Exponential Growth and Decay

Mathematical Modeling · Axiom Academy

LESSON Exponential Growth and Decay The simplest continuous model: when the rate of change is proportional to the current amount The exponential model captures a simple but powerful idea: the rate of change is proportional to the current amount . If you have more of something, it changes faster; if you have less, it changes slower. Here N(t) represents the quantity at time t, and k is the rate constant that determines how fast the process occurs. 2. Solving by Separation of Variables We can solve the exponential equation using separation of variables , a powerful technique for first-order ODEs. Start with the equation: dN/dt = kN Separate variables: dN/N = k dt Integrate both sides: ln|N| = kt + C Exponentiate: N = e^(kt+C) = Ae^(kt) Apply initial condition: N(0) = N_0 gives A = N_0 N_0 is the initial value at time t = 0, and e is Euler's number (approximately 2.718). 3. Growth vs Decay: The Sign of k The behavior of the solution depends entirely on the sign of the rate constant k : Population increases without bound as t increases. Larger k means faster growth. Quantity decreases toward zero. More negative k means faster decay. 4. Doubling Time and Half-Life Rather than using the abstract rate constant k, we often describe exponential processes by their characteristic times - concrete measures of how fast the process occurs. Time for the quantity to double. For growth processes. Time for the quantity to halve. For decay processes. Deriving the Half-Life Formula

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