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Exponential growth. 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 ...
Doubling time. The doubling time is the time it takes for a population to double in size/value. It is applied to population growth, inflation, resource extraction, consumption of goods, compound interest, the volume of malignant tumours, and many other things that tend to grow over time. When the relative growth rate (not the absolute growth ...
RGR is a concept relevant in cases where the increase in a state variable over time is proportional to the value of that state variable at the beginning of a time period. In terms of differential equations, if is the current size, and its growth rate, then relative growth rate is. . If the RGR is constant, i.e., , a solution to this equation is.
The exercise of working through this problem may be used to explain and demonstrate exponents and the quick growth of exponential and geometric sequences. It can also be used to illustrate sigma notation. When expressed as exponents, the geometric series is: 2 0 + 2 1 + 2 2 + 2 3 + ... and so forth, up to 2 63. The base of each exponentiation ...
The number e is a mathematical constant approximately equal to 2.71828 that can be characterized in many ways. It is the base of the natural logarithm function. It is the limit of as n tends to infinity, an expression that arises in the computation of compound interest. It is the value at 1 of the (natural) exponential function, commonly ...
Double exponential function. A double exponential function is a constant raised to the power of an exponential function. The general formula is (where a >1 and b >1), which grows much more quickly than an exponential function. For example, if a = b = 10: f (x) = 10 10x. f (0) = 10.
This behavior is referred to as a "decaying" exponential function. The time τ (tau) is referred to as the "time constant" and can be used (as in this case) to indicate how rapidly an exponential function decays. Here: t is time (generally t > 0 in control engineering) V 0 is the initial value (see "specific cases" below). Specific cases
Sub-exponential growth. A common explanation for the growth of epidemics holds that 1 person infects 2, those 2 infect 4 and so on and so on with the number of infected doubling every generation. It is analogous to a game of tag where 1 person tags 2, those 2 tag 4 others who've never been tagged and so on. As this game progresses it becomes ...