The base e and continuous growth

The base e and continuous growth

Build an intuitive understanding of the number e and the notation used for continuous exponential growth and decay. This precalculus practice uses compounding values, exact exponential expressions and carefully stated numerical approximations without requiring formal limits or derivatives.

Interpret the initial quantity and continuous rate constant, evaluate a hypothetical growth model, and convert a formula when time is measured in different units. Compare a continuous rate with the actual percentage change over a full time interval, and express the same model with a fixed discrete multiplier.

Prerequisites are exponential functions, percentages and exponent rules. Questions move from interpreting notation to coordinating rate, time and repeated change. Both single-select and multiselect formats require attention to assumptions, and each problem stands on its own.

Use a scientific calculator only where the prompt permits it; otherwise leave answers in exact form or use the supplied approximation. Detailed explanations distinguish a rate parameter from an effective interval change and show why continuous decay remains positive at finite times. Contexts are synthetic models, not empirical claims about any species or process.