Exponensiell

Exponensiell is an adjective used to describe processes or functions that change by a constant proportion of their current value. In mathematics and applied sciences, it is most often modeled by exponential functions. The continuous form is typically written as f(t) = A e^{k t}, where A is the initial amount and k is the continuous growth rate. If k > 0 the quantity grows exponentially; if k < 0 it decays exponentially. A discrete version uses f(n) = A r^{n} with r > 0.

Key properties include that the exponential function with base e, e^{x}, is its own derivative. Its inverse function is the natural logarithm. For general bases a^{x}, the derivative is a^{x} ln a. Exponential functions are strictly increasing when the base is greater than one, and strictly decreasing when the base is between zero and one.

Common measures associated with exponential processes include the doubling time, T_d = ln 2 / k for growth, and the half-life, T_{1/2} = ln 2 / |k| for decay. Graphs of exponential functions are smooth and convex, with curvature determined by the sign and magnitude of k.

Applications are widespread. Exponential models describe population growth, compound interest, radioactive decay, and chemical kinetics. They appear in physics and computer science as well as in probability distributions such as the exponential distribution, which models time between independent random events.

Historically, the term arises from exponentiation theory, with the base e (~2.718) playing a central role in calculus and analysis. The exponential function is denoted exp(x) in many contexts and remains a foundational concept across mathematics and the sciences.