A quantity that grows exponentially grows at a rate proportional to its size. Anything that grows by the same factor in a fixed period is growing exponentially. In the long run, exponential growth of any kind will overtake any polynomial growth. That is, for all n:
Growth rates may also be faster than exponential. For example:lim xn/ex = 0 x -> ∞
Tetration is iterated exponentiation, the first hyper operator after exponentiation. Some examples, using Knuth's up arrow notation,1↑↑2 = 11 = 1 2↑↑2 = 22 = 4 3↑↑2 = 33 = 27 4↑↑2 = 44 = 256 1↑↑3 = 111 = 1 2↑↑3 = 222 = 24 = 16 3↑↑3 = 333 = 327 = 7,625,597,484,987 1↑↑4 = 1111 = 1 2↑↑4 = 2222 = 216 = 65,536The Ackermann function is a simple example of a recursive computer function that takes two natural numbers as arguments and yields a natural number. Its value grows extremely quickly. Below is an example of it written as a recursive function in the Zeno programming language.function Ackermann(x, y : int) : int var a : int if (x = 0) then a := y + 1 elsif (y = 0) then a := Ackermann(x-1, 1) else a := Ackermann(x-1, Ackermann(x, y-1)) end if return a end function
The effect of compounding interest over many years has a substantial effect. The value of an investment in the future value is given by the function:
The parameters are defined as follows:A(t) = P(1 + r/n)nt
For example, an initial investment of $1,000 is invested at a 12% annual interest rate, compounded monthly. Using the above function, the amount accumulated after 5, 10, 15, and 20 years is:A is the future value P is the initial value or principal r is the annual interest rate n is the number of compounding periods per year t is the number of years
A(t) = 1000(1 + 0.12/12)12t = 1000(1.01)12t years calculation future value -------------------------------------- 5 1000(1.01)60 1816.70 10 1000(1.01)120 3300.39 15 1000(1.01)180 5995.80 20 1000(1.01)240 10892.55
More frequent compounding increases the rate of growth. As n approaches infinity, compounding becomes continuous. In this case, the function for computation of future value is:
Repeating the above example, for continuous compounding:A(t) = Pert
A(t) = 1000e0.12t years calculation future value -------------------------------------- 5 1000e0.60 1822.12 10 1000e1.20 3320.12 15 1000e1.80 6049.65 20 1000e2.40 11023.18
The nucleus of an atom that undergoes fission produces multiple neutrons, each of which can be absorbed by adjacent nuclei, causing them to fission in turn. The basic equation for neutron production is exponential:
If the probability of neutron absorption exceeds the probability of neutron escape, then k > 1 and the production rate of neutrons and induced nuclear fission increases exponentially. Otherwise, k < 1 and the production rate decreases to a low level sustained by nuclear decay.N(n) = Noe(k – 1)n
A quantity is said to be subject to exponential decay if it decreases at a rate proportional to its value. An important characteristic of exponential decay is the time required for the quantity to decrease to one half of its initial value. This time is called the half-life, and often denoted by the symbol t1/2. For exponential decay modeled as:
The equation describing half-life is found by evaluating t for:C(t) = Coe-rt
e-rt = 1/2 = 2-1 -rt1/2 = ln 1/2 = -ln 2 t1/2 = (ln 2)/r
Newton's law of cooling states: "The rate of heat loss is proportional to the difference in temperatures between the body and its surroundings." If an object at one temperature is exposed to a medium of another temperature, the temperature difference between the object and the medium follows exponential decay:
In the equation above:T(t) = Ts + (To - Ts)e-kt
t time k a constant greater than zero T(t) temperature at time t Ts temperature of the surroundings To initial temperature
An electromagnetic field or a particle flux can be absorbed by a material. The distance d into a material is the length that corresponds to a reduction of the initial intensity to e-1 of the field or flux at the entry surface of the material:
This curve has the same shape as the exponential decay curve and d is material dependant. It is the thickness of a particular material that is needed for attenuation of about 63%:P(x) = Poe-x/d
1 - e-1 = 0.63212
The currents and voltages that arise when the energy stored in an inductor or capacitor is suddenly released to the resistors in the circuit are referred to as the natural response of the circuit. The behavior of these currents and voltages depends only on the nature of the circuit, and not on external sources of excitation. The natural response can be described in terms of the voltage at the terminals of the resistor when the external source of power stops delivering energy to the circuit. The expression for the voltage across the resistor is:
Above, Vo is the initial voltage and T is the time constant of the circuit. The response of a circuit to the sudden application of a constant voltage or current source is referred to as the step response of the circuit. The inductor or capacitor is assumed to be completely discharged and begins to absorb energy after an instantaneous application of an external voltage or current. The expression for the voltage across the resistor is:v(t) = Voe-t/T
v(t) = Vo(1 - e-t/T)
The general logistic function is defined by the mathematical formula:
1 + m e-t/T
P(t) = a -----------
1 + n e-t/T
Above, the parameters a, m, n, and T are real numbers. The sigmoid function is a special
case of the logistic function with a = 1, m = 0, n = 1, T = 1:
1
P(t) = ------
1 + e-t
The name is due to the sigmoid shape of its graph. This function is also called the standard
logistic function. The sigmoid curve shows early exponential growth for negative t, which slows
to linear growth of slope 1/4 near t = 0, then approaches y = 1 with an
exponentially decaying gap.
AbCd Classics - free on-line books