hw4 - f ( x ) = e x in the interval [0 , 1]. What is the...

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Homework 4 – Math 104A, Winter 2009 Due on Wednesday, February 4th, 2009 Section 3.1: 1.a, 3.a, 5, 17, 22, and 26. In problem 26, check your result using any of the methods learnt in the previous chapter. Section 3.2: 4, and 10. Additional problem 1: Consider a function f : [ a, b ] R . We want to approximate the integral I = Z b a f ( x ) dx In order to do this, consider the points a , ( a + b ) / 2, and b . Construct the Newton interpolation polynomial of degree 2, P 2 ( x ) and then integrate the polynomial. This procedure will give you a formula of the form: I 2 = a 0 f ( a ) + a 1 f ± a + b 2 ² + a 2 f ( b ) (You have to determine the coefficients a 0 , a 1 , and a 2 ). Use this formula to approximate the integral of
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Unformatted text preview: f ( x ) = e x in the interval [0 , 1]. What is the error in your approximation? How could one get a more accurate formula? Additional problem 2: Some of you might be familiar with the formula: n X i =1 i := 1 + 2 + + n = n ( n + 1) 2 . We want to generalize this formula to the sum of squares. Assume that there exists a polynomial, p ( x ), such that n X i =1 i 2 := 1 + 2 2 + + n 2 = p ( n ) . Use Newtons interpolation to nd this polynomial. Check your answer by computing the sum of the rst 100 squares. 1...
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