Concept preview

What you will be able to do

1. 01Given the graph of a rate $f$, determine where the accumulation function $g(x)=\int_a^x f(t)\,dt$ is increasing or decreasing and locate its local and absolute extrema on a closed interval.
2. 02Determine the concavity and inflection points of $g$ from whether $f$ is increasing or decreasing, using the relationships $g'(x)=f(x)$ and $g''(x)=f'(x)$.
3. 03Connect the $f\to g$ relationship to the Unit 5 analysis of $f\to f'$, and evaluate accumulation values as signed areas read directly from the graph of $f$.

Start with the definition $g(x)=\int_a^x f(t)\,dt$. Because the Fundamental Theorem of Calculus tells us $g'(x)=f(x)$, the graph of $f$ is literally the graph of the derivative of $g$. So the first question — where is $g$ increasing or decreasing? — is answered by the SIGN of $f$. Where $f$ lies above the axis, $g'>0$ and the accumulated total is growing; where $f$ lies below the axis, $g'<0$ and the total is shrinking because you are adding negative signed area. Nothing about the value of $g$ itself matters for this; only whether $f$ is positive or negative.