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

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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 &gt; (c) n &gt; 14. (a) n &gt; (c) n &gt; (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 &gt; (27)(3/4) (12)(5 × 10−4 ) 1/2 (b) n &gt; ≈ 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 &gt; (c) n &gt; 16. (a) n &gt; (c) n &gt; 17. (a) n &gt; (c) n &gt; 18. (a) n &gt; (c) n &gt; 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 &gt; (1)(1) (12)(10−3 ) 1/2 (b) n &gt; (8)(e−1 ) (12)(10−6 ) 1/2 (b) n &gt; ≈ 35.9; n = 36 (8)(8) (12)(10−6 ) 1/2 (b) n &gt; 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&gt; (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 &gt; 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)...
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