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# chapter6 - i book 2005/4/7 22:31 page S-38#59 i i i Answers...

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± ± “book” — 2005/4/7 — 22:31 — page S-38 — #59 ± ± ± ± Answers to Selected Exercises for Chapter 6 Section 6.1 (page 437) 1. To two decimal places µ =0 . 29, with the exception of small regions around θ = π 4 , θ =4 π and θ =5 π , where µ . 28 to two decimal places. 3. Using a not-a-knot cubic spline, the maximum sound speed of water is a = 1554 . 07 m/s at a temperature of 78 . 536 C. 5. Using a not-a-knot cubic spline, minimum thermal resistance is achieved with an insulation thickness of 5.879 mm. 7. Using either a not-a-knot cubic spline or the 4th degree interpolating polyno- mial (a) ∂θ ∂u =11 . 33 m (b) =44 . 79 Section 6.2 (page 445) 5. (b) hf √√√ ( ξ ), where x 0 <ξ<x 0 +2 h . (c) h f ( x 0 ) 2 f ( x 0 + h )+ f ( x 0 +2 h ) h 2 error 1 2.952492 1.952492 0.1 1.106092 0.106092 0.01 1.010060 0.010060 0.001 1.001000 0.001000 7. (a) f ( x 0 ) f ( x 0 2 h ) 6 f ( x 0 h )+3 f ( x 0 )+2 f ( x 0 + h ) 6 h (b) The error term is h 3 12 f (4) ( ξ ), where x 0 2 h<ξ<x 0 + h ; consequently, provided f has four continuous derivatives near x 0 , the Fnite di±erence for- mula has rate of convergence O ( h 3 ). 9. (a) f √√ ( x 0 )= 2 h 2 µ f ( x 0 αh ) α ( α +1) f ( x 0 ) α 2 + f ( x 0 + h ) α +1 (b) ²or α ± = 1, the error term is h 3 ( α 1) f √√√ ( ξ ); whereas, for α = 1, the error term is h 2 12 f (4) ( ξ ). S-38

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± ± “book” — 2005/4/7 — 22:31 — page S-39 — #60 ± ± ± ± Selected answers for Section 6.3 S-39 11. f ( x ) Formula (a) Formula (b) Formula (c) 10 0 0 x 111 x 2 2 x 0 2 x 0 2 x 0 x 3 3 x 2 0 2 h 2 3 x 2 0 2 h 2 3 x 2 0 + h 2 13. (a) With h =1 ,f (1) 8; with h =0 . 1 (1) 4 . 31; with h . 01 (1) 4 . 0301; with h . 001 (1) 4 . 003001. This data suggests frst-order convergence. (b) With h (0) 2; with h . 1 (0) 1 . 01; with h . 01 (0) 1 . 0001; with h . 001 (0) 1 . 000001. This data suggests second-order convergence. (c) The rate o± convergence in part (a) is what one would expect ±rom the given ±ormula; the rate o± convergence is higher than expected in part (b) because f √√ (0) = 0. 15. (a) 20 ² 3 h + h 3 M 4 (b) h = 4 ± 80 ²/ 9 M . 020499 using M =3 Section 6.3 (page 453) 1. h 0 . 00034 3. Second column, 0.7117737509 and 0.7076386896; third column, 0.7070479668 5. First column, 0.6941218505; second column, 0.6944444443 and 0.6932539683; third column, 0.6931479014 7. First column, 0.3678794407 and 0.8824969025; second column, 0.9861930220; third column, 0.9863672489 11. (b) f √√ (0) 2 . 000000682, error = 6 . 82 × 10 7 13. (a) With h / 4 ( π ) ≈− 0 . 9896158370; with h / 8, f ( π ) 0 . 9973978671. The extrapolated value is f ( π ) 1 . 0051798971. (b) The error in the approximation associated with h / 4 is 0.0103841630, the error in the approximation associated with h / 8 is 0.0026021329, and the error in the extrapolated value is 0.0051798971. The extrapolated value is not a better approximation to f ( π ) because the original approximations are second-order accurate (0 . 0103841630 / 0 . 0026021329 4), which is bet- ter than expected. This better than expected per±ormance ±or the original approximations arises because f √√ ( π )=0 . Section 6.4 (page 465) 1. Trapezoidal Rule Error Error Bound ² 2 1 1 x dx 0.750000 0.056853 0.166667 ² 1 0 e x dx 0.683940 0.051819 0.083333 ² 1 0 1 1+ x 2 dx 0.750000 0.035398 0.166667 ² 1 0 tan 1 xdx 0.392699 0.046125 0.054127
± ± “book” — 2005/4/7 — 22:31 — page S-40 — #61 ± ± ± ± S-40 Answers to Selected Exercises for Chapter 6 3. Midpoint Rule Error Error Bound ± 2 1 1 x dx 0.666667 0.026481 0.083333 ± 1 0 e x dx 0.606531 0.025590 0.041667 ± 1 0 1 1+ x 2 dx 0.800000 0.014602 0.083333 ± 1 0 tan 1 xdx 0.463648 0.024823 0.027063 5. f ( x ) I ( f ) I 1 , open ( f ) 1 b ab

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