AP Calculus AB and BC glossary

Instantaneous rate of change

The instantaneous rate of change is how fast a quantity is changing at one exact instant, and it is precisely the derivative. It is the limit of average rates of change over intervals that shrink to nothing around that instant.

instantaneous rate=lim⁡Δx→0ΔyΔx=dydx\text{instantaneous rate} = \lim_{\Delta x \to 0}\frac{\Delta y}{\Delta x} = \frac{dy}{dx}

Units matter on free response. If the function is in gallons and the input in minutes, the derivative is in gallons per minute, and an answer without units routinely loses a point.

The mistake

Computing an average rate when the question says at. At a specific input means one derivative value; over an interval means a difference quotient.

Appears in: Unit 2: Defining the Derivative