Please help with this numerical differential equations question.

7.1. This exercise concerns deriving finite difference approximations of deriva- tives. (a) Derive a O(k2) approximation of y'(t;) that uses y(t;), y(t;-1), and y(t; +2).(b) Derive a O(k2) approximation of y'(t; ) that uses y(t;), y(t;+1), and y (t; +2). (c) Derive a O(k2) approximation of y'(t;) that uses y(t;), y(t;-1), and y(tj-2). Also, explain how your answer can also be obtained from one of the formulas in Table 7.1.This is listed in Table 7.1 as a centered difference approximation. Type Difference Approximation Truncation Term Forward y' ( t ; ) ~ y( t;+ 1 ) - y(t; ) k Tj = -sky"(nj) Backward [ y' ( t; ) ~ y(t;) - y(t;-1) k Tj = =ky"(nj) Centered [ y' (t ; ) ~ y(ti+1) - y(tj-1) 2k Ti = -5k?y"(n;) One-sided \\ y'(t;) ~ _y(tit2) +4y(titi)-3y(t;) 2k Tj = =kay"(n;) One-sided | y'(t; ) ~ sy(ti)-4y(ti-1) ty(t;-2) 2k Tj = 2kay'"(nj) Centered y"(t;) ~ y(titi)-2y(ti)ty(ti-1) K2 Tj BKly""(ni) Table 7.1 Numerical differentiation formulas. The exact relationships between the approximation and the derivative are illustrated in (7.9), (7.14), (7.16), and (7.20). Also, these formulas assume equally spaced points with step size k = tit1 - ti, and the point nj is located between the left- and rightmost points used in the formula