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Power Rule
In calculus, the power rule is used to differentiate functions of the form f(x) = x^r, whenever r is a real number. Since differentiation is a linear operation on the space of differentiable functions, polynomials can also be differentiated using this rule. The power rule underlies the Taylor series as it relates a power series with a function's derivatives. Statement of the power rule Let f be a function satisfying f(x)=x^r for all x, where r \in \mathbb. Then, :f'(x) = rx^ \, . The power rule for integration states that :\int\! x^r \, dx=\frac+C for any real number r \neq -1. It can be derived by inverting the power rule for differentiation. In this equation C is any constant. Proofs Proof for real exponents To start, we should choose a working definition of the value of f(x) = x^r, where r is any real number. Although it is feasible to define the value as the limit of a sequence of rational powers that approach the irrational power whenever we encounter such a power, or ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithmetic operations. It has two major branches, differential calculus and integral calculus; the former concerns instantaneous Rate of change (mathematics), rates of change, and the slopes of curves, while the latter concerns accumulation of quantities, and areas under or between curves. These two branches are related to each other by the fundamental theorem of calculus, and they make use of the fundamental notions of convergence (mathematics), convergence of infinite sequences and Series (mathematics), infinite series to a well-defined limit (mathematics), limit. Infinitesimal calculus was developed independently in the late 17th century by Isaac Newton and Gottfried Wilhelm Leibniz. Later work, including (ε, δ)-definition of limit, codify ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reciprocal Rule
In calculus, the reciprocal rule gives the derivative of the reciprocal of a function ''f'' in terms of the derivative of ''f''. The reciprocal rule can be used to show that the power rule holds for negative exponents if it has already been established for positive exponents. Also, one can readily deduce the quotient rule from the reciprocal rule and the product rule. The reciprocal rule states that if ''f'' is differentiable at a point ''x'' and ''f''(''x'') ≠ 0 then g(''x'') = 1/''f''(''x'') is also differentiable at ''x'' and g'(x) = \frac \left(\frac \right) = -\frac. Proof This proof relies on the premise that f is differentiable at x, and on the theorem that f is then also necessarily continuous there. Applying the definition of the derivative of g at x with f(x) \ne 0 gives \begin g'(x) = \frac d \left(\frac \right) & = \lim_ \left (\frac \right )\\ & = \lim_ \left( \frac \right)\\ & = \lim_ \left( - \frac \cdot \frac 1 \right).\end The limit o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Branch Point
In the mathematical field of complex analysis, a branch point of a multi-valued function (usually referred to as a "multifunction" in the context of complex analysis) is a point such that if the function is n-valued (has n values) at that point, all of its neighborhoods contain a point that has more than n values. Multi-valued functions are rigorously studied using Riemann surfaces, and the formal definition of branch points employs this concept. Branch points fall into three broad categories: algebraic branch points, transcendental branch points, and logarithmic branch points. Algebraic branch points most commonly arise from functions in which there is an ambiguity in the extraction of a root, such as solving the equation ''w''2 = ''z'' for ''w'' as a function of ''z''. Here the branch point is the origin, because the analytic continuation of any solution around a closed loop containing the origin will result in a different function: there is non-trivial monodromy. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alphonse Antonio De Sarasa
Alphonse Antonio de Sarasa was a Jesuit mathematician who contributed to the understanding of logarithms, particularly as areas under a hyperbola. Alphonse de Sarasa was born in 1618, in Nieuwpoort in Flanders. In 1632 he was admitted as a novice in Ghent. It was there that he worked alongside Gregoire de Saint-Vincent whose ideas he developed, exploited, and promulgated. According to Sommervogel, Alphonse de Sarasa also held academic positions in Antwerp and Brussels. In 1649 Alphonse de Sarasa published ''Solutio problematis a R.P. Marino Mersenne Minimo propositi''. This book was in response to Marin Mersenne's pamphlet "Reflexiones Physico-mathematicae" which reviewed Saint-Vincent's ''Opus Geometricum'' and posed this challenge: : Given three arbitrary magnitudes, rational or irrational, and given the logarithms of the two, to find the logarithm of the third geometrically. R.P. BurnR. P. Burn (2001) "Alphonse Antonio de Sarasa and Logarithms", Historia Mathematica 28:1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Grégoire De Saint-Vincent
Grégoire de Saint-Vincent - in latin : Gregorius a Sancto Vincentio, in dutch : Gregorius van St-Vincent - (8 September 1584 Bruges – 5 June 1667 Ghent) was a Flemish Jesuit and mathematician. He is remembered for his work on quadrature of the hyperbola. Grégoire gave the "clearest early account of the summation of geometric series." Margaret E. Baron (1969) ''The Origins of the Infinitesimal Calculus'', Pergamon Press, republished 2014 by ElsevierGoogle Books preview/ref> He also resolved Zeno's paradox by showing that the time intervals involved formed a geometric progression and thus had a finite sum. Life Gregoire was born in Bruges 8 September 1584. After reading philosophy in Douai, he entered the Society of Jesus 21 October 1605. His talent was recognized by Christopher Clavius in Rome. Gregoire was sent to Louvain in 1612, and was ordained a priest 23 March 1613. Gregoire began teaching in association with François d'Aguilon in Antwerp from 1617 to 20. Moving to Lou ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gottfried Wilhelm Leibniz
Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of mathematics. He wrote works on philosophy, theology, ethics, politics, law, history and philology. Leibniz also made major contributions to physics and technology, and anticipated notions that surfaced much later in probability theory, biology, medicine, geology, psychology, linguistics and computer science. In addition, he contributed to the field of library science: while serving as overseer of the Wolfenbüttel library in Germany, he devised a cataloging system that would have served as a guide for many of Europe's largest libraries. Leibniz's contributions to this vast array of subjects were scattered in various learned journals, in tens of thousands of letters and in unpublished manuscripts. He wrote in several languages, primarily in Latin, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isaac Newton
Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician, physicist, astronomer, alchemist, theologian, and author (described in his time as a "natural philosopher"), widely recognised as one of the greatest mathematicians and physicists and among the most influential scientists of all time. He was a key figure in the philosophical revolution known as the Enlightenment. His book (''Mathematical Principles of Natural Philosophy''), first published in 1687, established classical mechanics. Newton also made seminal contributions to optics, and shares credit with German mathematician Gottfried Wilhelm Leibniz for developing infinitesimal calculus. In the , Newton formulated the laws of motion and universal gravitation that formed the dominant scientific viewpoint for centuries until it was superseded by the theory of relativity. Newton used his mathematical description of gravity to derive Kepler's laws of planetary motion, account for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Blaise Pascal
Blaise Pascal ( , , ; ; 19 June 1623 – 19 August 1662) was a French mathematician, physicist, inventor, philosopher, and Catholic Church, Catholic writer. He was a child prodigy who was educated by his father, a tax collector in Rouen. Pascal's earliest mathematical work was on conic sections; he wrote a significant treatise on the subject of projective geometry at the age of 16. He later corresponded with Pierre de Fermat on probability theory, strongly influencing the development of modern economics and social sciences, social science. In 1642, while still a teenager, he started some pioneering work on calculating machines (called Pascal's calculators and later Pascalines), establishing him as one of the first two inventors of the mechanical calculator. Like his contemporary René Descartes, Pascal was also a pioneer in the natural and applied sciences. Pascal wrote in defense of the scientific method and produced several controversial results. He made important contribu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
John Wallis
John Wallis (; la, Wallisius; ) was an English clergyman and mathematician who is given partial credit for the development of infinitesimal calculus. Between 1643 and 1689 he served as chief cryptographer for Parliament and, later, the royal court. He is credited with introducing the symbol ∞ to represent the concept of infinity. He similarly used 1/∞ for an infinitesimal. John Wallis was a contemporary of Newton and one of the greatest intellectuals of the early renaissance of mathematics. Biography Educational background * Cambridge, M.A., Oxford, D.D. * Grammar School at Tenterden, Kent, 1625–31. * School of Martin Holbeach at Felsted, Essex, 1631–2. * Cambridge University, Emmanuel College, 1632–40; B.A., 1637; M.A., 1640. * D.D. at Oxford in 1654 Family On 14 March 1645 he married Susanna Glynde ( – 16 March 1687). They had three children: # Anne Blencoe (4 June 1656 – 5 April 1718), married Sir John Blencowe (30 November 1642 – 6 May 1726) in 1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gilles De Roberval
Gilles Personne de Roberval (August 10, 1602 – October 27, 1675), French mathematician, was born at Roberval near Beauvais, France. His name was originally Gilles Personne or Gilles Personier, with Roberval the place of his birth. Biography Like René Descartes, he was present at the siege of La Rochelle in 1627. In the same year he went to Paris, and in 1631 he was appointed the philosophy chair at Gervais College, Paris. Two years after that, in 1633, he was also made the chair of mathematics at the Royal College of France. A condition of tenure attached to this particular chair was that the holder (Roberval, in this case) would propose mathematical questions for solution, and should resign in favour of any person who solved them better than himself. Notwithstanding this, Roberval was able to keep the chair till his death. Roberval was one of those mathematicians who, just before the invention of the infinitesimal calculus, occupied their attention with problems which are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Evangelista Torricelli
Evangelista Torricelli ( , also , ; 15 October 160825 October 1647) was an Italian physicist and mathematician, and a student of Galileo. He is best known for his invention of the barometer, but is also known for his advances in optics and work on the method of indivisibles. The Torr is also named after him. Biography Early life Torricelli was born on 15 October 1608 in Rome, the firstborn child of Gaspare Torricelli and Caterina Angetti. His family was from Faenza in the Province of Ravenna, then part of the Papal States. His father was a textile worker and the family was very poor. Seeing his talents, his parents sent him to be educated in Faenza, under the care of his uncle, Giacomo (James), a Camaldolese monk, who first ensured that his nephew was given a sound basic education. He then entered young Torricelli into a Jesuit College in 1624, possibly the one in Faenza itself, to study mathematics and philosophy until 1626, by which time his father, Gaspare, had died. The uncl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pierre De Fermat
Pierre de Fermat (; between 31 October and 6 December 1607 – 12 January 1665) was a French mathematician who is given credit for early developments that led to infinitesimal calculus, including his technique of adequality. In particular, he is recognized for his discovery of an original method of finding the greatest and the smallest ordinates of curved lines, which is analogous to that of differential calculus, then unknown, and his research into number theory. He made notable contributions to analytic geometry, probability, and optics. He is best known for his Fermat's principle for light propagation and his Fermat's Last Theorem in number theory, which he described in a note at the margin of a copy of Diophantus' '' Arithmetica''. He was also a lawyer at the '' Parlement'' of Toulouse, France. Biography Fermat was born in 1607 in Beaumont-de-Lomagne, France—the late 15th-century mansion where Fermat was born is now a museum. He was from Gascony, where his father, Domin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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