Exponential growth in the context of "Derivative"

In economics, economic growth is an increase in the quantity and quality of the economic goods and services that a society produces. It can be measured as the increase in the inflation-adjusted output of an economy in a given year or over a period of time.

The rate of growth is typically calculated as real gross domestic product (GDP) growth rate, real GDP per capita growth rate or GNI per capita growth. The "rate" of economic growth refers to the geometric annual rate of growth in GDP or GDP per capita between the first and the last year over a period of time. This growth rate represents the trend in the average level of GDP over the period, and ignores any fluctuations in the GDP around this trend. Growth is usually calculated in "real" value, which is inflation-adjusted, to eliminate the distorting effect of inflation on the prices of goods produced. GDP per capita is the GDP of the entire country divided by the number of people in the country. Measurement of economic growth uses national income accounting.

The book An Essay on the Principle of Population was first published anonymously in 1798, but the author was soon identified as Thomas Robert Malthus. The book warned of future difficulties, on an interpretation of the population increasing in geometric progression (so as to double every 25 years) while food production increased in an arithmetic progression, which would leave a difference resulting in the want of food and famine, unless birth rates decreased.

While it was not the first book on population, Malthus's book fuelled debate about the size of the population in Britain and contributed to the passing of the Census Act 1800. This Act enabled the holding of a national census in England, Wales and Scotland, starting in 1801 and continuing every ten years to the present. The book's 6th edition (1826) was independently cited as a key influence by both Charles Darwin and Alfred Russel Wallace in developing the theory of natural selection.

Negative feedback (or balancing feedback) occurs when some function of the output of a system, process, or mechanism is fed back in a manner that tends to reduce the fluctuations in the output, whether caused by changes in the input or by other disturbances.

Whereas positive feedback tends to instability via exponential growth, oscillation or chaotic behavior, negative feedback generally promotes stability. Negative feedback tends to promote a settling to equilibrium, and reduces the effects of perturbations. Negative feedback loops in which just the right amount of correction is applied with optimum timing, can be very stable, accurate, and responsive.

The square metre (international spelling as used by the International Bureau of Weights and Measures) or square meter (American spelling) is the unit of area in the International System of Units (SI) with symbol m. It is the area of a square with sides one metre in length.

Adding and subtracting SI prefixes creates multiples and submultiples; however, as the unit is exponentiated, the quantities grow exponentially by the corresponding power of 10. For example, 1 kilometre is 10 (one thousand) times the length of 1 metre, but 1 square kilometre is (10) (10, one million) times the area of 1 square metre, and 1 cubic kilometre is (10) (10, one billion) cubic metres.

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 rate) is constant, the quantity undergoes exponential growth and has a constant doubling time or period, which can be calculated directly from the growth rate.

This time can be calculated by dividing the natural logarithm of 2 by the exponent of growth, or approximated by dividing 70 by the percentage growth rate (more roughly but roundly, dividing 72; see the rule of 72 for details and derivations of this formula).

Positive feedback (exacerbating feedback, self-reinforcing feedback) is a process that occurs in a feedback loop where the outcome of a process reinforces the inciting process to build momentum. As such, these forces can exacerbate the effects of a small disturbance. That is, the effects of a perturbation on a system include an increase in the magnitude of the perturbation. That is, A produces more of B which in turn produces more of A. In contrast, a system in which the results of a change act to reduce or counteract it has negative feedback. Both concepts play an important role in science and engineering, including biology, chemistry, and cybernetics.

Mathematically, positive feedback is defined as a positive loop gain around a closed loop of cause and effect.That is, positive feedback is in phase with the input, in the sense that it adds to make the input larger.Positive feedback tends to cause system instability. When the loop gain is positive and above 1, there will typically be exponential growth, increasing oscillations, chaotic behavior or other divergences from equilibrium. System parameters will typically accelerate towards extreme values, which may damage or destroy the system, or may end with the system latched into a new stable state. Positive feedback may be controlled by signals in the system being filtered, damped, or limited, or it can be cancelled or reduced by adding negative feedback.

Engelbart's law is the observation that the intrinsic rate of human performance is exponential. The law is named after Douglas Engelbart, whose work in augmenting human performance was explicitly based on the realization that although we use technology, the ability to improve on improvements (bootstrapping, "getting better at getting better") resides entirely within the human sphere.

Engelbart's Bootstrapping concept identifies the general, and particular, meaning of the observation with regards to rate and performance: a quantity, amount, or degree of something measured per unit of something else.That is, Engelbart's law is not limited to an increase in the acquisition, or use of, or quantity of knowledge, nor of the extent or depth of participation among individuals or teams, nor of the period-to-period change. The law is independent of the domain of performance and the quantity, amount, or degree on which one chooses a measure.

The Kardashev scale (Russian: шкала Кардашёва, romanized: shkala Kardashyova) is a method of measuring a civilization's level of technological advancement based on the amount of energy it is capable of harnessing and using. The measure was proposed by Soviet astronomer Nikolai Kardashev in 1964, and was named after him.

Kardashev first outlined his scale in a paper presented at the 1964 conference that communicated findings on BS-29-76, Byurakan Conference in the Armenian SSR, which he initiated, a scientific meeting that reviewed the Soviet radio astronomy space listening program. The paper was titled "Передача информации внеземными цивилизациями" ("Transmission of Information by Extraterrestrial Civilizations"). Starting from a functional definition of civilization, based on the immutability of physical laws and using human civilization as a model for extrapolation, Kardashev's initial model was developed. He proposed a classification of civilizations into three types, based on the axiom of exponential growth:

Cell proliferation is the process by which a cell grows and divides to produce two daughter cells. Cell proliferation leads to an exponential increase in cell number and is therefore a rapid mechanism of tissue growth. Cell proliferation requires both cell growth and cell division to occur at the same time, such that the average size of cells remains constant in the population. Cell division can occur without cell growth, producing many progressively smaller cells (as in cleavage of the zygote), while cell growth can occur without cell division to produce a single larger cell (as in growth of neurons). Thus, cell proliferation is not synonymous with either cell growth or cell division, despite these terms sometimes being used interchangeably.

Stem cells undergo cell proliferation to produce proliferating "transit amplifying" daughter cells that later differentiate to construct tissues during normal development and tissue growth, during tissue regeneration after damage, or in cancer.