For presentation purposes, we note that square root c = c0.5<\/sup>. In general,<\/p>\n Following the form of the Taylor polynomial gives, for square root, <\/p>\n P(x) = (c)0.5<\/sup> + 0.5c-0.5<\/sup>(x-c) -0.25c-1.5<\/sup>(x-c)2<\/sup>\/2! + 0.375c-2.5<\/sup>(x-c)3<\/sup>\/3! – 0.9375c-3.5<\/sup>(x-c)4<\/sup>\/4! + ….<\/p>\n The meaning of c<\/u><\/p>\n In the Taylor polynomial above, c is an “anchor value” at which you already know the output. Preferably it’s the closest value [to the one being evaluated] for which the exact answer is known.<\/p>\n Example: Evaluate square root 31 using a Taylor polynomial.<\/strong><\/p>\n Solution: closest to 31 is 36, so c=36. Then<\/p>\n P(31) = 360.5<\/sup> + 0.5(36)-0.5<\/sup>(31-36) – 0.25(36)-1.5<\/sup>(31-36)2<\/sup>\/2! + 0.375(36)-2.5<\/sup>(31-36)3<\/sup>\/3! – 0.9375(36)-3.5<\/sup>(31-36)4<\/sup>\/4! + ….<\/p>\n which becomes<\/p>\n 6 + (0.5\/6)(-5) – (0.25\/216)(-5)2<\/sup>\/2! + (0.375\/7776)(-5)3<\/sup>\/3! – (0.9375\/279936)(-5)4<\/sup>\/4! + ….<\/p>\n and then<\/p>\n 6 – 0.416666667 – 0.014467592 – 0.00100469393 – 0.00008721301476<\/p>\n =5.567773835<\/p>\n According to the calculator,<\/p>\n 310.5<\/sup> = 5.567765363<\/p>\n The difference between the values is 0.00000947215. Perhaps each term in the series gives an additional decimal place of accuracy.<\/p>\n HTH:)<\/p>\n Source:<\/p>\n Larson, Roland E. and Robert P. Hostetler. Calculus<\/u>, 3rd ed. Toronto: DC Heath and Company, 1989.<\/p>\n![](\"\/..\/images\/rat_exp.png\")
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