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# Science and Computers II PHY 308

Syracuse

GPA 3.64

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This 18 page Class Notes was uploaded by Clement Bernier on Wednesday October 21, 2015. The Class Notes belongs to PHY 308 at Syracuse University taught by Duncan Brown in Fall. Since its upload, it has received 38 views. For similar materials see /class/225621/phy-308-syracuse-university in Physics 2 at Syracuse University.

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Date Created: 10/21/15

LAGRANGE INTERPOLATION FORMULA WITH REMAINDER Idea We are in the situation where the values of a function are known at some nite number of nodes Then we can nd a polynomial that interpolates the function at these nodes If the function is smooth enough we can estimate the error of the approximation Theorem 01 Let f be di erentiable n1 times on 1 b Consider the interpolating polynomial p off at n 1 nodes zox1n 6 ab Emplieitly p 20 fjL where Ljx szm Then for any x 6 ab f p Rz where f 1 01 RW EOW 95k for some an 6 ab Proof Consider z xed Then we are going to nd g 6 ab which makes 01 hold De ne the number E by the equation f p EH 7z We want to show that E m for some g This is 170 k n1 tricky let gt ft 7pt 7 E HZOt 7 A direct differentiation shows that g 1t f 1t 7 En 1l Then it is enough to prove that g 1 0 for some g 6 ab This is done by noting that g 0 and 9zk 0 for h 0 1 If x coincides with an mm the whole proof is trivial Otherwise7 g has at least n 2 distinct zeros on abl Then 91 has at least n 1 distinct zeros on ab because 9 is differentiable on abapplication of Rolle7s theorem Continue this way to conclude that gm has a zero g 6 ab D gt moignv chwn 5 T iv f O mKZQ 3 0 0r g f xx a A Qm xhi wg La 56 w ruv C AVAVLL s hwy Ag Wye f E 3 w m Zn 9 x m 5L Of owwx fxw 9 gnvff jawaid ngmr w1 T Vi A2841 19L R Y Aha m qucrwaw gr a Q rlo 392 f 92 Dgca ww Mm OS 93 O e x AFAWW lt49 w 0 Q WE army if V Q L 4w mmwmcr ko r phatgwx 0V 9f mg waywva w 5 oh 23 9 E if E A WE a Em TCSnQW rim 9 4f mmgnm QWVELLR V s x x u ISL 52 5 ET an MA Av uni2 MOIgt20 8 aky lt3 Dwarf Nayv a hfPN hf hznAJ 0v allZ 5795 as m i or 5 3 ii 9 f r E Fm y 2ng Cmiw ltL mm 69w 5 mag93W VS 3 W M 0 k md 3m Zr 0 prmw E r L O 950 Z t g MW n 3 a 93 3 Q9in Tr F in Viva 930 t myfg 0 Q ix mp Zoi Em w ammo m A H V km Q fixer x my TL r g 4 Kb Q n tackyV 3 Ln WW ASAXJ Whamm AuJ 128 mm m x M an 4 6 950 Q37 A m 3 Am quot mf g Rsamp 0 Nmeb mgmz 3 231 8 wg Z02 r4 JEN fgmo w OWN ltQ m Af vx q KW 8 25 0mm Cg j 4954 Wang N J 050 gm 41v 0 6 929 A 4 30 a g 2A6 mhunib I Hairy u NY 0 a adv 3 8 m Q A Er x g NHL 198QO H 3 Equot h f I a If Fa a 2 g 5 99ch f f w a ow A FL 03 Ava OWYQN rwt v Cm Wig g u w T92 253LLT m me Mirwar Wm gm 255 Ed 033 592 mka Dr Ef gy v or gt50 20 y T E E0 19 u my ulw I ww n wmhuif h waw Wm AWEth Zgt Sm On 3 NY A A M UP Z0 Q WM k 37 mam 95 3mm 29 g 2 Armg w gt3 6 my w 02 65 Zm y mm 9g QAHQ R 289 a v mg chrmk 125w 5 3 ownm WFOPP Sgiumwl s in f 529 2 930 M tam gt45 lt9hnrynrmuwx w S a OrbLA Rwfn Wm 0 SCSSAL f E 6 E559 AggyV u m T Ya m Auk Mfr 75 53 05 WWW 49 07 mug 0 S r9 hf 7 I a 27 WW 65 45le wagAgni v 053 l NARRL WW mg Fr 6 Smacim f 93 a 78 95 m 35 98 0 592 a 49 m 6 23 g 86 WW wag w nlt5 lt 956 O A 3 w 36gt lt54 lt5 g gt gwcxnxni 6 350 JG M3 655rkmw 6 B 3 E j 30 g6 2 g g a T A g 1302 3163 A 33 E E 333 Aggy 3 3 5J9 3 L7 ow 3 2 5 0 g 5m 3 m w 28 30 3A ORALS k3 11 mosma a 1 y 7 APQ y k 3 w 1 w gm 3 mg g rtx 31f 4 MA So A5 XxM my 2 ATOM 52 53M a a L M RM Zf K V v MW 5 A AM WA 8M m m M5 HE m A A0 uni3 WEN lt23 Qocuaj i3 Le 2 N A 3A 4M5 AM mm A 4Ag W AME CTN ARM 2 ma A 3939 M w E 9V 922i 5 Q3 z 95 i 3amp3 a x 92w Smaip g 5 m rm 9 QQPEWP mar an w n gt E i PE gt3 90 i E i gtNCgtJ n W QNE i gtwCgtJ u r I 96 gt n gtan 4 g3 ALT gt9 v xygLAYJ i v NWZ N rml I v 535 gt n 4 J u Cans Aer hm chm be FAA Olewave Ox 3 ms 03M gymng Qtykwg A 3 Q 23 Q 333 39 33 Am T 302332 24 23 4 4 315 015 2 HM m Hm 2x023 t 3 33 H M W as W x m W M W w f 0123 gol sg quot 733 a quot Ski 1 31quot 2x 0415 is g 31032 The am 33 O39L39SQS RRCVCMCQSWI magma gas 84 39 ms a AM 4 M a AC4 LugChg A A 34032 RS 3 3 KEN New Qm b 473 F u fxtvx j v m o a w ngG 140 am A 7 x 2 w Z 2 O u 4 a WHW fx unw 12 w Z 367 gm A Ax UXU IQ g ij Wm w W 6 ww ggx M An g m3 w i m7 W2 040 mi u 30 3 AZ x w Mag 43w wwa 0 mm 3mm 3 b 3 3 6 mm gran twi any 2 id 1 vme 5 8 0H JJSEL O 4ro 24 Q6 u 73 54 n TJQJ 3 g 7 u Fi v lt24 9031 32 3 v2 d ngodg v w 28 26 52 0w 036 ll bJsN magi nxiy 3g 2 NEAQI Q a lt a M7 M5 NM EA cock 63 3Q kw w ae ujnv 68 6 SEN 0 16 3 mg a gx g g i j j 1 S m U36 a sec la 3L DL 2 5 mg ampL i 3 2 A 7 f r a 3x0 3 Lg 343 139Ham 139 adj quot gig Y met A E3 g x os Ne maci Men I M M Wd k a VG KN Q 714311 2363 740 w MN d R A m 03 m w a E Ho d web Kb H R a 5 h n 4 a 4 4 1 Or 0 quotwar Z 9 gowaorrx QM 9 Q F A5 of a A if rm 2 5 r 2 A 50 EC 3 a n 2 4 w Chem h r 2 A m 1 W it W 31 H a aw 5 e

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