Exponential Growth
Functions & Advanced AlgebraExponential growth occurs when a quantity increases by a constant percentage rate over equal time intervals, causing it to grow faster and faster.
Formula
A(t) = A_0 \cdot b^t, b > 1
Definition
Exponential growth happens when something keeps multiplying by the same number over and over. It starts slowly but then shoots up very fast.
Example
A bacteria population that doubles every hour: $1, 2, 4, 8, 16, 32, \ldots$ After $10$ hours, there are $1{,}024$. After $20$ hours: $1{,}048{,}576$. The growth accelerates rapidly.
Key Insight
A common saying: "The first doubling doesn't seem like much, but eventually it takes over." Exponential growth is deceptive because it looks slow at first and then becomes explosive.
Definition
Exponential growth follows the model $A(t) = A_0 \cdot b^t$, where $A_0$ is the initial amount, $b > 1$ is the growth factor, and $t$ is time. Equivalently, $A(t) = A_0 \cdot e^{rt}$ where $r > 0$ is the continuous growth rate.
Example
A $\$1{,}000$ investment earns $6\%$ annual interest (compounded annually): $A(t) = 1000 \cdot (1.06)^t$. After $12$ years: $A(12) = 1000 \cdot (1.06)^{12} \approx \$2{,}012$. The "Rule of 72": $72/6 = 12$ years to double.
Key Insight
Exponential growth has a constant doubling time: the time to double is $\ln(2)/r$ regardless of when you start measuring. This is a defining property that distinguishes it from polynomial growth.
Definition
Exponential growth is the solution to the ODE $dA/dt = rA$ ($r > 0$): $A(t) = A_0 e^{rt}$. It is the unique function (up to scaling) that is proportional to its own derivative. In discrete time, it arises in linear maps with eigenvalue $> 1$, and in population models as the Malthusian growth model.
Example
The Lotka-Volterra predator-prey model departs from pure exponential growth by introducing nonlinear coupling. Near the equilibrium, linearization shows exponential modes. Far from equilibrium, nonlinear terms dominate.
Key Insight
Exponential growth is ultimately limited in physical systems. The logistic growth model $A'(t) = rA(1 - A/K)$ introduces a carrying capacity $K$, causing growth to slow and stabilize. Near $t = 0$, it approximates exponential; near $K$, it stabilizes.