AP Calculus AB and BC glossary

Exponential growth

Exponential growth describes a quantity whose rate of change is proportional to its current amount. The differential equation says the derivative equals a constant times the quantity, and its solution is a constant multiple of e raised to that constant times time.

\frac{dy}{dt} = ky \implies y = Ce^{kt}

The sign of $k$ decides everything: positive means growth, negative means decay. The constant $C$ is the initial amount, because setting $t = 0$ leaves $y = C$ .

This model appears as population growth, radioactive decay, and continuously compounded interest. Recognizing the phrase proportional to the amount present is what signals it.

Unbounded

Pure exponential growth has no ceiling, which is why realistic population models switch to logistic growth once resources become limited.

Compound interest is this same function with money substituted for population. A balance growing at a fixed rate satisfies the identical differential equation, which is why interest compounds rather than adds: how compound interest works.

Appears in: Unit 7: Differential Equations