# Loss of accuracy (Matlab only)

1) Use Taylor expansions to show that f(t+h)-f(t-h)/(2h)=f '(t)+O(h^2)

2) Let f(x)=e^x and t=0. Plot the error (using Matlab) in the approximation of f '(t) by the finite difference in 1) for h=logspace(-1,-16,100). Explain your results

3) Use calculus to show that the optimal h to use in the approximation in 2) - the one that gives the smallest error- is h_opt is approximately CE^1/3

where E is approximately 2.2 x 10^-16 is a machine precision. Does this result agree with the data you plotted in 2)?

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This solution provides step-by-step explanations for how to solve the given calculus problems.