# chapter5 - i book 2005/4/7 22:31 page S-31#52 i i i Answers...

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“book” — 2005/4/7 — 22:31 — page S-31 — #52 Answers to Selected Exercises for Chapter 5 Section 5.1 (page 349) 1. (a) L 2 , 0 ( x ) = ( x 1)( x 2) ( 1 1)( 1 2) , L 2 , 1 ( x ) = ( x +1)( x 2) (1+1)(1 2) L 3 , 1 ( x ) = ( x +1)( x 1) (2+1)(2 1) 3. (a) L 6 , 0 ( x ) = ( x 1 . 6)( x 3 . 8)( x 4 . 5)( x 6 . 3)( x 9 . 2)( x 10 . 0) ( 1 . 6)( 3 . 8)( 4 . 5)( 6 . 3)( 9 . 2)( 10 . 0) L 6 , 2 ( x ) = x ( x 1 . 6)( x 4 . 5)( x 6 . 3)( x 9 . 2)( x 10 . 0) 3 . 8(3 . 8 1 . 6)(3 . 8 4 . 5)(3 . 8 6 . 3)(3 . 8 9 . 2)(3 . 8 10 . 0) L 6 , 5 ( x ) = x ( x 1 . 6)( x 3 . 8)( x 4 . 5)( x 6 . 3)( x 10 . 0) 9 . 2(9 . 2 1 . 6)(9 . 2 3 . 8)(9 . 2 4 . 5)(9 . 2 6 . 3)(9 . 2 10 . 0) 5. (a) P ( x ) = ( x π 4 )( x π 2 ) π 2 8 · 0 + x ( x π 2 ) π 2 16 · 2 2 + x ( x π 4 ) π 2 8 · 1 (c) sin( π/ 3) P ( π/ 3) = 0 . 850762; sin( π/ 6) P ( π/ 6) = 0 . 517428 (d) theoretical error bound: 3( π/ 12) 3 0 . 031079; actual error: 0.015264 7. Since there are four data points, the theorem guarantees a unique interpolat- ing polynomial of degree at most three; however, g is polynomial of degree four. 11. (a) 2 3 / 9 (b) 1/4 (c) answer will vary; may be greater than or less than value from part (a), but will be greater than value from part (b) 13. The polynomial appears to match the underlying character of the data for 1 . 5 x 5 . 45. 15. φ 5 10 15 20 25 30 35 40 c p 0.9963 0.9864 0.9713 0.9523 0.9306 0.9072 0.8829 0.8582 φ 45 50 55 60 65 70 75 80 c p 0.8339 0.8101 0.7871 0.7652 0.7443 0.7244 0.7055 0.6872 φ 85 90 95 100 c p 0.6695 0.6521 0.6347 0.6170 17. Using the interpolating polynomial, the viscosity of sulfuric acid with a 5% concentration is 0 . 0037 and with a 10% concentration is 0.1289. S-31

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“book” — 2005/4/7 — 22:31 — page S-32 — #53 S-32 Answers to Selected Exercises for Chapter 5 Section 5.2 (page 361) 1. (a) P 0 , 1 , 2 , 3 ( x ) = ( x x 0 ) P 1 , 2 , 3 ( x ) ( x x 3 ) P 0 , 1 , 2 ( x ) x 3 x 0 (b) P 0 , 1 , 2 , 3 ( x ) = ( x x 2 ) P 0 , 1 , 3 ( x ) ( x x 1 ) P 0 , 2 , 3 ( x ) x 1 x 2 (c) P 0 , 1 , 2 , 3 ( x ) = ( x x 1 ) P 0 , 2 , 3 ( x ) ( x x 0 ) P 1 , 2 , 3 ( x ) x 0 x 1 (d) P 0 , 1 , 2 , 3 ( x ) = ( x x 3 ) P 0 , 1 , 2 ( x ) ( x x 2 ) P 0 , 1 , 3 ( x ) x 2 x 3 3. 2 1 4 4 3.25 5 8 2.8 2.995 5. 1 3 0 1 1 1 3 0 0.75 2 1 9 2.25 1 7. x 2 + x + 2 9. P 1 (1 . 3) = 4 , P 2 (1 . 3) = 6 , P 1 , 2 (1 . 3) = 4 . 6 11. 0.382534 13. 1.723371 15. At 20 C, surface tension 486 .
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