Maclaurin uye manyowa vamwe mabasa

Kudzidza mazhinji masvomhu vanofanira kuziva kuti kuwanda ane simba akatevedzana ari penguva convergence vanoverengeka vedu, chinhu chinoramba uye mapfumo nhamba kane kusiyaniswa basa. The mubvunzo unomuka: Zvinoita nharo kuti kupiwa zvemasanga nerevane add (x) - pakuverengwa ane simba dzakatevedzana? Ndiko, pasi chii ezvinhu f-zvi- add (x) inogona hunomiririrwa simba dzakatevedzana? Kukosha nyaya iyi kuti zvinokwanisika kuti kutsiva anenge £ Theological F (x) pakuverengwa rokutanga mashoko mashomanana ane simba series, kuti chinhu polynomial. Akadaro yokutsiva basa iri nyore okuti - polynomial - zviri nyore uye kugadzirisa zvimwe zvinetso zviri kwemasvomhu pachisara kureva kugadzirisa integrals apo pakuverenga differential equations , etc ...

Zvinonzi akaratidza kuti vamwe add-II add (x), sezvamakaropafadzwa nomukaka zviri (n + 1) -th kuti inogona dzaona, kusanganisira ichangoburwa pedyo (α - R; X ₀ + R) ane pfungwa x = α wakanaka payakavakirwa ndiko:

mutoo uyu inonzi pashure yakakurumbira musayendisiti Brooke Taylor. A nhamba izvo rinobva yaimbovapo, anonzi Maclaurin akatevedzana:

A mutemo kuti zvinoita kuti pave nokuwedzera mune Maclaurin akatevedzana:

1. Sarudza nomukaka okutanga, wechipiri, newechitatu, ... hurongwa.
2. Change chii vari nomukaka pana x = 0.
3. Record Maclaurin Dzinotibatsira basa iri, uye zvadaro kuona penguva convergence.
4. Sarudza penguva (-R; R), uko residual chikamu fomura Maclaurin

R _N (x) -> 0 kuti N -> kusaverengeka. Kana munhu ariko, nayo basa add (x) anofanira kuva akaenzana Verengai Maclaurin akatevedzana.

zvino Funga Maclaurin akatevedzana nokuda nomumwe mabasa.

1. Nokudaro, wokutanga F (x) = d ^x. Chokwadi, kuti maitiro avo saka add-Ia akaita Rinobva akasiyana-siyana varayirwa, uye c ^{(k) (x)} = mae ^X, apo K akaenzana zvose zvakasikwa nhamba. Chinotsiva x = 0. Tinowana d ^{(k) (0)} = E ⁰ = 1, K = 1,2 ... Pachishandiswa ataurwa, wandei mae ^X Zvichaita sezvinotevera:

2. Maclaurin Dzinotibatsira mashandiro add (x) = chivi x. Pakarepo tsanangurai kuti add-zvi- nokuda nomukaka zvose zvisingazivikanwi uchava, kunze e ^'(x) = Kosi x = chivi (x + n / 2), ^{f' '(x)} = -sin x = chivi (x + 2 * n / ²⁾ ..., f ^{(k) (x)} = chivi (x + n *, k / 2), apo K akaenzana chero zvakanaka integer. Kureva, zvichiita nyore Masvomhu, tinogona kugumisa kuti akatevedzana kuti d (x) = chivi X achafanana ichi:

3. Zvino ngationei iju e-f (x) = Kosi x. Hazvina kuzivikanwa kuti nomukaka vose zvemasanga kurongeka, uye ^{| add (k) (x)} | = | Kosi (x + K * N / 2) | <= 1, K = 1,2 ... Zvakare, izvozvo vaita zvimwe Masvomhu, tinoona kuti akatevedzana kuti d (x) = Kosi X vachatarira seizvi:

Saka, isu zvawanyora anokosha zvinhu inogona kuwedzerwa mune Maclaurin series, asi ivo mukwanisi kuti Taylor akatevedzana vamwe mabasa. Zvino tichakupai wozvinyora pamwe. Zvinofanira kucherechedzwa kuti Taylor akatevedzana uye Maclaurin akatevedzana vari chikamu chinokosha mumusangano munhevedzano zvisarudzo yepamusoro masvomhu. Saka, Taylor akatevedzana.

1. wokutanga dzakatevedzana add-II add (x) = Dr (1 + x). Sezvaiitwa yapfuura mienzaniso, nokuti ichi isu add (x) = Dr (1 + x) inogona akapeta nhamba, uchishandisa mukuru muchimiro Maclaurin akatevedzana. asi nokuti chikamu ichi Maclaurin kunogona kuwanikwa nyore zvikuru. Kusanganisa munhu geometric series, tinowana nhamba kuti d (x) = Dr (1 + x) ari sample:

2. Zvino wechipiri, iyo ichava yokupedzisira munyaya ino, vachava nyaya d (x) = arctg x. Kuti X kungova penguva [-1; 1] chinoshanda manyowa:

Ndizvo zvose. Munyaya ino ndaita akaongorora kupfuura kushandiswa Taylor akatevedzana uye Maclaurin dziri dzakakwirira masvomhu, kunyanya mune zvoupfumi uye michina mumakoreji.