Math128_hw6

Math128_hw6 - Math 128A Spring 2007 Homework 6 Solution...

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Math 128A, Spring 2007 Homework 6 Solution 5.1.12 It’s enough to check exactness for 1, x , x 2 , x 3 , etc. The degree of precision is the ﬁrst n for which our rule is not exact for x n +1 . Z 1 0 1 dx = 1 1 4 1 + 3 4 1 = 1 Z 1 0 xdx = 1 / 2 1 4 0 + 3 4 2 / 3 = 1 / 2 Z 1 0 x 2 dx = 1 / 3 1 4 0 + 3 4 4 / 9 = 1 / 3 Z 1 0 x 3 dx = 1 / 4 1 4 1 + 3 4 8 / 27 = 2 / 9 so the degree is 2. 5.1.14 We have I ( f ) = Z 1 - 1 f ( x ) Z 1 - 1 [ f ( - 1 / 3)(3 / 2)(1 / 3 - x ) + f (1 / 3)(3 / 2)( x + 1 / 3)] dx = f ( - 1 / 3) + f (1 / 3) and we can check the degree is 2 by the method of problem 12. 5.1.15 When f ( x ) = 1, we have R 1 - 1 f ( x ) dx = 2 = 1 + 1 = f ( - β ) + f ( β ) for any β . When f ( x ) = x , we have R 1 - 1 f ( x ) dx = 0 = - β + β = f ( - β ) + f ( β ) for any β . Hence the degree is at least 1. To make it work for f ( x ) = x 2 , we need R 1 - 1 f ( x ) dx = 2 / 3 = f ( - β ) + f ( β ) = 2 β 2 , i.e. β = ± 1 / 3. This choice of β is also exact for f ( x ) = x 3 but not for

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Math128_hw6 - Math 128A Spring 2007 Homework 6 Solution...

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