lec38 - Lecture 38 18.01 Fall 2006 Lecture 38: Final Review...

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Unformatted text preview: Lecture 38 18.01 Fall 2006 Lecture 38: Final Review Review: Differentiating and Integrating Series. If f ( x ) = a n x n , then n =0 n +1 a n x f ( x ) = na n x n- 1 and f ( x ) dx = C + n + 1 n =1 n =0 Example 1: Normal (or Gaussian) Distribution. x x e- t 2 dt = 1- t 2 + (- 2! t 2 ) 2 + (- 3! t 2 ) 3 + dt x t 4 t 6 t 8 = 1- t 2 + 2!- 3! + 4!- ... dt x 3 1 x 5 1 x 7 = x- 3 + 2! 5- 3! 7 + ... x 2 Even though e- t dt isnt an elementary function, we can still compute it. Elementary functions are still a little bit better, though. For example: sin x = x- x 3! 3 + x 5! 5- = sin 2 = 2- ( / 3! 2) 3 + ( / 5! 2) 5- But to compute sin( / 2) numerically is a waste of time. We know that the sum if something very simple, namely, sin = 1 2 Its not obvious from the series expansion that sin x deals with angles. Series are sometimes com- plicated and unintuitive. Nevertheless, we can read this formula backwards to find a formula for . Start with sin = 1 ....
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lec38 - Lecture 38 18.01 Fall 2006 Lecture 38: Final Review...

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