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Taylor-Series - Deriving using a Taylor-MacLaurin...

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Deriving using a Taylor - MacLaurin Series (Newton’s Preferred Calculation Method) (Doug Ruby and Danielle Tarnow) In his Method of Fluxions and Infinite Series , Sir Isaac Newton invented a way to calculate the “fluxion of a fluent” (derivative of a vari able) and a way to find the “flowing quantity from a fluxion” (integral). While developed and written in Latin during the Plague years of 1665 - 66, much of this work was not published until 1742 (Beckmann, 1971). In this magnum opus , Newton calculated to 16 decimal places, devoting just one paragraph of four lines to the development. Newton used the formula (in modern notation): x dx x arcsin 1 1 2 (1) plus prior trigonometric knowledge that arcsin ½ = /6, and his knowledge of the binomial theorem to derive: 5 3 2 5 4 2 3 1 2 3 2 1 2 1 6 2 1 arcsin 6 (2) We will derive the same equation and explore its usefulness by using the Taylor- MacLaurin series for arcsin x. Starting with the general form of a Taylor series expansion for f(x) about x=a: ! 3 ) )( ( ! 2 ) )( ( ) )( ( ) ( ) ( 3 2 a x a f a x a f a x
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