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taylor - Dene the the N th remainder term of f written Rf...

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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 0 ( Q ) = f 0 ( 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 [ 0 .. 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 0 ] = h T N [ f ] i 0 . 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 0 ]( 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 0 : 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. Our TayThm-2 , (6a), is one of the many Integral forms of the remainder. Webpage http://www.math.ufl.edu/ squash/ Page 1 of 10
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Page 2 of 10 Taylor’s Theorem for R Prof. JLF King Proof. WLOG x =7. For natnum k and posint , define A k ( q ) := f ( k ) ( q ) k ! · [7 - q ] k , and B ( q ) := f ( ) ( q ) [ - 1]! · [7 - q ] - 1 . Now A 0 ( q ) = f ( q ) · 1, so d d q A 0 ( q ) = f 0 ( q ) = B 1 ( q ). And for k positive, d d q A k ( q ) equals 1 k ! h f ( k +1) ( q ) · [7 - q ] k + f ( k ) ( q ) · k · [7 - q ] k - 1 · [ 1] i note === B k +1 ( q ) - B k ( q ) .
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