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

1. What an Accumulation Function Is

An accumulation function is defined by

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

This means:

• $g(x)$ is the net signed area under $f$ from $x=a$ to $x=x$.
• Area above the x-axis counts positive.
• Area below the x-axis counts negative.

By the Fundamental Theorem of Calculus:

• $g'(x) = f(x)$
• $g''(x) = f'(x)$

That connection drives everything in this unit.

If you remember nothing else, remember this:

The graph of $f$ is the graph of $g'$.

So:

• If $f(x) > 0$, then $g'(x) > 0$ → g is increasing
• If $f(x) < 0$, then $g'(x) < 0$ → g is decreasing
• If $f(x) = 0$, then $g'(x)=0$ → possible critical point of g

Since $g''(x) = f'(x)$:

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

You are always analyzing $f$ to understand $g$.

2. Analyzing $g(x) = \int_a^x f(t)\,dt$ from Information About $f$

Most problems give you a graph, table, equation, or context for $f$ and ask about $g$.

A. Increasing and Decreasing

Because $g'(x)=f(x)$:

• $g$ increases where $f$ is above the x-axis.
• $g$ decreases where $f$ is below the x-axis.

For local extrema of $g$:

• Local max when $f$ changes from positive to negative.
• Local min when $f$ changes from negative to positive.

Zeros alone are not enough. You must check the sign change.

B. Concavity and Inflection Points

Because $g''(x)=f'(x)$:

• $g$ is concave up where $f$ is increasing.
• $g$ is concave down where $f$ is decreasing.

Inflection points of $g$ happen where $f$ changes from increasing to decreasing or vice versa.

Students often mix this up and look at where $f=0$. That tells you about extrema of $g$, not concavity.

C. Finding Values of $g(x)$

To find $g(x)$, interpret it as signed area.

If you’re given a graph, break it into geometric shapes and compute each piece.

In the graph above, you can see a triangle above the axis, a rectangle below the axis, and a semicircle above the axis. Each contributes positive or negative area to $g(x)$.

For example:

• Triangle area $=\frac{1}{2}bh$
• Rectangle area $=bh$
• Semicircle area $=\frac{1}{2}\pi r^2$

Add areas above the axis. Subtract areas below.

Be careful with:

• Reversing bounds (which flips the sign)
• Forgetting that below-axis area is negative

On FRQs, you must show the geometric work to earn full credit.

3. Working Across Representations

The AP rotates formats constantly. Same idea, different surface.

A. Graph of $f$

Most common situation.

In the diagram, you’re given the graph of $y=f(x)$ with shaded regions representing signed area.

Graph of $y=f(x)$ with signed area regions

When you see this:

• Treat it as $g'$.
• Sign of graph → increasing/decreasing of $g$.
• Slope of graph → concavity of $g$.
• Area → values of $g$.

Big trap: students describe the graph as if it were $g$. It’s not.

B. Table of Values

If given values of $f$:

• Positive entries → $g$ increasing there.
• Negative entries → $g$ decreasing.
• Changes in $f$ help you estimate concavity of $g$.
• Approximate $g(b)$ using Riemann sums if asked.

You cannot magically know exact values unless integration is possible.

C. Equation for $f$

If $f(x)$ is given explicitly:

• Solve $f(x)>0$ for increasing.
• Compute $f'(x)$ for concavity.
• Evaluate definite integrals directly for $g(x)$.

On no-calculator sections, expect manageable algebra.

D. Verbal Context

If $f$ is a rate (gallons/hour, people/minute, meters/second):

• $g$ is the total accumulated amount.
• Units of $g$ = (units of $f$) × (units of $x$).

Example: velocity in meters/second → accumulation gives total displacement in meters.

4. Absolute Extrema of an Accumulation Function

On a closed interval:

1. Find critical points where $f(x)=0$.
2. Evaluate $g$ at:
   • Endpoints
   • Those critical points
3. Compare values.

Do not assume the highest point of $f$ gives the max of $g$. Accumulation depends on total area, not height.

5. Common Mistakes

• Mixing up $f$, $g$, $g'$, and $g''$.
• Forgetting endpoints for absolute extrema.
• Calling every zero of $f$ a max or min of $g$.
• Using sign of $f$ to determine concavity of $g$.
• Dropping a negative when area is below the axis.