AP® Calculus AB · Unit 5 · Topic 5.9
Reading a Graph of the Derivative
A graph of f prime tells you the slope of f at every x-value, so you can read off where f rises, falls, and turns around — but it never tells you f's actual height.
The derivative graph is a report on slope, not on position. Where f' is positive, f is climbing; where f' is negative, f is falling. A zero of f' marks a spot where f's tangent line is flat, a candidate for a relative extremum. Whether that candidate is really a max, a min, or neither depends on whether f' actually changes sign there, not just on whether it touches zero.
To read extrema off the f' graph, track sign changes. If f' goes from negative to positive at x = c, f has a relative minimum there, because f was falling and starts rising. If f' goes from positive to negative, f has a relative maximum. If f' touches zero but stays the same sign on both sides, like at a point where the graph of f' just grazes the x-axis, f has no extremum there at all — it keeps doing what it was doing, just with a momentary pause in slope.
This graph-reading skill connects directly to the first and second derivative tests from earlier in the unit: you are applying those same sign-change rules, just visually instead of algebraically. It also sets up the next step, where concavity comes from watching whether f' itself is increasing or decreasing. One gap remains: a derivative graph fixes the shape of f completely but leaves its vertical position free, since any vertical shift of f has the exact same derivative.
Key terms
- relative extremum
- A point where f changes from increasing to decreasing or vice versa, giving a local high or low point.
- sign change
- The moment a function crosses from positive to negative or negative to positive, which for f' signals an extremum of f.
- critical point
- An x-value where f' equals zero or is undefined, marking a candidate for an extremum.
- vertical shift ambiguity
- The fact that functions differing only by an added constant share the same derivative graph.
Watch it explained
- How it worksHow do you read peaks, zeros, and sign changes off a derivative curve to describe f?
- Worked exampleFind the x-values of the relative extrema of f when f prime of x equals (x - 1)(x - 4)2.
- Common mistakeWhy isn't a high point on the derivative curve a high point on the function?
- Going furtherWhat extra piece of information do you need to pin down the actual graph of f?
Check yourself
End of topic.