AP Calculus AB and BC glossary

Absolute value function

The absolute value function returns the distance of a number from zero. It is continuous everywhere but has a corner at the origin, where the one-sided slopes are negative one and one, so it is not differentiable there.

∣x∣={xx≥0−xx<0\left|x\right| = \begin{cases} x & x \ge 0 \\ -x & x < 0 \end{cases}

It is the standard counterexample showing continuity does not imply differentiability. It also appears inside total distance integrals, where the absolute value of velocity is integrated.

The mistake

Integrating an absolute value by ignoring the bars. Split the integral where the inside changes sign and handle each piece with the right formula.

Appears in: Unit 1: Limits and Continuity, Unit 2: Defining the Derivative