Exponential growth

In mathematics, a quantity that grows exponentially is one that grows at a rate proportional to its size. This means that for any exponentially growing quantity, the larger the quantity gets, the faster it grows. But it also implies that the relationship between the size of the dependent variable and its rate of growth is governed by a strict law, of the simplest kind: direct proportion. It is proved in calculus that this law requires that the quantity is given by the exponential function, if we use the correct time scale. This explains the name.

Technical details

Let x be a quantity growing exponentially with respect to time t. By definition, the rate of change dx/dt obeys the differential equation:

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: !, rac{dx}{dt} = k x

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where k > 0 is the constant of proportionality (the average number of offspring per individual in the case of the population). (See logistic function for a simple correction of this growth model where k is not constant). The solution to this equation is the exponential function !, x(t)=x_0 e^{kt} -- hence the name exponential growth. The constant !, x_0 is determined by the initial size of the population.

Related Topics:
Logistic function - Exponential function

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In the long run, exponential growth of any kind will overtake linear growth of any kind (the basis of the Malthusian catastrophe) as well as any polynomial growth, i.e., for all α:

Related Topics:
Malthusian catastrophe - Polynomial

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:lim_{x ightarrowinfty} {x^lpha over Ce^x} =0

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There is a whole hierarchy of conceivable growth rates that are slower than exponential and faster than linear (in the long run). Growth rates may also be faster than exponential. The linear and exponential models are merely simple candidates but are those of greatest occurrence in nature.

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In the above differential equation, if k < 0, then the quantity experiences exponential decay.

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