AP Calculus AB and BC glossary

Graph of the derivative

Also called: Derivative graph

The graph of the derivative, f prime, describes the original function f. Where f prime is positive, f is increasing; where f prime is negative, f is decreasing. Zeros of f prime mark critical numbers, and where f prime is itself increasing, f is concave up. The height of f prime is the slope of f.

The key idea is that a value on the f′f' graph is a slope on the ff graph. When the f′f' graph is above the axis, ff is going uphill; a point where the f′f' graph crosses the axis is a critical number of ff, a candidate for a maximum or minimum.

Concavity of ff comes from the direction of the f′f' graph, not its sign. Where f′f' is increasing, ff is concave up; where f′f' has a peak or valley, ff has an inflection point. Reading this well means treating the f′f' graph as slopes twice over.

The mistake

Reading the f′f' graph as if it were ff. A peak on the f′f' graph is not a maximum of ff; it is where ff is steepest and, if f′f' turns there, an inflection point of ff. Always ask whether the curve in front of you is ff, f′f', or f′′f''.

Appears in: Unit 5: Analytical Applications