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Determine the coefficients to compute the second and third derivative of $f(x)=\sin \left(x^{2}\right)$ at $a=2$ using the points $\{a-2 h, a-h, a, a+h, a+2 h\} .$ Give the estimate of the second and third derivatives as functions of $h$. Determine the best value of $h$ for the greatest accuracy of the answer. How many digits accuracy can you expect with this choice of $h ?$
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