Mistake Master · AP Calculus · Unit 5 · Step-Through Animation
A Falling Curve Can Bend Upward, and Most of Them Do
You'll learnto read concavity as the sign of f″ — equivalently, as whether f′ is going up or down — to keep that reading apart from rising and falling, and to split a concavity chart at the gaps in the domain as well as at the roots of f″.
Direction was Topic 5.3 and it belongs to f′. Bending is this topic and it belongs to f″. A function can be falling while it curves upward and rising while it curves downward, which is the whole reason the two questions are asked separately — and nothing about the sign of f′ settles the sign of f″. There is a second phrasing worth more than the first: f is concave up exactly where f′ is increasing. That one says what concavity means in plain terms, the slopes are getting bigger, and it is the sentence that lets you read bending straight off a graph of the derivative.
8 STEPS · 6 QUICK CHECKS · CONCAVE UP MEANS f′ IS INCREASING · POSITION IS DIRECTION, SHAPE IS CONCAVITY · v1
One cubic, f(x) = x³ − 3x² − 9x + 5, and later four other functions chosen because each one settles a different pairing of direction with bending. Every panel shares the same x axis, so a vertical line cuts them all at the same input.
The question
Which way is the graph bending? Not which way is it going — that is a different question, answered by a different derivative, and the two answers are independent of each other.
Watch for
The slope readout in step 2. It climbs from −12 back toward zero while staying negative the whole time. The curve above it is falling and bending upward at the same moment.
Mistake Master