html5-img
1 / 3

Example 2 (a) Estimate by the Midpoint, Trapezoid and Simpson's

Example 2 (a) Estimate by the Midpoint, Trapezoid and Simpson's Rules using the regular partition P of the interval [ 0, /4 ] into 8 subintervals. (b) Find bounds on the errors of those approximations.

hilde
Télécharger la présentation

Example 2 (a) Estimate by the Midpoint, Trapezoid and Simpson's

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Example 2 (a)Estimate by the Midpoint, Trapezoid and Simpson's Rules using the regular partition P of the interval [0, /4] into 8 subintervals. (b) Find bounds on the errors of those approximations. Solution The partition P = {0, /32, /16, 3/32, /8, 5/32, 3/16, 7/32, /4} determines 8 subintervals, each of width /32. The Midpoint Rule for g(x) = tan x uses: L1=g(/64).0491 on [0,/32], L2=g(3/64)  .1483 on [/32, /16], L3=g(5/64)  .2505 on [/16, 3/32], L4=g(7/64)  .3578 on [3/32, /8], L5=g(9/64)  .4730 on [/8, 5/32], L6=g(11/64)  .5994 on [5/32, 3/16], L7=g(13/64)  .7417 on [3/16, 7/32], L8=g(15/64)  .9063 on [7/32, /4]. By the Midpoint Rule: To bound the error, we must first bound g//(x) on [0, /4]: g/(x) = sec2x, g//(x)= 2sec2x tan x and | g//(x)|  (2)(2)(1)=4 on [0, /4]. By Theorem 3.8.9(a) with a=0, b= /4, K=4, n=8:

  2. P = {0, /32, /16, 3/32, /8, 5/32, 3/16, 7/32, /4} The Trapezoid Rule for g(x) = tan x uses: L1=½[g(0)+g(/32)] .0492 on [0,/32], L2=½[g(/32)+g(/16)] .1487 on [/32, /16], L3=½[g(/16)+g(3/32)]  .2511on [/16, 3/32], L4=½[g(3/32)+g(/8)]  .3588 on [3/32, /8], L5=½[g(/8)+g(5/32)]  .4744 on [/8, 5/32], L6=½[g(5/32)+g(3/16)]  .6011 on [5/32, 3/16], L7=½[g(3/16)+g(7/32)]  .7444 on [3/16, 7/32], L8=½[g(7/32)+g(/4)]  .9103 on [7/32, /4]. By the Trapezoid Rule: By Theorem 3.8.9(b) with a=0, b= /4 , K=4, n=8:

  3. P = {0, /32, /16, 3/32, /8, 5/32, 3/16, 7/32, /4} g//(x)= 2sec2x tan x By Simpson’s Rule with x = /32 and g(x) = tan x: To bound the error on this estimate, we must first bound g(4)(x) on [0, /4]. g(3)(x) = (4sec x)(sec x tan x)(tan x) + (2sec2 x )(sec2 x) = 4sec2 x tan2 x + 2sec4 x, g(4)(x) = (8sec x)(sec x tan x)(tan2 x) + (4sec2 x)(2tan x)(sec2 x) + (8sec3 x)(sec x tan x) = 8sec2 x tan3 x + 16sec4 x tan x. Hence |g (4)(x)|  8(2)(1)+16(4)(1) = 80 on [0, /4]. By Theorem 3.8.16 with a=0, b= /4, M=80, n=8:

More Related