notes15 - Using Taylor’s Theorem This lecture will be...

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Unformatted text preview: Using Taylor’s Theorem This lecture will be about finding power series for all of the common functions we know about. The way that we accomplish this is by using the fact that for any smooth function f ( x ) we can find it’s power series centered at x = a by computing all the derivatives at a and dividing by a factorial, i.e., a k = f ( k ) ( a ) k ! Let’s start with, what turns out, to be the easiest one to compute. f ( x ) = e x . Remember, once we find a formula for f ( k ) (0) k ! Taylor’s theorem says f ( x ) = ∞ X k =0 f ( k ) (0) k ! x k Math 9C Fall 2011 (UCR) Pro-Notes October 29, 2011 1 / 18 Example 1 Find a power series for the function f ( x ) = e x centered at x = 0 The first thing we need to do is find all the derivatives of f ( x ) and then evaluate them at x = 0. This is easy though because the derivatives of f ( x ) = e x are just e x . So this says f ( k ) ( x ) = e x . In other words f ( k ) (0) k ! = e k ! = 1 k ! . So by Taylor’s Theorem e x = ∞ X k =0 x k k ! = 1 + x + x 2 / 2! + x 3 / 3! + x 4 / 4! + ··· Math 9C Fall 2011 (UCR) Pro-Notes October 29, 2011 2 / 18 Example 2 Find a power series expansion for f ( x ) = sin( x ) centered at x = 0. We need to know all the derivatives of sin( x ) evaluated at 0. So let’s start f ( x ) = sin( x ) f ( x ) =- cos( x ) f 00 ( x ) =- sin( x ) f 000 ( x ) =- cos( x ) This is all we need to compute because after the 3rd derivative it just...
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This note was uploaded on 02/22/2012 for the course MATH 9c taught by Professor C during the Fall '11 term at UC Riverside.

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notes15 - Using Taylor’s Theorem This lecture will be...

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