AP Calculus AB and BC glossary

Average velocity

Average velocity over a time interval is displacement divided by elapsed time, the change in position from start to end divided by how long it took. Graphically it is the slope of the secant line joining the two endpoints of the position graph.

Average velocity measures net motion per unit time. Over an interval [a,b][a, b] it divides the displacement, the signed change in position, by the length of the time interval.

average velocity=s(b)−s(a)b−a\text{average velocity} = \frac{s(b) - s(a)}{b - a}

This is exactly the slope of the secant line through the endpoints of the position graph, so average velocity is a case of average rate of change. The Mean Value Theorem then guarantees at least one instant where the instantaneous velocity equals this average.

The mistake

Computing average velocity from total distance instead of displacement. A particle that leaves and returns to the same spot has average velocity zero over that trip, no matter how far it travelled. Total distance belongs to average speed, a different quantity.

Appears in: Unit 4: Contextual Applications