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Exponential growth is a process that increases quantity over time at an ever-increasing rate. It occurs when the instantaneous rate of change (that is, the derivative) of a quantity with respect to time is proportional to the quantity itself. Described as a function, a quantity undergoing exponential growth is an exponential function of time ...
Exponential decay. A quantity undergoing exponential decay. Larger decay constants make the quantity vanish much more rapidly. This plot shows decay for decay constant ( λ) of 25, 5, 1, 1/5, and 1/25 for x from 0 to 5. A quantity is subject to exponential decay if it decreases at a rate proportional to its current value.
The stretched exponential function. is obtained by inserting a fractional power law into the exponential function. In most applications, it is meaningful only for arguments t between 0 and +∞. With β = 1, the usual exponential function is recovered. With a stretching exponent β between 0 and 1, the graph of log f versus t is ...
Half-life (symbol t½) is the time required for a quantity (of substance) to reduce to half of its initial value. The term is commonly used in nuclear physics to describe how quickly unstable atoms undergo radioactive decay or how long stable atoms survive. The term is also used more generally to characterize any type of exponential (or, rarely ...
Bacterial growth. Growth is shown as L = log (numbers) where numbers is the number of colony forming units per ml, versus T (time.) Bacterial growth is proliferation of bacterium into two daughter cells, in a process called binary fission. Providing no mutation event occurs, the resulting daughter cells are genetically identical to the original ...
A logistic function or logistic curve is a common S-shaped curve ( sigmoid curve) with the equation. where. , the value of the function's midpoint; , the supremum of the values of the function; , the logistic growth rate or steepness of the curve. [1] Standard logistic function where. For values of in the domain of real numbers from to , the S ...
To the right is the long tail, and to the left are the few that dominate (also known as the 80–20 rule ). In statistics, a power law is a functional relationship between two quantities, where a relative change in one quantity results in a relative change in the other quantity proportional to a power of the change, independent of the initial ...
The Hubbert peak theory says that for any given geographical area, from an individual oil-producing region to the planet as a whole, the rate of petroleum production tends to follow a bell-shaped curve. It is one of the primary theories on peak oil. Choosing a particular curve determines a point of maximum production based on discovery rates ...