AP Calculus AB and BC glossary

Cusp

A cusp is a sharp point on a graph where the one-sided slopes grow without bound in opposite directions. The function stays continuous there but is not differentiable, because the tangent slope is not a finite number.

The function $f(x) = x^{2/3}$ has a cusp at the origin. Its derivative $\frac{2}{3}x^{-1/3}$ goes to $-\infty$ from the left and $+\infty$ from the right, so the graph comes to a point rather than a wedge.

Cusp versus corner

At a corner the one-sided slopes are finite and merely unequal. At a cusp they are infinite and opposite in sign. Both block differentiability.

Appears in: Unit 2: Defining the Derivative