АвтоАвтоматизацияАрхитектураАстрономияАудитБиологияБухгалтерияВоенное делоГенетикаГеографияГеологияГосударствоДомДругоеЖурналистика и СМИИзобретательствоИностранные языкиИнформатикаИскусствоИсторияКомпьютерыКулинарияКультураЛексикологияЛитератураЛогикаМаркетингМатематикаМашиностроениеМедицинаМенеджментМеталлы и СваркаМеханикаМузыкаНаселениеОбразованиеОхрана безопасности жизниОхрана ТрудаПедагогикаПолитикаПравоПриборостроениеПрограммированиеПроизводствоПромышленностьПсихологияРадиоРегилияСвязьСоциологияСпортСтандартизацияСтроительствоТехнологииТорговляТуризмФизикаФизиологияФилософияФинансыХимияХозяйствоЦеннообразованиеЧерчениеЭкологияЭконометрикаЭкономикаЭлектроникаЮриспунденкция

Taylor’s formula

Читайте также:
  1. Getting ready for the round table. Group the following conversational formulas in A into categories in B.
  2. Total probability formula

Suppose we’re working with a function f(x) that is continuous and has n+1 continuous derivatives on an interval about x = 0. We can approximate f near 0 by a polynomial of degree n:

• For n = 0, the best constant approximation near 0 is which matches f at 0.

• For n = 1, the best linear approximation near 0 is . Note that matches f at 0 and matches at 0.

• For n = 2, the best quadratic approximation near 0 is . Note that , , and match , , and , respectively, at 0.

Continuing this process,

 

.

This is the Taylor polynomial of degree n about 0 (also called the Maclaurin series of degree n). More generally, if f has n+1 continuous derivatives at x = a, the Taylor series of degree n about a is

.

This formula approximates f (x) near a. Taylor’s Theorem gives bounds for the error in this approximation:

Taylor’s Theorem:

Suppose f has n+1 continuous derivatives on an open interval containing a. Then for each x in the interval,

,

where the error term satisfies for some c between a and x.

This form for the error , derived in 1797 by Joseph Lagrange, is called the Lagrange formula for the remainder. The infinite Taylor series converges to f,

,

if and only if .


1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 30 | 31 | 32 | 33 | 34 | 35 | 36 | 37 | 38 | 39 | 40 | 41 | 42 | 43 | 44 | 45 | 46 | 47 | 48 | 49 | 50 | 51 | 52 | 53 | 54 | 55 | 56 | 57 | 58 | 59 | 60 | 61 | 62 | 63 | 64 | 65 | 66 | 67 | 68 | 69 | 70 | 71 | 72 |

Поиск по сайту:



Все материалы представленные на сайте исключительно с целью ознакомления читателями и не преследуют коммерческих целей или нарушение авторских прав. Студалл.Орг (0.003 сек.)