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Class Note for MATH 1432 at UH 2

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This 6 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at University of Houston taught by a professor in Fall. Since its upload, it has received 13 views.

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Date Created: 02/06/15
Lecture 25Section 115 Taylor Polynomials in 1 Taylor Series in z Jiwen He 1 Taylor Polynomials 11 Taylor Polynomials Taylor Polynomials Taylor Polynomials The nth Taylor polynomial at 0 for a function f is H 71 Pm W How if 22 if 715 Pu is the polynomial that has the same value as f at 0 and the same rst n derivatives Pn0 f07Pl0 f07Pn0 f 07 7135 0 f 0A Best Approximation Pn provides the best local approximation of near 0 by a polynomial of degree Snl P0z 160 1311 f0 f017 H P 21 W f ltogtz Taylor Polynomials of the Exponential fr e n 13711 fO fggmx JU f 012 I I 39 f 0172 I 7 7 167 e Hxe 7 llzer ffltogt 1ff9lt5gt1 Hold k r P1X 3 y l l 2 3 x P3 P1 Taylor Polynomials of ef Poz17 P1z 117 1321 1z 12 13 P3z1zj 3 i 12 I Pnz1z Hl Taylor Polynomials of the Sine sinz f 0 f 0 71 Has 39le minEE71ZQSquot y P1X P5X 2 I 7139 77 x fx sin x 2 P3X P7X Taylor Polynomials of f sins P0l 0 P103 P203 33 3 P3433 7 3133 3 7 3 5 P433 7 P433 7 3 7 3 5 7 P7ltscgt P8ltscgt 7 x7 7 7 7 7 1 2 Remainder Term Remainder Term Remalnder Term De ne the nth remainder by Rnc f 7 P7433 that is f Pnc Then Taylor s 7 f if and only if 711133012433 7 0 If f has 71 1 continuous derivatives on an open interval I that contains 0 then for each a E I 1 z R7431 7 flt 1gtta 7 t dt Lagrange Formula for t e Re ainder For some number 0 between 0 and 3 R as fn1c n1 n 1 3 Taylor Polynomials of the Exponential fE e n n 1 C Pnrf0f0zmf 50 Rnltxgt7f ltgt n1 v Taylor Polynomials of the Exponenntial fE e n 1 E2 E3 E 7 T 7 i i i fzie Pnz71z23l 7 Remainder Term For each real E7 RAE A 0 as n A 00 Proof Let J be the interval that joins 0 to E and let M 6 Note that f 1t e for all n7 then lfltn1 Ml W A n l anIl S M 0 as naool Taylor Polynomials of the Sine 39 PnI 0f0E Ti WW gnfaslmwlt 71 Taylor Polynomial toef the Sine sinE7 P7E P3E E 7 Remainder Term For each real E7 RAE A 0 as n A 00 Vk fkt icost or isin t then max lfltn1gt S ll 6 w a n l anIl S 0 as naool 2 Taylor Series 21 Taylor Series Taylor Series Taylor Polynomial and the Remainder lf is in nitely differentiable on interval I containing 07 then PAE RAE VE E I W Whack f0 f 0z 160 kl n 7 fltn15 n1 7 1 E n1 n RAE 7 n1 E or RAE 7 n 0 f 7 t dt Taylor Series n f 100 k lf RAE A 0 as n A 007 then PAE Z k E A lex In this case7 160 can be expanded as a Taylor series in E and Write k 1 E f 0113 k0 4 Taylor Series of the Exponential fx e n fk0 k fltzgt PM Rnltzgt Pnltzgt ZTz k0 k f 0zk lim If lim Rm H0 then 161 Z k k0 Taylor Series of the Exponential fx e 00 1k 12 13 eHlz n forallrealz 0 Number e 7 171 1 1 1 eiggi i m Taylor Series of the Sine fx sinz fPR Wkiwk 1 n1 n1 nzik k z 0 0 fk0 k lf hm RAE A 07 then Z I lim naoo 160 kl naoo Taylor Series of the Sine fx sinz m Dk 2k1 3 5 7 s1nz mz 17 51 forallrealx Number sin 1 m 411 1 1 1 1 71 7 sm gemm 3151 71 Taylor Series of the Cosine fx cosz k fltzgt PM Rnltzgt Pnltzgt Zfomzk k0 00 k If lim Rm H0 then 111 Zfomzk lim PM 160 1 mm Taylor Series of the Cosine fx cosz 0 71 k 2 4 5 cosz IQkliiiii forallrealz k0 2k 21 41 6 Numbercosl 00 k 7 71 7 1 1 1 0051 W WIW Taylor Series of the Logarithm fx lnl In 7P R P 7 fk0 k fltzgt 7 nltzgt we nltzgt 7 k z 00 k If lim Rm so then 161 Zfomzk lim PM k0 Taylor Series of the Logarithm fx ln1 1 00 71 k1 2 3 4 lnlzzkziin forilltz l Number ln2 00 k 7 71 1 1 1 12 Z k 1 23 22 Numerical Calculations Outline Contents 1 Taylor Polynomials lll Taylor Polynomials l l l l l l l l l l l l l l l l l l l l l l l l l l 12 Remainder Term l l l l l l l l l l l l l l l l l l l l l l l l l l l 2 Taylor Series 21 Taylor Series l l l l l l l l l l l l l l l l l l l l l l l l l l l l l 22 Numerical Calculations l l l l l l l l l l l l l l l l l l l l l l l

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