{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

taylors_thm

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

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

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

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

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### 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
Ask a homework question - tutors are online