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# Class Note for MATH 1432 at UH

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Date Created: 02/06/15

Lecture 25 Section 115 Taylor Polynomials in x Taylor Series in X Jiwen He Department of Mathematics University of Houston jiwenhe mathuhedu httpmathuhedumjiwenheMath1432 1 P300 6 x fx e P2 5 P x 4 1 l Jiwen He University of Houston Math 1432 Section 26626 Lecture 25 April 15 2008 1 13 m m szbv vo ymmu um um Wm 39 5 aw n pngpwuwmyg m m szbv pu ymm 21 u a 2 mm 39 5 w m amnwmyd m 39TD PM mewme Msihegmev uezs 39 2mm 09 me m n dam10w aw n m m szbv pu ymm 21 u a 2 mm 39 5 w m amnwmyd m 39TD PM mewme Msihegmev uezs 39 2mm 09 me m n dam10w Mn mu aw n m m szbv pu ymm 21 u a 2 mm 39 5 w m amnwmyd m 39TD PM mewme Msihegmev uezs 39 2mm 09 me m n dam10w PAD 39U PL TU m m szbv pu ymm 21 u a 2 mm 39 5 w m amnwmyd m 39TD PM mewme Msihegmev uezs 39 2mm 09 me m n dam10w PA J quotDIPWD WU m m szbv pu ymm 21 u a 2 mm 39 5 w m amnwmyd m 39TD PM mewme Msihegmev uezs 39 2mm 09 me m n dam10w PA J quotDIPWD WU WW quotquot W m m szbv pu ymm 21 u a 2 mm 39 5 w m mu wwyd m 39TD PM mewme Msihegmev uezs 39 2mm 09 me m n dam10w P U PAD 39U PL quotU1 PM vmvdsihe km W mmmm m 39 POMmmM a gamp m m szbv pu ymm 21 u a 2 mm 39 5 w m mu wwyd m 39TD PM mewme Msihegmev uezs 39 2mm 09 me m n dam10w PA J quotDIPWD WU WW quotquot W L m m szbv pu ymm 21 u a 2 mm 39 5 w m mu wwyd m 39TD PM mewme Msihegmev uezs 39 2mm 09 me m n dam10w PA J quotDIPWD WU WW quotquot W m m szbv pu ymm 21 u a 2 mm 39 5 w m mu wwyd m 39TD PM mewme Msihegmev uezs 39 2mm 09 me m n dam10w MEI IU PLU quotDIPWD WU WW quotquot W p pmvdsihe km W mmmm m w my u by 2 POMmmM omega 5 n L Taylor Polynomials m Taylor Polynomials of the Exp 0 Hquot 0 13X r0 f 0x 23 X2 f Xn y PM Taylor Polynomials of fx ex fmex P2 P1x 1 Pom 3 W1 1 2 3 Jiwen He University of Houston Math 1432 Section 26626 Lecture 25 April 15 2008 3 l3 Taylor Polynomials w H Taylor Polynomials of the Exp y PM Taylor Polynomials of fx eX fxe PM 5 4 P1x 3 P2 3 y 1 1 2 3 P3 P1 Jiwen He University of Houston Math 1432 Section 26626 Lecture 25 April 15 2008 3 l3 Taylor Polynomials w H Taylor Polynomials of the Exp P2 3 y P3 P1 Jiwen He University of Houston P306 flx P2x P1X Taylor Polynomials of fx eX Math 1432 Section 26626 Lecture 25 April 15 2008 313 Taylor Polynomials w H Taylor Polynomials of the Exp P2 3 y P3 P1 P306 ex P2X Jiwen He University of Houston Taylor Polynomials of fx eX P1X Math 1432 Section 26626 Lecture 25 April 15 2008 313 Taylor Polynomials w H Taylor Polynomials of the Exp P2 3 y P3 P1 P306 ex P2X Jiwen He University of Houston Taylor Polynomials of fx eX P1X Math 1432 Section 26626 Lecture 25 April 15 2008 313 Taylor Polynomials w H Taylor Polynomials of the Exp P2 3 y P3 P1 P306 ex P2X Jiwen He University of Houston Taylor Polynomials of fx eX P1X Math 1432 Section 26626 Lecture 25 April 15 2008 313 Taylor Polynomials w H Taylor Polynomials of the Exp fX eX7 fX eX7 fIX ex ex mo 1 Mo 1 Taylor Polynomials of fx ex fx 2 ex P2 3 y 1 P3 P1 Jiwen He University of Houston Math 1432 Section 26626 Lecture 25 April 15 2008 313 Taylor Polynomials w H Taylor Polynomials of the Exp P1X P2 3 y 1 1 2 3 P3 P1 Jiwen He University of Houston Math 1432 Section 26626 Lecture 25 April 15 2008 3 l3 Taylor Polynomials 1 151 Taylor Polynomials of the Exp fX ex fX ex flX ex 39 39 39 7 ex f0 1 f 0 1 f 0 1 fltquotgt0 1 1 PM Taylor Polynomials of fx ex 6 fxe P20 P2 3 y 1 1 2 3 P3 P1 Jiwen He University of Houston Math 1432 Section 26626 Lecture 25 April 15 2008 3 l3 Taylor Polynomials 1 151 Taylor Polynomials of the Exp fX ex fX ex flX ex 39 39 39 7 ex f0 1 f 0 1 f 0 1 fltquotgt0 1 1 PM Taylor Polynomials of fx ex 6 fxe P20 P2 3 y 1 1 2 3 P3 P1 Jiwen He University of Houston Math 1432 Section 26626 Lecture 25 April 15 2008 3 l3 Taylor Polynomials w H Taylor Polynomials of the Exp rltngtx ex P2 3 y P3 P1 flx 6 Jiwen He University of Houston P1X Math 1432 Section 26626 Lecture 25 April 15 2008 313 Taylor Polynomials 1 151 Taylor Polynomials of the Exp fXeX7 fXex fx X7 7 fnXex e f0 1 f 0 1 f 0 1 fltquotgt0 1 1 PM Taylor Polynomials of fx ex 6 fmex P20 5 P0X 17 4 PM P1 X 1 l X P2 3 y 1 1 2 3 P3 P1 Jiwen He University of Houston Math 1432 Section 26626 Lecture 25 April 15 2008 3 l3 Taylor Polynomials 151 Taylor Polynomials of the Exp fX ex fX ex f X X7 7 fnX ex e f0 1 f 0 1 f 0 1 fltquotgt0 1 1 PM Taylor Polynomials of fx ex 6 fmex P20 5 P0X 17 4 PM P1 X 1 l X P 3 P2X1X 7 2 1 Pom 3 y 1 1 2 3 P3 P1 Jiwen He University of Houston Math 1432 Section 26626 Lecture 25 April 15 2008 Taylor Polynomials 151 Taylor Polynomials of the Exp fX ex fX ex f X exa 39 39 39 7 ex f0 1 f 0 1 f 0 1 fltquotgto 1 1 PM Taylor Polynomials of fx ex 6 fxe Pm 5 2 P0X Z 1 4 PM P1 X 1 l X P 3 P2X1X 7 2 2 3 X X 1 Pom 33 X 1 X 3 y 1 1 2 3 P3 P1 Jiwen He University of Houston Math 1432 Section 26626 Lecture 25 April 15 2008 Taylor Polynomials 151 Taylor Polynomials of the Exp fX ex fX ex f X exa 39 39 39 7 ex f0 1 f 0 1 f 0 1 fltquotgto 1 1 PM Taylor Polynomials of fx ex 6 fxe Pm 5 2 P0X Z 1 4 PM P1 X 1 l X P 3 P2X1X 7 2 2 3 X X 1 Pom 33 X 1 X 3 y 1 1 2 3 P3 P1 Jiwen He University of Houston Math 1432 Section 26626 Lecture 25 April 15 2008 Taylor Polynomials 151 Taylor Polynomials of the Exp ax eX r x eX r x eX rltngtx eX f0 1 f 0 1 f 0 1 fltquotgt0 1 1 PM Taylor Polynomials of fx ex 6 fmex P20 5 P0X Z 1 4 PM P1 X 1 l X P 3 P2X1X 7 2 1 X3 1 pom P3X 1X v 3 2 1 i E 3 1 39 2 3 7 PnX1 i Xquot39 Jiwen He University of Houston Math 1432 Section 26626 Lecture 25 April 15 2008 3 l3 Taylor Polynomials i l Taylor Polynomials of the Sine f sinX n PnXf0fOX X2HfnfOXn I Taylor Polynomials of fX sinX P1X P5X V Q 2 7Z39 7139 x fx sin x 2 P3X P7X Jiwen He University of Houston Math 1432 Section 26626 Lecture 25 April 1539 2008 4 13 Taylor Polynomials i l Taylor Polynomials of the Sine f sinX n PnXf0fOX X2HfnfOXn I fX sinX Taylor Polynomials of fX sinX Jiwen He University of Houston Math 1432 Section 26626 Lecture 25 April 1539 2008 4 13 Taylor Polynomials i l Taylor Polynomials of the Sine f sinX n PnXf0fOX X2HfnfOXn I fX sin X f X cosx Taylor Polynomials of fX sinX Jiwen He University of Houston Math 1432 Section 26626 Lecture 25 April 1539 2008 4 13 Taylor Polynomials i l Taylor Polynomials of the Sine f sinX n PnXf0fOX X2HfnfOXn I fX sin X7 f x cosx fX sinX Taylor Polynomials of fX sinX Jiwen He University of Houston Math 1432 Section 26626 Lecture 25 April 1539 2008 Taylor Polynomials i l Taylor Polynomials of the Sine f sinX Pngtlt f0 flox fZ mx2 lm I RX SlnX fX COSX fX sinX fHX cosx Taylor Polynomials of fX sinX Jiwen He University of Houston Math 1432 Section 26626 Lecture 25 April 1539 2008 Taylor Polynomials i l Taylor Polynomials of the Sine f sinX n PnXf0fOX X2HfnfOXn I x sinX f x cosx f x sinX fHx cosx f0 0 Taylor Polynomials of fX sinX Jiwen He University of Houston Math 1432 Section 26626 Lecture 25 April 1539 2008 Taylor Polynomials i l Taylor Polynomials of the Sine f sinX PnX f0 f OX y WW I ax sinX r x cosx f x sinX fHx cosx 260 o f O 1 Taylor Polynomials of fX sinX Jiwen He University of Houston Math 1432 Section 26626 Lecture 25 April 1539 2008 Taylor Polynomials i l Taylor Polynomials of the Sine f sinX PnX 0 f 0X X2 W I ax sinX r x cosx f x sinX fHx cosx f0 0 Wm 1 f 0 o Taylor Polynomials of fX sinX Jiwen He University of Houston Math 1432 Section 26626 Lecture 25 April 1539 2008 Taylor Polynomials i l Taylor Polynomials of the Sine f sinX PnX 0 f 0X X2 W I fX sin X fX cosx fX sinX fHX COSX f0 0 Wm 1 f 0 o fHO 1 Taylor Polynomials of fX sinX Jiwen He University of Houston Math 1432 Section 26626 Lecture 25 April 1539 2008 Taylor Polynomials i l Taylor Polynomials of the Sine f sinX n PnXfO l fOX X2H Xn l fX sin X fX cosx fX sinX7 fHX QSX7 f00 f 0 1 f 00 f 0 1 flt4gtoo Taylor Polynomials of fX sinX Jiwen He University of Houston Math 1432 Section 26626 Lecture 25 April 1539 2008 Taylor Polynomials i l Taylor Polynomials of the Sine f sinX n PnXfO l fOX X2H Xn l fX sin X fX cosx fX sinX7 fHX QSX7 f00 f 0 1 f 00 f 0 1 flt4gtoo Taylor Polynomials of fX sinX Jiwen He University of Houston Math 1432 Section 26626 Lecture 25 April 1539 2008 Taylor Polynomials i l Taylor Polynomials of the Sine f sinX n f 0 PnX fO l fOXl X2 n fX sin X f x cosx7 fHX sinX7 fHX COSX7 H fO 0 f 0 1 f o o fHO 1 f40 0 Taylor Polynomials of fX sinX P0X I O x i P1xP2XX Jiwen He University of Houston Math 1432 Section 26626 Lecture 25 April 1539 2008 Taylor Polynomials i l Taylor Polynomials of the Sine f sinX n f 0 PnX fO l fOXl X2 n fX sin X f x cosx7 fHX sinX7 fHX COSX7 H fO 0 f 0 1 f o o fHO 1 f40 0 Taylor Polynomials of fX sinX y P0X I 0 2 1 5 P100 P2X X7 X3 J 39 X P3XP4XX a Jiwen He University of Houston Math 1432 Section 26626 Lecture 25 April 1539 2008 Taylor Polynomials i l Taylor Polynomials of the Sine f sinX n f 0 PnX fO l fOXl X2 n fX sin X f x cosx7 fHX sinX7 fHX COSX7 H fO 0 f 0 1 f o o fHO 1 f40 0 Taylor Polynomials of fX sinX P0X 0 2 1 5 P1X P2X X J I i X P3X P4X X 5 fx sin x X3 X5 P500 PW Z X a a Jiwen He University of Houston Math 1432 Section 26626 Lecture 25 April 1539 2008 Taylor Polynomials i l Taylor Polynomials of the Sine f sinX n f 0 n PnXf0fOXX2H fX sin X f x cosx7 fHX sinX7 fHX COSX7 H fO 0 f 0 1 f o o fHO 1 f40 0 Taylor Polynomials of fX sinX P0X 0 2 1 5 P1X P2X X J I i X P3X P4X X 5 fx sin x X3 X5 P500 P600 Z X a a P7XP8XX l l Jiwen He University of Houston Math 1432 Section 26626 Lecture 25 April 1539 2008 4 13 Da neihe m renumda by m 39x PAU 3g TE n 3 m 2 3T 3 u 3g 3 Egg 5 252 Q u 3s 354 a 3g TE n 3 m 2 3T 3 u 3g 3 Egg 5 252 E 2 1wa4 a 3g TE n 3 m 2 3T 3 u 3g 3 Egg 5 252 133 E a 31 34 a 3g TE n 3 m 2 3T 3 u 3g 3 Egg 5 252 Mum m mm by m 39x 47m m s 39x m W hen mmpqx Mum nkmxm w 39 mg n 1zunimmu5devmiw5 on n Oven mm mm mums u gw ngvTem Mum m mm by m 39x 47m I mug Mam Thequot max Ix haww nm b z 5va ram w 39 mg n 1zunimmu5devmiw5 on in Wequot mam mm u m m x e A vhf N39m avh Wigwmwv m s m z pm R x mquot mm m mm by m a 7 mman mm amps w 39 mg n 1zunimmu5devmiw5 on n Oven mm mm mums u 09quot mm x e A f WWW 7 equot 4 F Jgnng gammy lE anrnmav m s m z pm R x 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