Dy 1 c dx 2 y x2 c 2 c 20 a y0 1 yn1

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: ≈ 0.000002814 2. exact value = 2 (a) 1.998377048, |EM | ≈ 0.001622952 (b) 2.003260982, |ET | ≈ 0.003260982 (c) 2.000072698, |ES | ≈ 0.000072698 4. exact value = sin(1) ≈ 0.841470985 (a) 0.841821700, |EM | ≈ 0.000350715 (b) 0.840769642, |ET | ≈ 0.000701343 (c) 0.841471453, |ES | ≈ 0.000000468 1 ln 5 ≈ 0.804718956 2 (a) 0.801605339, |EM | ≈ 0.003113617 (b) 0.811019505, |ET | ≈ 0.006300549 (c) 0.805041497, |ES | ≈ 0.000322541 6. exact value = 327 Chapter 9 7. f (x) = √ 1 15 x + 1, f (x) = − (x + 1)−3/2 , f (4) (x) = − (x + 1)−7/2 ; K2 = 1/4, K4 = 15/16 4 16 27 (1/4) = 0.002812500 2400 243 (15/16) ≈ 0.000126563 (c) |ES | ≤ 180 × 104 (a) |EM | ≤ (b) |ET | ≤ 27 (1/4) = 0.005625000 1200 √ 3 105 −9/2 x ; K2 = 3/4, K4 = 105/16 8. f (x) = 1/ x, f (x) = x−5/2 , f (4) (x) = 4 16 27 (3/4) = 0.008437500 2400 243 (105/16) ≈ 0.000885938 (c) |ES | ≤ 180 × 104 (a) |EM | ≤ (b) |ET | ≤ 27 (3/4) = 0.016875000 1200 9. f (x) = sin x, f (x) = − sin x, f (4) (x) = sin x; K2 = K4 = 1 π3 (1) ≈ 0.012919282 2400 π5 (1) ≈ 0.000170011 (c) |ES | ≤ 180 × 104 (a) |EM | ≤ (b) |ET | ≤ π3 (1) ≈ 0.025838564 1200 10. f (x) = cos x, f (x) = − cos x, f (4) (x) = cos x; K2 = K4 = 1 1 (1) ≈ 0.000416667 2400 1 (1) ≈ 0.000000556 (c) |ES | ≤ 180 × 104 (a) |EM | ≤ (b) |ET | ≤ 1 (1) ≈ 0.000833333 1200 11. f (x) = e−x , f (x) = f (4) (x) = e−x ; K2 = K4 = e−1 8 (e−1 ) ≈ 0.001226265 2400 32 (e−1 ) ≈ 0.000006540 (c) |ES | ≤ 180 × 104 (a) |EM | ≤ (b) |ET | ≤ 8 (e−1 ) ≈ 0.002452530 1200 12. f (x) = 1/(2x + 3), f (x) = 8(2x + 3)−3 , f (4) (x) = 384(2x + 3)−5 ; K2 = 8, K4 = 384 8 (8) ≈ 0.026666667 2400 32 (384) ≈ 0.006826667 (c) |ES | ≤ 180 × 104 (a) |EM | ≤ 13. (a) n > (c) n > 14. (a) n > (c) n > (27)(1/4) (24)(5 × 10−4 ) (243)(15/16) (180)(5 × 10−4 ) (27)(3/4) (24)(5 × 10−4 ) (243)(105/16) (180)(5 × 10−4 ) (b) |ET | ≤ 1/2 ≈ 23.7; n = 24 8 (8) ≈ 0.053333333 1200 (27)(1/4) (12)(5 × 10−4 ) 1/2 (b) n > (27)(3/4) (12)(5 × 10−4 ) 1/2 (b) n > ≈ 33.5; n = 34 1/4 ≈ 7.1; n = 8 1/2 ≈ 41.1; n = 42 1/4 ≈ 11.5; n = 12 ≈ 58.1; n = 59 Exercise Set 9.7 15. (a) n > (c) n > 16. (a) n > (c) n > 17. (a) n > (c) n > 18. (a) n > (c) n > 328 1/2 (π 3 )(1) (24)(10−3 ) (π 5 )(1) (180)(10−3 ) (π 3 )(1) (12)(10−3 ) 1/2 (b) n > (1)(1) (12)(10−3 ) 1/2 (b) n > (8)(e−1 ) (12)(10−6 ) 1/2 (b) n > ≈ 35.9; n = 36 (8)(8) (12)(10−6 ) 1/2 (b) n > 1/4 ≈ 6.4; n = 8 1/2 (1)(1) (24)(10−3 ) ≈ 6.5; n = 7 (1)(1) (180)(10−3 ) ≈ 1.5; n = 2 ≈ 350.2; n = 351 (32)(e−1 ) (180)(10−6 ) ≈ 15.99; n = 16 ≈ 1632.99; n = 1633 (32)(384) (180)(10−6 ) ≈ 495.2; n = 496 1/4 1/2 (8)(8) (24)(10−6 ) ≈ 9.1; n = 10 1/4 1/2 (8)(e−1 ) (24)(10−6 ) ≈ 50.8; n = 51 ≈ 2309.4; n = 2310 1/4 ≈ 90.9; n = 92 19. 0.746824948, 0.746824133 20. 1.137631378, 1.137630147 21. 2.129861595, 2.129861293 22. 2.418388347, 2.418399152 23. 0.805376152, 0.804776489 24. 1.536963087, 1.544294774 25. (a) 3.142425985, |EM | ≈ 0.000833331 (b) 3.139925989, |ET | ≈ 0.001666665 (c) 3.141592614, |ES | ≈ 0.000000040 26. (a) 3.152411433, |EM | ≈ 0.010818779 (b) 3.104518326, |ET | ≈ 0.037074328 (c) 3.127008159, |ES | ≈ 0.014584495 27. S14 = 0.693147984, |ES | ≈ 0.000000803 = 8.03 × 10−7 ; the method used in Example 5 results in a value of n which ensures that the magnitude of the error will be less than 10−6 , this is not necessarily the smallest value of n. 28. (a) greater, because the graph of e−x is concave up on the interval (1, 2) 2 (b) less, because the graph of e−x is concave down on the interval (0, 0.5) 2 29. f (x) = x sin x, f (x) = 2 cos x − x sin x, |f (x)| ≤ 2| cos x| + |x| | sin x| ≤ 2 + 2 = 4 so K2 ≤ 4, n> (8)(4) (24)(10−4 ) 1/2 ≈ 115.5; n = 116 (a smaller n might suffice) 30. f (x) = ecos x , f (x) = (sin2 x)ecos x − (cos x)ecos x , |f (x)| ≤ ecos x (sin2 x + | cos x|) ≤ 2e so K2 ≤ 2e, n > 31. f (x) = √ (1)(2e) (24)(10−4 ) x, f (x) = − 1/2 ≈ 47.6; n = 48 (a smaller n might suffice) 1 , lim |f (x)| = +∞ 4x3/2 x→0+ 329 Chapter 9 √ √ √ √ x sin x + cos x 32. f (x) = sin x, f (x) = − , lim+ |f (x)| = +∞ x→0 4x3/2 π 3 1 + cos2 x dx ≈ 3.820187623 33. L = 0 35. 1 + 1/x4 dx ≈ 2.146822803 34. L = 1 t (s) 0 5 10 15 20 v (mi/hr) 0 40 60 73 84 v (ft/s) 0 58.67 88 107.07 123.2 20 v dt ≈ 0 36. 20 [0 + 4(58.67) + 2(88) + 4(107.07) + 123.2] ≈ 1604 ft (3)(4) t 0 1 2 3 4 5 6 7 8 a 0 0.02 0.08 0.20 0.40 0.60 0.70 0.60 0 8 8 [0 + 4(0.02) + 2(0.08) + 4(0.20) + 2(0.40) + 4(0.60) + 2(0.70) + 4(0.60) + 0] (3)(8) ≈ 2.7 cm/s a dt ≈ 0 180 v dt ≈ 37. 0 180 [0.00 + 4(0.03) + 2(0.08) + 4(0.16) + 2(0.27) + 4(0.42) + 0.65] = 37.9 mi (3)(6) 1800 (1/v )dx ≈ 38. 0 4 2 4 2 4 1 1800 1 + + + + + + ≈ 0.71 s (3)(6) 3100 2908 2725 2549 2379 2216 2059 16 16 r2 dy ≈ π πr2 dy = π 39. V = 0 0 16 [(8.5)2 + 4(11.5)2 + 2(13.8)2 + 4(15.4)2 + (16.8)2 ] (3)(4)...
View Full Document

Ask a homework question - tutors are online