Question

this is numerical analysis please do a and b

3. Consider the trapezoidal rule (T) and Simpson's rule (S) for approximating the integral of a relatively smooth function f on an interval (a, b), for which the following error local estimates are known to hold: (6 - a)"}" (n), for some 7 € (a, b), 12 [ f(z)de –T(S) = [ f(a)der – 5(8) = f(), for some 5 € (a, b), where 8 = (b -a)/2. (a) Given a uniform partition a = to <$<< ein = b of an interval [a, b] with an even number n of intervals, derive formulas for the composite trapezoidal and Simpson's rules both based on the values of the function f at the same nodes, i.e., f(xo), f(x),..., f(x). (b) Use the local estimates provided to derive error estimates for the composite rules, in terms of h = Xi+1-3and certain quantities related to the relevant derivatives off.

Answer 1

Suppose I = (fr) dx Diriding the range of integration la b] uto in each of width hoje, equal Jouts To 2, Loth, n, xo + 2h, b=an nothh In- 2. T 3. jul de j Xu Yr di. 1 2 Mesin Newton's forward Formula, da 1. por u by what youlutlu yo. 2 Where u=x-a or u=2-10 th h dx = hodu. - 2 du dz ܐܟ

u=0 to n и -14) Ayo t I lüt ander 3] 14 New Limits (more they pour les 1 - defy + uayos (u²4 sty (18+24) By - 19 hing a lot 1 (1 - 1) . LO This is called Newton - Cote's Quadrature Formula which can be used to derive Trapezoidal Rule Simpson's Rules, Boole's Rule and Weddle's lule. (1) Trapezoidal Rule In - 110 " and approximating He straight fines and lan, y> joining (dur, yn=) sy); in. Puthing n=1 in fequence of n. ein A (2, 49) land and co a=% 6:24

To Mo, id; 1. Jedr = h Tg. + 1 Ayo . ka2 you 1 49-yes} 2 h (ory.) Suidely for IL , x , I . . yadr = h ly go) - In: 2 ann Ini Adding, 1 3 /ged - delo 1 od 1 ly yu) + 2 lyseynt - You)] This is Trapezoidal Rule Lamben's One - Thind Tule 1-21 eq" ID and approximating ye by a quence (a) quadrati cinsesjonsigo

and le (e, yo), C. y.) and leny.); (s, ya), y (hines, Yuan (on, you) auld (z. haryan for Tre, and I = Lynda the eyne 3-1)4 الطلاق - 3 h 2y + 2y = y) } (y z yo) | ch 1 3 yoxty y for bodele jeune Similarly for 3 ♡ Gebas, ced, hogy Liyede ch 3

Adding, ada I. Întret to 1 h Lly a yn) + 4 ly, ayu uys.. tyna) + 2(yue yo yenyn) This is simpson's One - Third Rule and is used when a is and even number