Calc09_2day1 - Brook Taylor 1685 - 1731 9.2: Taylor Series...

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Unformatted text preview: Brook Taylor 1685 - 1731 9.2: Taylor Series Brook Taylor was an accomplished musician and painter. He did research in a variety of areas, but is most famous for his development of ideas regarding infinite series. Greg Kelly, Hanford High School, Richland, Washington Suppose we wanted to find a fourth degree polynomial of the form: ( 29 2 3 4 1 2 3 4 P x a a x a x a x a x = + + + + ( 29 ( 29 ln 1 f x x = + at x = that approximates the behavior of If we make , and the first, second, third and fourth derivatives the same, then we would have a pretty good approximation. ( 29 ( 29 P f = ( 29 2 3 4 1 2 3 4 P x a a x a x a x a x = + + + + ( 29 ( 29 ln 1 f x x = + ( 29 ( 29 ln 1 f x x = + ( 29 ( 29 ln 1 f = = ( 29 2 3 4 1 2 3 4 P x a a x a x a x a x = + + + + ( 29 P a = a = ( 29 1 1 f x x = + ( 29 1 1 1 f = = ( 29 2 3 1 2 3 4 2 3 4 P x a a x a x a x = + + + ( 29 1 P a = 1 1 a = ( 29 ( 29 2 1 1 f x x = - + ( 29 1 1 1 f = - = - ( 29 2 2 3 4 2 6 12 P x a a x a x = + + ( 29 2 2 P a = 2 1 2 a = - ( 29 2 3 4 1 2 3 4 P x a a x a x a x a x = + + + + ( 29 ( 29 ln 1 f x x = + ( 29...
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Calc09_2day1 - Brook Taylor 1685 - 1731 9.2: Taylor Series...

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