Maths Question:

(a) The following finitedifference formula approximates the second derivative of a func tion 91:3}: 2y[3: + 2h} 591:1: + l1} + 4mm} yfm E1} E12 " where h :5 l} is the step size. (i) Use Taylor series expansions to nd the leadingiorder term in the truncation error for this approximation. [6] (ii) The following table shows the values of a function x) at the given values of m: numm Use the formula given in part {a} to numerically estimate the second derivative of this function at 1.- : ill for h = {1.2 and for h = 0.1. Give your answers rounded to 4 decimal places. [4] (iii) If it is known that the exact value of y"(l}.1} is 2.5535. determine the errors of the numerical results of part (ii). What happens with the error as it changes from {1.2 to 0.1? Does this behaviour agree with the result of part (i)? [4] (iv) Apply extrapolation [i.e._ one step of Richardson extrapolation} to the results of part (ii) to obtain an improved estimate of y"[.1]l. [4] {b} Given the initialvalue problem y' = 92 + 1-2, ym) = 1.. compute numerically {i} y at 3: = [1.], I12 and 0.3 employing the Taylor series method, calculating the terms up to and including I4. [7] [Give your answers rounded to 4 decimal places]