Final_Problems copy 2.pdf - Study Problems for the Final...

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Study Problems for the Final, Math 104 A1Instructor: Prof. Hector D. Ceniceros1. Suppose that we would like to approximateR10f(x)dxbyQ[f] =Z10P2(x)dx,(1)whereP2(x) is the polynomial of degree at most two which interpolatesfat 0, 1/2,and 1.(a) WriteP2(x) in Lagrange form and prove thatQ[f] =16f(0) + 4f12+f(1).(2)(b) Consider now a general interval [a, b] and the integralRbaf(x)dx. Do the changeof variablesx=a+ (b-a)tto transform the integral to one in [0,1] and use (2)to obtain the more general quadrature (Simpson’s Rule):QS[f] =b-a6f(a) + 4fa+b2+f(b).(3)(c) Explain why the quadrature is exact, i.e.QS[f] =I[f] whenfis a polynomial ofdegree 2 or less and prove that it is actually exact for polynomials of degree 3 orless.(d) If we use an interpolating polynomial of much higher degree than 2 and equidis-tributed nodes, does the accuracy of the quadrature improve for a general contin-uous integrand? Explain.2. (a) Compute an approximate value for ln 1.

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