pracfinal - Math 104A Practice Final Problems 1 Find the natural(free cubic spline that interpolates the function values f(0 = 1 f(1 = 2 and f(2 = 9 in

pracfinal - Math 104A Practice Final Problems 1 Find the...

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Math 104A Practice Final Problems 1. Find the natural (free) cubic spline that interpolates the function valuesf(0) = 1,f(1) = 2, andf(2) = 9 in the interval [0,2].2. Letf(0.1) = 1.10517,f(0.2) = 1.22140, andf(0.3) = 1.34986. a) Find a first order approxi-mation tof0(0.2). b) Find a second order approximation tof0(0.2).3. Suppose thatN(h) is an approximation toMfor everyh >0 and thatM=N(h) +K1h+K2h2+K3h3+· · ·for some constantsK1,K2,K3,. . .. Use the valuesN(h),N(h3), andN(h9) to produce andO(h3) approximation toM.4. The (simple) trapezoidal rule applied toR40f(x)dxgives the value 2 and Simpson’s rule givesthe value 1. What isf(2)?5. Derive a quadrature formula for approximatingZ30f(x)dxin terms off(0),f(1),f(2) andf(3).6. a) Define the degree of accuracy or precision of a quadrature formula. b) What’s the degreeof accuracy for the Composite Trapezoidal rule?c) What’s the degree of accuracy for theComposite Simpson’s rule?7. Consider the integralZ10exx3dxApproximate this integral using a) the Composite Trapezoidal rule with
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