AP Calculus AB and BC glossary

Exponential decay

Exponential decay describes a quantity falling at a rate proportional to how much is left, which makes the derivative equal to a negative constant times the amount. The solution is an exponential with a negative exponent, and it has a constant half-life.

dydt=−ky  ⟹  y=y0e−kt,k>0\frac{dy}{dt} = -ky \implies y = y_{0}e^{-kt}, \quad k > 0

Half-life is independent of where you start, which is the characteristic signature of exponential behaviour. It is what distinguishes radioactive decay from a quantity falling at a constant rate.

The mistake

Confusing it with a constant rate of decrease. Exponential decay slows down as the amount shrinks; a constant rate would reach zero in finite time.

Appears in: Unit 7: Differential Equations