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Calculus (from Latin , "pebble" or "little stone") is a branch of mathematics that includes the study of Limits , Derivative s, Integral s, and Infinite Series , and constitutes a major part of modern university education. Calculus has widespread applications in Science , Engineering and is used to solve complex and expansive problems for which Algebra alone is insufficient. It builds on Analytic Geometry and Mathematical Analysis and includes two major branches, Differential Calculus and Integral Calculus , that are related by the Fundamental Theorem Of Calculus . HISTORY '' was one of the more famous inventors and contributors of calculus, with relation to his laws of motion and other mathematical physics concepts.]] Development See Also: History of calculus The history of calculus falls into several distinct periods, most notably the . See Eudoxus (c. 408−355 BC) used the Method Of Exhaustion , which prefigures the concept of the limit, to calculate areas and volumes. Archimedes (c. 287−212 BC) developed this idea further, inventing Heuristics which resemble Integral Calculus .Archimedes, ''Method'', in ''The Works of Archimedes'' ISBN 978-0-521-66160-7 The Method Of Exhaustion was rediscovered in China by Liu Hui in the 3rd century AD, who used it to find the area of a circle. It was also used by Zu Chongzhi in the 5th century AD, who used it to find the volume of a Sphere . In the modern period, independent discoveries in calculus were being made in early 17th century Japan , by mathematicians such as Seki Kowa , who expanded upon the Method Of Exhaustion . In Europe, the second half of the 17th century was a time of major innovation. Calculus provided a new opportunity in Mathematical Physics to solve long-standing problems. Several mathematicians contributed to these breakthroughs, notably John Wallis and Isaac Barrow . James Gregory proved a special case of the Second Fundamental Theorem Of Calculus in 1668. '' was originally accused of Plagiarism of Sir Isaac Newton's unpublished works, but is now regarded as an independent inventor and contributor towards calculus.]] Leibniz and Newton pulled these ideas together into a coherent whole and they are usually credited with the independent and nearly simultaneous invention of calculus. Newton was the first to apply calculus to general Physics and Leibniz developed much of the notation used in calculus today; he often spent days determining appropriate symbols for concepts. The basic insight that both Newton and Leibniz had was the Fundamental Theorem Of Calculus . When Newton and Leibniz first published their results, there was Great Controversy over which mathematician (and therefore which country) deserved credit. Newton derived his results first, but Leibniz published first. Newton claimed Leibniz stole ideas from his unpublished notes, which Newton had shared with a few members of the Royal Society. This controversy divided English-speaking mathematicians from continental mathematicians for many years, to the detriment of English mathematics. A careful examination of the papers of Leibniz and Newton shows that they arrived at their results independently, with Leibniz starting first with integration and Newton with differentiation. Today, both Newton and Leibniz are given credit for developing calculus independently. It is Leibniz, however, who gave the new discipline its name. Newton called his calculus the " The Science Of Fluxions ". Since the time of Leibniz and Newton, many mathematicians have contributed to the continuing development of calculus. In the 19th century, calculus was put on a much more rigorous footing by mathematicians such as Cauchy , Riemann , and Weierstrass . It was also during this period that the ideas of calculus were generalized to Euclidean Space and the Complex Plane . Lebesgue further generalized the notion of the integral. Calculus is a ubiquitous topic in most modern high schools and universities, and mathematicians around the world continue to contribute to its development. UNESCO -World Data on Education {Link without Title} Significance While some of the ideas of calculus were developed earlier, in Egypt , Greece , China , India , Iraq, Persia , and Japan , the modern use of calculus began in Europe , during the 17th century, when Isaac Newton and Gottfried Wilhelm Leibniz built on the work of earlier mathematicians to introduce the basic principles of calculus. This work had a strong impact on the development of Physics . Applications of differential calculus include computations involving Velocity and Acceleration , the Slope of a curve, and Optimization . Applications of integral calculus include computations involving Area , Volume , Arc Length , Center Of Mass , Work , and Pressure . More advanced applications include Power Series and Fourier Series . Calculus can be used to compute the trajectory of a shuttle docking at a space station or the amount of snow in a driveway. Calculus is also used to gain a more precise understanding of the nature of space, time, and motion. For centuries, mathematicians and philosophers wrestled with paradoxes involving division by zero or sums of infinitely many numbers. These questions arise in the study of Motion and Area . The Ancient Greek Philosopher Zeno gave several famous examples of such Paradoxes . Calculus provides tools, especially the Limit and the Infinite Series , which resolve the paradoxes. Foundations In mathematics, ''foundations'' refers to the Rigorous development of a subject from precise axioms and definitions. Working out a rigorous foundation for calculus occupied mathematicians for much of the century following Newton and Leibniz and is still to some extent an active area of research today. There is more than one rigorous approach to the foundation of calculus. The usual one is via the concept of Limits defined on the Continuum of Real Number s. An alternative is Nonstandard Analysis , in which the real number system is augmented with Infinitesimal and Infinite numbers. The foundations of calculus are included in the field of Real Analysis , which contains full definitions and Proof s of the theorems of calculus as well as generalizations such as Measure Theory and Distribution Theory . PRINCIPLES Limits and Infinitesimals See Also: Limit (mathematics) Calculus is usually developed by manipulating very small quantities. Historically, the first method of doing so was by Infinitesimal s. These are objects which can be treated like numbers but which are, in some sense, "infinitely small". On a number line, these would be locations which are not zero, but which have zero distance from zero. No non-zero number is an infinitesimal, because its distance from zero is positive. Any multiple of an infinitesimal is still infinitely small, in other words, infinitesimals do not satisfy the Archimedean Property . From this viewpoint, calculus is a collection of techniques for manipulating infinitesimals. This viewpoint fell out of favor in the 19th century because it is difficult to make the notion of an infinitesimal precise. However, the concept was revived in the 20th century with the introduction of Non-standard Analysis , which provided solid foundations for the manipulation of infinitesimals. In the 19th century, infinitesimals were replaced by Limit s. Limits describe the value of a Function at a certain input in terms of its values at nearby input. They capture small-scale behavior, just like infinitesimals, but using ordinary numbers. From this viewpoint, calculus is a collection of techniques for manipulating certain limits. Infinitesimals get replaced by very small numbers, and the infinitely small behavior of the function is found by taking the limiting behavior for smaller and smaller numbers. Limits are easy to put on rigorous foundations, and for this reason they are the standard approach to calculus. Derivatives This gives an exact value for the slope of a straight line. If the function is not a straight line, however, then the change in ''y'' divided by the change in ''x'' varies, and we can use calculus to find an exact value at a given point. (Note that ''y'' and ''f''(''x'') represent for the same thing: the output of the function.) A line through two points on a curve is called a secant line. The slope, or rise over run, of a secant line can be expressed as: : where the Coordinate s of the first point are (''x'', ''f''(''x'')) and ''h'' is the horizontal distance between the two points. To determine the slope of the curve, we use the ''limit'': : Working out one particular case, we find the slope of the squaring function at the point where the input is 3 and the output is 9 (i.e. , so ). : The slope of the squaring function at the point (3, 9) is 6, that is to say, it is going up six times as fast as it is going to the right. Integrals See Also: Integral Integral calculus is the study of the definitions, properties, and applications of two related concepts, the ''indefinite integral'' and the ''definite integral''. The process of finding the value of an integral is called ''integration''. In technical language, integral calculus studies two related Linear Operator s. The indefinite integral is the ''antiderivative'', the inverse operation to the derivative. F is an indefinite integral of ''f'' when ''f'' is a derivative of F. (This use of upper- and lower-case letters for a function and its indefinite integral is common in calculus.) The definite integral inputs a function and outputs a number, which gives the area between the graph of the input and the X-axis . The technical definition of the definite integral is the Limit of a sum of areas of rectangles, called a Riemann Sum . A motivating example is the distances traveled in a given time. : If the speed is constant, only multiplication is needed, but if the speed changes, then we need a more powerful method of finding the distance. One such method is to approximate the distance traveled by breaking up the time into many short intervals of time, then multiplying the time elapsed in each interval by one of the speeds in that interval, and then taking the sum (a Riemann Sum ) of the approximate distance traveled in each interval. The basic idea is that if only a short time elapses, then the speed will stay more or less the same. However, a Riemann sum only gives an approximation of the distance traveled. We must take the limit of all such Riemann sums to find the exact distance traveled. If ''f(x)'' in the diagram on the left represents speed as it varies over time, the distance traveled between the times represented by ''a'' and ''b'' is the area of the shaded region s. To approximate that area, an intuitive method would be to divide up the distance between ''a'' and ''b'' in to a number of equal segments, the length of each segment represented by the symbol ''Δx''. For each small segment, we can choose one value of the function ''f''(''x''). Call that value ''h''. Then the area of the rectangle with base ''Δx'' and height ''h'' gives the distance (time ''Δx'' multiplied by speed ''h'') traveled in that segment. Associated with each segment is the average value of the function above it, ''f(x)''=h. The sum of all such rectangles gives an approximation of the area between the axis and the curve, which is an approximation of the total distance traveled. A smaller value for ''Δx'' will give more rectangles and in most cases a better approximation, but for an exact answer we need to take a limit as ''Δx'' approaches zero. The symbol of integration is , an elongated ''S'' (which stands for "sum"). The definite integral is written as: : and is read "the integral from ''a'' to ''b'' of ''f''-of-''x'' with respect to ''x''." The indefinite integral, or antiderivative, is written: :. Since the derivative of the function ''y'' = ''x''&2 + ''C'' is ''y'' ' = 2''x'' (where ''C'' is any constant) :. Fundamental theorem See Also: Fundamental theorem of calculus The Fundamental Theorem Of Calculus states that differentiation and integration are inverse operations. More precisely, it relates the values of antiderivatives to definite integrals. Because it is usually easier to compute an antiderivative than to apply the definition of a definite integral, the Fundamental Theorem of Calculus provides a practical way of computing definite integrals. It can also be interpreted as a precise statement of the fact that differentiation is the inverse of integration. The Fundamental Theorem of Calculus states: If a function ''f'' is Continuous on the interval ''b'' and if ''F'' is a function whose derivative is ''f'' on the interval (''a'', ''b''), then : Furthermore, for every ''x'' in the interval (''a'', ''b''), : This realization, made by both Newton and Leibniz , who based their results on earlier work by Isaac Barrow , 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 Antiderivative s. 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. APPLICATIONS of the Nautilus Shell is a classical image used to depict the growth and change related to calculus]] Calculus is used in every branch of the Physical Science s, in Computer Science , Statistics , Engineering , Economics , Business , Medicine , and in other fields wherever a problem can be Mathematically Modeled and an Optimal solution is desired. and Einstein's theory of General Relativity are also expressed in the language of differential calculus. Calculus can be used in conjunction with other mathematical disciplines. For example, it can be used with Linear Algebra to find the "best fit" linear approximation for a set of points in a domain. In the realm of medicine, calculus can be used to find the optimal branching angle of a blood vessel so as to maximize flow. In Analytic Geometry , the study of graphs of functions, calculus is used to find high points and low points (maximums and minimums), slope, Concavity and Inflection Points . In economics, calculus allows for the determination of maximal profit by providing a way to easily calculate both Marginal Cost and Marginal Revenue . Calculus can be used to find approximate solutions to equations, in methods such as Newton's Method , Fixed Point Iteration , and Linear Approximation . For instance, spacecraft use a variation of the Euler Method to approximate curved courses within zero gravity environments. SEE ALSO Lists
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