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:
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.
The presentation of the material
First, let us define a power series.
Function can be decomposed in a power series on the interval if there exists a power series converging to in this interval. If the function is expanded in a power series in some neighborhood of the point , then this Taylor series.
Let the function is infinitely differentiable on the interval and all its derivatives are limited in the aggregate to the interval, i.e. there exists a number , such that for all and for all the following inequality holds: . Then the Taylor series converges to for all . We will give the decomposition in a Taylor series for elementary functions.
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.
If the function has derivative on the interval with endpoints and , then for any positive number , there is a point in lying between and , such that
This is the formula of Taylor with the residual member in the General form.
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.
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”.
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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