Question

121, C. Consider the integral - dr. Use the theorems from Section 4.4 to get upper and lower bounds for the signed error (estimate - actual value) of these three methods, as a function of n. Note that this is a pencil-and paper calculation, and h should be eliminated in favor of n.

Answer 1

Answer :- Consider the Integral I re dx , comparing with pan food da, we get LO Xo = 1, xn=e, f00-5 xn= xornh let n = 10 - 2 = 1 + loh =) = 10h =) h= 1110 = 0. 1 The interval [1, e] is divided into the subintertels 1,1.1, 1.2.1.3, 1.4, 1.5, 1.6, 1.7, 1.8, 1.9,20 The following teebie gives the values of f(x) in the given interval values. स 2) - 4x 1.0000 1.1 0.9091 1.2 0-8333 1.3 0 7692 1.4 0.7143 0.6667 1.6 0-6250 1. 7 I 0.5882 1.8 05556 1.9 05263 2.0 | 0.5000 (1) Trapezoidal Rule por ful) da = b[16) + f(n) + 2 (f(x) + 406) + + f(xn-b)] 1.5 10

..}d2 = 0,3 [at 1+0.5+ 2 CO-9001+ 0.8333 + 07692 +0.7143 to 6667 + 0.6250 +0.5882 +0.5556 to 5263) [o I = 3! [1.5+2 (6-1877)] = 0.698770 = 0,6938 ] simpson's lande fox) dx = ☆ fzo) + 80Xn) + (f(x) + f(x3) + + f(x0-)) + a cf(x) + fixu) t -t flan2) | +0-5263 = 0,! [ 1+0.5 + 460-9091 +0.769€ + 0-6667 +0.5882) t2 C0.8333 +0.7143 +0.6250 +0.5556) = 0,1 11,5 + 4C304595) + 2 (2.7282) = 0,1 (1.5+ 18,9380 + 5.4564] = 0.69 31
=0.6931
[3 ] Midpoint sule we have ² 1 dx, xn=2, xo=1, T Let n=4, we get xn= not nh .-) 2 = 1 +4h =) 1=an a h= 114 The subintervals consists of: [1, 514 ] [ $14, 312 ] , [3/2,74] [714, 2] We evaluate the function at these points; f (x+34) = f( 17974) – 4 (915) = 819 = 0,8889 | (24+x2) = f(5747312 = R(148) = 890 = 0 7272 f(x2 +213) = f ( 312 +74) – †(1318) - 8113 = 0.6154 +(x3 +24) – † (Hute) = f (1518) = 8715 70-5333 Mid point sules pen fa doc = h + (207,29) + f(21742) + + f(xml +1) 7 8 at 0.7272 +0-6154 to 53 = 0.6912

Also, ļ fw dz = ge / dx = [ log x] = log a- log 1 = log 2-0 a dog a =10 69311 True relue Error Estimation Treepezoidal sule error = estimate - true value = 0.6438 -0.6931 = 0.0007 - - & Simpson's Y3 rule error = estimate - true relue = 0.6432 -0.6931 = 0 (3) Midpoint sule error = estimate - true relue =0.6912 -0.6931 = -0-0019