Topic 6.5 Interpreting the Behavior of Accumulation Functions Involving Area: analyze extrema and concavity of an accumulation function using the graph of the integrand.

What this topic is asking

The College Board (Topic 6.5) applies the analytical toolkit of Unit 5 to an accumulation function $g(x) = \int_a^x f(t)\,dt$, using the graph of the integrand $f$. Since $g' = f$ and $g'' = f'$, the sign and slope of $f$ control the behavior of $g$.

The translation from f to g

A worked area analysis

Computing values of g as areas

Beyond behavior, the exam asks for values of $g$ at specific points, which you read as signed areas under $f$. For example $g(4) = \int_0^4 f(t)\,dt$ is the net signed area of the region between $f$ and the axis from $0$ to $4$, computed by adding areas above the axis and subtracting areas below. When $f$ is made of line segments and simple shapes, this is straightforward geometry. Combining value computations (signed area) with behavior analysis (sign and slope of $f$) is the full version of this topic, and it appears as a multi-part free-response question almost every year.

Why g lags one level behind f

The key mental model is that $g$ sits one level above $f$: $f$ plays the role of $g$'s first derivative, and $f'$ plays the role of $g$'s second derivative. So features you would normally read from $f'$ and $f''$ when analyzing a function $f$ are instead read from $f$ and $f'$ when analyzing $g$. The most common error is treating the graph of $f$ as if it were the graph of $g$ itself, for instance saying $g$ has a maximum where $f$ has a maximum. It does not: $g$ has a maximum where $f$ crosses zero from positive to negative. Keeping the level shift straight is the whole skill.

Absolute extrema of an accumulation function

To find the absolute maximum or minimum of $g(x) = \int_a^x f(t)\,dt$ on a closed interval, treat it as a candidates-test problem on $g$. The critical points of $g$ are where $g'(x) = f(x) = 0$, that is, where the graph of $f$ crosses the axis. Evaluate $g$ (as a signed area) at those critical points and at the endpoints of the interval, then compare. The largest value is the absolute maximum and the smallest the absolute minimum. This ties the accumulation-function analysis directly to the Unit 5 optimization framework, with the integrand $f$ playing the role of $g'$ throughout. The only extra work is computing the candidate values of $g$ as signed areas rather than from a formula.

A common multi-part exam structure

These accumulation questions usually come as a multi-part free response built on one graph of $f$. A typical sequence asks for a value of $g$ at a point (signed area), then where $g$ is increasing or decreasing (sign of $f$), then the relative extrema of $g$ (sign changes of $f$), and finally the concavity or a point of inflection of $g$ (slope of $f$). Working the parts in this order, and justifying each with the correct feature of $f$ or $f'$, earns the marks systematically. The recurring pitfall across all parts is the level confusion, so before answering each part, restate which feature of $f$ controls the requested feature of $g$.