AP Calculus AB and BC glossary

Average acceleration

Also called: Average rate of change of velocity

Average acceleration over a time interval is the change in velocity divided by the elapsed time, which is the slope of the secant line joining the endpoints of the velocity graph. Instantaneous acceleration is a different quantity: it is the derivative of velocity at a single instant.

Average acceleration is a two point calculation across an interval $[t_1, t_2]$ , so it needs the velocity only at the two endpoints and never the acceleration function itself. Its units are velocity units per unit of time, such as meters per second per second.

a_{\text{avg}} = \frac{v(t_2) - v(t_1)}{t_2 - t_1}

Two theorems tie this secant slope back to the acceleration function. When the velocity function is differentiable on the interval, the Mean Value Theorem guarantees an instant $c$ between $t_1$ and $t_2$ where the instantaneous acceleration $v'(c)$ equals it exactly, which fails on a piecewise linear velocity graph with a corner. The Fundamental Theorem goes further: the average value of the acceleration function on the interval, $\frac{1}{t_2 - t_1}\int_{t_1}^{t_2} a(t)\,dt$ , evaluates to the same number, so average acceleration and average value of acceleration always agree.

The mistake

Dividing change in position by elapsed time and calling the answer average acceleration. The quotient $\frac{s(t_2) - s(t_1)}{t_2 - t_1}$ is a secant on the position graph, so it is the average velocity. Average acceleration is the secant slope one derivative up, on the velocity graph, and a units check catches the slip in seconds.

Appears in: Unit 4: Contextual Applications, Unit 8: Applications of Integration