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Инфоурок / Математика / Конспекты / Разработка лекции на тему " Taylor series"

Разработка лекции на тему " Taylor series"


  • Математика

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Theme of lecture: Taylor series

Aim of the lesson: to provide students with a General definition of the Taylor series, to make the decomposition in a Taylor series and to introduce the Taylor’s formula.

Type of the lesson: combined

The stages of a lesson:

  1. Greeting

Good afternoon students. Today we start a new topic. Please be very careful and to carry out tasks which I will give during the lesson. the theme of the lesson: the Taylor Series. The Taylor formula.



  1. The presentation of the material



First, let us define a power series.

C:\Users\Азема\Downloads\20160314_193127_resized.jpg



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Function image84.gif (951 bytes)can be decomposed in a power series on the interval image85.gif (1116 bytes) if there exists a power series converging to image84.gif (951 bytes) in this interval. If the function is expanded in a power series in some neighborhood of the point image86.gif (886 bytes), then this Taylor series.

Let the function image78.gif (951 bytes) is infinitely differentiable on the interval image73.gif (1116 bytes) and all its derivatives are limited in the aggregate to the interval, i.e. there exists a number image87.gif (953 bytes), such that for all image88.gif (1171 bytes) and for all image89.gif (977 bytes) the following inequality holds: image90.gif (1179 bytes). Then the Taylor series converges to image91.gif (951 bytes) for all image92.gif (1174 bytes) . We will give the decomposition in a Taylor series for elementary functions.

image93.gif (1642 bytes)

image94.gif (1874 bytes)

image95.gif (1780 bytes)

image96.gif (1708 bytes)

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The Taylor formula is used in the proof of a large number of theorems in the differential calculus. Speaking of lax, the Taylor formula shows the behavior of the function in the neighborhood of a point.

Theorem:

If the function f(x) has n+1 derivative on the interval with endpoints a and x, then for any positive number p , there is a point \xi in lying between a and x, such that

f(x) = \sum_{k=0}^n {f^{(k)} (a) \over k!} (x - a)^k + \left({x - a \over x - \xi}\right)^p{(x - \xi)^{n+1}\over n! p}f^{(n+1)}(\xi).



This is the formula of Taylor with the residual member in the General form.

  1. Initial consolidation of



Нave any questions? If not then let's solve some problems from the textbook. On page 54, task number: 435 -440, 4 students can solve it on the Board.



  1. Summing up the lesson



Today we got acquainted with one type of power series near Taylor. You can see more Various forms of the remainder term of the Taylor formula and Examples the decomposition in a Taylor series of the function a large number of variables in the textbooks

Il'in V. A., Sadovnichii V. A., B. H. Sendov “Mathematical analysis”;

Kamynin L. I. “Mathematical analysis”.



  1. Give homework



Secondsperminute topic, refer to the additional materials

and solve tasks from the Demidovich on page 55 under the numbers 452 -460.

Good luck! Have a nice day.














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Дата добавления 21.03.2016
Раздел Математика
Подраздел Конспекты
Просмотров105
Номер материала ДВ-544075
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