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mth.122.handout.12

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

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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 (0) x. Degree 2: The Taylor Polynomial of degree 2 approximating f ( x ) for x near zero is: f ( x ) f (0) + f 0 (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 0 + 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 0 = f (0) C 1 = f 0 (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 (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 X 2 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

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

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