Πολυώνυμα και Σειρές Taylor 1. Motivation Why do we use approximations? –They are made up of the simplest functions – polynomials. –We can differentiate.

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1 Πολυώνυμα και Σειρές Taylor 1

2 Motivation Why do we use approximations? –They are made up of the simplest functions – polynomials. –We can differentiate and integrate them very easily. –We can use them in cases where we do not know the actual function! 2

3 Motivation We can easily compute expressions like: 0.6 a b 3

4 For a function f (x) that is differentiable at x=x 0, the tangent is a close approximation of the function in a neighborhood of the tangent point x 0. We call the equation of the tangent the linearization of the function. 4

5 Ambition: Approximating of differential functions more precisely by polynomials of higher degrees. 5

6 Polynomials are “nice” functions. We want to construct, for any function, a degree n polynomial that matches a given function “perfectly” at some base point, x 0. But what does it mean to say “matches perfectly”? Suppose f is the function of interest and p is the degree n polynomial to be constructed. We need: 6

7 Σειρές και Πολυώνυμα Taylor Το θεώρημα του Taylor απαιτεί όχι τη μορφή της συνάρτησης και των παραγώγων της, αλλά την τιμή της συνάρτησης και των παραγώγων της σε ένα σημείο. Έτσι, αν γνωρίζουμε την τιμή της συνάρτησης και των παραγώγων της σε ένα σημείο, μπορούμε να βρούμε την τιμή της συνάρτησης σε κάποιο άλλο σημείο. ΠΡΟΫΠΟΘΕΣΗ : Πρέπει να υπάρχουν όλες οι παράγωγοι και να είναι συνεχείς, μεταξύ του γνωστού σημείου x και του σημείου x+h όπου επιθυμούμε να υπολογίσουμε την τιμή της συνάρτησης, δηλ. απαιτείται στο διάστημα [x, x+h] : –να υπάρχουν οι παράγωγοι και να είναι συνεχείς –Να υπάρχει η παράγωγος (n+1) τάξης και να είναι συνεχής στο (x,x+h) 7

8 The Taylor Series Formula Μany different functions can be expressed as power series. A general power series can be expressed as are constants. As with a polynomial, we often don't bother to write terms that have a coefficient of 0, but we can imagine that every power series has every one of these terms. The first term of a power series is called the constant term. The second term of a power series is called the linear term or x term, and has the form c 1 x for some coefficient c 1. You can obtain the coefficient c 1 by taking the derivative of the series and then substituting x=0. 8

9 In general, taking the derivative of a power series "demotes" each of the coefficients by one step. The following formula relates the coefficients of a power series to the values of the derivatives at x=0 : The following calculation illustrates this pattern: 9

10 10

11 Taylor Polynomials Τhe Taylor polynomial of degree for a function which is times differentiable at 11

12 Taylor series 12 which has derivatives of all orders at a point is given by Taylor series The Taylor series for a function

13 Compare 13 Taylor polynomial Taylor series

14 A Taylor Polynomial 14 is a (finite) sum with a defined degree.

15 A Taylor Series 15 is an infinite sum, i.e., a sequence of partial sums. Each member of the sequence of partial sums is itself a Taylor polynomial. A Taylor series is a sequence of Taylor polynomials.

16 Any smooth function can be approximated as a polynomial. Taylor series provides a mean to predict a function value at one point x in terms of the function and its derivatives at another point a. Taylor’s Series Note: This is the same expression except that a and x are replaced by x i and x i+1 respectively.

17 If we let h = x i+1 - x i, we can rewrite the Taylor series and the remainder as Taylor’s Series h is called the step size.

18 Taylor Series Approximation Example: More terms used implies better approximation f(x) = 0.1x 4 - 0.15x 3 - 0.5x 2 - 0.25x + 1.2 x = h

19 Taylor Series Approximation Example: Smaller step size implies smaller error f(x) = 0.1x 4 - 0.15x 3 - 0.5x 2 - 0.25x + 1.2 Reduced step size Errors

20 Convergence of Taylor Series The Taylor series converges fast (few terms are needed) when x is near the point of expansion. If |x-a| is large then more terms are needed to get a good approximation. 20

21 Assumptions For a Taylor polynomial, you need derivatives up to order For a Taylor series, you need derivatives of all orders. It’s important to understand the difference between expressing a function as an infinite series and approximating a function by using a finite number of terms of series. You can think of a power series as a polynomial with infinitely many terms (Taylor polynomial). In practice, however, adding up an infinite number of terms simply isn’t possible. Nevertheless, you can approximate the value of f(x) by adding a finite number from the appropriate Taylor series. Generally speaking, a higher-degree polynomial results in a better approximation. 21

22 While it is beautiful that certain functions can be represented exactly by infinite Taylor series, it is the inexact Taylor series that do all the work… In practical terms, we would like to be able to use Taylor polynomials to approximate functions over the intervals of convergence of the Taylor series, and we would like to keep the error of the approximation within specified bounds… The error results from truncating the series down to a polynomial (that is, cutting it off after some number of terms), is truncation error. Every truncation splits a Taylor series into equally significant pieces: the Taylor polynomial that gives us the approximation, and the remainder that tells us whether the approximation is any good… 22

23 Terminology Maclaurin seriesThe Taylor series for at is called the Maclaurin series for The Taylor series is named after the English mathematician Brook Taylor (1685–1731). The Maclaurin series is named for the Scottish mathematician Colin Maclaurin (1698–1746). –This is despite the fact that the Maclaurin series is really just a special case of the Taylor series. 23

24 TAYLOR & MACLAURIN SERIES 24

25 TAYLOR & MACLAURIN SERIES The figure shows the graph of sin x together with its Taylor (or Maclaurin) polynomials 25

26 TAYLOR & MACLAURIN SERIES Notice that, as n increases, T n (x) becomes a better approximation to sin x. 26

27 Deriving Taylor Series from a Basic List 27

28 Using Taylor’s Theorem 28 Find the Taylor series for f that is based at x = π/4.

29 Taylor Series for f (x) = sin(x) nf (n) (x)f (n) ( )a n = f (n) ( )/n! 0 1 2 3 4 29

30 Find the Taylor polynomial of order four for the function at x = 0, and use it to approximate the value of the function at x = 0.2.

31 Find the Taylor polynomial of order four for the function at x = 0, and use it to approximate the value of the function at x = 0.2.

32 Find the Taylor polynomial of order four for the function at x = 0, and use it to approximate the value of the function at x = 0.2. How close is this approximation to the actual function value?

33 Intervals of Validity A series converges if the terms of the infinite series become smaller and smaller; This is when the partial sum of the series approaches a limit; A series is only valid for values of x where the series converges; We call this the interval of validity. We can only use a series within its interval of validity. 33

34 Intervals of Validity How do we calculate the interval of validity? This is difficult to do!!! However, we know the intervals of validity for some well known series (see your formulae book); So we can find the interval of validity for a series by transforming it into the form of one of the well known series. 34

35 The taylor(f,n,a) function gives the first n-1 terms in the Taylor series for the function defined in the expression f, evaluated at the point x  a. If the parameter a is omitted the function returns the series evaluated at x  0. >>syms x >>f = exp(x); >>taylor(f,4) ans = 1+x+1/2*x^2+1/6*x^3 >>taylor(f,3,2) ans = exp(2)+exp(2)*(x-2)+1/2*exp(2)*(x-2)^2 Taylor series in MATLAB 35

36 Άσκηση Να βρεθεί η τιμή sin(2) χρησιμοποιώντας σειρές Taylor, χρησιμοποιώντας μέχρι 4 ης τάξης παράγωγο. 36