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Nagyprojekt
K\[EAcute]sz\[IAcute]tette: Bezd\[AAcute]ny D\[AAcute]niel P\[EAcute]ter, \
Bujdos\[OAcute] L\[AAcute]szl\[OAcute]\
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Feladat sorsz\[AAcute]ma, sz\[ODoubleDot]vegez\[EAcute]se: 
1. FELADAT
K\[EAcute]sz\[IAcute]tsen oktat\[OAcute]anyagot, ami az egyv\[AAcute]ltoz\
\[OAcute]s f\[UDoubleDot]ggv\[EAcute]nyek Taylor k\[ODoubleDot]zel\[IAcute]t\
\[EAcute]s\[EAcute]t mutatja be adott x0 hely \
k\[ODoubleDot]rnyezet\[EAcute]ben. Mutassa
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\[AAcute]s,
parit\[AAcute]s, valamint a megfelel\[ODoubleAcute] Taylor polinomok \
kapcsolat\[AAcute]ra.\
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 "\nEdmonton, Middlesex, Anglia, 1685. aug. 18. - London, 1731. dec. 29.\n\
Angol matematikus\nCambridge-ben tanult matematik\[AAcute]t. 1712-ben a Royal \
Society tagj\[AAcute]v\[AAcute] v\[AAcute]lasztott\[AAcute]k, 1714 \[EAcute]s \
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titk\[AAcute]ri poz\[IAcute]ci\[OAcute]j\[AAcute]t is bet\[ODoubleDot]lt\
\[ODoubleDot]tte. Matematikai anal\[IAcute]zissel foglalkozott, legjelent\
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\[UDoubleDot]ggv\[EAcute]nyek hatv\[AAcute]nysorba fejt\[EAcute]s\[EAcute]t \
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Nev\[EAcute]t viseli a matematikai anal\[IAcute]zisben alapvet\[ODoubleAcute] \
szerepet j\[AAcute]tsz\[OAcute] Taylor-sor."
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\[ODoubleDot]zel\[IAcute]t\[ODoubleAcute] \[EAcute]rt\[EAcute]kei \[IAcute]gy \
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\[ODoubleDot]rt\[EAcute]n\[ODoubleAcute] fel\[IAcute]r\[AAcute]s\[AAcute]t a \
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fejt\[EAcute]s\[EAcute]nek nevezz\[UDoubleDot]k.\n",
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\[Ellipsis] sokadfok\[UAcute] f\[UDoubleDot]ggv\[EAcute]nyekkel \
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f\[UDoubleDot]ggv\[EAcute]ny hatv\[AAcute]nysorba fejt\[EAcute]sei \
l\[AAcute]that\[OAcute]ak n = 1, 3, 5, 7, 9, 11 \[EAcute]s 13 fokig bez\
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k\[ODoubleDot]r\[UDoubleDot]l, a megadott rendig.Pl.Series[Tan[x], {x, 0, 5}] \
hat\[AAcute]s\[AAcute]ra a tangens f\[UDoubleDot]ggv\[EAcute]nyt a 0 pont k\
\[ODoubleDot]r\[UDoubleDot]l Taylor - sorba fejtj\[UDoubleDot]k, 5 - \
\[ODoubleDot]d rendig. Az eredm\[EAcute]ny : x + x^3/3 + 2 x^5/15 + O[x]^6. \
Itt az O[x]^6 jel\[ODoubleDot]l egy \"nagyord\[OAcute] x^6-os\" tagot. Ett\
\[OTilde]l a Normal utas\[IAcute]t\[AAcute]s \
seg\[IAcute]ts\[EAcute]g\[EAcute]vel szabadulhatunk meg, \[IAcute]gy : \
Normal[Series[ ...]]. Az eredm\[EAcute]ny egy polinom lesz."
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\[AAcute]ban haszn\[AAcute]lt jel\[ODoubleDot]l\[EAcute]sm\[OAcute]d, amely t\
\[ODoubleDot]m\[ODoubleDot]ren fejezi ki egy adott \
f\[UDoubleDot]ggv\[EAcute]ny n\[ODoubleDot]veked\[EAcute]s\[EAcute]nek m\
\[EAcute]rt\[EAcute]k\[EAcute]t. Informatik\[AAcute]ban \
\[AAcute]ltal\[AAcute]ban egy algoritmus fut\[AAcute]sidej\[EAcute]nek \
jellemz\[EAcute]s\[EAcute]re haszn\[AAcute]ljuk: a vizsg\[AAcute]lt f\
\[UDoubleDot]ggv\[EAcute]ny a fut\[AAcute]shoz sz\[UDoubleDot]ks\[EAcute]ges \
id\[ODoubleAcute]t adja meg a bemenet hossz\[AAcute]nak f\[UDoubleDot]ggv\
\[EAcute]ny\[EAcute]ben. Leegyszer\[UDoubleAcute]s\[IAcute]tve, egy O(n^2) \
fut\[AAcute]sidej\[UDoubleAcute] algoritmus k\[EAcute]tszer akkora \
m\[EAcute]ret\[UDoubleAcute] bemenetre n\[EAcute]gyszer annyi ideig fog \
futni, h\[AAcute]romszor akkora bemenetre kilencszer annyi ideig. M\[IAcute]g \
maga a fut\[AAcute]sid\[ODoubleAcute]t le\[IAcute]r\[OAcute] \
f\[UDoubleDot]ggv\[EAcute]ny f\[UDoubleDot]gg az \
implement\[AAcute]ci\[OAcute] r\[EAcute]szleteit\[ODoubleAcute]l \[EAcute]s a \
futtat\[AAcute]shoz haszn\[AAcute]lt architekt\[UAcute]r\[AAcute]t\[OAcute]l, \
az algoritmus nagy ord\[OAcute]ja csak az algoritmus alapelv\[EAcute]t\
\[ODoubleAcute]l f\[UDoubleDot]gg. "
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Bizonyos f\[UDoubleDot]ggv\[EAcute]nyek, a val\[OAcute]s sz\[AAcute]mok tetsz\
\[OAcute]leges intervallum\[AAcute]n k\[ODoubleDot]zel\[IAcute]thet\
\[ODoubleAcute]k, tetsz\[ODoubleAcute]leges pontoss\[AAcute]ggal a \
Taylor-polin\[OAcute]mjukkal, ilyenek p\[EAcute]ld\[AAcute]ul az exponenci\
\[AAcute]lis \[EAcute]s logaritmikus f\[UDoubleDot]ggv\[EAcute]nyek, valamint \
az \[ODoubleDot]sszes polin\[OAcute]m f\[UDoubleDot]ggv\[EAcute]ny.
Sajnos a legt\[ODoubleDot]bb f\[UDoubleDot]ggv\[EAcute]nyre ez nem teljes\
\[UDoubleDot]l, de sok k\[ODoubleDot]z\[UDoubleDot]l\[UDoubleDot]k lok\
\[AAcute]lisan k\[ODoubleDot]zel\[IAcute]thet\[ODoubleAcute].\
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Wolfram Demonstration Keres\[EAcute]si kulcsszavak: Taylor-polinom
El\[ODoubleAcute]ad\[AAcute]s jegyzet: Kalkulus2, Matematikai modellek
Google-k\[EAcute]pek\
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