mth.122.handout.12

mth.122.handout.12 - MTH 122 — Calculus II Essex County...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

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

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

Unformatted text preview: MTH 122 — Calculus II Essex County College — Division of Mathematics and Physics 1 Lecture Notes #12 — Sakai Web Project Material 1 Taylor Polynomials In the last class we actually generated several Taylor 2 polynomials and everyone should be clear that our example followed a pattern, as follows: Degree 1: The Taylor Polynomial of degree 1 approximating f ( x ) for x near zero is: f ( x ) ≈ f (0) + f (0) x. Degree 2: The Taylor Polynomial of degree 2 approximating f ( x ) for x near zero is: f ( x ) ≈ f (0) + f (0) x + f 00 (0) 2! x 2 . Certainly if we continue this process an easy pattern emerges. For example if we have an n th degree polynomial of the form, f ( x ) ≈ C + C 1 x + C 2 x 2 + C 3 x 3 + C 4 x 4 + ··· + C n- 1 x n- 1 + C 9 x n , and we follow the same process outlined in the prior worksheet, we’ll get: C = f (0) C 1 = f (0) 1! C 2 = f 00 (0) 2! C 3 = f 000 (0) 3! C 4 = f (4) (0) 4! . . . = . . . C n = f ( n ) (0) n ! Finally we have a Taylor polynomial of degree n approximating f ( x ) for x near 0 is, f ( x ) ≈ f (0) + f (0) x + f 00 (0) 2! x 2 + f 000 (0) 3! x 3 + f (4) (0) 4! x 4 + ··· + f ( n ) (0) n ! x n . 1 This document was prepared by Ron Bannon ( [email protected] ) using L A T E X2 ε . Last revised January 10, 2009. 2 Brooke Taylor was an English Mathematician (1685–1731), but these approximating polynomials were known prior to Taylor’s exposition on the subject. 1 1.1 Examples 1. Find the Taylor polynomial of degree 9 about x = 0 for the function f ( x ) = e x . Work: This one is pretty easy, mainly because the derivative of e x never changes. So we have: e x = 1 + x + x 2 2! + x 3 3! + x 4 4! + x 5 5! + x 6 6! + x 7 7! + x 8 8! + x 9 9! . Here’s a graph of both f ( x ) = e x (in black) and the ninth degree polynomial (in red). You should notice that the fit is not perfect, but it looks damn good for x >- 3. Although not obvious, the higher degree for this polynomial the better the fit becomes....
View Full Document

Page1 / 9

mth.122.handout.12 - MTH 122 — Calculus II Essex County...

This preview shows document pages 1 - 3. Sign up to view the full document.

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