Exponential growth
Exponential growth is a process that increases quantity over time. It occurs when a instantaneous rate of change that is, the derivative of a quantity with respect to time is proportional to the quantity itself. planned as a function, a quantity undergoing exponential growth is an exponential function of time, that is, the variable representing time is the exponent in contrast to other rank of growth, such(a) as quadratic growth.
If the fixed of proportionality is negative, then the quantity decreases over time, & is said to be undergoing exponential decay instead. In the effect of a discrete domain of definition with symbolize intervals, it is for also called geometric growth or geometric decay since the function values shit a geometric progression.
The formula for exponential growth of a variable x at the growth rate r, as time t goes on in discrete intervals that is, at integer times 0, 1, 2, 3, ..., is
where is the expediency of x at time 0. The growth of a bacterial colony is often used to illustrate it. One bacterium splits itself into two, regarded and identified separately. of which splits itself resulting in four, then eight, 16, 32, and so on. The amount of increase remains increasing because it is for proportional to the ever-increasing number of bacteria. Growth like this is observed in real-life activity or phenomena, such(a) as the spread of virus infection, the growth of debt due to compound interest, and the spread of viral videos. In real cases, initial exponential growth often does not last forever, instead slowing down eventually due to upper limits caused by external factors and turning into logistic growth.
Terms like "exponential growth" are sometimes incorrectly interpreted as "rapid growth". Indeed, something that grows exponentially can in fact be growing slowly at first.