6.5 Interpreting the Behavior of Accumulation Functions Involving Area

An accumulation function does not just have a derivative; it also has graphical behavior that can be analyzed. If ${F(x)=\int_a^x f(t)\,dt}$, then F'(x)=f(x), so the sign of f(x) tells you whether F(x) is increasing or decreasing. If f(x)>0, then F is increasing because positive area is being added. If f(x)<0, then F is decreasing because negative signed area is being added. So the graph of the original rate function controls the behavior of the accumulation function.

You should also connect the size of f(x) to the steepness of F(x). A large positive value of f(x) means the accumulation function rises quickly, while a large negative value means it falls quickly. If f(x)=0, then F'(x)=0, so the accumulation function has a horizontal tangent there. This lesson is really about reading one graph to infer the behavior of another related function. It blends integral meaning, derivative meaning, and function analysis all at once.