Midterm_ProblemsSM.pdf - Study Problems for the Midterm Math 104 A 1 Instructor Prof Hector D Ceniceros 1 Suppose you have a quadrature Qh[f to

# Midterm_ProblemsSM.pdf - Study Problems for the Midterm...

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Study Problems for the Midterm , Math 104 A 1 Instructor: Prof. Hector D. Ceniceros 1. Suppose you have a quadratureQh[f] to approximate the definite integralI[f] =Zbaf(x)dx(1)and you know that for sufficiently smallhthe errorEh[f] =I[f]-Qh[f] satisfiesEh[f] =c4h4+R(h),(2)wherec4is a constant andR(h)/h40 ash0.(a) What is the rate of convergence ofQhand what does it mean for the error (ifhis halved what happens to the corresponding error)?(b) Use (2) to find a computable estimate of the error,˜E[f].(c) Give a way to check that ifhis sufficiently small for that estimate of the error tobe accurate.(d) Use˜E[f] (or equivalently Richardson’s extrapolation) to produce a more accuratequadrature fromQh.2. Suppose that we would like to approximateR10f(x)dxbyQ[f] =Z10P2(x)dx,(3)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).(4)(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 (4)