Question

Problem 2 (hand-calculation): Consider the function f(x) tabulated in table 1. Apply improved trapezoid rule to estimate the integral, If) J ) dz, by using the following number of subintervals, n (a) n-3. Use grid points at i0, 4, 8 and 12 (b) n- 6. Use grid points at i0, 2,4, 6, 8, 10 and 12 (c) n = 12, Use all grid points For each part, compute the integral, T(f) and the corresponding absolute error Er(f), and the error ratio. Here, the error ratio in part (b) is computed by dividing the error in part (a) by the error in part (b) and so on. Use the results in parts (a-c) to populate table 2 Based on the grid spacing and the error ratio, what is the asymptotic order of the absolute error of the improved trapezoid rule? Take a photo of your work. Include all pages in a single photo named problem2.jpg. Set the following in your homework script: figure(2); imshow(imread('problem2.jpg)); p2 - "See figure 2' 0.1 0.2 0.3 0.4 0.5 0.6 0.8 0.9 1.0 Table 1: An unknown function f(x) is evaluated at various grid points ri. The exact solution to the integral of the function is given by I(f-as f(z) dz = 0.742584 error ra NIA Table 2: Table to report results in problems 2 through 4

Answer 1

golu tion Consid㎝ the funcHon f(x) tabulated in table xo.oo ape3oid sule ToApmPBove à trap 3 3 1.3

E T,h (f) - (1-3- 0. 2 i2 3 K grid Points ose o,2,416,8 12 oc have h- An 3- .3

0. LI +2 (o.oo& { oooo \ + o.062s+ o. 2401 to. 6 S 6) h).54 36 1-44 1.3) - f (o C ven au gsid Pointi using tSimidooly Solve that 30.U

+ 2.856) and tle crnov 아.tie f (1.3 )-fl(0.1) 1.3)-北0, 1 ) 172 8 iuy t-1 13 3 32 128 じ4882