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HW1 - Page 496 problem 11 f x = x < x < is an even function...

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Page 496, problem 11: π π π < < - - = x x x f , ) ( is an even function. ( 29 ( 29 ( 29 ( 29 ( 29 n n n nx nx nx n nx dx nx x a x x dx x a ) 1 ( 1 1 2 ) sin( ) cos( 1 ) sin( 2 cos 2 2 2 1 1 2 0 0 0 2 0 0 - - = + - = - = = - = - = π π π π π π π π π π π π π π Plots of f and some partial sums: Page 496, problem 15: < < < < - = 2 0 0 0 2 2 ) ( x if x if x f is neither even nor odd, but if we write f(x)=1+g(x), the function g(x) is odd. Therefore ( 29 ( 29 1 1 2 2 sin 1 2 2 , 2 sin 1 ) ( 1 ) ( 2 0 1 - - = - = + = + = = n n n n n dx x n B x n B x g x f π π π Plots of f and some partial sums:
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p. 499, #9 ( 29 ( 29 ( 29 ( 29 1 1 1 1 2 2 2 1 cos 1 1 inx inx inx in in n n n inx n x c xe dx e e dx e e in in in in i n in n i f x e n π π π π π π π π π π π π π π - - - - - - - =-∞ = = - + = - - = - = - = - p. 505, #4 Since the function is odd all the ' n a s are 0. Repeated integration by parts gives ( 29 ( 29 ( 29 ( 29 ( 29 3 3 2 2 3 4 cos 3 sin 6 cos 6sin sin n x nx n x nx nx nx nx x nx dx n - + + - = so that ( 29 ( 29 ( 29 3 3 2 3 4 3 0 2 2 6 12 2 sin cos 1 n n n n b x nx dx n n n n π π π π π π π - + = = = - -
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( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 6 7 * 7 2 1 * * 2 2 * * 2 3 * * 2 4 * * 2 5 2 7 2 1 674.8 7 2 1 454.7 3 2 336.4 4 3 265.6 5 4 219.0 x dx E b E E b E E b E E b E E b π π π π π π π π π - = = - = = - = = - = = - = = - = p. 505, #5 The function is even, hence all the
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