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a) Explain how we arrive at the formula for Simpson's rule (standard formula) using the Lagrange Interpolating Polynomial of degree 2. Ignore the error term, and do not compute any integral.
b) We define the Composite Simpson's Rule by splitting the interval [a,b] into smaller sub-intervals, applying Simpson's Rule on those sub-intervals, and then summing up the results. Write down Composite Simpson's Rule applied to the sub-intervals [x0,x2], [x2, x4],....[x2-n, x2n], where xk= a + k(b-a) / 2n.
c) What is the error term for Composite Simpson's Rule?
d) Richard Extrapolation: Explain how we can use multiple instances of Composite Simpson's Rule (with point-spacing h, h/2, h/4....) to generate a scheme with an error term = O(h^6)
Find the maximum and minimum concavitiy
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Prove that the subgroup of A4 generated by any element of order 2 and and any element of order 3 is all of A4. Prove that if x and y are distinct 3-cycles in S4 with x != y^-1 (x not equal to the inverse of y), then the subgroup of S4 generated by x..
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