

• It represents the signed area under $f(t)$ from $t=a$ to $t=x$.
• The lower limit $a$ is constant.
• The upper limit is the variable.

By the Fundamental Theorem of Calculus,

• $g'(x) = f(x)$
• $g''(x) = f'(x)$

So the function you’re given inside the integral, $f$, is actually the derivative of $g$.

That idea drives everything.

Visualizing Accumulation

Here’s what that definition looks like geometrically. In the graph below, the red curve represents $f$, and $g(x) = \int_a^x f(t)\,dt$ measures the accumulated signed area from the fixed point $a$ to the moving point $x$.

Accumulation function $g(x)=\int_a^x f(t)\,dt$

As $x$ moves to the right, more area is added, or subtracted if the graph is below the axis. That changing total area is the value of $g(x)$.

How Behavior of g Comes From f

Since $g'(x) = f(x)$, the graph of $f$ tells you everything about the behavior of $g$.

Increasing and Decreasing

• If $f(x) > 0$, then $g'(x) > 0$ → $g$ is increasing.
• If $f(x) < 0$, then $g$ is decreasing.
• If $f(x) = 0$, that’s a critical point of $g$.

You are reading the graph of $f$ as if it were the derivative of another function.

Relative Max and Min

Relative extrema occur where:

1. $f(x) = 0$, and
2. $f(x)$ changes sign.

Sign change patterns:

• $+ \to -$ → relative maximum of $g$
• $- \to +$ → relative minimum of $g$

Just like first derivative tests, except the derivative is already given to you.

Concavity

Since $g''(x) = f'(x)$, concavity depends on whether $f$ is increasing or decreasing.

• If $f$ is increasing → $g$ is concave up.
• If $f$ is decreasing → $g$ is concave down.

So now you’re analyzing the slope of the graph of $f$.

Points of Inflection

Inflection points of $g$ occur where:

• $g''(x)$ changes sign
• meaning $f'(x)$ changes sign
• meaning the slope of $f$ changes from positive to negative or vice versa.

Important trap:
A point where $f(x)=0$ is not automatically an inflection point. The slope of $f$ must change sign.

Using Area to Find Function Values

Sometimes you’re told a value like $f(c)=k$ and given a graph of $f'$. You use:

$$f(b) = f(a) + \int_a^b f'(x)\,dx$$

The integral gives net change.

• Area above the x-axis → positive change
• Area below the x-axis → negative change

On tests, the graph often forms geometric shapes. For example:

Geometric areas on a graph of $f'(x)$

From $x=0$ to $x=2$, the rectangle gives $+4$. From $x=2$ to $x=4$, the triangle gives $-2$. The net change is $+2$.

Break regions into rectangles, triangles, semicircles, etc. Keep signs correct. If you reverse bounds, the sign flips.

A clean setup that earns full credit on FRQs looks like:

$$f(\text{starting value}) + \int_{\text{start}}^{\text{end}} f'(x)\,dx = f(\text{end})$$

Students lose points by skipping that equation line.

Graph, Table, and Equation Representations

You’ll see accumulation functions represented in three main ways.

1. Graph of $f$

• Treat the graph as $g'(x)$.
• Use sign for increasing/decreasing.
• Use slope for concavity.
• Use geometry for values.

This is extremely common on non-calculator FRQs.

2. Table of Values

If a table gives $f(x)$:

• That’s $g'(x)$.
• Signs determine behavior.

If it gives $f'(x)$:

• That controls concavity of $f$.

Always pause and ask yourself:
“What level of derivative am I looking at?”

3. Equation

Example:

$$G(x)=\int_1^x 3t^2\,dt$$

Then:

• $G'(x)=3x^2$
• Since $3x^2 \ge 0$, $G$ is increasing everywhere.
• $G''(x)=6x$, so concavity depends on sign of $x$.

When it’s algebraic like this, you analyze normally after taking the derivative.