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History of Calculus

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History of Calculus History of Calculus Development · First steps were taken by Greek mathematicians, when Archimedes (around 225BC) constructed an infinite sequence of triangles starting with one of area A and continually adding further triangles between those already there and the parabola to estimate the area of a parabola. This is the first known infinite series sum (2, pg 1) · Archimedes used the method of exhaustion to find an approximation to the area of a circle, an early example of integration (2, pg 1) · Late 16th and early 17th century, scholars known as natural philosophers were trying to use math to understand the universe (1, pg 42) · The most famous natural philosopher was Galileo Galilei (1564-1642) who blended observation and experimentation with mathematical analysis (1, pg 42) · Kepler (1571-1630) used conic sections to describe the solar system, in that planets moved around the sun in elliptical orbits, and he formulated how fast each planet moved (1, pg 42) · Harriot (1560-1621) developed algebra and applied math to optics and navigation (1, pg 43) · Descartes brought together algebra and geometry, and studied comets and light (1, pg 43) · All of these ideas and studies brought velocities and infinite divisibility of space and time into the forefront (1, pg 43) · Equally important was the questions of finding tangents to curves and areas of figures (1, pg 43) · Cavalieri (1598-1647) worked with the principle of indivisibles, that is cutting a planar region into infinite sets of line segments or cutting solid figures into an infinite set of planar regions (1, pg 43) · Cavalieri showed that the integral of x^n from 0 to a was a^(n+1) / (n+1) by showing a number of values of n and inference (2, pg 2) · Roverval was more rigorous, and looked at the area between a curve and a line as being made up of an infinite number of narrow rectangular strips, and approximated the integral of x^m from 0 to 1 (2, pg 2) · Fermat investigated max and min by considering when the tangent to the curve was parallel to the x axis (2, pg 2) · This idea of estimation by Cavalieri was also done by Descartes (method of finding normals in 1637 (2, pg 2) ) and Fermat (1, pg 43) · Barrow gave a method of tangents to a curve where the tangent is given as the limit of a chord as the points approach each other (2, pg 3) · Torricelli and Barrow considered the problem of motion with variable speed. The inverse relationship of the integral and derivative were familiar to both (2, pg 3) · The idea of estimation led to development of calculus by Newton (who was influenced by Descartes and DeBeaune (2, pg 2) ) and Liebniz (1, pg 43) · Newton wrote a tract in 1666 on fluxions with horizontal and vertical velocities, or derivatives, and in discusses antidifferentiation (2, pg 3) · Newton calculated the series expansion for sin, cos and exponential functions, although they are now called Taylor or Maclaurin series (2, pg 3) · Newton (1642-1727) and Leibniz (1646-1716) independently discovered a general method for finding areas and volumes (1, pg 43) Calculus, Page 1 · Newton emphasized flowing quantities, and fluxions, which were their rates of flow (1, pg 44) · Liebniz used the idea of infinitesimal quantities, and defined the differential of x to be dx, which is the change in x in an infinitesimal small amount of time (1, pg 44) · Crucial was the idea that one could develop a kind of recipe for computing these changes, a method of calculation called calculus (1, pg 44) · Leibniz emphasized the importance of being able to solve problems without having to think, just apply the rules (1, pg 44) · Leibniz used the notations of d and the integral symbol in 1675 (2, pg 4) · Johann Bernoulli (1667-1748) taught calculus to a L'Hospital by writing a series of letters, which was the first calculus book, 'authored' by L'Hospital · Euler (1707-1783) wrote on every aspect of math and physics, among other subjects. He developed the calculus into a powerful instrument and applied it to all sorts of complicated problems. He wrote textbooks and emphasized the idea of a function in his precalculus book (1, pg 45) · Laplace (1749-1827) wrote books of celestial mechanics and probability (1, pg 46) · Lagrange (1736-1813) worked on all math areas, but his most important book focused on mechanics/physics The Limit (3, Limits) · Every major concept of calculus is defined in terms of limits · It is a product of the late 18th are early 19th century · Limits were first needed for the resolution of the four paradoxes of Zeno · Fermat developed an algebraic method for finding tangents, and let E stand for a small number and then discarded it, essentially taking the limit as E tends to 0 · The need for limits was not recognized in find areas of bounded regions · Newton failed to acknowledge the role of the limit · Newton recognized the limit must be the starting point to the all these problems · Lagrange reformulated mechanics in terms of calculus, and made a heroic flawed effort to make calculus purely algebraic by eliminating limits The Derivative (3, Derivatives) · The origin resides in tangent problems · Greeks originated problems of motion and velocity · Galilei established the principle that math was the indispensable tool for studying motion and science · Ideas of tangents reappeared as part of analytic geometry in the 17t century · Newton developed fluxional calculus, where motion was the fundamental basis for curves, tangents and related calculus phenomena Calculus, Page 2 The Fundamental Theorem of Calculus (3, Fundamental Theorem) · There is a faint connection between tangents and quadratures in Fermat and Descartes description of curves in analytic geometry · Gregory came close by considering the area between a curve and the axis, rewriting this as a new function, then finding the tangent line to this new curve · Barrow (1663-1669) displayed an understanding that tangents and quadratures were inverse operations · Newton used the Fundamental Theorem to find exact values for many areas, and began to think in algebraic terms · Lebiniz claimed that in finding the area represented by the integral of y, one should find a curve Y so that y=dY (Y is the antiderivative) · There was quite a dispute between Leibniz and Newton over ownership The Integral (3, Integrals) · Originated with quadrature and cubature problems, finding area and volume · Eudoxus is credited with the method of exhaustion, a technique approximating the area of a region with an increasing number of polygons · Archimedes used the method of exhaustion to find the quadrature of the parabola · During medieval times calculus idea were applied to motion problems · Fermat devised a technique for finding areas under higher parabolas · St Vincent determined the area under the hyperbola xy=1 by using narrow inscribed and circumscribed rectangles of different widths · The term integral was coined by Johann Bernoulli, a consequence of Newton and Leibniz Fundamental Theorem of Calculus they were regarded as reverse differentiation · Cauchy defined the integral of any continuous function on interval [a,b] to be the limit of the sums of thin rectangular areas · Reimann generalized Cauchys definition of the integral to arbitrary functions on the interval [a,b] 1. Math Through The Ages by Berlinghoff 2. http://www-history.mcs.st-andrews.ac.uk/history/HistTopics/The_rise_of_calculus.html 3. http://occawlonline.pearsoned.com/bookbind/pubbooks/thomas_awl/chapter1/medialib/custom3/topics.htm (Limits, Integrals, Derivatives, Fundamental Theorem) 4. http://www.scit.wlv.ac.uk/~cm1993/maths/mm2217/hc.htm Calculus, Page 3.
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