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Calculus is a branch of mathematics, developed from algebra and geometry, built on two major complementary ideas. The first idea, called differential calculus, is about a vast generalization of the slope of a line. It is a theory about rates of change, defining differentiation. It permits velocity, acceleration, and the slope of a curve at a given point all to be discussed on a common conceptual basis. The second idea, called integral calculus, is about a vast generalization of area. It is a theory about accumulation of small, even infinitesimal, quantities, defining integration. Though originally motivated by area, it includes related concepts such as volume and even distance. The two concepts differentiation and integration define inverse operations in a sense made precise by the fundamental theorem of calculus. Therefore, in teaching calculus either may in fact be given priority, but the usual educational approach (nowadays) is to introduce differential calculus first. Often what determines whether calculus or simpler mathematics is required to solve any given problem is not what ultimately needs to be accomplished. Rather, it is whether the requisite formula is provided or not. For example, finding the circumference of a circle does not require calculus provided the following is given: <math>C = 2 \pi r \,\!<math> However, if one has only a related formula such as for the area of a circle, <math>A = \pi r^2 \,\!<math> then calculus must be used to derive the formula for the circumference. For students studying calculus, this formula is usually the final answer to the problem, and no further input is requested. It must be realized, though, that calculus is not about formulas. The subject applies in many situations where the relevant functions and answers do not have formulas, which is the usual situation in real world applications. More precisely, any function whose graph is smooth enough to have tangent lines can be investigated with differential calculus. And any function whose graph has no breaks can be approached with the integral calculus.
HistorySee main article History of calculus Though the origins of integral calculus are generally regarded as going no farther back than to the ancient Greeks, there is evidence that the ancient Egyptians may have harbored such knowledge amongst themselves as well. (See Moscow Mathematical Papyrus.) Eudoxus is generally credited with the method of exhaustion, which made it possible to compute the area and volume of regions and solids. Archimedes developed this method further, while also inventing heuristic methods which resemble modern day concepts somewhat. Leibniz and Newton are usually designated the inventors of calculus, mainly for their discovery of the fundamental theorem of calculus and work on notation. There has been considerable debate about whether Newton or Leibniz was first to come up with the important concepts of the calculus. The truth of the matter will likely never be known. Leibniz' greatest contribution to calculus was his notation; he would often spend days trying to come up with the appropriate symbol to represent a mathematical idea. This controversy between Leibniz and Newton was unfortunate, however, in that it divided English-speaking mathematicians from those in Europe for many years. This set back British analysis (i.e. calculus-based mathematics) for a very long time. Newton's terminology and notation was clearly less flexible than that of Leibniz, yet it was retained in British usage until the early 19th century, when 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 had; however, Leibniz was the first to publish. Today, both Leibniz and Newton are considered to have discovered calculus independently. Lesser credit for the development of calculus is given to Barrow, Descartes, de Fermat, Huygens, and Wallis. Specifically, de Fermat is sometimes described as the "father" of differential calculus. A Japanese mathematician, Kowa Seki, lived at the same time as Leibniz and Newton and also elaborated some of the fundamental principles of integral calculus, though this was not known in the West at the time, and he had no contact with Western scholars. [1] (http://www2.gol.com/users/coynerhm/0598rothman.html) Differential calculusMain article derivative The derivative measures the sensitivity of one variable to small changes in another variable. A hint is the formula
Your speed (a derivative) in a car tells you about your change in location, relative to changes in time. Your speed may be changing; the calculus deals with this more 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), but for intervals of time that are very short. 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 difference quotients. The derivative of a function gives information about small pieces of its graph. It is directly relevant to finding the maxima and minima of a function — because at those points the graph is flat. Another application of differential calculus is Newton's method, an algorithm to find zeroes of a function by approximating the function by its 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 × Acceleration, has meaning in calculus because acceleration is a derivative. 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. Integral calculusMain article integral The definite integral evaluates the cumulative effect of many small changes in a quantity. The simplest instance is the formula Distance = Speed x Time 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 speed. More formally, we say that the definite integral of a function on an interval is a 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, you can approximate the area of a region by chopping it up into many very tiny 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.) 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 you know your speed at every instant in time for an hour you should be able to figure out how far you go during that hour. The definite integral of your speed tells you how. Many of the functions that are integrated are densities. If for example the pollution density along a river (tons per mile) is known, then the integral of that density can tell you 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. FoundationsThe rigorous foundation of calculus is based on the notions of functions and limits. 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 consists of rigorous definitions and of proofs of the theorems of calculus, as well as generalisations such as measure theory and functional analysis. Fundamental theorem of calculusThe fundamental theorem of calculus states that differentiation and integration are, in a certain sense, inverse operations. More precisely, antiderivatives can by calculated with definite integrals. It was this realization by both Newton and Leibniz which was key to the explosion of analytic results after their work became known. 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. The fundamental theorem also provides a method to compute many definite integrals algebraically, without actually performing the limit processes, by finding antiderivatives. It also allows us to solve some differential equations, equations that relate an unknown function to its derivatives. Differential equations are ubiquitous in the sciences. 1st Fundamental theorem of Calculus: derivative(integral(f(x)) in terms of x is x. 2nd Fundamental theorem of Calculus: integral(f(x)) where b is the upper limit and a is the lower limit equals F(b) - F(a) where F(x) is the anti-derivative of f(x). ApplicationsThe development and use of calculus has had wide reaching effects on nearly all areas of modern living. It underlies nearly all of the sciences, and especially physics. Almost all modern developments such as building techniques, aviation, and nearly all other technologies make fundamental use of calculus. Calculus has been extended to differential equations, vector calculus, calculus of variations, complex analysis, time scale calculus infinitesimal calculus, and differential topology. See also
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