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Lecture35-rev

# Lecture35-rev - MAT A26 Lecture 35 1 Taylor Polynomials...

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MAT A26 Lecture 35 1 Taylor Polynomials Taylor polynomials are generalizations of the constant and linear approxima- tions we studied last fall. Let us see how to obtain them. Fix a R . Suppose that f ( x ) is n +1 times diﬀerentiable on the interval I = ( a - h, a + h ) for some h > 0. Let x I be chosen arbitrarily and then ﬁx it (so for now we will treat x as a constant!) Then by FTC we may write f ( x ) - f ( a ) | {z } = f 0 ( x ) the const. approx. = Z x a f 0 ( x ) dx (= the error E 0 ( x )) Now integrate by parts “stupidly”, taking u = f 0 ( t ) and dv = dt so v = t - x . (This is ok since x is a constant and so dv = dt as required.) Then f ( x ) - f ( a ) | {z } = E 0 = Z x a f 0 ( t ) dt = f 0 ( t )( t - x ) | x a - Z x a ( t - x ) f 00 ( t ) dt = f 0 ( x )( x - x ) - f 0 ( a )( a - x ) - Z x a ( t - x ) f 00 ( t ) dt = 0 + f 0 ( a )( x - a ) - Z x a ( t - x ) f 00 ( t ) dt Transposing the ﬁrst term on the right gives f ( x ) - ( f ( a ) + f 0 ( a )( x - a )) | {z } = f 1 ( x ) , the linear approx. = - Z x a ( t - x ) f 00 ( t ) dt 1

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Now repeat this: integrating by parts “stupidly”, taking u = f 00 ( t ) and dv = ( t - x ) dt so v = 1 2 ( t - x ) 2 . Then f ( x ) - ( f ( a ) + f 0 ( a )( x
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Lecture35-rev - MAT A26 Lecture 35 1 Taylor Polynomials...

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