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hmwk1sol

# hmwk1sol - b →∞-xe(1 7 x | b Z ∞ e(1 7 x dx = 7 1 2...

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Solutions to Homework 1, ISyE 3232 Fall 2006 Problem 1 A = { 1 , 3 , 5 , 7 , ... } , A B = { 8 , 10 , 12 , 14 } , A B = { 2 , 4 , 6 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 16 , 18 , ... } , B A = { 2 , 4 , 6 , 16 , 18 , ... } . Problem 2 A = [2 , 9), A B = [7 / 5 , 2), A B = [0 , 4), A B = [2 , 4). Problem 3 n =0 (1 / 9) n = 1 1 - 1 / 9 = 9 / 8 n =0 (1 / 9) n n ! = e 1 / 9 Problem 4 5! = 120, ( 9 7 ) = 36. Problem 5 n =2 4 n is infinite. This is a p -series, with p = 1. n =2 4 n 2 however, is finite. This is a p -series, with p = 2. Problem 6 (a) 3 n =0 (1) n = . (b) 3 n =0 ( - 1) n is undefined. Recall that for a sequence of numbers { a n } n =0 , n =0 a n = lim m →∞ m n =0 a n = lim m →∞ s m where s m = m n =0 a n . Notice that for even m , s m = 1, but for odd m , s m = 0. Therefore, the sequence s m does not converge to a limit, which means that the series is undefined. Problem 7 2 0 6 x 5 dx = x 6 | 2 0 = 64 where f ( x ) | b a = f ( b ) - f ( a ). 0 (1 / 7) e - (1 / 7) x dx = lim b →∞ - e - (1 / 7) x | b 0 = 1 . 0 (1

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Unformatted text preview: b →∞-xe-(1 / 7) x | b + Z ∞ e-(1 / 7) x dx = 7 . 1 2 Problem 8 Z 2 Z x 5 xydydx = Z 2 5 x 3 2 dx = (5 / 8)(16) = 10 . Z ∞ Z ∞ e-(10 x +9 y ) dydx = 1 / 90 . Problem 9 To ﬁnd the value of x such that e-x/ 6 = 1 / 3, take the natural log of both sides and solve for x : once you do that, you’ll see that x = 6 . 5917. Problem 10 Here d dt Z x 4 x 2 3 xtdt = Z x 4 x 2 ± ∂ ∂t 3 xt ² dt + 3 x ( x 4 )(4 x 3 )-3 x ( x 2 )(2 x ) = Z x 4 x 2 3 xdt + 12 x 8-6 x 4 = 3 2 x 8-3 2 x 4 = 27 2 x 8-15 2 x 4 ....
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