Maths Question:

3. (a) Consider the following points: (-2, -24). (0, -2), (1, -3), (4, 18), so that To = -2, x1 = 0, 12 = 1, 13 = 4 and yo = -24, y1 = -2, y2 = -3, y3 = 18. Use Lagrangian interpolation to construct the cubic polynomial Pa() passing through these points. Simplify your answer so that it is written in the form Pa(x) = bo+ br+ bar? + baz3 for some bo, b1; by, by E R. [11] (b) (i) Let p3(x) be the cubic interpolating polynomial to f(x) that fits f(r) at r = I;, Titl, Ti42, Ti+3, where Tit1 = rith, Tit2 = r;+2h and rita = 1;+3h (h > 0). By integrating this polynomial from r = r; tor = Tits (where its = c; +4h), derive the following numerical integration formula In for the integral I = [ (rita f(x) dx: In = Ah 3 [2fi+1 - fitz + 2 fita], where we use the notations fit1 = f(Ti+1). fi42 = f(1:42), fita = f(~;43). [6] (ii) Use Taylor series expansions at ar = >; to find the leading-order term in the truncation error for the numerical integration formula derived above. [8]