Maclaurin me whāwehe o etahi mahi

Hono ako pāngarau matatau kia mōhio e te moni o te raupapa mana i roto i te wā o te pūtahitanga o te maha o tatou, ko te maha tonu, me te mutunga o nga wa, he mahi rerekëtanga. whakatika te pātai: te reira taea ki te tautohe i homai he mahi f noho (x) - ko te moni o te raupapa mana? Ko, i raro i te mea tikanga i te f f-mau (x) e taea te māngai e te raupapa mana? Ko te faufaa o tenei take ko e ko e taea ki te whakakapi āhua £ Theological f (x) ko te moni o te tuatahi torutoru ngā o te raupapa mana, e ko te pūrau. he mahi taua whakakapinga Ko te tino māmā faaiteraa - pūrau - he watea, me te i roto i te whakaoti etahi raruraru i roto i te tātari pāngarau, ara i roto i whakaoti pāwhaitua ka tātai whārite pārōnaki , etc ...

Kei te whakamatauria reira, e hoki etahi f-ii f (x), e taea te tātai ai i te pärönaki o te (n + 1) -th tikanga, tae atu i te hou i roto i te takiwā o (α - R; x ₀ + R) o te pūwāhi x = α tātai ataahua ko:

ingoa tenei tātai te i muri i te pūtaiao rongonui Brooke Taylor. ka karanga te maha o ahu e te i te tetahi o mua, ko te raupapa Maclaurin:

He ture e hanga taea reira ki te whakaputa i roha i roto i te raupapa Maclaurin:

1. Te whakatau pärönaki o tuatahi, tuarua, tuatoru, ... kia.
2. Tātaihia te mea e pärönaki i x = 0.
3. Record raupapa Maclaurin mo tenei mahi, a ka ki te whakatau i te wā o te pūtahitanga.
4. Whakatau wā (-R: R), i reira te wahi wäriu o te tātai Maclaurin

R _n (x) -> 0 mō te n -> mutu-. Ki te vai kotahi, me te mea mahi f (x) e rite ana ki te moni o te raupapa Maclaurin.

inaianei Whakaaroa te raupapa Maclaurin mo te mahi takitahi.

1. Ko te kupu, ki te tuatahi kia f (x) = e ^x. Ko e mo'oni, e ratou āhuatanga kia kua ahu f-Ia he momo o whakahau, me f ^{(k) (x)} = e ^x, kei hea he rite ki katoa k nga tau māori. Whakakapia x = 0. whiwhi tatou f ^{(k) (0)} = e ⁰ = 1, k = 1,2 ... I runga i te i mua, he maha o e ^x ka waiho te reira e whai ake:

2. raupapa Maclaurin mo te f pānga (x) = hara x. whakapūtā tonu taua f-mau mo pārōnaki unknown katoa e whai, haunga f ^'(x) = cos x = hara (x + n / 2), ^{f' '(x)} = -sin x = hara (x + 2 * n / ²⁾ ..., f ^{(k) (x)} = hara (x + n * k / 2), kei hea he rite ki tetahi tau tōpū pai k. Ko, hanga tātaitanga ohie, e nehenehe tatou e faaoti e te raupapa mō f (x) = ka hara x rite tenei:

3. Na kia whakaaro o iju f-f (x) = cos x. Ko reira unknown hoki pärönaki katoa o te tikanga te noho, a ^{| f (k) (x)} | = | Koha (x + k * n / 2) | <= 1, k = 1,2 ... ano, ka meinga reira etahi tātai, kitea tatou e te raupapa mō f (x) = cos x ka titiro rite tenei:

Na, kua whakarārangitia e matou nga āhuatanga tino nui e taea te whakawhānui i roto i te raupapa Maclaurin, engari kīnaki i ratou i te raupapa Taylor mo etahi mahi. Na ka whakarārangi tatou ratou rite te pai. Me mahara hoki käore te reira e raupapa Taylor me raupapa Maclaurin ko te wāhanga nui o te raupapa awheawhe o whakatau i roto i te pāngarau teitei. Na, raupapa Taylor.

1. Ko te tuatahi ko te raupapa o f-ii f (x) = ln (1 + x). Ka rite ki i roto i nga tauira o mua, mo tenei matou f (x) = ln (1 + x) e taea te takai he maha, te whakamahi i te puka whānui o te raupapa Maclaurin. engari mo tenei āhuatanga e taea te whiwhi Maclaurin nui māmā. Tuitui i te raupapa āhuahanga, whiwhi tatou he maha hoki f (x) = ln (1 + x) o te tauira:

2. Na ko te tuarua, e ka e whakamutunga i roto i tenei tuhinga, ka waiho te raupapa mō f (x) = arctg x. Hoki x no ki te wā [-1: 1] he whāwehe whaimana:

Ko te katoa. I roto i tenei tuhinga i ruritia e ahau te raupapa Taylor tino whakamahia, me te raupapa Maclaurin i roto i te pāngarau teitei, ngā i roto i te kareti ōhanga, me te hangarau.