MATH 5610/6860PRACTICE FINAL EXAMNote: This exam is longer and more difficult than the actual final.Problem 1. Consider a smooth function f with the following values.x 1 2 4 5f(x) 0 3 -3 0(a) Use divided differences to find the polynomial p(x) interpolating f (x)(at all four nodes) in Newton form.(b) Assuming all derivatives of f are available, give an expression of theinterpolation error f(t) − p(t) for some t ∈ [1, 5].Problem 2. Find the cubic spline S(x) for x ∈ [0, 1] satisfying the conditionsS(0) = 0, S(1) = 1, S0(0) = S0(1) = 0.Problem 3. Derive the centered difference formula for approximating f0(x)with an error term involving a higher order derivative of f.Problem 4. (K&C 7.1.15) Derive a numerical differentiation formula of or-der O(h4) by applying Richardson’s extrapolation tof0(x) =12h[f(x + h) − f(x − h)] −h26f000(x) −h4120f(5)(x) − · · ·What is the error in terms of h4?Problem 5. (K&C 7.2.10) Use the Lagrange interpolation polynomial toderive a quadrature formula of the formZ10f(x)dx ≈ Af(1/3) + Bf(2/3).Transform this formula to one for integration over [a, b].Problem 6. (K&C 7.5.2) The trapezoid rule can be written in the formI ≡Zvuf(x)dx = T (u, v) −12(v − u)3f00(ξ).(a) Let w = (u + v)/2 and assume f is twice continuously differentiable.Find the constant C in the error term belowI = T (u, w) + T (w, v) + C(v − u)3f00(˜ξ).(b) Assuming f00(˜ξ) ≈ f00(ξ) find an approximation to the error term inthe equation from (a) in terms of T (u, w), T (w, v) and T(u, v).12 MATH 5610/6860 PRACTICE FINAL EXAMProblem 7. Find the LU factorization (with L being a unit lower triangularmatrix) of the matrixA =1 1 21 2 11 1 1Problem 8. The first five Legendre polynomials (monic and orthogonal on[-1,1]) are:p0(x) = 1p1(x) = xp2(x) = x2−23p3(x) = x3−35xp4(x) = x4−67x2+335Find p4using p0, · · · , p3.Problem 9. Find the polynomial of the form p(x) = a + bx that best ap-proximates f(x) = x3in [0, 1], where the norm is induced by the product(f, g) =Z10f(x)g(x)dx.Problem 10. Let xj= 2πj/N and Ej(x) = exp[2iπjx/N], j = 0, . . . , N − 1.Recall the pseudo-inner product(f, g)N=1NN−1Xj=0f(xj)g(xj),and its fundamental property(En, Em) =(1 if n − m is divisible by N0 otherwise.Consider the trigonometric polynomialp =N−1Xk=0ckEk.Show the discrete Parseval’s identityN−1Xj=0|p(xj)|2= NN−1Xj=0|ck|2.Hint: For j = 0, . . . , N − 1 we have,|p(xj)|2= N−1Xk=0ckEk(xj)!