Dictionary
a hard lump produced by the concretion of mineral salts found in hollow organs or ducts of the body "renal calculi can be very painful" an incrustation that forms on the teeth and gums the branch of mathematics that is concerned with limits and with the differentiation and integration of functions
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Wikipedia
For other uses of the term calculus see calculus (disambiguation)''Calculus Calculus is a central branch of mathematics, developed from algebra and geometry, and built on two major complementary ideas. One concept is differential calculus which studies rates of change of one quantity relative to change in another, usually illustrated by the slope of a line. The other key concept is integral calculus and studies the accumulation of quantities, such as areas under a curve, linear distance traveled, or volume displaced. They are both inverse operations, which is elucidated by the fundamental theorem of calculus. Examples of typical differential calculus problems are finding: The acceleration and speed of a free-falling body at a particular moment. Change in profitability over time of a growing business at a particular point in time.Examples of integral calculus problems include finding the following quantities: The amount of water pumped by a pump with a set power input but varying conditions of pumping losses and pressure. The amount of parking lot plowed by a snowplow of given power with varying rates of snowfall.Today, calculus is used in every branch of the physical sciences and engineering, in economics and business, in medicine, and as a general method in endeavors where the goal is an optimum solution to a problem that can be given in mathematical form. From a mathematical standpoint, it is used in conjunction with limits which, roughly speaking, allows controlling or accurately describing an otherwise uncontrollable output.
Differential calculus - mainDerivative The derivative measures the sensitivity of one variable to small changes in another variable. Consider the formula::for an object moving at constant speed. The speed of a car, as measured by the speedometer, is the derivative of the car's displacement as a function of time. Calculus is a mathematical tool for dealing with this complex but natural and familiar situation. Differential calculus determines the ''instantaneous speed'' at any given specific ''instant'' in time, not just ''average speed'' during an ''interval'' of time. The formula Speed = Distance/Time applied to a single instant is the meaningless quotient "zero divided by zero". This is avoided, however, because the quotient Distance/Time is not used for a single instant (as in a still photograph). Rather a formula is developed for the quotient Distance/Time in which division by zero can be avoided, by a method called "taking the Limit (mathematics)limit".The derivative answers the question: as the elapsed time ''approaches'' zero, what does the average speed computed by Distance/Time ''approach''? In mathematical language, this is an example of taking a limit. More formally, differential calculus defines the instantaneous rate of change (the derivative) of a mathematical function's value, with respect to changes of the variable. The derivative is defined as a limit of a difference quotient.The derivative of a function, if it exists, gives information about small pieces of its graph. It is useful for finding the maxima and minima of a function — because at those points the graph is flat (i.e. the slope of the graph is zero). Another application of differential calculus is Newton's method, an algorithm to find root (mathematics)zeroes of a function by approximating the graph of the function by tangent lines. Differential calculus has been applied to many questions that are not first formulated in the language of calculus.The derivative lies at the heart of the physical sciences. Newton's law of motion, !Force = Mass &t imes; Acceleration,? involves calculus because acceleration is a derivative. (See Differential equation.) Maxwell's theory of electromagnetism and Einstein's theory of gravity (general relativity) are also expressed in the language of differential calculus, as is the basic theory of electrical circuits and much of engineering. It is also applied to problems in biology, economics, and many other areas.
Integral calculus - mainIntegral The definite integral evaluates the cumulative effect of many small changes in a quantity. The simplest instance is the formula :for calculating the distance a car moves during a period of time when it is traveling at ''constant'' speed. The distance moved is the cumulative effect of the small distances moved in each of the many seconds the car is on the road. The calculus is able to deal with the natural situation in which the car moves with ''changing'' speed.Integral calculus determines the exact distance traveled during an interval of time by creating a series of better and better approximations, called ''Riemann sums'', that ''approach'' the exact distance. More formally, we say that the definite integral of a function on an interval is a limit (mathematics)limit of Riemann sum approximations.Applications of integral calculus arise whenever the problem is to compute a number that is in principle (approximately) equal to the sum of the solutions of many, many smaller problems. The classic geometric application is to area computations. In principle, the area of a region can be approximated by chopping it up into many very tiny square (geometry)squares and adding the areas of those squares. (If the region has a curved boundary, then omitting the squares overlapping the edge does not cause too great an error.) area (geometry)Surface areas and volumes can also be expressed as definite integrals.Many of the functions that are integrated are rates, such as a speed. An integral of a rate of change of a quantity on an interval of time tells how much that quantity changes during that time period. It makes sense that if one knows their speed at every instant in time for an hour (i.e. they have an equation that relates their speed and time), then they should be able to figure out how far they go during that hour. The definite integral of their speed presents a method for doing so.Many of the functions that are integrated represent densities. If, for example, the pollution density along a river (tons per mile) is known in relation to the position, then the integral of that density can determine how much pollution there is in the whole length of the river. Probability, the basis for statistics, provides one of the most important applications of integral calculus.
Foundations - The rigorous foundation of calculus is based on the notions of a function (mathematics)function and of a limit (mathematics)limit; the latter has a theory ultimately depending on that of the real numbers as a continuum (mathematics)continuum. Its tools include techniques associated with elementary algebra, and mathematical induction. The modern study of the foundations of calculus is known as real analysis. This includes full definitions and proofs of the theorems of calculus. It also provides generalisations such as measure theory and distribution theory.
Fundamental theorem of calculus - The fundamental theorem of calculus states that differentiation and integration are, in a certain sense, inverse operations. More precisely, antiderivatives can be calculated with definite integrals, and ''vice versa''.This connection allows us to recover the total change in a function over some interval from its instantaneous rate of change, by integrating the latter. This realization, made by both Isaac Newton Newton and Leibniz, was key to the massive proliferation of analytic results after their work became known. The fundamental theorem provides an algebraic method of computing many definite integrals—without performing limit processes—by finding formulas for antiderivatives. It is also a prototype solution of a differential equation. Differential equations relate an unknown function to its derivatives, and are ubiquitous in the !sciences.1st? Fundamental Theorem of Calculus: If a function ''f'' is continuous functioncontinuous on the interval ''a'', - ''b'' and ''F'' is an antiderivative of ''f'' on the interval ''a'', - ''b'', then :2 nd? Fundamental Theorem of Calculus: If ''f'' is continuous on an open interval ''I'' containing ''a'', then, for every ''x'' in the interval, :
Applications - The development and use of calculus has had wide reaching effects on nearly all areas of modern living. It underlies nearly all of the sciences, especially physics. Virtually all modern developments such as building techniques, aviation, and other technologytechnologies make fundamental use of calculus. Many algebraic formulas now used for ballistics, heating and cooling, and other practical sciences were worked out through the use of calculus. In a handbook, an algebraic formula based on calculus methods may be applied without knowing its origins.The success of calculus has been extended over time to differential equations, vector calculus, calculus of variations, complex analysis, and differential topology.
History - mainHistory of calculus The origins of integral calculus are generally regarded as going back no farther than to the time of the Ancient Greeceancient Greeks, circa 200 B.C., though there is some evidence that the Ancient Egyptancient Egyptians may have had some hint of the idea at a much earlier date. (See Moscow and Rhind Mathematical PapyriMoscow Mathematical Papyrus.) The Hellenic mathematician Eudoxus of CnidusEudoxus is generally credited with the method of exhaustion, which made it possible to compute the areas of regions and the volumes of solids. Archimedes developed this method further, and invented heuristic methods which resemble modern calculus. Of all the mathematicians of the ancient world, he was the closest to discovering integral calculus, but never made the breakthrough, and after him study of calculus did not advance appreciably for more than a thousand years.An Indian mathematiciansIndian mathematician, Bhaskara (1114-1185), developed a number of ideas that can now be seen to be forerunners of calculus, including the idea now known as "Rolle's theorem". He was the first to conceive of differential calculus. The 14th century Indian mathematician Madhava (mathematician)Madhava, along with other mathematicians of the Kerala school, studied infinite series, convergence, differentiation, and iterative methods for solution of non-linear equations. Jyestadeva of the Kerala school wrote the first calculus text, the Yuktibhasa, which explores methods and ideas of calculus repeated in Europe only by the seventeenth century.Calculus, towards the end of the early modern period and into the first years of the eighteenth century, was a time of major innovation in Europe, making accessible answers to old questions, and providing a new method in mathematical physics. Several mathematicians contributed to this breakthrough, notably Wallis and Barrow. James Gregory proved a result equivalent to the Fundamental Theorem of Calculus in 1668. Gottfried LeibnizLeibniz and Isaac NewtonNewton pulled these ideas together into a coherent whole and they are usually credited with the independent and nearly simultaneous creation of calculus. Newton was the first to apply calculus to physics and Leibniz developed much of the notation used in calculus today; he often spent days determining appropriate symbols for concepts. It was generations after Newton and Leibniz that Cauchy and other mathematicians finally put calculus on a rigorous basis, with the definition of the limit (mathematics)limit.There has been considerable debate about whether Newton or Leibniz was first to come up with the important concepts of calculus. The truth of the matter is that the ideas of calculus were a part of the mathematical knowledge of their day, and they independently put those pieces together in different but coherent ways. The mathematical proofs of much of what they did came later, with Cauchy and others. This controversy between Leibniz and Newton was unfortunate in that it divided English-speaking mathematicians from those in Europe for many years, setting back British analysis (calculus-based mathematics) for a long time. Newton's terminology and notation was retained in British usage until the early 19th century, long after it had been replaced by Leibniz's notation everywhere else. The work of the Analytical Society successfully saw the introduction of Leibniz's notation in Great Britain. It is now thought that Newton had discovered several ideas related to calculus earlier than Leibniz; but Leibniz published first. Today, both are given equal credit. Lesser credit for ideas that led to the development of calculus is given to René DescartesDescartes, Isaac BarrowBarrow, Pierre de Fermatde Fermat, Christian HuygensHuygens, and John WallisWallis.
See also - Calculus with polynomials Differential geometry List of calculus topics List of important publications in mathematics#Calculus Important publications in calculus Mathematics Nonstandard analysis Precalculus (education)
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