{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 0 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "" 0 18 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 261 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 266 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal " -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Plot" -1 13 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal " -1 256 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 2 2 2 2 2 1 1 1 1 } 1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 257 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 258 1 {CSTYLE "" -1 -1 "Times" 1 24 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 258 "" 0 "" {TEXT 257 16 "Newtonova metoda" } {TEXT 256 0 "" }}{PARA 256 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 179 "Newtonova metoda (t\351 \236 Newton-Raphsonova metoda nebo metoda linearizace) je metoda pro a proximaci ko\370en\371 funkc\355, spojit\375ch v\350etn\354 jejich prv n\355 derivace. C\355lem tohoto worksheetu je " }{TEXT 264 19 "grafick \351 zn\341zorn\354n\355" }{TEXT -1 78 " aplikace Newtonovy metody pro danou funkci jedn\351 prom\354nn\351 na dan\351m intervalu." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 109 "Pro danou k \370ivku y = f(x) hled\341me ty hodnoty argumentu x, pro n\354\236 pla t\355 f(x) = 0. Tyto hodnoty x se naz\375vaj\355 " }{TEXT 265 7 "ko \370eny." }{TEXT -1 24 " V\375po\350et za\350\355n\341 zad\341n\355m \+ " }{TEXT 266 16 "nult\351 aproximace" }{TEXT -1 81 " ko\370ene x, kter ou ozna\350\355me xn. Graficky lze kroky Newtonovy metody popsat takt o:" }}{PARA 0 "" 0 "" {TEXT -1 45 "1. Je f(xn) dostate\350n\354 bl\355 zko k hodnot\354 nula?" }}{PARA 0 "" 0 "" {TEXT -1 33 " Jestli \236e ano, jsme hotovi." }}{PARA 0 "" 0 "" {TEXT -1 19 " Jestli \236e ne:" }}{PARA 0 "" 0 "" {TEXT -1 39 "2. Nakresli te\350nu ke k \370ivce v bod\354 xn." }}{PARA 0 "" 0 "" {TEXT -1 61 "3. Najdi x-ovo u sou\370adnici xt pr\371se\350\355ku t\351to te\350ny s osou x." }} {PARA 0 "" 0 "" {TEXT -1 30 "4. Polo\236 hodnotu xn rovnu xt." }} {PARA 0 "" 0 "" {TEXT -1 18 "5. Jdi na krok 1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 69 "V\232imn\354te si, \236e \+ tento postup m\371\236e b\375t ne\372sp\354\232ny v n\354kolika p\370 \355padech:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 102 "1. Sm\354rnice te\350ny je rovna nule ( te\350na je rovnob \354\236n\341 s osou x ), tak\236e \236\341dny pr\371se\350\355k xt ne existuje." }}{PARA 0 "" 0 "" {TEXT -1 119 "2. Sm\354rnice te\350ny je bl\355zko k nule, tak\236e pr\371se\350\355k xt je tak daleko od odha du xn, \236e nem\341 smysl ve v\375po\350tu pokra\350ovat." }}{PARA 0 "" 0 "" {TEXT -1 91 "3. Hodnoty f(xn) se nep\370ibli\236uj\355 k nule v\371bec, nebo se k nule p\370ibli\236uj\355 jen velmi pomalu." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 370 "M\371 \236e-li postup b\375t ne\372sp\354\232n\375 z tolika d\371vod\371, pr o\350 se tedy metoda pou\236\355v\341? D\371vodem je, \236e za p\370ed pokladu vhodn\351 volby nult\351 aproximace (tj. je-li nult\341 aproxi mace dostate\350n\354 bl\355zko ko\370ene), konverguj\355 hodnoty xn \+ ke ko\370enu extr\351mn\354 rychle. Zpravidla se po\350et platn\375ch \+ desetinn\375ch m\355st v aproximaci xn p\370i ka\236d\351m kroku zdvoj n\341sobuje. Takto rychl\341 konvergence se naz\375v\341 kv" }{TEXT 262 9 "adratick\341" }{TEXT -1 263 ". N\355\236e uveden\341 procedura \+ pro animaci je pokusem o ilustraci v\232ech v\375\232e uveden\375ch pr obl\351m\371 a doporu\350en\355m, jak se jim vyhnout. V t\354chto \372 vah\341ch budou vyu\236\355v\341ny dva dal\232\355 parametry: Maxim \341ln\355 po\350et povolen\375ch krok\371 iterace a p\370\355pustn \375 interval hodnot aproximac\355 xn." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 61 "Tento p\370\355kaz ru\232 \355 v\232echny d\370\355ve stanoven\351 hodnoty prom\354nn\375ch." }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 118 "Tato procedura t ransformuje aritmetick\375 v\375raz jedn\351 prom\354nn\351 na procedu ru. Je-li v\375raz ji\236 procedurou, nic se nem\354n\355. " }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 292 "vytvorfunkci:= proc(f :: \{ algebraic,procedure\})\n local x;\n if f::procedure then\n f \n else\n x:= remove(type, indets(f, name), realcons);\n i f nops(x) <> 1 then\n ERROR(`V\355ce ne\236 jedna nezn\341m \341 v `, f)\n else \n codegen[makeproc](f, x[1])\n \+ fi\n fi\nend:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 75 "Procedura Kresli_Newt_met prov\341d\355 animaci \370 e\232en\355 rovnic Newtonovou metodou " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7966 "Kresli_Newt_met:= proc(F :: \{algebraic,procedure\} \n ,x0 :: realcons\n ,xrange :: rang e(realcons)\n ,nsteps :: posint\n ) \n # F je funkce. Lze je vyj\341d\370it jako funkci nebo jako aritme tick\375 v\375raz.\n #\n # x0 je nult\341 aproximace ko\370ene x v Newtonov\354 metod\354.\n #\n # xrange je obor hodnot x pro zobra zuj\355c\355 okno. \n # Je specifikov\341n jako a..b.\n #\n # n steps je maxim\341ln\355 p\370\355pustn\375 po\350et iterac\355 Newton ovy metody.\n \n local\n f # kopie F \n ,a, b \+ # koncov\351 body xrange\n ,ylow, yhigh # koncov\351 body yr ange\n ,n # po\350et pr\341v\354 proveden\375ch krok\371 Newtonovy iterace\n ,x # prom\354nn\341\n ,xn \+ # n-t\341 Newtonova approximace ko\370ene\n ,yn # f(x n)\n ,`f'` # derivace f\n ,`f'(xn)` # `f'`(xn)\n \+ ,T # funkce, jej\355m\236 grafem je te\350na\n ,txtx, t xty # pozice pro tisk textu\n ,plotopts # u\236ivatelem dodan \351 volby pro tisk\n \n #mo\236nosti tisku\n ,Curve #Gr af dan\351 funkce\n ,ToCurve #\332se\350ka z osy x ke k\370ivc e\n ,TanLine #Graf te\350ny\n ,Xmarks #Znak \"X\" pro bod dotyku na k\370ivce\n ,XAxis #Zn\341zorn\354n\355 osy x \n ,Frame #tabulka vzor\371 pro animaci\n ;\n \n # o p (zkratka slova operand) je p\370\355kaz pro odd\354len\355 jednotliv \375ch operand\371 z v\375raz\371 # V tomto p \370\355pad\354 jsou odd\354leny 1. a 2. \350\341st z xrange a yrange \n \n a:= evalf(op(1,xrange));\n b:= evalf(op(2,xrange));\n\n \+ if evalf(x0) <= a or evalf(x0) >= b then\n ERROR(`Nult\341 apro ximace nen\355 v p\370\355pustn\351m oboru.`)\n fi;\n\n # Vlo\236 \+ voliteln\351 argumenty do intern\355 formy\n plotopts:= `plot/option s2d`(args[5..-1]);\n \n f:= vytvorfunkci(F);\n plot(f, xrange); \n # vlo\236 p\370\355padn\351 specifikace barvy do CURVES\n Curve := CURVES(op(op(1,%)), op(select(x->op(0,x)=COLOUR or op(0,x)=COLOR, % )));\n \n # Z\341vorky [] jsou zde u\236ity vn\354 seznamu. \n # Pro pot\370ebu p\370\355kaz\371 min a max.\n yn:= op(remove(`=`, ma p(P->P[2], map(op, [op(select(type, Curve, listlist))])), FAIL));\n\n \+ # Vlo\236 obor y. N\341soben\355 1.1 vytvo\370\355 okraje. \n yhig h:= 1.1*max(yn);\n ylow:= 1.1*min(yn);\n\n # Zkontroluj, zda zn \341zorn\354n\355 bude obsahovat pr\371se\350\355k s osou x.\n if yl ow*yhigh > 0 then\n #\"cat\" provede z\370et\354zen\355 dan\375ch \370et\354zc\371.\n print\n (cat\n (`Tato fun kce osu x v`\n ,` dan\351m oboru neprotne.`\n ,` P\370esto budu hledat pr\371se\350\355k xt te\350ny s osou x.`\n \+ )\n );\n\n # Zv\354t\232i obor y tak, aby obsahov al osu x.\n if ylow > 0 then\n ylow:= -.1*yhigh \n e lif yhigh < 0 then \n yhigh:= -.1*ylow \n fi \n fi;\n \+ \n # Najdi sou\370adnice lev\351ho horn\355ho bodu oboru. \n txtx: = .9*a+.1*b;\n txty:= .9*yhigh+.1*ylow;\n\n # Nepo\236aduj tisk te xtu p\370\355li\232 bl\355zko k ose x.\n if abs(txty)/(yhigh-ylow) < .1 then txty:= .8*yhigh+.2*ylow fi;\n\n # Tiskni osu x \350ern\354 \+ (RGB,0,0,0) jako \372se\350ku.\n XAxis:= CURVES([[a,0], [b,0]], COLO R(RGB,0,0,0));\n\n # Inicializace prom\354nn\351 TanLine pro prvn \355 krok. \n # A\350koliv osa x pravd\354podobn\354 nen\355 te\350 na, bude\n # nakreslena, tak\236e z n\341sleduj\355c\355 inicializac e nevznik\341 \236\341dn\351 nebezpe\350\355.\n TanLine:= XAxis;\n\n # Za\350\355n\341 program Newtonovy metody.\n\n # Definuj derivac i f jako funkci. \n `f'`:= unapply(diff(f(x), x), x);\n\n # Initi alizuj Newtonovu itera\350n\355 posloupnost.\n xn:= evalf(x0);\n\n \+ # Tento for cyklus je po\350\355t\341n jednou v ka\236d\351m kroku Ne wtonovy iterace. Nen\355 nutn\354 po\350\355t\341n \+ # nsteps kr\341t. P\370\355kaz \"break\" m\371\236e cyklus ukon\350 it d\370\355ve.\n\n for n to nsteps do\n\n # Vypo\350\355tej f( xn). P\370\355kaz evalf p\370evede hodnotu f(xn) do \n # desetin n\351ho tvaru. To zabr\341n\355 p\370\355li\232 komplikovan\375m \n \+ # z\341znam\371m t\351to hodnoty.\n yn:= evalf(f(xn));\n\n \+ # Nakresli mod\370e (RGB,0,0,1), \350\341rkovan\354 LINESTYLE(3) \+ \372se\350ku mezi osou x\n # a k\370ivkou. V\232imni si, \236e \+ \372se\350ku lze nakreslit zad\341n\355m jej\355ch \n # koncov \375ch bod\371.\n ToCurve:= CURVES([[xn,0], [xn,yn]], COLOR(RGB,0 ,0,1), LINESTYLE(3));\n\n # Nakresli znak \"X\" na k\370ivce.\n \+ Xmarks:= TEXT([xn,yn], 'X');\n\n # Existuj\355 dva zp\371soby animace \372pln\351 Newtonovy iterace.\n # Prvn\355 zaznamen\341 a nakresl\355 sou\370adnice bod\371 dotyku te\350en. Druh\375\n \+ # zaznamen\341 a nakresl\355 rovnici te\350ny. \n Frame[2*n-1]:= \n [TanLine\n ,ToCurve\n ,Xmarks\n # Tiskni sou\370adnice (xn,yn)\n ,TEXT([txtx,txty]\n \+ ,cat(`Bod dotyku: (`\n # Druh\375 parametr je po \350et vyti\232t\354n\375ch \350\355slic. \n # V\375 po\350ty jsou st\341le v re\236imu \n # pohybliv \351 \370\341dov\351 \350\341rky.\n ,convert(evalf(xn ,4), name)\n ,`, `\n ,convert(evalf( yn,4), name)\n ,`).`\n ) \n \+ ,ALIGNRIGHT\n )\n ];\n\n # Test konver gence. P\370\355kaz fnormal zm\354n\355 hodnotu \350\355sla\n # n a nulu, je-li \"bl\355zko k\" nule vzhledem k nastaven\355 parametru\n # Digits.\n if fnormal(yn) = 0 then\n print\n \+ (`(Aproximace) ko\370ene x`[n]=\n cat(convert(evalf( xn, Digits-2), symbol)\n ,` nalezena. Pro v\354t\232 \355 p\370esnost zv\354t\232i Digits.`\n )\n \+ );\n # V p\370\355pad\354 \372sp\354\232n\351 konvergence zv \354t\232uji dvakr\341t\n # p\370edposledn\355 vzor, nebo\235 \+ nem\341m z\341jem vid\354t dal\232\355 te\350nu \n Frame[2*n]: = Frame[2*n-1];\n\n # P\370\355kaz break je pokynem vystoupit \+ z cyklu for\n break\n fi;\n\n # Najdi analytick\351 \+ vyj\341d\370en\355 rovnice te\350ny \n # a potom nakresli jej\355 graf zelen\354 (RGB,0,1,0). V\232imni si, \236e derivace je \n # vypo\350tena v bod\354 xn. V\232imni si zp\371sobu pou\236it\355 p \370\355kazu unapply:\n # M\354n\355 v\375raz na funkci.\n ` f'(xn)`:= evalf(`f'`(xn));\n T:= unapply(yn+`f'(xn)`*(x-xn), x); \n TanLine:= CURVES([[a,T(a)], [b,T(b)]], COLOR(RGB,0,1,0));\n\n \+ # Tiskni rovnici te\350ny.\n Frame[2*n]:= \n [TanLine\n ,ToCurve\n ,Xmarks\n ,TEXT([txtx,txty], cat(` Te\350na : y = `, convert(evalf(T(x), 4), name), `.`), ALIGNRIGHT)\n ];\n \n # Testuj, zda te\350na nen\355 p\370\355li\232 \"divok\341\" \+ \n if fnormal(`f'(xn)`) = 0 then\n print\n (ca t(`Te\350na je (t\351m\354\370) horizont\341ln\355. Nelze pokra\350ova t.`\n ,` Zadej jinou nultou aproximaci a pokra\350uj.` \n )\n );\n break\n fi;\n \+ \n # Prove\357 krok Newtonovy iterace\n xn:= xn - yn/`f'(xn) `;\n\n # Ov\354\370, zda nov\341 hodnota xn le\236\355 v p\370 \355pustn\351m oboru xrange.\n if xn < a or xn > b then\n \+ print\n (`x`[n]=\n cat(convert(evalf(xn,4),sym bol)\n ,` je mimo dan\375 obor. Zv\354t\232i obor`\n \+ ,` a pokra\350uj ve v\375po\350tu.`\n )\n \+ );\n break\n fi\n od;\n\n # Ov\354\370, zd a ji\236 byl proveden maxim\341ln\355 po\350et iterac\355.\n # V kla dn\351m p\370\355pad\354 je \n # n=nsteps+1. Toto je \"\232pr\375m\" Mapleovsk\351ho p\370\355kazu cyklu for.\n if n = nsteps+1 then\n \+ print\n (`x`[n-1]= convert(evalf(xn,Digits-2), symbol)\n \+ ,'f'(`x`[n-1])= \n cat(convert(yn,symbol)\n \+ ,`. S po\236adovanou p\370esnost\355 nebyl nalezen \236\341dn\375 ko \370en.`\n ,` Zv\354t\232i p\370\355pustn\375 po\350et it erac\355 a pokra\350uj ve `\n ,` v\375po\350tu.`\n \+ )\n )\n fi;\n\n # Najdi po\350et krok\371. Jestli\236 e cyklus skon\350il po maxim\341ln\355m po\350tu iterac\355, \n # pa k n=nsteps+1.\n n:= min(2*n, 2*nsteps-1);\n\n # plotopts obsahuj \355 v\232echno, co se zaznamen\341v\341 v ka\236d\351m tisku (\"backg round\")\n plotopts:= Curve, XAxis, VIEW(a..b, ylow..yhigh), AXESSTY LE(FRAME), plotopts;\n \n # Vytvo\370\355 se struktura pro animaci \n PLOT\n (ANIMATE\n (# Dopl\362 \"Prvn\355 krok\" a \" Posledn\355 krok\" k prvn\355mu a posledn\355mu kroku. \n [o p(Frame[1]), TEXT([.3*a+.7*b, .7*ylow+.3*yhigh], `Prvn\355 krok`, ALIG NRIGHT), plotopts]\n\n # V\232echny vnit\370n\355 kroky.\n \+ ,seq([op(Frame[k]), plotopts], k= 2..n-1)\n\n # Posled n\355 krok.\n ,[op(Frame[n]), TEXT([.3*a+.7*b, .7*ylow+.3*yhi gh], `Posledn\355 krok`, ALIGNRIGHT), plotopts]\n )\n )\n end: \n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 667 "Po spu\232t\354n\355 \+ procedury Kresli_Newt_met se objev\355 obr\341zek grafu funkce f(x) a \+ po\350\341te\350n\355 aproximace. Kliknut\355m lev\351ho tla\350\355tk a my\232i na obr\341zku se na obrazovce objev\355 nov\341 li\232ta s i konami podobn\354 jako pro audio CD player. Jedna z ikon, t\370et\355 zleva, zn\341zorn\355 v\375po\350et po kroc\355ch tak, \236e ka\236d \351 kliknut\355 vyvol\341 zobrazen\355 dal\232\355ho kroku. Pro zopak ov\341n\355 v\375po\350tu lze kliknout na ikonu pro zp\354tn\375 chod \+ (\350tvrt\341 ikona zleva). Kliknut\355m druh\351 ikony zleva lze dos \341hnout animace cel\351ho v\375po\350tu v opa\350n\351m po\370ad\355 a kliknut\355 p\341t\351 ikony zp\371sob\355 animaci v\375po\350tu od za\350\341tku do konce. V\232echny tyto ikony jsou dosa\236iteln\351 \+ kliknut\355m prav\351ho tla\350\355tka na obr\341zku a vyu\236it\355m \+ objeviv\232\355 se nab\355dky." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 174 "Procedura Kresli_Newt_met m\341 \350ty \370i parametry. Prvn\355m z nich je funkce, ktr\341 m\371\236e b\375t definov\341na jako funkce nebo jako v\375raz. Druh\375 parametr defin uje nultou aproximaci ko\370ene " }{XPPEDIT 18 0 "x[0];" "6#&%\"xG6#\" \"!" }{TEXT -1 169 ". T\370et\355 ud\341v\341 obor prom\354nn\351 x ve tvaru a..b a \350tvrt\375 stanovuje maxim\341ln\355 p\370\355pustn \375 po\350et iterac\355. Tyto \350ty\370i parametry lze doplnit dal \232\355mi ve tvaru parametr\371 p\370\355kazu " }{TEXT 259 8 "displa y." }{TEXT -1 12 " Nap\370\355klad " }{TEXT 260 22 "scaling= constrai ned " }{TEXT -1 6 "nebo " }{TEXT 261 12 "axes= normal" }{TEXT -1 8 " nebo " }{TEXT 263 22 "font= [HELVETICA, 12]." }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 149 "Obvykle \+ se jako prvn\355 p\370\355klad u\236it\355 Newtonovy metody uv\341d \355 v\375po\350et odmocniny ze dvou. Za t\355mto \372\350elem je t \370ba naj\355t funkci f(x), jej\355m\236 ko\370enem je " }{XPPEDIT 18 0 "x = sqrt(2);" "6#/%\"xG-%%sqrtG6#\"\"#" }{TEXT -1 11 ". Fuknce \+ " }{XPPEDIT 18 0 "f(x) = x-sqrt(2);" "6#/-%\"fG6#%\"xG,&F'\"\"\"-%%sq rtG6#\"\"#!\"\"" }{TEXT -1 108 " v\232ak za t\355mto \372\350elem nen \355 vhodn\341, nebo\235 p\370i ka\236d\351m vy\350\355slen\355 hodnot y f(x) byx bylo nutno nezn\341mou hodnotu " }{XPPEDIT 18 0 "sqrt(2); " "6#-%%sqrtG6#\"\"#" }{TEXT -1 56 " po\350\355tat. Proto je vhodn \354j\232\355 pou\236\355t nap\370\355klad funkce " }{XPPEDIT 18 0 "f (x) = x^2-2;" "6#/-%\"fG6#%\"xG,&*$F'\"\"#\"\"\"F*!\"\"" }{TEXT -1 1 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 129 "Za \+ nultou aproximaci zvolme hodnotu 1. Je z\370ejm\351, \236e ko\370en l e\236\355 v intervalu [0,2]. 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