taylors_thm

taylors_thm - Harvey Mudd College Math Tutorial: Taylor's...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
Harvey Mudd College Math Tutorial: Taylor’s Theorem 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 P n ( x ) of degree n : For n = 0, the best constant approximation near 0 is P 0 ( x ) = f (0) which matches f at 0. For n = 1, the best linear approximation near 0 is P 1 ( x ) = f (0) + f 0 (0) x. Note that P 1 matches f at 0 and P 0 1 matches f 0 at 0. For n = 2, the best quadratic approximation near 0 is P 2 ( x ) = f (0) + f 0 (0) x + f 00 (0) 2! x 2 . Note that P 2 , P 0 2 , and P 00 2 match f , f 0 , and f 00 , respectively, at 0. Continuing this process, P n ( x ) = f (0) + f 0 (0) x + f 00 ( x ) 2! x 2 + ... + f ( n ) (0) n ! x n . 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 n X k =0 f ( k ) ( a
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 4

taylors_thm - Harvey Mudd College Math Tutorial: Taylor's...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online