Calc09_2day1 - 9.2 Taylor Series Brook Taylor was an...

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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
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Suppose we wanted to find a fourth degree polynomial of the form: ( 29 2 3 4 0 1 2 3 4 P x a a x a x a x a x = + + + + ( 29 ( 29 ln 1 f x x = + at 0 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 0 0 P f =
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( 29 2 3 4 0 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 0 ln 1 0 f = = ( 29 2 3 4 0 1 2 3 4 P x a a x a x a x a x = + + + + ( 29 0 0 P a = 0 0 a = ( 29 1 1 f x x = + ( 29 1 0 1 1 f = = ( 29 2 3 1 2 3 4 2 3 4 P x a a x a x a x = + + + ( 29 1 0 P a = 1 1 a = ( 29 ( 29 2 1 1 f x x ′′ = - + ( 29 1 0 1 1 f ′′ = - = - ( 29 2 2 3 4 2 6 12 P x a a x a x ′′ = + + ( 29 2 0 2 P a ′′ = 2 1 2 a = -
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( 29 2 3 4 0 1 2 3 4 P x a a x a x a x a x = + + + + ( 29 ( 29 ln 1 f x x = + ( 29 ( 29 3 1 2 1 f x x ′′′ = + ( 29 0 2 f ′′′ = ( 29 3 4 6 24 P x a a x ′′′ = + ( 29 3 0 6 P a ′′′ = 3 2 6 a = ( 29 ( 29 ( 29 4 4 1 6 1
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