Maclaurin le li bole kapele 'ea ba bang ba mesebetsi

Ho ithuta thuto ea lipalo e tsoetseng pele e lokela ho ba hlokomela hore chelete ya letoto matla ka karohano ya convergence ba 'maloa ba rōna, ke e tsoelang pele le se nang moeli palo ea makhetlo a mosebetsi farologantswng. Potso ena e hlaha: e ho ka khoneha ho pheha khang ea hore fuoa hatellang mosebetsi o f (x,) - ke chelete ea letoto matla? Ke hore, ka tlas'a hore na maemo a sa f e-saleme o f (x,) ka emeloa ke letoto matla? Bohlokoa ba taba ena ke hore ho ka etsahala ho nka sebaka tse ka bang £ thuto ea bolumeli o f (x,) ke bala mantsoe a 'maloa pele ba letoto matla, e le hore ke homogeneous polynomial. Jwalo Phetolo mosebetsi e bonolo haholo polelo e reng - homogeneous polynomial - ke e loketseng le ho rarolla mathata a itseng ka and analysis thuto ea lipalo, e leng ho rarolla integrals ha bala differential ditekanyo , joalo-joalo ...

Ho bo pakoa, hore e le hore ba bang ba o f ii o f (x,), moo ho derivatives ya (n g + 1) -th odara ka balwa, ho akarelletsa le tsa morao-rao sebakeng se haufi le (α - R; : x ₀ + R) ya ntlha ya x = α hlokang leeme moralo ke:

moralo ona e mong ea bitsoang ka mor'a le rasaense ea tummeng Brooke Taylor. Ba bangata ba e tsoa ho e mong nakong e fetileng, e bitsoa Maclaurin letoto lena:

puso ea A e etsang hore ho ka khoneha ho hlahisa tjaleho letotong Maclaurin:

1. Lekanya derivatives ea pele, ea bobeli, ea boraro, ... odara.
2. A bale seo ke derivatives ka: x = 0.
3. Record Maclaurin letoto la lihlooho tsa mosebetsi ona, 'me ntan'o ba ho fumana hore na karohano ya convergence.
4. Lekanya karohano (-R; R), moo karolo e barrel ya moralo Maclaurin

R _{n g} (x,) -> 0 bakeng n g -> egoist. Haeba e 'ngoe teng, ho mosebetsi o f (x,) e lokela ho ba ba lekanang le chelete ea letoto la lihlooho tse Maclaurin.

Nahana hona joale letoto la lihlooho tse Maclaurin bakeng sa mesebetsi ea motho ka mong.

1. Ka hona, pele ho F (x,) = ea ^e-x. Ya e le hantle, e le hore le litšobotsi tse ba bona e le f e-ia e nkiloeng mefuta e fapaneng ya ditaelo, le f ^{(K) (x,)} = ea ^e-x, moo bzlmeet o lekana le tsohle linomoro tsa tlhaho. Ka nkang sebaka sa x = 0. Re fumana o f ^{(K) (0)} = ea e ⁰ = 1, bzlmeet = 1,2 ... E Thehiloe ho se boletsoeng ka holimo, a mangata a ea ^e-x E tla ba ka mokgwa o latelang:

2. Maclaurin letoto la lihlooho bakeng sa mosebetsi o f (x,) = na le sebe sa x. Hang-hang bolela ka ho toba hore o f-saleme bakeng derivatives tsohle tsejoeng tla ba le, ntle ho f e ^'(x,) = cos: x = sebe (x, + n / 2), ^{o f' '(x,)} = -sin: x = sebe (x, + 2 * n / ²⁾ ..., o f ^{(K) (x,)} = sebe (x, + 'n' * bzlmeet / 2), moo bzlmeet o lekana le palotlalo efe kapa efe e ntle. Ke, ho etsa dipalelo bonolo, re ka fihlela qeto ea hore letoto la lihlooho tsa f-(x,) = na le sebe sa x tla e kang ena:

3. Joale a re hlahlobeng iju f e-o f (x,) = cos: x. Ho sa tsejoeng tsa derivatives kaofela odara hatellang, 'me ^{| f-(K) (x,)} | = | Cos (x, + bzlmeet * n / 2) | <= 1, bzlmeet = 1,2 ... Hape, ho eka entse ba bang ba dipalelo, re fumana hore letoto la lihlooho tsa f-(x,) = cos: x tla sheba e kang ena:

Ho joalo, re ho tse thathamisitsoeng likarolo bohlokoa ka ho fetisisa e ka atolosoa letotong Maclaurin, empa ba tlatsana le Taylor letoto la lihlooho tse ho ba bang mesebetsi. Joale re tla thathamisa bona hammoho. Ho lokela ho hlokomeloe hore Taylor letoto la lihlooho le Maclaurin letoto la lihlooho tse karolo ea bohlokoa ea ho kokoano letoto la liqeto tse thuto ea lipalo e phahameng. Ho joalo, Taylor letoto lena.

1. Ea pele ke letoto la f e-II o f (x,) = Ln (1 + x e). Jwalo ka mehlala e fetileng, kaha sena se re o f (x,) = Ln (1 + x,) ka a phuthilweng maloa, a sebelisa mofuta o akaretsang oa Maclaurin letoto lena. empa bakeng sa tšobotsi ena Maclaurin ka fumanoa bonolo haholo. Ya go kopanya letoto thutatekanyo, re fumana tse 'maloa bakeng sa-f (x,) = Ln (1 + x,) ya sampole ea:

2. 'Me oa bobeli, e leng tla ba ho qetela a sehloohong sena a, e tla ba le letoto la lihlooho tse ho f-(x,) = arctg: x. Bakeng sa x bao e leng ea karohano [-1; 1] ke le utloahalang li bole kapele:

Ke tsohle. A sehloohong sena a ke buisanoeng le bona phuputsong e fetisisa e sebediswa Taylor letoto la lihlooho le Maclaurin letoto la lihlooho tse ka thuto ea lipalo e phahameng, haholo-holo ka, lik'holeje moruo le botekgeniki.