Maclaurin ndi kuwonongeka kwa ena ntchito

Kuphunzira masamu apamwamba ayenera kudziwa kuti Uwerenge angapo mphamvu mu imeneyi ya mgwirizano wa anthu angapo mwa ife, ndi mosalekeza ndi malire kangapo ntchito kusiyanitsidwa. funso n'lakuti: ndizotheka amanena kuti anapatsidwa umasinthasintha ntchito f (x) - ndi Uwerenge mndandanda mphamvu? Ndiko kuti, kodi zinthu f-imathanso f (x) akhoza umaimiridwa ndi mndandanda mphamvu? Kufunika kwa nkhani imeneyi ndi kuti n'zotheka m'malo pafupifupi £ Theological f (x) ndi Uwerenge mawu oyamba a angapo mphamvu kuti ndi polenomiyo. Chotero m'malo ntchito ndi wosavuta mawu - polenomiyo - yabwino ndi kuthetsa mavuto ena pomaliza masamu, ndicho kuthetsa integrals pamene kuwerengetsa masiyanidwe maikwezhoni , etc ...

Iwo imatsimikizirika kuti kwa ena f-II f (x), mmene opangidwa kuchokera kwa (n + 1) -th kuti akhoza kuchita masamu, kuphatikizapo atsopano pafupi ndi (α - R; × ₀ + R) mfundo × = α chilungamo ndiyo njira:

chilinganizo Izi chikutchedwa wasayansi wotchuka Brooke Taylor. zingapo zomwe lachokera ku imodzi yapita, akutchedwa Maclaurin mndandanda:

ulamuliro A lomwe limachititsa kutulutsa Kukula mndandanda Maclaurin:

1. Kudziwa opangidwa kuchokera kwa woyamba, wachiwiri, wachitatu, ... dongosolo.
2. Kuwerengetsa kodi opangidwa kuchokera ku × = 0.
3. Lembani Maclaurin mndandanda kwa nchito imeneyi, ndipo kenako kudziwa imeneyi ya mgwirizano.
4. Kudziwa imeneyi (-R; R), pamene gawo yotsalira ya chilinganizo Maclaurin

R _N (x) -> 0 chifukwa N -> osawerengeka. Ngati wina alipo, unagwira ntchito f (x) ayenera kukhala wolingana Uwerenge mndandanda Maclaurin.

Taonani tsopano mndandanda Maclaurin ntchito zimene munthu.

1. Motero, woyamba f (x) = mauthenga ^x. N'zoona kuti chikhalidwe chawo kotero f-Ia ali linachokera zosiyanasiyana malamulo, f ^{(k) (x)} = mauthenga ^×, kumene K ndiyofanana ndi onse manambala zachilengedwe. Wogwirizira × = 0. Timapeza f ^{(k) (0)} = mauthenga ⁰ = 1, K = 1,2 ... Kuchokera pa zimene takambiranazi, chiwerengero cha mauthenga ^× Zidzakhala motere:

2. Maclaurin mndandanda kwa ntchito f (x) = tchimo x. Pomwepo mwachindunji kuti f-imathanso kwa ena ofanana onse osadziwika adzayenera kusiya f ^'(x) = Ko × = tchimo (× + N / 2), ^{f' '(x)} = -sin × = tchimo (× + 2 * N / ²⁾ ..., f ^{(k) (x)} = tchimo (× + N * K / 2), kumene K ndiyofanana ndi inteja abwino alionse. Kuti, kukonza kuwerengetsera yosavuta, tinganene kuti zino chifukwa f (x) = tchimo × adzakhala ngati izi:

3. Tsopano tiyeni tione iju f-f (x) = Ko x. Sizikudziwika kuti ena ofanana onse kuti lachabechabe, ndipo ^{| f (k) (x)} | = | Ko (× + K * N / 2) | <= 1, K = 1,2 ... kachiwiri, atacita ena anapezazo, ife tikupeza kuti mndandanda kwa f (x) = Ko × lidzaonekera izi:

Kotero, ife kutchulidwa zinthu zofunika kwambiri kuti angathe kukodzedwa mu mndandanda Maclaurin, koma iwo amathandizana ndi Taylor mndandanda ena ntchito. Tsopano ife kutchula amenewa. Iwo ayenera kukumbukira kuti Taylor mndandanda ndipo Maclaurin mndandanda ndi mbali yofunika ya msonkhano mndandanda wa zochita masamu apamwamba. Choncho, Taylor zino.

1. Choyamba ndi mndandanda wa f-II f (x) = Ln (1 + x). Monga zitsanzo yapita, chifukwa ichi ife f (x) = Ln (1 + x) apangidwe angapo, ntchito mawonekedwe onse a Maclaurin zino. koma chifukwa cha nkhani zimenezi Maclaurin akhoza kuwapeza mosavuta kwambiri. Kaphatikizidwe angapo zojambula timapeza nambala ya f (x) = Ln (1 + x) wa chitsanzo:

2. Ndipo lachiwiri, amene adzakhala omaliza m'nkhani ino, adzakhala angapo kwa f (x) = arctg x. Pakuti × a imeneyi [-1; 1] chomveka kuwonongeka:

Ndizo zonse. M'nkhaniyi ine analiyang'ana kwambiri ntchito Taylor mndandanda ndipo Maclaurin mndandanda wa masamu apamwamba, makamaka makoleji zachuma ndi ukadaulo.