Exponential Growth and Decay
Exponential growth and decay describe processes in which a quantity changes at a rate proportional to its current value. This relationship appears throughout science, economics, and everyday life: populations of bacteria, the decay of radioactive isotopes, the cooling of a hot drink, and the balance of a savings account all follow exponential models. Because the rate of change depends on the quantity itself, these situations are naturally described using differential equations. The key differential equation is dN/dt = kN, where N is the quantity of interest, t is time, and k is the proportionality constant. When k > 0 the quantity grows without bound; when k < 0 it shrinks toward zero. Solving this equation using separation of variables yields the familiar exponential function N(t) = N₀e^(kt), where N₀ is the initial value. This solution connects the abstract machinery of calculus directly to real, measurable phenomena. Understanding exponential models requires both algebraic skill and conceptual insight. Students must be comfortable setting up and solving separable differential equations, interpreting the sign and magnitude of k, and applying the natural logarithm to find unknown times or rates. Mastery of this topic prepares students for further study in biology, chemistry, physics, and finance, where exponential change is ubiquitous.
- Setting Up the Differential Equation
- Solving by Separation of Variables
- Applying the Initial Condition
- Finding the Growth or Decay Constant k
- Interpreting the Model
Setting Up the Differential Equation
The phrase 'rate of change is proportional to the current amount' translates directly into the differential equation dN/dt = kN. Here dN/dt is the instantaneous rate of change of N with respect to time t, and k is a constant called the growth rate (if k > 0) or decay rate (if k < 0). For example, a bacterial colony that doubles every hour obeys this equation with k > 0, while carbon-14 in a fossil obeys it with k < 0.
Solving by Separation of Variables
Rearrange the equation so that all N terms are on one side and all t terms on the other: dN/N = k dt. Integrate both sides: ∫(1/N) dN = ∫k dt, giving ln|N| = kt + C, where C is the constant of integration. Exponentiating both sides yields |N| = e^(kt + C) = e^C · e^(kt). Since N represents a positive quantity (a population, a mass, etc.), we write N = Ae^(kt) where A = e^C is a positive constant.
Applying the Initial Condition
An initial condition fixes the value of A. If N(0) = N₀ (the amount at time t = 0), substitute t = 0 into N = Ae^(kt): N₀ = Ae^0 = A. Therefore A = N₀ and the complete solution is N(t) = N₀e^(kt). This is the general exponential growth or decay formula.
Finding the Growth or Decay Constant k
Often k is not given directly but can be found from a second data point. If N(t₁) = N₁, then N₁ = N₀e^(kt₁). Solving for k: k = (1/t₁) · ln(N₁/N₀). A special case is the half-life T_{1/2} used in radioactive decay: when N = N₀/2, we get 1/2 = e^(kT_{1/2}), so T_{1/2} = −ln(2)/k (note k < 0 makes T_{1/2} positive). Similarly, doubling time T_d = ln(2)/k for growth.
Interpreting the Model
The exponential function N(t) = N₀e^(kt) is never zero for finite t, which is both a strength and a limitation of the model. It accurately describes early-stage growth or decay but breaks down when external constraints appear (e.g., limited resources cap population growth, giving rise to logistic models). Always check whether the exponential model's assumptions — constant k, no limiting factors — are justified for the situation at hand.
The Key Equation
- N(t) — Quantity at time t (varies (e.g., individuals, grams, dollars)): The amount of the substance, population, or value at time t.
- N_0 — Initial quantity (same as N(t)): The value of N at t = 0, the starting point of the model.
- e — Euler's number (dimensionless): The base of the natural logarithm, approximately 2.71828.
- k — Growth/decay constant (per unit time (e.g., hr⁻¹, yr⁻¹)): Positive k indicates exponential growth; negative k indicates exponential decay. Its magnitude determines how fast the change occurs.
- t — Time (seconds, hours, years, etc.): The independent variable measuring elapsed time from the initial moment.
Worked Example
Problem. A sample of iodine-131 used in medical imaging has an initial mass of 50 mg. Iodine-131 has a half-life of 8 days. (a) Find the decay constant k. (b) Write the equation for the mass M(t) remaining after t days. (c) How much iodine-131 remains after 20 days? Round to two decimal places.
Answer. The decay constant is k ≈ −0.0866 per day, the model is M(t) = 50e^(−0.0866t) mg, and approximately 8.84 mg remains after 20 days.
Origins · 1 minLeonhard Euler and how this was discovered
Euler introduced the number e (now called Euler's number, ≈ 2.71828) as the natural base of exponential functions. His systematic study of exponential and logarithmic functions in his 1748 work Introductio in analysin infinitorum established e as a fundamental constant and clarified why the derivative of e^x equals e^x — the property that makes exponential functions the natural solutions to growth and decay differential equations. Euler continued to produce groundbreaking mathematics even after losing sight in both eyes, composing entire papers entirely from memory and mental calculation.
Related Concepts
- Separation of Variables — The exponential growth/decay formula is derived by separating variables in the differential equation dN/dt = kN, making this technique the essential prerequisite for understanding where the formula comes from.
- Logistic Growth — Logistic growth extends the exponential model by adding a carrying capacity, making it more realistic for populations that cannot grow without limit.
- The Natural Logarithm — Solving for unknown times or rates in exponential models requires applying the natural logarithm, the inverse of e^x.
- The Chain Rule — Verifying that N(t) = N₀e^(kt) satisfies dN/dt = kN uses the chain rule to differentiate e^(kt).