Maclaurin oo kala daatay ee hawlaha qaar ka mid ah

Barashada xisaabta sare waa inay ogaadaan in wadarta taxanaha awood bareeg ah ee wada of badan oo innaga mid ah, waa tiro joogto ah aan xad lahayn iyo jeer hawl kala duwan. Su'aasha soo baxdo, waxaa suurto gal ah inuu ku doodo in la siin ah shaqo aan loo aabo yeelin f (x) - waa wadarta taxanaha awood ah? Taasi waa, waxa xaaladaha f-wey ku f (x) matali karaan by taxane xoog ah? muhiimadda ay leedahay arrintan waa in ay suurto gal tahay inuu bedeli doono ku dhowaad £ fiqi f (x) waa wadarta ugu horeysay marka la eego dhowr ka mid ah taxanaha awood, waa polynom. Noocan oo kale ah function bedelka waa hadal fudud ilaa xad - polynom - haboon iyo xalinta dhibaatooyinka qaarkood ee falanqaynta xisaabta, kuwaas oo in xalinta integrals marka la xisaabinayo isla'egyada kala duwan , iwm ...

Waxaa caddeeyeen in f gaar ah f-ii (x), kuna waari dhexdeeda ku taagayo oo ka mid ah (n + 1) Si -th loo xisaabin karaa, oo ay ku jiraan ugu dambeeyay oo goobta joogay ee (α - R; x ₀ + R) of a dhibic x = α formula caddaalad ah waa:

formula Tan waxaa loo magacaabay ka dib markii saynisyahan caan Brooke Taylor. Tiro ka mid ah kaas oo la soo jeeda mid ka mid ah la soo dhaafay, waxaa la yiraahdaa taxane Maclaurin ah:

Xeer dhigaya suurto gal ah in ay soo saaraan fidinta ee taxanahan Maclaurin ah:

1. Go'aaminta taagayo ee koowaad, labaad, saddexaad, ... si.
2. Xisaabi waa maxay taagayo at x = 0.
3. Record Maclaurin taxanaha shaqo this, ka dibna si loo ogaado bareeg ka mid ah wada.
4. Go'aaminta bareeg (-R, R), halkaas oo qayb ka mid ah haraaga ah ee formula Maclaurin

R _n (x) -> 0 n -> xad la'aan. Haddii mid ka mid ah ay ka jirto, waxaa shaqo f (x) Waa in loo siman yahay si wadarta taxanaha Maclaurin ah.

Haddaba taxane Maclaurin ee hawlaha shakhsiga.

1. Sayidka, marka hore in la f (x) = e ^x. Dabcan, in sifooyinka ay si f-IA ayaa soo minguuriyey noocyo kala duwan oo amar, iyo f ^{(k) (x)} = e ^x, halkaas oo k waa loo siman yahay oo dhan ku tirooyinka caadiga ah. Oo bedel ku x = 0. Waxaan aad u hesho f ^{(k) (0)} = e ⁰ = 1, k = 1,2 ... Iyadoo lagu saleynayo ku qorani, tiro ka mid ah e ^x Waxa ay noqon doontaa sida soo socota:

2. Maclaurin taxane for f ku function (x) = dembi x. Isla markiiba u sheeg in f-xaggey dhan taagayo aan la garanayn waxay yeelan doonaan, oo ayan ku jirin f ^'(x) = sababtoo ah x = dembiga (x + n / 2), ^{f' '(x)} = -sin x = dembiga (x + 2 * n / ²⁾ ..., f ^{(k) (x)} = dembiga (x + n * k / 2), halkaas oo k waa loo siman yahay si abyoonaha wax wanaagsan. Taasi waa, samaynta xisaabo fudud, waxaannu ku tirinnaa karaa in taxanaha ah ee f (x) = dembi x noqon doonaan sida tan:

3. Hadda aynu ka fiirsan iju f-f (x) = sababtoo ah x. Waa la aqoon waayo, dhammaan taagayo ah si aan loo aabo yeelin, iyo ^{| f (k) (x)} | = | Koos (x + k * n / 2) | <= 1, k = 1,2 ... Mar kale, waxa isagoo ka dhigay xisaabaha qaar ka mid ah, waxaan ka heli in taxanaha ah ee f (x) = sababtoo ah x eegi doonaa sida tan:

Sidaas daraaddeed, waxaannu ku qoran sifooyinka ugu muhiimsan in loo ballaarin karo ee taxanahan Maclaurin ah, laakiin waxa ay dhamaystir u taxane Taylor ee hawlaha qaar ka mid ah. Hadda waxaan sidoo iyaga qor doonaa. Sidoo kale waa in la ogaadaa in taxane Taylor oo taxane Maclaurin waa qayb muhiim ah oo ka mid ah taxanaha waxbarasho ee go'aanada xisaabta sare. Sidaas daraaddeed, taxane Taylor.

1. Marka koowaad waa taxane ah f-ii f (x) = lihida (1 + x). Sida tusaalooyinka hore, waayo, waannu f this (x) = lihida (1 + x) karaa hanta dhowr ah, oo isticmaalaya foomka guud ee taxanaha Maclaurin. laakiin habkaani Maclaurin waxaa laga heli karaa inta badan ka sahlan. Is-dhexgalka taxane ah joomateri, waxaan ka heli tiro for f (x) = lihida (1 + x) ee tusaalaha ah:

2. Oo labaad, kaas oo noqon doona finalka ee qodobkan, wuxuu noqon doonaa taxane ah oo loogu talagalay f (x) = arctg x. Waayo, x tirsan bareeg ah [-1, 1] waa kala daatay oo ansax ah:

Taasi oo dhan. In this article waxaan waraystay taxanaha Taylor ugu used oo taxane Maclaurin xisaabta sare, gaar ahaan jaamacadaha dhaqaale iyo mid farsamo.