Maclaurin da kuma bazuwar na wasu ayyuka

Karatu m lissafi kamata ka sani cewa Naira Miliyan Xari da wani ikon jerin a cikin tazara na haduwa da dama da mu, shi ne a ci gaba da kuma Unlimited yawan sau da wani bambancin aiki. The tambaya: Shin yana yiwuwa a yi jayayya cewa ba wani sabani aiki f (x) - shi ne Naira Miliyan Xari da wani ikon jerin? Wannan shi ne, a karkashin abin da yanayi da f-tions f (x) za a iya wakilta wani ikon jerin? Muhimmancin wannan batu shi ne cewa shi ne mai yiwuwa a sauya kamar £ tauhidin f (x) ne Naira Miliyan Xari da farko 'yan sharuddan mai ikon jerin, cewa shi ne mai polynomial. Irin wannan canji aiki ne quite sauki magana - polynomial - shi ne m, kuma a warware wasu matsaloli a cikin ilmin lissafi analysis, wato a warware integrals lokacin kirga bambanci lissafai , da dai sauransu ...

An tabbatar da, cewa ga wasu f-ii f (x), cikinsa da Kalam da (n + 1) -th domin za a iya lasafta, ciki har da latest a cikin kusanci (α - R. x ₀ + R) na wani batu x = α adalci dabara ne:

Wannan dabara ne mai suna bayan sanannen masanin kimiyya Brooke Taylor. A yawan wanda aka samu daga baya daya, da aka kira wani Maclaurin jerin:

A mulkin da ya sa ya yiwu don samar da fadada a wani Maclaurin jerin:

1. Ƙayyade Kalam na farko, na biyu, na uku, ... domin.
2. Lissafi abin da suke Kalam a x = 0.
3. Record Maclaurin jerin ga wannan aiki, sa'an nan domin sanin tazara na haduwa.
4. Ƙayyade tazara (-R. R), inda saura kashi na dabara Maclaurin

R _n (x) -> 0 for n -> rashin iyaka. Idan daya wanzu, aiki f (x) dole ne daidai da Naira Miliyan Xari da Maclaurin jerin.

La'akari da yanzu da Maclaurin jerin ga mutum ayyuka.

1. Saboda haka, na farko da za a F (x) = e ^x. Hakika, cewa su halaye haka f-IA ya samu dama umarni, kuma f ^{(k) (x)} = e ^x, inda k shi ne daidai to duk da na halitta lambobi. Canza x = 0. Mun samu f ^{(k) (0)} = e ⁰ = 1, k = 1,2 ... Bisa ta gabatar ba, a yawan e ^x Yana zai zama kamar haka:

2. Maclaurin jerin ga aikin f (x) = zunubi x. Nan da nan saka da cewa f-tions ga duk ba a sani ba Kalam zai yi, kuma f ^'(x) = cos x = zunubi (x + n / 2), ^{f' '(x)} = -sin x = zunubi (x + 2 * n / ²⁾ ..., f ^{(k) (x)} = zunubi (x + n * k / 2), inda k shi ne daidai wani m lamba. Wannan shi ne, yin sauki lissafin, za mu iya kammala da cewa, jerin ga f (x) = zunubi x zai zama kamar wannan:

3. Yanzu bari la'akari iju f-f (x) = cos x. Shi ne ba a sani ba ga duk Kalam na sabani da oda, da kuma ^{| f (k) (x)} | = | Cos (x + k * n / 2) | <= 1, k = 1,2 ... Sa'an nan, shi ya sanya wasu lissafin, mun sami cewa jerin ga f (x) = cos x zai yi kama da wannan:

Saboda haka, za mu jera mafi muhimmanci siffofin da cewa za a iya fadada a wani Maclaurin jerin, amma sun gaba da Taylor jerin ga wasu ayyuka. Yanzu za mu jera su kamar yadda kyau. Ya kamata kuma a lura da cewa, Taylor jerin kuma Maclaurin jerin ne wani muhimmin ɓangare na bitar jerin yanke shawara a cikin mafi girma lissafi. Saboda haka, Taylor jerin.

1. Na farko shi ne jerin f-ii f (x) = Ln (1 + x). Kamar yadda a baya misalai, domin wannan mun f (x) = Ln (1 + x) za a iya folded da dama, ta amfani da janar nau'i na Maclaurin jerin. amma ga alama wannan Maclaurin za a iya samu sauƙin. Hadawa da wani lissafi da jerin, za mu sami lambar don f (x) = Ln (1 + x) da samfurin:

2. Kuma na biyu, wanda zai zama karshe a cikin wannan labarin, za a yi wani jerin for f (x) = arctg x. Domin x na da tazara [-1. 1] yake da inganci bazuwar:

Shi ke nan. A wannan labarin na surveyed mafi used Taylor jerin kuma Maclaurin jerin a cikin mafi girma lissafi, musamman a fannin tattalin arziki da fasaha kolejoji.