Approximate calculations and modelling

An equation containing a derivative is a differential equation; its solution is a function that turns it into a true equality. If the rate of change of a process is proportional to the quantity itself (growth of bacteria, radioactive decay), we get y′ = ky, with solution y = y₀eᵏᵗ, where y₀ = y(0) is the initial condition. For k > 0 there is growth, for k < 0 decay; with half-life T we can write y = y₀ · 2^(−t/T). In cooling, the rate of change of a body’s temperature is proportional to the difference between body and surroundings: T′ = −k(T − T_env), with solution T = T_env + Ce^(−kt) (k > 0). To build a model, rewrite the condition as “rate ∝ amount”, find the proportionality coefficient from one data point, and then compute the value at the requested time. A model is an idealisation: if resources are limited, growth does not stay exponential.