Maclaurin a me decomposition o kekahi mana

Këia holomua makemakika ë he mähele wale i ka huina o ka mana puke i loko o ka wā mawaena o ka hoʻohui o ka helu o mākou, o ke ana a me kona palena helu o ka manawa i ka differentiated kuleana pili i. Ke ninau aicieeaao: ua mea hiki, e hoʻopaʻapaʻa ana i haawi mai ai i ākeʻakeʻa kumu papa f (x) - o ka huina o ka mana puke? Ia mea, ma lalo o ka rula o ka [illegible] [illegible]-ō (m) hiki ke poe ma ka mana moʻo? Ka nui o kēia pukana o ia ia mea hiki e puku i ka hola £ Theological f (x) o ka huina o ka mua mau hua'ōlelo o ka mana puke, i mea he polynomial. Oia ka hope papa mea he mea 'aoʻao - polynomial - mea pono, a ma ka hoʻoponopono i kekahi mau pilikia i loko o makemakika Ka Ikepili, ʻo ia i loko o ka hoʻoponopono integrals ka wā helu ana palena pau nā helu kaulike , a pela aku ...

Ua Ua hoao, i no kekahi. F-ii ai f (x), ai na nā mea loaʻa o ka (N + 1) -th mea hiki ke pōpilikia, a me ka polokalamu i loko o ka wahi o ka (α - R; m ₀ + R) o ka wahi m = α maikai haʻilula mea:

Kēia haʻilula ua kapaia ma hope o ke nä känaka 'epekema kaulana Brooke Taylor. Ua kapaʻia ka helu ana i ua loko mai o ka mua kekahi, he Maclaurin moʻo:

A noho aliʻi i hana ia e hiki ai i ka paka? Anoe? I loko o ka Maclaurin moʻo:

1. Kau nā mea loaʻa o ka mua, ka lua, ke kolu, ... kauoha.
2. Ho omaulia mea i nā mea loaʻa ma m = 0.
3. Record Maclaurin moʻo no keia hana, a laila, e hooholo i ka wā mawaena o ka hoʻohui.
4. Ho oholo wā mawaena (-R; R), ma ke koena hapa o ka haʻilula Maclaurin

R _N (m) -> 0 no N -> infinity. Inā kekahi nei, pono e like me ka huina o ka Maclaurin puke ia papa f (x).

E hoomanao i kēia manawa i ka Maclaurin moʻo no ke kanaka i hana '.

1. Ke'ī mai, ka mea mua e e f (x) = E ^m. O ka holo ana, i ko lakou mau ano ai [illegible]-Ia i loko i keʻano o nā kauoha, a me ka [illegible] ^{(k) (x)} = e ka ^m, ma ke k mea like me nā mea a pau i ka nui maoli. Ae pani m = 0. E loaa. F ^{(k) (0)} = E ⁰ = 1, K = 1,2 ... Ma muli o ka keia, he helu ana e ka ^m Ka mea, e like penei:

2. Maclaurin moʻo no ka papa f (x) = hewa m. Koke ke koho 'ia. F-ō no nā mea a pau i ike ole nā mea loaʻa, e i, he okoa na ^{[illegible]' (x)} = cos He m = hewa (m + n / 2), ^{[illegible] '' (x)} = -sin He m = hewa (m + 2 * n / ²⁾ ..., f ^{(k) (x)} = hewa (m + n. * ke k / 2), kahi ho mea like i kekahi maikaʻi helu. Ia mea, e na mea helu, ua hiki ke manao nei makou, o ka hopena ua no f (x) = hewa m, e e like me keia:

3. Ano, e ka noonoo iju [illegible]-f (x) = cos m. Heʻikeʻole i nā mea a pau nā mea loaʻa o ākeʻakeʻa kumu o ka papa, a me ka ^{| f (k) (m)} | = | Cos (m + ke k * N / 2) | <= 1, K = 1,2 ... Eia hou, ia ua i kekahi ho, ua loaʻa i ka moʻo no f (x) = cos m, e nana me keia:

No laila, ua i helu i nā hiʻona mea nui i hiki e hoomaka ana ma ka Maclaurin puke, akā, e hoʻokō i ka Taylor moʻo no kekahi mau hana. Ano, e helu mākou ia e like me ka pono. Ua E hoi e kaulana ia Taylor moʻo a me Maclaurin moʻo mea nui i kekahi hapa o ka Workshop moʻo o ka olelo hooholo i kiekie ae makemakika. No laila, Taylor moʻo.

1. Ka mua o ka moʻo o ka [illegible]-ii ai f (x) = Ln (1 + m). Like i loko o ka mua examples, no keia mākou f (x) = Ln (1 + m) hiki ke pelupelu ia i ka helu, ka hoʻohana 'ana i ka nui kino o Maclaurin moʻo. akā, no keia hiʻona Maclaurin hiki e loaa nui maʻalahi. Ka hoʻononiakahiʻana i ka geometric puke, ua loaa i ka helu no ka f (x) = Ln (1 + m) o ka hāpana.

2. A o ka lua, a e e paa i loko o kēia 'atikala, e lilo ia i ka moʻo no f (x) = arctg m. No ka m no a hiki i ka wā mawaena [-1; 1] mea i pololei ia decomposition:

Ia ka leo o nā mea a pau. Ma kēia 'atikala i anamanaʻo au i ka loa hoʻohana' Taylor moʻo, a Maclaurin moʻo i kiekie ae makemakika, pakahi aku la ia i loko o ka waiwai a me ka oaoieei kulanui.