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Unformatted text preview: Taylor’s theorem in several guises Jonathan L.F. King University of Florida, Gainesville FL 32611-2082, USA [email protected] Webpage http://www.math.ufl.edu/ ∼ squash/ 24 November, 2010 (at 00:53 ) Preliminaries. A polynomial p () is an “ N- topped polynomial ” if Deg( p ) < N . So x 2- 2 x and x + √ 7 are 3-topped, but x 3 + x- 5 is not. The set of N-topped polynomials is an N- dimensional VS. ♥ 1 Taylor polynomials The setting is an open interval J ⊂ R . Use Diff N = Diff N ( J → R ) for the set of N-times dif- ferentiable fncs. This is a superset of C N = C N ( J → R ); those, whose N th derivative is continuous . For posint N , a point Q ∈ J and f ∈ Diff N- 1 , define “ the N th Taylor polynomial of f , cen- tered at Q ” 1a : to be the unique N-topped polynomial p () whose zero th through [ N- 1] st derivatives, at Q , agree with those of f . I.e, these N equations hold: p ( Q ) = f ( Q ) , p ( Q ) = f ( Q ) , p 00 ( Q ) = f 00 ( Q ) , ..., p ( N- 1) ( Q ) = f ( N- 1) ( Q ) . One easily checks that p ( x ) must be the RhS of (1b), below. That is, T N ( x ) = T f N,Q ( x ) := X k ∈ [ ..N ) f ( k ) ( Q ) k ! · [ x- Q ] k 1b : is the N th Taylor polynomial of f . ♥ 1 Abbreviations: VS for vector space ; FTC for Funda- mental Theorem of Calculus ; posint for positive integer . Define the “ the N th remainder term of f ” , written R f N,Q or just R N , by f ( x ) =: T N ( x ) + R N ( x ) . 1c : Properties of T N . Use T N,Q [ f ] as a synonym for T f N,Q , with the same convention for the R N operator and friends. Often times either the fnc, f , or the center-point, Q , is implicit, and we drop it from the notation. For a scalar α and f,g ∈ Diff N- 1 , note that T N [ α · f ] = α · T N [ f ] and T N [ f + g ] = T N [ f ] + T N [ g ] . 2 : I.e, T N is a linear operator from Diff N- 1 to the VS of polynomials. Also, for f ∈ Diff N , T N [ f ] = h T N [ f ] i . 3 : That is, T N commutes with differentiation; in symbols T N D . Letting I denote the identity operator, we have that I = T N + R N . Since I is linear and commutes with D , so is/does R N . Courtesy FTC , integrating (3) gives Z x Q T N,Q [ f ]( t ) · d t = T N,Q [ f ]( x )- T N,Q [ f ]( Q ) = T f N,Q ( x )- f ( Q ) . There is a corresponding statement about the re- mainder term. Taylor’s Theorem for R We produce two estimates ♥ 2 of the remainder term. In the results below, N ∈ Z + . 4 : Lem. Fix an f ∈ Diff N ( J → R ) . At each x ∈ J , function q 7→ T f N,q ( x ) is differentiable, with value d d q T f N,q ( x ) = f ( N ) ( q ) · [ x- q ] N- 1 [ N- 1]! . 4 : And d d q R f N,q ( x ) =- d d q T f N,q ( x ) . ♦ ♥ 2 The TayThm-1 estimate, (5a), is called Lagrange’s form of the N th remainder term....
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