Taylor Polynomials inx Let fx)be differentiable at x=0.Then the polynomial P(x)=f(0)+f'(0)x is agood"approximation of fwhen x is small.Note that P(O)=f(O)and P'(0)=f'(0) If f)is twice differentiable at x=0.Then we get a better approximation B()=f0+rOx+/0 Again,we have P(0)=f(0),P'(0)=f'(0)andP"(0)=f"(0) Main Meny
Main Menu Taylor Polynomials in x Let f(x) be differentiable at x=0. Then the polynomial P(x) f (0) f '(0)x 1 = + is a “good ” approximation of f when x is small. Note that (0) (0) 1 P = f '(0) '(0) 1 and P = f If f(x) is twice differentiable at x=0. Then we get a better approximation "(0) . 2 1 ( ) (0) '(0) 2 2 P x = f + f x + f x Again, we have (0) (0), 2 P = f '(0) '(0) 2 P = f and "(0) "(0) 2 P = f
If f)is n-th differentiable at x=0.Then we can form the polynomial B=f0+0x+r0r+5/ox4+ox 31 P has the value as f at 0 and the same first n derivatives Pn(0)=f(0),Pn'(0)=f'(0)P,"(0)=f"(0),,pPnm(0)=(0) The polynomials P,P2,P3,....P are called the Taylorpolynomials off. Example f(x)=e* Main Meny cme5时
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. If f(x) is n-th differentiable at x=0. Then we can form the polynomial (0) . ! 1 (0) 3! 1 "(0) 2! 1 ( ) (0) '(0) 2 (3) 3 (n) n n f x n P x = f + f x + f x + f x ++ Pn has the value as f at 0 and the same first n derivatives: P (0) f (0), n = P '(0) f '(0), n = "(0) "(0), , (0) (0) (n) (n) n n P = f P = f The polynomials P1 , P2 , P3 , …, Pn are called the Taylor polynomials of f. Example x f (x) = e
Example f(x)=sin x Example f(x)=cosx Main Menu C 007e
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Example f (x) = sin x Example f (x) = cos x
Taylor Polynomials in x;Taylor Series in x THEOREM 12.6.1 TAYLOR'S THEOREM Iffhas n+1 continuous derivatives on an open interval that contains0, then for eachx∈I =⑩+0x+9+9士R® with R(x)= ()(x-t"dt.We call R(x)the remainder. Main Meny own6
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Taylor Polynomials in x; Taylor Series in x
Taylor Polynomials in x;Taylor Series in x COROLLARY 12.6.2 LAGRANGE FORMULA FOR THE REMAINDER The remainder in Taylor's theorem can be written R(x)= fn+D(c) (+1)! with c some number between 0 and x. The following estimate for R(x)is an immediate consequence of Corollary 12.6.2: (12.6.3) Rn(x)I≤ +1 (n+1)月 whereJ is the closed interval that joins 0 to x. Main Menu cme5时
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Taylor Polynomials in x; Taylor Series in x The following estimate for Rn (x) is an immediate consequence of Corollary 12.6.2: where J is the closed interval that joins 0 to x
Example f(x)=e* Main Meny C 007e
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Example x f (x) = e
Example f(x)=sin x Main Meny o
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Example f (x) = sin x
Taylor Polynomials in x;Taylor Series inx Taylor Series inx By definition 0!=1.Adopting the convention thatf(o)=f,we can write Taylor polynomials P(-=f0+f0x+f0r++@x 21 n! inΣnotation R)=2f@ k! In this case,we say that f(x)can be expanded as a Taylor series in x and write (12.6.4) f(x)= ox 司 k! Main Menu 5
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Taylor Polynomials in x; Taylor Series in x Taylor Series in x By definition 0! = 1. Adopting the convention that f (0) = f , we can write Taylor polynomials in Σ notation: In this case, we say that f (x) can be expanded as a Taylor series in x and write ( ) ( ) ( ) ( ) ( ) ( ) 2 0 0 0 0 2! ! n n n f f P x f f x x x n = + + + + ( ) ( ) ( ) 0 0 . ! k n k n k f P x x = k =
Taylor Polynomials in x;Taylor Series in x (12.6.5) =1+x+ 2+3 for all real x. 、(-1) 21=x x3 x5 x7 (12.6.6) sinx= (2k+1) 3+5-7元 十… for all real x. Main Meny C 007e
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Taylor Polynomials in x; Taylor Series in x
(12.6.7) cOSx= x24=1 x2 x4 x6 (2k 2+4- for all real x. 61 Main Meny own6
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved