Show that if approximating the first derivative of a differentiable function f with the centered difference formula, f ′(xi) ≈ f (xi+1) − f (xi−1) 2h the error is O(h2). State any assumptions that you make.
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Central difference error analysis Taylor expansion can be used to derive higher order derivatives. Please derive the central difference approximation for second derivative for f(x) at point x_i, f''(x_i) = (f(x_{i+1}) + f(x_{i-1}) - 2f(x_i)) / h^2 Please show that the error is O(h^2). Extra credit: Can you derive forward difference, backward difference for the second derivative? What is the error for each of them?
Adi S.
Use the Taylor series expansion to derive the following backward finite-difference formula to calculate the second derivative with the accuracy of O(h): f''(x_i) = (f(x_i) - 2f(x_{i-1}) + f(x_{i-2})) / h^2 and derive the following backward finite-difference formula to calculate the first derivative with the accuracy of O(h^2) f'(x_i) = (3f(x_i) - 4f(x_{i-1}) + f(x_{i-2})) / 2h
Show that the following forward difference approximation to the first derivative is second order: f'(x) = (-3f_i + 4f_{i+1} - f_{i+2}) / (2Δx) + E Also develop an estimate for the error E that is associated with this approximation. Again, there are no codes/figures to provide, only hand-written work. 4. Come up with a centered finite difference approximation to the first derivative that is 6th order. Then, using the function sin(x), demonstrate graphically that this approximation is indeed 6th order by calculating the derivative at x = π/4 starting with a Δx = 0.2 and reducing it.
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