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# Class Note for MATH 1432 at UH

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This 5 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 26Section 116 Taylor Polynomials and Taylor Series inr7a Jiwen He 1 Taylor Polynomials in x a 11 Taylor Polynomials in z 7 a Taylor Polynomials in Powers of z 7 a Taylor Polynomials in Powers of z 7 a The nth Taylor polynomial in z 7 a for a function f is a n a Pnltzgt faf az7a f ltz7agt2f7 ltz7agtn Pn is the polynomial that has the san ze value as f at a and the same rst n derivatives H H Best Ef g gxf t a faLPnW f at quot 39 7Plna f 0a Pn provides the best local approximation of near a by a polynomial of degree lt n P01 ab 1311 fa f ar a 1321 fa f az f 22 z 7 a Taylor s Theorem and Remainder Term Taylor s Theorem If f has n1 continuous derivatives on an open interval I that contains a7 then for each I E I low a n M 7 fa f ltagtltz 7 a To 7 a Rnltzgt was 7 g f fltn1gtlttgtltz 7 t dt Lagrange Formula for the Remainder n 1 5 RN 7 18 z 7 an where c is some number between a and 1 Estimate for the Remainder Term 7 n1 was gleawWw J a71 or 17a 1 12 Taylor Series in z 7 a Taylor Series in z 7 a Taylor Polynomial and the Remainder lf is in nitely differentiable on interval I containing a then 7 7 fna 7 n M i faf ltagtltz agt n z a Rnltzgt S ltIgleaflfnltlgt J a71 or La Taylor Series in z 7 a If RAE 7gt 0 as n 7gt 007 then Pnz 7gt 7 7 fna 7 n f17fafaz agt n z a Sigma Not at ion 00 k a n k a 1 2f kl 4016 JLIEOZprWWV 160 k0 Taylor Polynomials and Taylor Series in z 7 a of fx e Pnltzgt fa f az ea f a 7 agt2 e a fltkgtltzgt7e72 fltkgtltagt7e Vk012 Taylor Polynomials in z 7 a of the Exponential fx e a a Pnz ea eaz 7 a z 7 a2 z 7 a i Taylor Series in z 7 a of the Exponential fx e 00 1 a e7 e Hz7akeaJre x7az7a 7 Vzi 0 Expansion of e in z 7 a Taylor Series in z 7 a by Translation Another way to expand in in powers ofz 7a is to expand fta in powers 0ft andthensettz7ai Taylor Series in z 7 a of the Exponential fx e a a e e eaz7az7a2z7an7 Vzi i mi 1 Expand e in powers oft ta a z a Do tk a 00 e e e e Zge Z 2 Settz7a 00 160 160 as a 7 7 k e 7e Z Iz a 7 for all real Xi gt7 k lti PS 13 Powers in x 7 a by Translation Expansion of fx in x 7 a as Expansion of ft a in t Taylor Series in x 7 a by Translation One way to expand in in powers ofx 7 a is to expand ft a in powers oft and then set It x 7 at This is the approach to take when the expansion in t is either known or is readily available Example 1 Expand eTQ in powers of x 7 3 1 Expand ft 3 in powers of t 00 00 k flttgt et32 632622 632 2 W2 632 1 tic k 0 kl k OZkkl 2 Set It x 7 3 7 T eE2 es2 2 7 3k7 for all real xi 160 Expansion of lnx in x 7 a7 a gt 0 Taylor Series in x 7 a by Translation One way to expand in in powers ofx 7 a is to expand ft a in powers oft andthensettx7al Taylor Series in x 7 a of the Logarithm fx lnx l l l lnxlna7x7a7jx7a2 x7a377 0ltx 2al 1 Expand lntr a in powes oft a 71k1 1t71 1t 71 1 1t 71 Ari tk71 Jrikly ltk n aina a inan ainak1 k a inak1kak 2i Settx7a7llt 177altt a70ltta 2a7 00 4161 k lnxlnak77x7a7 for0ltx 2al Expansion of sinx in x 7 7r Taylor Series in x 7 7r of the Sine fx sinx l l l sinx 7x77r7x77r375x77r5ix77r7 Vxl 1 Expand sint 7r in powers oft sint 7r sintcosrr costsinrr 7 sint 7 i Dk 7 21644 i 1k1t2k1 160 2k1 160 2k1 2 Set It x 7 7r 00 71 k1 Sim I 70 for all real 1 Expansion of cosz in z 7 7r Taylor Series in z 7 7r of the Cosine fx cosz 1 1 1 cosz7171770277177r47z77r577 V11 11 Expand cost 7T 11n powers 0amp1 cost 7r costcosw 7 sintsinTr 700st 00 411 2 7 m 41111 2 1 mythgwtk 21 Settz77r 41 cosz Z 7 70211 for all real 11 k0 Expansion of cos2 I in z 7 7r Taylor Series in z 7 7r of fx cos2 I 71k22k71 2 7 7 2k cos 171 2k 1 7r V11 11 Expand cos2t 7r in powers oft cos2t7r g cos2t7r cos2t27r 1 1 1 1 0 411 2k 00 4112211 1 2k 7 7 2t 7 7 7 2t 1 t 22 S 22 21M 1 1 2k 21 Settz77r 0quot 71k22k71 cos21 1ZWI7W for all real 11 k1 Expansion of 1 7 z 1 in z and Related Expansion of 1 7 z 1 in z and Geometric Series 00 17171zk1zz27 z lt11 160 Expand 17 21 1 in powers ofz 21 11 Expand ft 7 2 in powers oft ft72 7 172172 1 5721 1 7 17 EMA 7 ilt tgtk 71ilt gt k 160 k0 21Settz271lt tlt17 lttig79ltt72ltl he Expansion of E1 7 Ig m in z and Related Expansion of 171 m in z for m gt 1 1 oo 1717m7 k1km71zk1 m71lk Expand 17 21 3 in powers of z 21 11 Expand ft 7 2 in powers oft 7 ft7 2 7 17 2t 72 3 7 5 721W 7 l 17 2173 53 5 1 53 2 k 2171 k Zltk 1k 2 31gt 7 120 1k 7 mm k0 01H 21 Settz2 00 21671 M 7 lt17 2W 7 Zltk1gtltk 5113 x 2 160 Outline Contents 1 Taylor Polynomials 111 Taylor Polynomials 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 112 Taylor Series 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Powers in z 7 a 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 3

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