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| [[File:Exponential.svg|thumb|300px|right|The graph illustrates how exponential growth (green) surpasses both linear (red) and cubic (blue) growth. {{legend|green|Exponential growth}} {{legend|red|Linear growth}} {{legend|blue|Cubic growth}}]]
| | We understand the distinguished BMI or Body Mass Index Calculation so popular to assess whether you're usual, obese or obese? Well, what do you certainly know about it? Do we think it's important?<br><br>Doctors plus dietitians utilize body mass index (BMI) and measurement of the waist to assess a person's risk of developing certain diseases, for instance, diabetes plus heart condition. If your BMI is greater than 25, you may be obese. BMI equal to or greater than 30 indicates weight.<br><br>In 1972, Ancel Keys gave the concept its more familiar name, Body Mass Index, or BMI. In the 1980s, JS Garrow plus JD Webster suggested arbitrary zones for the BMI--including "overweight" and "obese". And the rest is history.<br><br>In addition, guys normally have more muscle than ladies plus ladies have more fat inside their body than guys. For this reason, women should have their fat plus height ratio calculated by a [http://safedietplansforwomen.com/bmi-calculator bmi calculator females] for females.<br><br>> 905-409-8031 or bodyconcepts1@uniserve.com Would you like a Professional Personal Trainer Specialist? - If thus I am offering we my Personal Training services for about half price. Instead of $68 per session we pay just $39 per session, for Toronto & GTA's #1 Personal Trainer services! - For a Limited time.<br><br>In purchase to determine perfect weight for kids plus teenagers (between age group 2-20 years), there is a particular reference tool called the BMI percentile chart. In case, when the percent value falls at 80, it signifies the kid is having more weight than 80 % of the kids of the same gender and age group. According to the chart, a child's body mass index falling inside 95 percentile or above indicates he is overweight, whilst those with a BMI above 85 percentile have a risk of becoming overweight. On the other hand, when a kid's BMI percentile is 5 or lower, he/she is underweight.<br><br>If you don't fall in the normal range, then get oneself checked with other (omit) techniques to figure out the amount of body fat. This will provide a better perspective plus aid you to do or keep up a perfect fat. |
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| '''Exponential growth''' occurs when the growth rate of the value of a mathematical function is [[proportionality (mathematics)|proportional]] to the function's current value. [[Exponential decay]] occurs in the same way when the growth rate is negative. In the case of a discrete [[Domain of a function|domain]] of definition with equal intervals it is also called '''geometric growth''' or '''geometric decay''' (the function values form a [[geometric progression]]).
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| The formula for exponential growth of a variable ''x'' at the (positive or negative) growth rate ''r'', as time ''t'' goes on in discrete intervals (that is, at integer times 0, 1, 2, 3, ...), is
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| :<math>x_t = x_0(1+r)^t</math>
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| where ''x''<sub>0</sub> is the value of ''x'' at time 0. For example, with a growth rate of ''r'' = 5% = 0.05, going from ''any'' integer value of time to the next integer causes ''x'' at the second time to be 1.05 times (i.e., 5% larger than) what it was at the previous time.
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| ==Examples==
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| {{unsourced|section|date=August 2013}}
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| [[File:E.-coli-growth.gif|right|frame|150px|[[Bacteria]] exhibit exponential growth under optimal conditions.]]
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| * [[Biology]]
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| ** The number of [[microorganism]]s in a [[microbiological culture|culture]] will increase exponentially until an essential nutrient is exhausted. Typically the first organism [[cell division|splits]] into two daughter organisms, who then each split to form four, who split to form eight, and so on.
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| ** A virus (for example [[SARS]], or [[smallpox]]) typically will spread exponentially at first, if no artificial [[immunization]] is available. Each infected person can infect multiple new people.
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| ** [[World population|Human population]], if the number of births and deaths per person per year were to remain at current levels (but also see [[logistic growth]]). For example, according to the United States Census Bureau, over the last 100 years (1910 to 2010), the population of the United States of America is exponentially increasing at an average rate of one and a half percent a year (1.5%). This means that the doubling time of the American population (depending on the yearly growth in population) is approximately 50 years.<ref>2010 Census Data. “U.S. Census Bureau.” 12 Nov. 2011. http://2010.census.gov/2010census/data/index.php</ref>
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| ** Many responses of living beings to [[Stimulus (physiology)|stimuli]], including human [[perception]], are [[logarithm]]ic responses, which are the inverse of exponential responses; the [[loudness]] and [[frequency]] of [[sound]] are perceived logarithmically, even with very faint stimulus, within the limits of perception. This is the reason that exponentially increasing the [[brightness]] of [[Visual perception|visual stimuli]] is perceived by humans as a linear increase, rather than an exponential increase. This has [[survival value]]. Generally it is important for the organisms to respond to stimuli in a wide range of levels, from very low levels, to very high levels, while the [[Accuracy and precision|accuracy]] of the [[Approximation|estimation]] of differences at high levels of stimulus is much less important for survival.
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| ** Genetic complexity of [[life on Earth]] has doubled every 376 million years. Extrapolating this exponential growth backwards indicates life began 9.7 billion years ago, potentially predating the [[Earth]] by 5.2 billion years.<ref>{{cite |url=http://phys.org/news/2013-04-law-life-began-earth.html |title=Researchers use Moore's Law to calculate that life began before Earth existed |author=Bob Yirka |date=2013-04-18 |accessdate=2013-04-22 |publisher=phys.org}}</ref><ref>{{cite journal |title=Larger than Life Indeed |publisher=Economic Times |date=April 2013 |accessdate=2013-04-23 |url=http://economictimes.indiatimes.com/opinion/cosmic-uplink/larger-than-life-indeed/articleshow/19685998.cms}}</ref>
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| * [[Physics]]
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| ** [[Avalanche breakdown]] within a [[dielectric]] material. A free [[electron]] becomes sufficiently accelerated by an externally applied [[electrical field]] that it frees up additional electrons as it collides with [[atom]]s or [[molecule]]s of the dielectric media. These ''secondary'' electrons also are accelerated, creating larger numbers of free electrons. The resulting exponential growth of electrons and ions may rapidly lead to complete [[dielectric breakdown]] of the material.
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| ** [[Nuclear chain reaction]] (the concept behind [[nuclear reactors]] and [[nuclear weapons]]). Each [[uranium]] [[atomic nucleus|nucleus]] that undergoes [[Nuclear fission|fission]] produces multiple [[neutron]]s, each of which can be [[absorption (chemistry)|absorbed]] by adjacent uranium atoms, causing them to fission in turn. If the [[probability]] of neutron absorption exceeds the probability of neutron escape (a [[function (mathematics)|function]] of the [[shape]] and [[mass]] of the uranium), ''k'' > 0 and so the production rate of neutrons and induced uranium fissions increases exponentially, in an uncontrolled reaction. "Due to the exponential rate of increase, at any point in the chain reaction 99% of the energy will have been released in the last 4.6 generations. It is a reasonable approximation to think of the first 53 generations as a latency period leading up to the actual explosion, which only takes 3–4 generations."<ref>{{cite web|url=http://nuclearweaponarchive.org/Nwfaq/Nfaq2.html|title=Introduction to Nuclear Weapon Physics and Design|publisher=Nuclear Weapons Archive|last=Sublette|first=Carey|accessdate=2009-05-26}}</ref>
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| ** [[Positive feedback]] within the linear range of electrical or electroacoustic [[Amplifier|amplification]] can result in the exponential growth of the amplified signal, although [[resonance]] effects may favor some [[component frequencies]] of the signal over others.
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| ** [[Heat transfer]] experiments yield results whose best fit line are exponential decay curves.
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| * [[Economics]]
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| ** [[Economic growth]] is expressed in percentage terms, implying exponential growth. For example, U.S. GDP per capita has grown at an exponential rate of approximately two percent since World War 2.{{cn|date=August 2013}}
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| * [[Finance]]
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| ** [[Compound interest]] at a constant interest rate provides exponential growth of the capital. See also [[rule of 72]].
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| ** [[Pyramid scheme]]s or [[Ponzi scheme]]s also show this type of growth resulting in high profits for a few initial investors and losses among great numbers of investors.
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| * [[Computer|Computer technology]]
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| ** [[Clock rate|Processing power]] of computers. See also [[Moore's law]] and [[technological singularity]] (under exponential growth, there are no singularities. The singularity here is a metaphor.).
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| ** In [[computational complexity theory]], computer algorithms of exponential complexity require an exponentially increasing amount of resources (e.g. time, computer memory) for only a constant increase in problem size. So for an algorithm of time complexity 2<sup>''x''</sup>, if a problem of size ''x'' = 10 requires 10 seconds to complete, and a problem of size ''x'' = 11 requires 20 seconds, then a problem of size ''x'' = 12 will require 40 seconds. This kind of algorithm typically becomes unusable at very small problem sizes, often between 30 and 100 items (most computer algorithms need to be able to solve much larger problems, up to tens of thousands or even millions of items in reasonable times, something that would be physically impossible with an exponential algorithm). Also, the effects of [[Moore's Law]] do not help the situation much because doubling processor speed merely allows you to increase the problem size by a constant. E.g. if a slow processor can solve problems of size x in time t, then a processor twice as fast could only solve problems of size x+constant in the same time t. So exponentially complex algorithms are most often impractical, and the search for more efficient algorithms is one of the central goals of computer science today.
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| ** [[History of the Internet|Internet traffic growth]].<sup>[[Citation needed|[Citation needed]]]</sup>
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| ==Basic formula==
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| A quantity ''x'' depends exponentially on time ''t'' if
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| :<math>x(t)=a\cdot b^{t/\tau}\,</math> | |
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| where the constant ''a'' is the initial value of ''x'',
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| :<math>x(0)=a\, ,</math>
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| the constant ''b'' is a positive growth factor, and ''τ'' is the [[time constant]]—the time required for ''x'' to increase by one factor of ''b'':
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| :<math>x(t+\tau)=a \cdot b^{\frac{t+\tau}{\tau}} = a \cdot b^{\frac{t}{\tau}} \cdot b^{\frac{\tau}{\tau}} = x(t)\cdot b\, .</math>
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| If ''τ'' > 0 and ''b'' > 1, then ''x'' has exponential growth. If ''τ'' < 0 and ''b'' > 1, or ''τ'' > 0 and 0 < ''b'' < 1, then ''x'' has [[exponential decay]].
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| Example: ''If a species of bacteria doubles every ten minutes, starting out with only one bacterium, how many bacteria would be present after one hour?'' The question implies ''a'' = 1, ''b'' = 2 and ''τ'' = 10 min.
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| :<math>x(t)=a\cdot b^{t/\tau}=1\cdot 2^{(60\text{ min})/(10\text{ min})}</math>
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| :<math>x(1\text{ hr})= 1 \cdot 2^6 =64.</math>
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| After one hour, or six ten-minute intervals, there would be sixty-four bacteria.
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| Many pairs (''b'', ''τ'') of a [[dimensionless]] non-negative number ''b'' and an amount of time ''τ'' (a [[physical quantity]] which can be expressed as the product of a number of units and a unit of time) represent the same growth rate, with ''τ'' proportional to log ''b''. For any fixed ''b'' not equal to 1 (e.g. ''e'' or 2), the growth rate is given by the non-zero time ''τ''. For any non-zero time ''τ'' the growth rate is given by the dimensionless positive number ''b''.
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| Thus the law of exponential growth can be written in different but mathematically equivalent forms, by using a different [[exponentiation|base]]. The most common forms are the following:
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| :<math>x(t) = x_0\cdot e^{kt} = x_0\cdot e^{t/\tau} = x_0 \cdot 2^{t/T}
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| = x_0\cdot \left( 1 + \frac{r}{100} \right)^{t/p},</math>
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| where ''x''<sub>0</sub> expresses the initial quantity ''x''(0).
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| Parameters (negative in the case of exponential decay):
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| * The ''growth constant'' ''k'' is the [[frequency]] (number of times per unit time) of growing by a factor ''e''; in finance it is also called the logarithmic return, [[continuous compounding|continuously compounded return]], or [[Compound interest#Force of interest|force of interest]].
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| * The ''[[e-folding|e-folding time]]'' τ is the time it takes to grow by a factor ''e''.
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| * The ''[[doubling time]]'' ''T'' is the time it takes to double.
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| * The percent increase ''r'' (a dimensionless number) in a period ''p''.
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| The quantities ''k'', τ, and ''T'', and for a given ''p'' also ''r'', have a one-to-one connection given by the following equation (which can be derived by taking the natural logarithm of the above):
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| :<math>k = \frac{1}{\tau} = \frac{\ln 2}{T} = \frac{\ln \left( 1 + \frac{r}{100} \right)}{p}\,</math>
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| where ''k'' = 0 corresponds to ''r'' = 0 and to τ and ''T'' being infinite.
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| If ''p'' is the unit of time the quotient ''t/p'' is simply the number of units of time. Using the notation ''t'' for the (dimensionless) number of units of time rather than the time itself, ''t/p'' can be replaced by ''t'', but for uniformity this has been avoided here. In this case the division by ''p'' in the last formula is not a numerical division either, but converts a dimensionless number to the correct quantity including unit.
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| A popular approximated method for calculating the doubling time from the growth rate is the [[rule of 70]],
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| i.e. <math>T \simeq 70 / r</math>.
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| ==Reformulation as log-linear growth==
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| If a variable ''x'' exhibits exponential growth according to <math>x(t)=x_0(1+r)^t</math>, then the log (to any base) of ''x'' [[linear function|grows linearly]] over time, as can be seen by taking [[logarithm]]s of both sides of the exponential growth equation:
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| :<math>\log x(t) = \log x_0 + t \cdot \log (1+r).</math>
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| This allows an exponentially growing variable to be modeled with a [[Nonlinear regression#Linearization|log-linear model]]. For example, if one wishes to empirically estimate the growth rate from intertemporal data on ''x'', one can [[linear regression|linearly regress]] log ''x'' on ''t''.
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| ==Differential equation==
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| The [[exponential function]] <math>\scriptstyle x(t)=ae^{kt}</math> satisfies the [[linear differential equation]]:
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| :<math> \!\, \frac{dx}{dt} = kx</math>
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| saying that the growth rate of ''x'' at time ''t'' is proportional to the value of ''x''(''t''), and it has the [[initial value]]
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| :<math>x(0)=a.\,</math>
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| For ''a'' > 0 the differential equation is solved by the method of [[separation of variables]]:
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| :<math>\frac{dx}{dt} = kx</math>
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| :<math>\Rightarrow \frac{dx}{x} = k\, dt</math>
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| :<math>\Rightarrow \int \frac{dx}{x} = \int k \, dt</math>
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| :<math>\Rightarrow \ln x = kt + \text{constant}\, .</math>
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| Incorporating the initial value gives:
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| :<math>\ln x = kt + \ln a\,</math>
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| :<math>\Rightarrow x = ae^{kt}\, </math>
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| The solution also applies for ''a'' ≤ 0 where the logarithm is not defined.
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| For a [[nonlinear]] variation of this growth model see [[logistic function]].
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| ==Difference equation==
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| The [[difference equation]]
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| :<math>x_t = a \cdot x_{t-1}</math>
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| has solution
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| :<math>x_t = x_0 \cdot a^t,</math>
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| showing that ''x'' experiences exponential growth.
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| ==Other growth rates==
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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:
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| :<math>\lim_{t\rightarrow\infty} {t^\alpha \over ae^t} =0.</math>
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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). See [[Degree of a polynomial#The degree computed from the function values]].
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| Growth rates may also be faster than exponential.
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| In the above differential equation, if ''k'' < 0, then the quantity experiences [[exponential decay]].
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| ==Limitations of models==
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| Exponential growth models of physical phenomena only apply within limited regions, as unbounded growth is not physically realistic. Although growth may initially be exponential, the modelled phenomena will eventually enter a region in which previously ignored [[negative feedback]] factors become significant (leading to a [[logistic growth]] model) or other underlying assumptions of the exponential growth model, such as continuity or instantaneous feedback, break down.
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| {{further2|[[Limits to Growth]], [[Malthusian catastrophe]], [[Apparent infection rate]]}}
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| ==Exponential stories==
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| ===Rice on a chessboard===
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| {{see also|Wheat and chessboard problem}}
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| According to an old legend, vizier Sissa Ben Dahir presented an Indian King Sharim with a beautiful, hand-made [[chessboard]]. The king asked what he would like in return for his gift and the courtier surprised the king by asking for one grain of rice on the first square, two grains on the second, four grains on the third etc. The king readily agreed and asked for the rice to be brought. All went well at first, but the requirement for 2<sup> ''n'' − 1</sup> grains on the ''n''th square demanded over a million grains on the 21st square, more than a million million (aka [[Orders of magnitude (numbers)#1012|trillion]]) on the 41st and there simply was not enough rice in the whole world for the final squares. (From Swirski, 2006)<ref name=Porritt-2005>{{cite book|last=Porritt|first=Jonathan|title=Capitalism: as if the world matters|year=2005|publisher=Earthscan|location=London|isbn=1-84407-192-8|page=49}}</ref>
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| For variation of this see [[second half of the chessboard]] in reference to the point where an exponentially growing factor begins to have a significant economic impact on an organization's overall business strategy.
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| ===Water lily===
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| French children are told a story in which they imagine having a pond with [[Nymphaeaceae|water lily]] leaves floating on the surface. The lily population doubles in size every day and if left unchecked will smother the pond in 30 days, killing all the other living things in the water. Day after day the plant seems small and so it is decided to leave it to grow until it half-covers the pond, before cutting it back. They are then asked on what day half-coverage will occur. This is revealed to be the 29th day, and then there will be just one day to save the pond. (From Meadows ''et al''. 1972)<ref name=Porritt-2005/>
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| ==See also==
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| <div style="-moz-column-count:3; column-count:3;">
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| * [[Albert Allen Bartlett]]
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| * [[Arthrobacter]]
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| * [[Asymptotic notation]]
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| * [[Bacterial growth]]
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| * [[Bounded growth]]
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| * [[Cell growth]]
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| * [[Exponential algorithm]]
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| * [[EXPSPACE]]
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| * [[EXPTIME]]
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| * [[Hausdorff dimension]]
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| * [[Hyperbolic growth]]
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| * [[Information explosion]]
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| * [[Law of accelerating returns]]
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| * [[List of exponential topics]]
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| * [[Logarithmic growth]]
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| * [[Logistic curve]]
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| * [[Malthusian growth model]]
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| * [[Menger sponge]]
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| * [[Moore's law]]
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| </div>
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| ==References==
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| {{reflist}}
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| ===Sources===
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| * Meadows, Donella H., Dennis L. Meadows, Jørgen Randers, and William W. Behrens III. (1972) ''[[The Limits to Growth]]''. New York: University Books. ISBN 0-87663-165-0
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| * Porritt, J. ''Capitalism as if the world matters'', Earthscan 2005. ISBN 1-84407-192-8
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| * Swirski, Peter. ''Of Literature and Knowledge: Explorations in Narrative Thought Experiments, Evolution, and Game Theory''. New York: Routledge. ISBN 0-415-42060-1
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| * Thomson, David G. ''Blueprint to a Billion: 7 Essentials to Achieve Exponential Growth'', Wiley Dec 2005, ISBN 0-471-74747-5
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| * Tsirel, S. V. 2004. [http://www.mmsed.narod.ru/articles/artTsirel.ps On the Possible Reasons for the Hyperexponential Growth of the Earth Population]. ''Mathematical Modeling of Social and Economic Dynamics'' / Ed. by M. G. Dmitriev and A. P. Petrov, pp. 367–9. Moscow: Russian State Social University, 2004.
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| ==External links==
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| * [http://www.webwinder.com/wwhtmbin/jexpont.html Exponent calculator] — This calculator enables you to enter an exponent and a base number and see the result.
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| * [http://consumptiongrowth101.com/ExponentialGrowthCalculator.php Exponential Growth Calculator] — This calculator enables you to perform a variety of calculations relating to exponential consumption growth.
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| * [http://www.youtube.com/watch?v=hM1x4RljmnE Understanding Exponential Growth] — video clip 8.5 min
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| * [http://www.slideshare.net/amenning/growth-in-a-finite-world-sustainability-and-the-exponential-function Growth in a Finite World – Sustainability and the Exponential Function] — Presentation
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| * [http://www.energybulletin.net/media/2004-08-29/dr-albert-bartlett-arithmetic-population-and-energy Dr. Albert Bartlett: Arithmetic, Population and Energy] — streaming video and audio 58 min
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| [[Category:Ordinary differential equations]]
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| [[Category:Exponentials]]
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| [[Category:Mathematical modeling]]
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