morework_14

morework_14 - Morework Solutions Math 213 – Spring 2008...

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Unformatted text preview: Morework Solutions Math 213 – Spring 2008 Week 14 5.4#13: f ( x ) = sin 3 ( x ) Sometimes, we don’t need to compute the integrals for a n and b n ! Provided we remember our trigonometric identities... sin 3 ( x ) = sin( x ) sin 2 ( x ) = sin( x ) 1 2- 1 2 cos(2 x ) = 1 2 sin( x )- 1 2 sin( x ) cos(2 x ) = 1 2 sin( x )- 1 4 sin(3 x )- 1 4 sin(- x ) = 3 4 sin( x )- 1 4 sin(3 x ) . 5.4#19: Derive the equations Z ‘- ‘ cos nπx ‘ cos kπx ‘ dx = ( when k 6 = n ‘ when k = n (1) Z ‘- ‘ cos nπx ‘ sin kπx ‘ dx = 0 (2) Z ‘- ‘ sin nπx ‘ sin kπx ‘ dx = ( when k 6 = n ‘ when k = n (3) First observe that if m is any integer, positive or negative, then Z ‘- ‘ cos mπx ‘ dx = ( when m 6 = 0 2 ‘ when m = 0 (4) 1 Indeed, when m = 0, the integrand is simply 1 and the integral R ‘- ‘ 1 dx = 2 ‘ . When m 6 = 0, Z ‘- ‘ cos mπx ‘ dx = ‘ mπ sin mπx ‘ ‘ x =- ‘ = ‘ mπ sin( mπ )- ‘ mπ sin(- mπ ) = 0 ....
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This note was uploaded on 05/09/2008 for the course MATH 2130 taught by Professor Dorais during the Spring '08 term at Cornell.

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morework_14 - Morework Solutions Math 213 – Spring 2008...

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