1 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. CHAPTER 4 4.1 (a) For this case xi = 0 and h = x. Thus, 2)0(")0(')0()(2xfxffxf 1)0(")0(')0(0 efff 21)(2xxxf (b) 231()26ii i ixx x xihhfx e e h e e for xi = 0.2, xi+1 = 1 and h = 0.8. True value = e–1 = 0.367879. zero order: 818731.0)1(2.0ef %55.122%100367879.0818731.0367879.0t first order: 163746.0)8.0(818731.0818731.0)1( f %49.55%100367879.0163746.0367879.0t second order: 42574.028.0818731.0)8.0(818731.0818731.0)1(2f%73.15%100367879.042574.0367879.0t third order: 355875.068.0818731.028.0818731.0)8.0(818731.0818731.0)1(32f %26.3%100367879.0355875.0367879.0t 4.2 Use the stopping criterion: s = 0.5102–2% = 0.5% True value: cos(/3) = 0.5 zero order: 13cos %100%1000.510.5t first order: 451689.023/13cos2 %66.9t %4.121%1000.45168910.451689a second order:2 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. 501796.0243/451689.03cos4 %359.0t %986.9%100501796.0451689.0501796.0a third order: 499965.07203/501796.03cos6 %00709.0t %366.0%100499965.0501796.0499965.0a Since the approximate error is below 0.5%, the computation can be terminated. 4.3 Use the stopping criterion: s = 0.5102–2% = 0.5% True value: sin(/3) = 0.866025… zero order: 047198.133sin %92.20%1000.866025047198.10.866025t first order: 855801.063/047198.13sin3 %18.1t %36.22%100855801.0047198.1855801.0a second order: 866295.01203/855801.03sin5 %031.0t %211.1%100866295.0855801.0866295.0a third order: 866021.050403/866295.03sin7 %000477.0t %0316.0%100866021.0866295.0866021.0a Since the approximate error is below 0.5%, the computation can be terminated. 4.4 (a) 01232(0) 1 2(1) 1 2(2) 1 2(3) 1357(1) (1) (1) (1)arctan 2(0) 1 2(1) 1 2(2) 1 2(3) 1 357xx x x xxxxx (b) Use the stopping criterion: s = 0.5102–2% = 0.5%3 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. True value: arctan(/6) = 0.482347907 zero order: arctan 0.52359966 0.482348 0.523599100% 8.55%0.482348t first order: 47575.004785.0523599.03)6/(66arctan3 1. 3 7 %t 0.47575 0.523599100% 10.06%0.47575a second order: 35/6 /6arctan 0.47575 0.007871 0.4836266 3 5 0.26%t 0.48362 0.47575100% 1.63%0.48362a third order: 357/6 /6 /6arctan 0.48362 0.00154 0.48207966357 0.06%t 0.482079 0.48362100% 0.32%0.482079a fourth order: 3579/6 /6 /6 /6arctan 0.482079 0.000329 0.48240866 3 5 7 9 0.012%t 0.482408 0.482079100% 0.07%0.482408a Since the approximate error is below 0.5%, the computation can be terminated. 4.5 True value: f(3) = 554. zero order: 62)1()3( ff %191.111%10055462)(554t first order: 78)2(7062)13)(1('62)3( ff %921.85t second order: 3544213878)13(2)1("78)3(2ff %101.36t third order: 55486150354)13(6)1(354)3(3)3(ff %0t Thus, the third-order result is perfect because the original function is a third-order polynomial. 4.6 True value: f(2.5) = ln(2.5) = 0.916291... zero order:4 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. 0)1()5.2( ff %100%1000.91629100.916291t first order: 5.1)5.1(10)15.2)(1(')1()5.2( fff %704.63%1000.9162915.10.916291t second order: 375.05.1215.1)15.2(2)1("5.1)5.2(22ff %074.59%1000.916291375.00.916291t third order: 5.15.162375.0)15.2(6)1(375.0)5.2(33)3(ff %704.63%1000.9162915.10.916291t fourth order: 234375.05.12465.1)15.2(24)1(5.1)5.2(44)4(ff %421.74%1000.916291234375.00.916291t Thus, the process seems to be diverging suggesting that a smaller step would be required for convergence. 4.7 True value: 2837)2(12)2(75)2('71275)('22fxxxf function values: xi–1 = 1.8 f(xi–1) = 50.96 xi = 2
View Full Document