Understand and use exponential growth and decay; use in modelling (examples may include the use of e in continuous compound interest, radioactive decay, drug concentration decay, exponential growth as a model for population growth); consideration of limitations and refinements of exponential models

Exponential growth and decay describe processes where a quantity changes at a rate proportional to its current value. This leads to functions of the form N(t) = N₀ e^{kt} (growth, k>0) or N(t) = N₀ e^{-kt} (decay, k>0). The constant e ≈ 2.71828 is the base of natural logarithms and arises naturally when growth is continuous. In A-Level Mathematics, you will learn to set up and solve differential equations like dN/dt = kN, derive the exponential model, and apply it to real-world contexts such as population growth, radioactive decay, drug concentration in the bloodstream, and continuous compound interest.

Understanding exponential models is crucial because many natural and financial phenomena follow this pattern. For example, the decay of a radioactive substance is proportional to the number of atoms present, leading to a constant half-life. In finance, continuous compound interest uses A = Pe^{rt}, where the growth is compounded infinitely often. These models are powerful but have limitations: they assume unlimited resources (growth) or constant decay rate (decay), which may not hold in reality. Refinements include logistic growth (carrying capacity) or multi-compartment models in pharmacokinetics.

This topic builds on your knowledge of indices, logarithms, and differentiation. You will need to manipulate exponential equations, solve for unknowns using natural logs, and interpret parameters like the growth/decay constant k. In exams, you will often be given a scenario and asked to form an exponential model, then use it to make predictions or find half-life/doubling time. Mastery of this topic is essential for further study in science, economics, and engineering.