M1511106More

M1511106More - 7 t in n 11 ~|...

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Unformatted text preview: 7 t in n 11 ~| gtrrfii‘ErJ/gflflriflifl If” huffing-) tr“ 45H:- JHEM Taylor Series Calculatnrs, Camputers, and Guided Missiles Thought-aut- this bank we have used uaiculatnra tn evaluate and graph functiumi such 35 s"; In 1'. and sin :t. l-iuw dz: calm-1.1mm and .THT cnmputcrs evaluate such functiuns? Muril' mtmlnlnm use :1 prim- dure call-3d CDRDIC {Eur Cflurdinat-e Rutatimi Dlgital Cnmputzfl that iii particularly HLIllL'Ul m ll'w EimilL-d circuitry Hi In mlt‘nlatnr.‘ Cnmputt'rri. 1m thL1 nti‘lur I‘Iflr'ILL L'EIIL‘ulfltt' vfllLLL-H uf lllL'fi-l' iUHt'Iinl'H-i wing scrim: lit-w ltw “'I‘nyrlur .‘i-Et'iL'H" Himwn ltt'ltlhfl which i.-; dis- cussed in this Hurtinn. 1 I. i‘ -l L _ I . .1- .t f.‘ - l 'l' 1 'l" 'i' 'I By taking L’nttugl‘l It-rnw, Vfl‘lLIL‘H Hi I" LTEIII'I l‘r-u i'i'IIIII'ILl In any tlrgruv ul- acturfll'}? tffiinfi 0111}? H1? basic upL'rfllim'IH L‘Jl nt‘li‘liiinn. HLIiilmL'tiun, multiplicatiun, and divisitm. fictually; mutt-I. cmnpulnn: nudity Ii‘lifi ['IHICL'dHI'L' in twn way“. The final mndit'imlinn iH called .I'ru'lgr‘ mfm'tinu, which :I'thluu-H tin.- sim ut the number that is substituted intu ranch lurmulflh'. 'l'hu m-mun- liul idea i5 that a number like :"--"" can Ir".- Laxpnfiriud :m r" rm“ {Lw adclir-i."r t‘xptmunts}. H” the t'nn'nuln is. uvdluulud M u fifth and 1hr: I'L'Hull n‘l'lJil'tPliL’d by t' In ubtnin I"':"". In this way. tl'lL' HI..'Fil.'!-'r is ILHL‘ll. uni}! ft‘lr I'Vflltlcfi bciwut‘t‘l - l 1'll'lil l. The sutnnd Inmliiimlinn is It: fllIL'r tilt? Eta-affidan Hf il'lLL writ-.5 Slighll}; giving wlml i5 railed .111 f't'flflttmifl’d St‘rlt’fi. rm that the L'I'l‘l'll‘ i5 mum uniinrm tun-r this Interval. "1 split: 0!" llk‘fit‘ pmcu‘lurufi. ncuisitmill urmrh‘ slip thrill-ugh. Fur example, ant: prlpuinr purnnnnl mmputrr umluntus In |.tH1l ma til-[HHTW‘i-Il-fii {tlfiing Halli-1";er t'fll lL‘LI HAEICA} ur .15 tillllll‘iLNS-‘I? {1:5— ing UHAHC}, L'mlh HE wi‘LiL‘l‘I an: wrung in “11‘ liIE-il i'uw durinnml plant's, the mrrurst value Ewing tttHJtILH‘i'illttfi. l-Ii-un such snmll urmm EEITI l'tn'WL‘ mainr t‘unscthncufi. lI‘I i‘H'l when lrml im'uilud Kuwait, EH Amtrimn fiflldifl'rfi were killed by a] Stud IriiHHilLu in Fm rt hutmlfiu an incumwt mmputcr pmgmm gmumtul am urmr in 1h:- sixth tlt'i'i- mull plan: alluwtng ll‘u.‘ Haul in Hlip thmugh liu' 'nlrinl ll‘llh‘fiill.‘ [lu— Eensusyfilumi lnln it'luL‘linn [ti ll‘IiH H-‘L‘L'lillfl Wu TL'I'LIFI‘I In int'iliilu :IL'I'ii‘H. lull willi llu- l1‘l'l11htll llw Ht'i'ir-w Imw lit-in}; .I'imn'liwrri. Wl' lwgin I11; lli‘at'llfihlll]; punu'r urnrw. llu'n Ht‘lt't'llllifilllg In 'liml'm' rtrria'ri. 'Wl'llt'l‘l mm: llv ll‘lIttlHlll Hi .Ir- "inliniti'lv lung" lintlnr puiynmninls. 'I'.1_1..'|nr ‘iL'l'lL'fl Ile tu n-In.l.r|t.1l1t'»' 't-lllli'tlt' m " l'I'Ir' llll'llh-I' I.II."|.1ll"u 'NiL'L" Khlldill I 1'ILE_~.L.|.I1I|1 .u-‘ul Lulu-IL. .u'i. Iii-Lining: "l. IIIIIEIIIII'I' null iulli Ill.'|1l'tr L'IIIHI‘ILIIMIHIII LI1 lill'llit'lllulr't' l-'I|ll.llll1r1'\-u.” ill'tl."‘.l" JIH'MIIIlI' i:'|-|H. CHM-"Hi Arlmfllnn, Magnl I‘ll-'l lF-N: Hnlwrl HiL'Ltl. 'IlEI't-ullthll I'l'tur .lnul llw I'.11|.'III| [til-null,- " 'rl.-'I."I.'l Nr'.-|-_-.I lull. I‘M]. Fl l|.. 1ILJl'tllnl1nt I'I1.' ||1:' “in Il'h' |I1r lllllliklrl.” .Iml r'I.|1|-|I|'ii h'I.I.I|I1Ir1I.ItIu. l'llll..llll.'l|"lll.l. MACLAURIN and TAYLOR POLYNDMIALS Maelaurin peiynemials fur f(x) = sinx (centered at r: = G}. Source: aleutus: istb Editien by Larsen, Hestetler, Edwards, p.625: Heughten Mifflin. I'Ei‘QS. .3. KM] = J.‘ - 3-! bruise, the graph at Fll mere tier.er IEEHIiJiE'S the sine tunctien. Ill Taylor polynomials fer flx] = In x (centered at r: = l). Seuree:Caleu1us: Sixth Editien by Larsen. Hestetler. Edwards, p.599; Heugliten Mil‘f‘lin. l993. Figure 1H4 campfires the graphs 13F P. . F]. H... Um] J'Z1 Will'l II“: gl'ilpit Itl'j'it'i " Int. Hutu tital atertrx = I the graphs are nuar'ly :indislirtgnishnhie. I"t1r institute. HINDI] =: —{1-.i1153-5i¢ altti Iltififli 1' -tl.|[ifi_'ihl. _'In'we.1hegraph ei H. humans a better and belle approximation at the graph pt Hr] = In 1 near it = I. It'll- ' FUN WITH e" We begin with the Maelaurin series Far 3": 2 3| 4 5 I I I X I1+I+E+—+_+'}§?+n This helds for any enmplea value efo! (2: is often used instead UfI.) New. plug in (i9) fer a, where 9 represents an arbitrar}J reai number, pessiny representing an angle measure in radians. . 2 . 3 . 4 5 . , t3 t9 :6 £3 3”] = 1 + 'l‘ 1% + 1% + [TI-IL + lgzl‘ +... _ Fa? £393 e94 has I + £5 + + + T + 'l'm it Remember that the pewers of 1‘ cycle between 11 i', —I, and —t'. 3 at flz—tg—+E—+ 95 =l+l9“2“i 3! 4! ’3 New, separate the real terms and the terms in t'. 2 4 a s = [I — 9— + 9 + [£3 — '9 + ‘3 2! E ’5 £5 32 n94 - 6‘3 I95 = — + 'E!‘ hm] + i'[9I — + —... h--—*f—IF L—nr—i' This is cos 3. This is sin 9. We new have Euler’s fermula: em = cactil + faintil The right side is the pelar {trig} Term of a complex number with medulus l and argument {standard angle) 6 in the cemplcx piane. Plug in 9211:: e” 2 cos tr + t' sin a: =—1 =1: e” = —1 cl" +1 2 CI ( Euler‘s Theorem ] “[ltl connects the five most impertant constants of mathematics [and else the three most important mathematical epcraliens ~— addittem multiplication, and expeaentiatien}. These ti vc constants symbolize the tent majer branches cf classical mathematics: arithmetic It]I and 1]; algebra l i]; geernetqr [tr]; and analysis Ie].” {Eli Maer, e: The Stage ef a Number, p.16fl} “...perhaps the mcst cempaet and fameus of all fennulas.... It appeals equally tn the mystic, the SCiEflliSL “1'3 PhilDSGPher. U‘IE- mathematieian." {Kasaer and Newman. Mathematics and the Imaginatinn, I940}. ...
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M1511106More - 7 t in n 11 ~|...

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