(5 pts) Derive a centered difference approximation for f'(x) using the equally spaced nodes xj-2, xj-1,...

 

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(5 pts) Derive a centered difference approximation for f'(x) using the equally spaced nodes xj-2, xj-1, xj, xj+1, +2, with corresponding node spacing 5x, and corresponding data fj-2, fj-1, fj fj+1 fj+2. Determine the order of accuracy of your approximation again using f(x) = cos(x). (5 pts) Derive a centered difference approximation for f" (x;) using the equally spaced nodes xj 2, xj 1, xj, j+1, xj+2, with corresponding node spacing da, and corresponding data fj-2, fj-1, fj, fj+1, fj+2. Determine the order of accuracy of your approximation again using f(x) = cos(x). Note, in order to measure the error in our approximation, we look at points ; E [0,2] where da = 2/N so that 2j xj 2j, j = 0,..., N. We plot the maximum of the absolute value of the error in our approximation over our chosen interval for increasing choices of N. (5 pts) Derive a centered difference approximation for f'(x) using the equally spaced nodes xj-2, xj-1, xj, xj+1, +2, with corresponding node spacing 5x, and corresponding data fj-2, fj-1, fj fj+1 fj+2. Determine the order of accuracy of your approximation again using f(x) = cos(x). (5 pts) Derive a centered difference approximation for f" (x;) using the equally spaced nodes xj 2, xj 1, xj, j+1, xj+2, with corresponding node spacing da, and corresponding data fj-2, fj-1, fj, fj+1, fj+2. Determine the order of accuracy of your approximation again using f(x) = cos(x). Note, in order to measure the error in our approximation, we look at points ; E [0,2] where da = 2/N so that 2j xj 2j, j = 0,..., N. We plot the maximum of the absolute value of the error in our approximation over our chosen interval for increasing choices of N.