Tutorial_10.pdf - ESO 208A Computational Methods in Engineering Tutorial 10 Numerical Differentiation 1 Derive finite difference approximations for f j

Tutorial_10.pdf - ESO 208A Computational Methods in...

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ESO 208A: Computational Methods in Engineering Tutorial 10 Numerical Differentiation 1. Derive finite difference approximations for j f and j f in terms of j f , 1 j f and 2 j f using Taylor’s series (or the method of undetermined coefficients). What are the orders of inaccuracies of these approximations? 2. The location of an object at various times was measured as follows: Time ( t ; s) 0 1 2 3 4 5 6 7 8 9 Distance ( x ; m) 1 1.55 2.32 3.58 5.79 9.68 16.49 28.22 48.2 81.92 Estimate the speed and acceleration of the object at 5 seconds by using - (i) Forward difference, O(h 2 ) (ii) Backward difference, O(h 2 ) (iii) Central difference, O(h 2 ) and (iv) Richardson extra- polation, O(h 6 ) using three central differences of O(h 2 ). Estimate the true percentage error if the object location is given by 0 5 2 0 1 . t x e . t . 3. Consider the function   3 x Sinx x f (a) Obtain finite difference approximations of f with first order backward difference, second order central difference and 4 th order central difference. Evaluate f

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