Question

6. [4 points] Consider the following finite-difference approximations of first derivative; on a uniformly spaced grid with spacing Ax. Standard notation introduced in class is used.point \left.\frac{d f}{d x}\right|_{x_{j}} \approx \frac{f_{j+1}-f_{j-1}}{2 \Delta x} \left.\frac{d f}{d x}\right|_{x_{j}} \approx \frac{f_{j+2}-f_{j-2}}{4 \Delta x} . (a) [2 points] Using appropriate Taylor series expansions, find out the leading order truncation terms in each of the expressions above. O [1 point] If the step size is decreased by a factor of 10, how will the error, in predicting-the value of first-order derivative, change and by how much using each formula? c) [1 point] Which formula will provide more accurate result? Show all steps and box final answers.

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