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Taylor Error 2

# Taylor Error 2 - n z n n z n z = = g z = 1 29 1 1.2.001 1 n...

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Taylor Series Error I) Lagrange Error Bound ( 1) ( 1) ( )( ) ( 1)! n n f z x a n + + - + = error bound Where a is where the series is centered z is a value between a and x (z giving the largest value for ( 1) ( ) n f z + . (z is usually a or x) -For sin or cos ( 1) ( ) n f z + = 1, (even if all z’s give smaller values). Polynomial value ± error bound = range of possible values of the series Ex 1: Find the 3 rd degree polynomial approximation for x e at 1, centered at 0. Find the range of possible values for x e at 1, centered at 0. Use the Lagrange error equation. ! n x x e n = 2 3 3 ( ) 1 2 6 x x P x x = + + + 3 8 (1) 3 P = (3 1) (3 1) ( )(1) (3 1)! f z + + + 0<z<1 z = 1 (3 1) (1) f e + = (4) (1) (4)! 24 e e = Range: 8 3 24 e ±

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Ex 2: Find the 4 th degree Maclaurin polynomial approximation for cos(x) where a = 0, evaluated at 1. Find the Lagrange error bound. cos(x) = ( ) 2 ( 1) (2 )! n n x n - 2 4 4 ( ) 1 2 4! x x P x = - + 4 37 (1) 24 P = (2 2) (2 2) ( )( ) (2 2)! n n f z x n + + + 6 ( ) 1 f z = (6) (1)(1) 1 1 (6)! 6! 720 = = Ex 3: What degree of Taylor Polynomial for ln(1.2) might have an error < .001 ( 1) ( 1) ( )( ) ( 1)! n n f z x a n + + - + ( 1) ( 1) ( 1)! ( ) n n n n f x x + - - = 1 1 1 1 1 ! (1.2 1) ! (.2) 1 .2 ( 1)! ( 1)! 1 n n n n n n
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Unformatted text preview: n z n n z n z + + + + +- = = + + + g z = 1 ( 29 1 1 .2 .001 1 n n + = < + n = 2.5 n = 3 II) Alternating Series Error In a series 1 1 ( 1) n n n a ∞ + =-∑ : The upper bound of the error is found by the n + 1 term. error < 1 n a + Ex 1: What degree polynomial for the function ( 1) ! n n n x n ∞ =-∑ has an error less than 1/4 for x = 1 2 2 ( ) 1 2 x P x x = - + Third term = 1/6 < 1/4 III) Actual Error If the actual value of the function is available, the error of a polynomial estimation can be found by subtracting the polynomial value from the actual value. f(x) - ( ) n P x = error Ex. 1: What is the error for the fourth degree polynomial approximation of cosx when x = 4 π 2 4 4 ( ) 1 2 24 x x P x = -+ 4 ( ) .707429 4 P = cos( ) .707107 4 = Error = 4 cos( ) ( ) .000322 4 4 P-=...
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Taylor Error 2 - n z n n z n z = = g z = 1 29 1 1.2.001 1 n...

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