1 CHAPTER 23 23 1 xi 2 xi 1 xi xi 1 xi 2 f x 0 965925826 0 866025404 0 707106781 0 5 0 258819045 x 0 261799388 0 523598776 0 785398163 1 047197551 1 308996939 true sin 4 0 70710678 The results are summarized as first order 0 79108963 11 877 0 60702442 14 154 0 69905703 1 138 Forward Backward Centered second order 0 72601275 2 674 0 71974088 1 787 0 70699696 0 016 23 2 xi 2 xi 1 xi xi 1 xi 2 truth f x 1 322219295 1 361727836 1 397940009 1 431363764 1 462397998 x 21 23 25 27 29 log10 e 0 017371779 25 The results are summarized as first order 0 016711878 3 799 0 018106086 4 227 0 017408982 0 214 Forward Backward Centered second order 0 017309258 0 360 0 017281994 0 517 0 017371197 0 003 23 3 xi 2 xi 1 xi xi 1 xi 2 x 1 8 1 9 2 2 1 2 2 f x 6 049647464 6 685894442 7 389056099 8 166169913 9 025013499 Both the first and second derivatives have the same value PROPRIETARY MATERIAL The McGraw Hill Companies Inc All rights reserved No part of this Manual may be displayed reproduced or distributed in any form or by any means without the prior written permission of the publisher or used beyond the limited distribution to teachers and educators permitted by McGraw Hill for their individual course preparation If you are a student using this Manual you are using it without permission 2 truth e 2 7 389056099 The results are summarized as First derivative Second derivative first order 7 401377351 0 166750 7 395215699 0 083361 second order 7 389031439 0 000334 7 389047882 0 000111 23 4 The true value is sin 4 0 70710678 D 3 0 25882 0 965926 0 58477 2 1 047198 D 6 0 258819 0 965926 0 67524 2 0 523599 D 4 1 0 67524 0 58477 0 70539 3 3 23 5 The true value is 1 x 1 5 0 2 D 2 1 94591 1 098612 0 211824 2 2 D 1 1 791759 1 386294 0 202733 2 1 D 4 1 0 202733 0 211824 0 199702 3 3 23 6 The true value f 0 8 0 3 18 0 2 12 12 Equation 23 9 can be used to compute the derivative as x0 0 5 x1 1 x2 2 f x0 1 125 f x1 24 f x2 48 2 0 1 2 2 0 0 5 2 2 0 0 5 1 24 48 0 5 1 0 5 2 1 0 5 1 2 2 0 5 2 1 0 9 24 9 6 13 5 Centered difference f 0 1 125 f 0 24 12 18 1 1 23 7 At x xi Eq 23 9 is PROPRIETARY MATERIAL The McGraw Hill Companies Inc All rights reserved No part of this Manual may be displayed reproduced or distributed in any form or by any means without the prior written permission of the publisher or used beyond the limited distribution to teachers and educators permitted by McGraw Hill for their individual course preparation If you are a student using this Manual you are using it without permission 3 f x f xi 1 2 xi xi xi 1 2 xi xi 1 xi 1 f xi xi 1 xi xi 1 xi 1 xi xi 1 xi xi 1 2 xi xi 1 xi xi 1 xi 1 xi 1 xi For equispaced points that are h distance apart this equation becomes f xi 1 f x f xi 1 2 x xi h xi h h h f xi i f xi 1 h 2h h h 2h h f xi 1 f xi 1 f xi 1 f xi 1 0 2h 2h 2h 23 8 a x 0 5 0 25 0 0 25 0 5 xi 2 xi 1 xi xi 1 xi 2 f x f x 17 125 16 0156 15 13 9844 12 875 12 875 8 13 9844 8 16 0156 17 125 4 12 0 25 Analytical f x 3 x 2 4 3 0 4 4 f x 12 875 16 13 9844 30 15 16 16 0156 17 125 12 0 25 2 0 Analytical f x 6 x 6 0 0 Therefore for this case the results are exact b x 0 2 0 3 0 4 0 5 0 6 xi 2 xi 1 xi xi 1 xi 2 f x f x 0 039203 0 08598 0 14737 0 219396 0 297121 0 297121 8 0 219396 8 0 08598 0 039203 0 674504 12 0 1 Analytical f 0 4 2 x cos x x 2 sin x 2 0 4 cos 0 4 0 4 2 sin 0 4 0 674542 t 0 674542 0 674504 100 0 0056 0 674542 f x 0 297121 16 0 219396 30 0 14737 16 0 08598 0 039203 12 0 1 2 1 071654 Analytical f 0 4 2 cos x x 2 cos x 4 x sin x 2 cos 0 4 0 4 2 cos 0 4 4 0 4 sin 0 4 1 071683 t 1 071683 1 071654 100 0 0027 1 071683 PROPRIETARY MATERIAL The McGraw Hill Companies Inc All rights reserved No part of this Manual may be displayed reproduced or distributed in any form or by any means without the prior written permission of the publisher or used beyond the limited distribution to teachers and educators permitted by McGraw Hill for their individual course preparation If you are a student using this Manual you are using it without permission 4 c f x 0 786843 1 100778 1 557408 2 338254 4 131729 x 2 2 5 3 3 5 4 xi 2 xi 1 xi xi 1 xi 2 f x 4 131729 8 2 338254 8 1 100778 0 786843 1 092486 12 0 5 tan x 3 2 1 tan 1 2 1 1 14184 3 3 1 14184 1 092486 t 100 4 3223 1 14184 Analytical f 3 f x 4 131729 16 2 338254 30 1 557408 16 1 100778 0 786843 12 0 5 2 1 127902 2 tan x 3 tan x 3 2 1 2 tan 1 tan 1 2 1 1 18554 9 9 1 18554 1 127902 t 100 4 8617 1 18554 d x f x Analytical f 3 xi 2 0 6 0 62948 xi 1 0 8 0 540569 1 0 479426 xi 1 1 2 0 433954 xi 2 1 4 0 398355 xi f x 0 398355 8 0 433954 8 0 540569 0 62948 0 25908 12 0 2 cos x 2 sin x 2 cos 1 2 sin 1 2 0 26003 4 2 4x3 2 2x2 0 26003 0 25903 t 100 0 365 0 26003 Analytical f 1 f x 0 398355 16 0 433954 30 0 479426 16 0 540569 0 62948 12 0 2 2 Analytical f 1 t …
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