Note for MATH 129 with Professor Rychlik at UA
Note for MATH 129 with Professor Rychlik at UA
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Date Created: 02/06/15
Taylor polynomial approximation with rigorous error bounds derived using integration by parts BY MAREK RYCHLIK November 8 2008 1 Motivation Our textbook is vague about the error of the Taylor approximations A very nice argument based on integration by parts gives the required estimates In this note I present this argu ment 2 Derivation of the Taylor formula Let fx be de ned for I close to a point a and have a suf cient number of derivatives for the argument below to be valid We have the following identity which follows from the Fundamental Theorem of Calculus x M fa mm a We transform the integral using integration by parts using virtually the same method as it is used to obtain the integral of lnx we insert 1 which is the derivative of a linear function and apply integration by parts frfaT lt714gtf tgtdtfaizitgt ameE eltzetgtflttgtdt After easy substitutions of the limits and flipping both minus signs this equation becomes fltzgt fa Mam eagt f H mm We observe that the first Taylor polynomial T1x fa f a z 7 a and thus we may rewrite this equation as 101 T1I Edi where E1z is the error term and it admits the following explicit representation 1 E1z z 7 t f tdt a We apply integration by parts again to the error term this time considering I 7 t as the derivative of a quadratic polynomial in t 7 E d 17732 7 17732 E E 17732 m Elm 71 e 7 f tth T1 t a a T1 wt Again after easy substitutions of the limits we obtain 06 7 2 E1z fTV Nz 7 a2 fquottm Combining this equation with fx T1x E1 we see that we fa f a meanwmw 7 JWAM 2 SECTION 3 This equation is equivalent to fr T2I E2I where x 2 E2ltzgt 7 W mm Now we see the emerging pattern after successive integrations by parts we obtain n ma 1 If n 101 f kf I7ak 7 n f 1tdti k0 1 Hence fI Tum EM where the error term is given explicitly by an integral EN 7 E L X f 1tdt Theorem 1 Let f be a function with n1 continuous derivatives and let a and I be two values in an interval entirely contained in the domain of Then fltzgt L z 71 E L X fltn1gtlttgtdt k0 1 This is a Taylor Formula with Remainder in lntegral Form 3 A more convenient form for the error term The expression for the error term given above is hard to use in practice in determining the size of the error in real Taylor approximations We may use the substitution for t where t is substi tuted with a new variable 8 such that when 0 S s S l t will vary between a and z t a z a Hence dt z 7 a ds With this substitution the error term can be represented as this integral A1 W fltn1asx 7 a z 7 a ds After some cleaning up this expression becomes Enzzian1ln1l7s f 1asz7ads 0 17a 1i n1 n1 Thus looks very much like the next omitted term of the Taylor expansion with f 1a replaced with some constant Cn1 which is represented by this integral 1 01 n1 17 snfltn1gta sz 7 a as In summary 0 Theorem 2 Let f be a function with n1 continuous derivatives and let a and I be two values in an interval entirely contained in the domain of Then there is a constant Cn1 such that fka On 1 n 1 161 k z akn1xx a CAUCHY FORM or THE ERROR TERM 3 The constant Cn1 is given explicitly by the following integral 01 A1n117 snfltn1gta 5z 7 a as 4 Cauchy form of the error term Moreover we may thing of Cn1 as a weighed average value of f 1t where t is between a and 1 Indeed we may write Cn1 1 98 h8 d8 where 98 n ll 7 8 1 and h8 fltn1a 8z 7 a We see that 98 2 0 and A 98d8 7 I 78 1li Let m and M be such real numbers that m g f 1t g Mi Thus m is a lower bound and M is an upper bound for all values t between a and I then 1 98md8 1 g8h8dtSl 98Md8 by the Comparison Theoremi Evaluating the two integrals not involving h yields m S Cn1 S M Finally if f 1t is continuous then the Intermediate Value Theorem says that for some value 6 between a and I we have fn19 Cn To apply the Intermediate Value Theorem we set m and M to the extreme values of f 1 n1 m 02151211 f a8z a M n1 7 oggl f a8r a The Intermediate Value Theorem states that if f 1 is continuous then every value between m and M is a value of f 1a 01 7 a for some a between 0 and 1 Clearly the value 6 a 01 7 a is between a and It This leads to the following representation of the error Theorem 3 Let f be a function with n1 continuous derivatives and let a and I be two values in an interval entirely contained in the domain of Then there exists a number 9 between a and z inclusive such that n k a n 1 x 2 f k I 7ak7f 6 I 7a 1 k0 n ll This is a well known Taylor Formula with Cauchy Remainder
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