2207 the range is 10551 y 14902 38 a 39 a the

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Unformatted text preview: he circle of radius 1 centered at (a, a2 ); therefore, the family of all circles of radius 1 with centers on the parabola y = x2 . (b) All parabolas which open up, have latus rectum equal to 1 and vertex on the line y = x/2. 24. (a) x = f (1 − t), y = g (1 − t) y 25. 2 1 -2 -1 1 -1 -2 2 x Supplementary Exercises 1 26. 40 Let y = ax2 + bx + c. Then 4a + 2b + c = 0, 64a + 8b + c = 18, 64a − 8b + c = 18, from which b = 0 32 6 and 60a = 18, or finally y = x− . 10 5 y 27. 2 1 x -1 1 2 -1 -2 28. 29. (b) R = R0 is the R-intercept, R0 k is the slope, and T = −1/k is the T -intercept (c) 1.1 = R0 (1 + 20/273), or R0 = 1.025 √ d = (x − 1)2 + ( x − 2)2 ; d = 9.1 at x = 1.358094 −1/k = −273, or k = 1/273 (d) (a) T = 126.55◦ C y 2 1 1 30. x 2 y d = (x − 1)2 + 1/x2 ; d = 0.82 at x = 1.380278 2 1.8 1.6 1.4 1.2 1 0.8 0.5 1 1.5 2 2.5 3 31. w = 63.9V , w = 63.9πh2 (5/2 − h/3); h = 0.48 ft when w = 108 lb 32. (a) (b) w = 63.9πh2 (5/2 − h/3); at h = 5/2, w = 2091.12 lb (b) N = 80 when t = 9.35 yrs (c) W 220 sheep 4000 3000 2000 1000 1 33. 2 3 4 h 5 N (a) 200 150 100 50 t 10 20 30 40 50 41 34. Chapter 1 (a) (b) T = 17◦ F, 27◦ F, 32◦ F (b) T = 3◦ F, −11◦ F, −18◦ F, −22◦ F (c) T v = 35, 19, 12, 7 mi/h 10 v 20 35. (a) WCI 20 v 10 20 30 40 50 -20 36. The domain is the set of all x, the range is −0.1746 ≤ y ≤ 0.1227. 37. The domain is the set −0.7245 ≤ x ≤ 1.2207, the range is −1.0551 ≤ y ≤ 1.4902. 38. (a) 39. (a) The potato is done in the interval 27.65 < t < 32.71. 91.54 min. (b) v (b) As t → ∞, (0.273)t → 0, and thus v → 24.61 ft/s. (d) No; but it comes very close (arbitrarily close). 25 20 15 10 5 t 1 (c) 2 3 4 5 For large t the velocity approaches c. (e) 3.013 s CHAPTER 1 HORIZON MODULE 0.25, 6.25 × 10−2 , 3.91 × 10−3 , 1.53 × 10−5 , 2.32 × 10−10 , 5.42 × 10−20 , 2.94 × 10−39 , 8.64 × 10−78 , 7.46 × 10−155 , 5.56 × 10−309 ; 1, 1, 1, 1, 1, 1, 1, 1, 1, 1; 4, 16, 256, 65536, 4.29 × 109 , 1.84 × 1019 , 3.40 × 1038 , 1.16 × 1077 , 1.34 × 10154 , 1.80 × 10308 1. (a) 2. 2, 2.25, 2.2361111, 2.23606798, 2.23606798, . . . 3. (a) 111 1 1 1 ,,, , , 2 4 8 16 32 64 4. (a) yn+1 = 1.05yn (b) y0 =$1000, y1 =$1050, y2 =$1102.50, y3 =$1157.62, y4 =$1215.51, y5 =$1276.28 (c) yn+1 = 1.05yn for n ≥ 1 (b) (d) yn = 1 2n yn = (1.05)n 1000; y15 =$2078.93 Horizon Module 1 5. (a) 42 x1/2 , x1/4 , x1/8 , x1/16 , x1/32 (b) They tend to the horizontal line y = 1, with a hole at x = 0. 1.8 0 3 0 6. (a) (b) 1 2 2 3 3 5 5 8 8 13 13 21 21 34 34 55 55 89 89 144 1, 2, 3, 5, 8, 13, 21, 34, 55, 89 ; each new numerator is the sum of the previous two numerators. (c) (d) F0 = 1, F1 = 1, Fn = Fn−1 + Fn−2 for n ≥ 2. (e) the positive solution (a) y1 = cr, y2 = cy1 = cr2 , y3 = cr3 , y4 = cr4 (b) yn = crn (c) 7. 144 233 377 610 987 1597 2584 4181 6765 10946 , , , , , , , , , 233 377 610 987 1597 2584 4181 6765 10946 17711 If r = 1 then yn = c for all n; if r < 1 then yn tends to zero; if r > 1, then yn gets ever larger (tends to +∞). 8. The first point on the curve is (c, kc(1 − c)), so y1 = kc(1 − c) and hence y1 is the first iterate. The point on the line to the right of this point has equal coordinates (y1 , y1 ), and so the point above it on the curve has coordinates (y1 , ky1 (1 − y1 )); thus y2 = ky1 (1 − y1 ), and y2 is the second iterate, etc. 9. (a) 0.261, 0.559, 0.715, 0.591, 0.701 (b) It appears to approach a point somewhere near 0.65. CHAPTER 2 Limits and Continuity EXERCISE SET 2.1 1. (a) −1 (d) 1 (b) 3 (e) −1 (c) does not exist (f ) 3 2. (a) 2 (d) 2 (b) 0 (e) 0 (c) (f ) 3. (a) 1 (b) 1 (c) 1 (d) 1 (e) −∞ (f ) +∞ 4. (a) 3 (b) 3 (c) 3 (d) 3 (e) +∞ (f ) +∞ 5. (a) 0 (b) 0 (c) 0 (d) 3 (e) +∞ (f ) +∞ 6. (a) 2 (b) 2 (c) 2 (d) (e) −∞ (f ) +∞ 7. (a) −∞ (d) undef 8. (a) +∞ (b) +∞ (c) +∞ (d) 9. (a) −∞ (b) −∞ (c) −∞ (d) 1 3 (b) +∞ (e) 2 does not exist 2 (c) does not exist (f ) 0 (e) 0 (f ) −1 (e) undef 1 (f ) 2 10. (a) 1 (d) −2 (b) −∞ (e) +∞ (c) does not exist (f ) +∞ 11. (a) 0 (d) 0 (b) 0 (e) does not exist (c) 0 (f ) does not exist 12. (a) 3 (d) 3 (b) 3 (e) does not exist (c) 3 (f ) 0 13. for all x0 = −4 15. (a) At x = 3 the one-sided limits fail to exist. (b) At x = −2 the two-sided limit exists but is not equal to F (−2). (c) At x = 3 the limit fails to exist. (a) At x = 2 the two-sided limit fails to exist. (b) At x = 3 the two-sided limit exists but is not equal to F (3). (c) At x = 0 the two-sided limit fails to exist. 16. 14. 43 for all x0 = −6, 3 Exercise Set 2.1 17. (a) 44 2 1.5 1.1 1.01 1.001 0 0.5 0.9 0.99 0.999 0.1429 0.2105 0.3021 0.3300 0.3330 1.0000 0.5714 0.3690 0.3367 0.3337 1 The limit is 1/3. 0 2 0 (b) 2 1.5 1.1 1.01 1.001 1.0001 0.4286 1.0526 6.344 66.33 666.3 6666.3 50 The limit is +∞. 1 2 0 (c) 0 0.5 0.9 0.99 0.999 0.9999 −1 −1.7143 −7.0111 −67.001 −667.0 −6667.0 0 0 1 The limit is −∞. -50 18. (a) −0.25 −0.1 −0.001 −0.0001 0.0001 0.001 0...
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