The following question:

1 Problem 1 (50 points) Consider the function f(a) = sin 2x te (")" defined in a ( [0..2x]. Partition the domain into N intervals with a spacing h = 2x/N. (i) Write down the final expression for the matrix F, where o = Ff is the discrete Fourier transform for N=8. (ii) Consider the mid-points of your original discretization. ic. (20.5, $1.5, 125,..., IN .5 ) = N 1 10.5h, 1.5h, 2.5h, .... (N -0.5)h}. Use the expression f(x) = 23 Naked to interpolate the values of the function at these mid-points. Tabulate f(x) and f(x) at these mid-points. (iii) At the mid-points of your original discretization, use the expression # f(x) = - > N 2 to determine the second derivative. Tabulate arf(x) and #f(x) at these mid-points. (iv) Plot f(x) for different values of N. Take N = {4, 8, 32, 64, 128} and compare it to the exact function. (v) For all of these cases (N = {4, 8, 32, 64, 128}), plot the 'energy spectrum' of f(r), Le. k vs oz. Use a log-log plot