Calculus Selected Answers Textbook Notes

496 Appendix A Selected Answers 1.3.3. {x | x 6= 1 and x 6= −1} 1.3.4. {x | x < 0} 1.3.5. {x | x ∈ R}, i.e., all x A 1.3.6. {x | x ≥ 0} 1.3.7. {x | h − r ≤ x ≤ h + r} 1.3.8. {x | x ≥ 1 or x < 0} Selected Answers 1.3.9. {x | −1/3 < x < 1/3} 1.3.10. {x | x ≥ 0 and x 6= 1} 1.3.11. {x | x ≥ 0 and x 6= 1} 1.1.3. (−2/3)x + (1/3) 1.1.4. y = 2x + 2, 2, −1 1.1.5. y = −x + 6, 6, 6 1.1.6. y = x/2 + 1/2, 1/2, −1 1.1.7. y = 3/2, y-intercept: 3/2, no x-intercept 1.1.8. y = (−2/3)x − 2, −2, −3 1.1.9. yes 1.1.10. y = 0, y = −2x + 2, y = 2x + 2 1.1.11. y = 75t (t in hours); 164 minutes 1.1.12. y = (9/5)x + 32, (−40, −40) 1.1.13. y = 0.15x + 10 1.1.14. 0.03x + 1.2 1.1.15. (a) y = ( 0 0 ≤ x < 100 (x/10) − 10 100 ≤ x ≤ 1000 x − 910 1000 < x 1.1.16. y ( = 0.15x 0 ≤ x ≤ 19450 0.28x − 2528.50 19450 < x ≤ 47050 0.33x − 4881 47050 < x ≤ 97620 1.1.17. (a) P = −0.0001x + 2 (b) x = −10000P + 20000 1.1.18. (2/25)x − (16/5) 2 1.2.1. (a) x + y = 9 (b) (x − 5)2 + (y − 6)2 = 9 (c) (x + 5)2 + (y + 6)2 = 9 (d) x2 + (y − 3)2 = 9 (e) x2 + (y + 3)2 = 9 (f) (x − 3)2 + y 2 = 9 2.3.8. 172 2.3.9. 0 2.3.10. 2 2.3.11. does not exist √ 2.3.12. 2 2.3.14. 512 1.3.15. V = r(50 − πr ), {r | 0 < r ≤ p 50/π} 1.3.16. A = 2πr 2 +2000/r, {r | 0 < r < ∞} 2.1.1. −5, −2.47106145, −2.4067927, −2.400676, −2.4 2.1.2. −4/3, −24/7, 7/24, 3/4 3 + 3∆x + ∆x2 →3 2.1.4. 1 + ∆x 2.1.5. 3.31, 3.003001, 3.0000, 3 + 3∆x + ∆x2 → 3 1.2.2. (a) ∆x = 2, ∆y = 3, m = 3/2, √ y = (3/2)x − 3, 13 (b) ∆x = −1, ∆y = 3, m = −3, √ y = −3x + 2, 10 (c) ∆x = −2, ∆y = −2, m = 1, √ y = x, 8 2.3.7. 3 1.3.13. {x | x ≥ 3}, {x | x ≥ 0} 2.1.3. −0.107526881, −0.11074197, −1 −1 → −0.1110741, 3(3 + ∆x) 9 2 2.3.6. 0 2.3.13. 3a2 2 1.1.1. (2/3)x + (1/3) 2.3.5. 1/6 1.3.12. R 1.3.14. A = x(500 − 2x), {x | 0 ≤ x ≤ 250} 1.1.2. y = −2x 2.3.4. undefined 2.1.6. m 2.3.15. −4 2.3.16. 0 2.3.18. (a) 8, (b) 6, (c) dne, (d) −2, (e) −1, (f) 8, (g) 7, (h) 6, (i) 3, (j) −3/2, (k) 6, (l) 2 p 2.4.1. −x/ 169 − x2 2.4.2. −9.8t 2.4.3. 2x + 1/x2 2.4.4. 2ax + b 2.4.5. 3x2 2.4.8. −2/(2x + 1)3/2 2.4.9. 5/(t + 2)2 2.4.10. y = −13x + 17 2.4.11. −8 2.2.1. 10, 25/2, 20, 15, 25, 35. 2.5.2. Yes; M = 1 2.2.2. 5, 4.1, 4.01, 4.001, 4 + ∆t → 4 2.5.3. Yes; M = 1 2.2.3. −10.29, −9.849, −9.8049, −9.8 − 4.9∆t → −9.8 2.5.4. No 1.2.6. (x + 2/7) + (y − 41/7) = 1300/49 2.3.1. 7 2.5.6. −0.5 or 1.3 or 3.2 1.3.2. {x | x 6= −1} 2.3.3. 0 2 2 1.3.1. {x | x ≥ 3/2} 495 2.3.2. 5 2.5.5. No 2.5.7. 0.2 or 1.3 3.1.1. 100x99 497 Appendix A Selected Answers √ −1 20 625 − x2 √ 3.4.4. − x21 x19 625 − x2 3.4.5. y = 17x/4 − 41/4 3.1.2. −100x−101 3.1.3. −5x −6 3.1.4. πxπ−1 3.1.5. (3/4)x 3.1.6. −(9/7)x 3.2.3. −30x + 25 3.2.4. 6x2 + 2x − 8 3.2.5. 3x2 + 6x − 1 p 3.2.6. 9x2 − x/ 625 − x2 3.2.7. y = 13x/4 + 5 3.2.8. y = 24x − 48 − π 3 3.2.9. −49t/5 + 5, −49/5 3.2.11. n X kak xk−1 k=1 3.2.12. x3 /16 − 3x/4 + 4 3.3.1. 3x2 (x3 − 5x + 10) + x3 (3x2 − 5) 3.3.2. (x2 + 5x − 3)(5x4 − 18x2 + 6x − 7) + (2x + 5)(x5 − 6x3 + 3x2 − 7x + 1) √ √ x x 625 − x2 √ 3.3.3. −√ 2 x 625 − x2 √ −1 20 625 − x2 √ 3.3.4. − x21 x19 625 − x2 3.3.5. f ′ = 4(2x − 3), y = 4x − 7 3.4.1. x3 3x2 x3 (3x2 − 5) − − 5x + 10 (x3 − 5x + 10)2 2x + 5 − 3.4.2. 5 x − 6x3 + 3x2 − 7x + 1 2 4 2 (x + 5x − 3)(5x − 18x + 6x − 7) (x5 − 6x3 + 3x2 − 7x + 1)2 3/2 3.4.3. 2 3.5.23. −8(4x − 1)(2x − x + 3) x 1 + √ √ (625 − x2 )3/2 2 x 625 − x2 3.5.25. 3(8x − 2)/(4x2 − 2x + 1)2 3.5.26. −3x2 + 5x − 1 3.5.27. 6x(2x − 4)3 + 6(3x2 + 1)(2x − 4)2 3.5.28. −2/(x − 1)2 3.5.29. 4x/(x2 + 1)2 3.5.30. (x2 − 6x + 7)/(x − 3)2 3.5.31. −5/(3x − 4)2 3.5.32. 60x4 + 72x3 + 18x2 + 18x − 6 3.5.17. 5(x + 8) 3.5.18. −3(4 − x) 2 4.5.2. − sin x cos(cos x) 4.5.3. tan x + x sec2 x √ 2 x tan x 3.5.35. 56x6 + 72x5 + 110x4 + 100x3 + 60x2 + 28x + 6 4.5.4. sec2 x(1 + sin x) − tan x cos x (1 + sin x)2 3.5.36. y = 23x/96 − 29/96 4.5.6. − csc x cot x 3.5.38. y = 13x/2 − 23/2 3.5.39. y = 2x − 11 √ √ 3 5 20 + 2 5 3.5.40. y = p √ x+ p √ 5 4+ 5 5 4+ 5 4.1.1. 2nπ − π/2, any integer n 1 + 3x2 3.5.15. 3(x + x3 )2/3 4 4.5.1. cos2 x − sin2 x 3.5.33. (5 − 4x)/((2x + 1)2 (x − 3)2 ) 3.5.37. y = 3 − 2x/3 300x (100 − x2 )5/2 4x3 + 4x 3.5.16. 4x(x + 1) + p 2 1 + (x2 + 1)2 q p 2 (x2 + 1)2 + 1 + (x2 + 1)2 √ √ √ 4.4.1. sin( x) cos( x)/ x sin x √ 4.4.2. √ + x cos x 2 x cos x 4.4.3. − 2 sin x (2x + 1) sin x − (x2 + x) cos x 4.4.4. sin2 x − sin x cos x 4.4.5. p 1 − sin2 x 3.5.34. 1/(2(2 + 3x)2 ) 3x2 − 2x + 1/x2 3.5.13. p 2 x3 − x2 − (1/x) 2 4.3.7. 2 −3 3.5.24. 1/(x + 1)2 3.5.3. 6(x2 + 1)2 x p p 169 − x2 − x2 / 169 − x2 3.5.4. p 3.5.5. (2x − 4) 25 − x2 − √ (x2 − 4x + 5)x/ 25 − x2 p 3.5.6. −x/ r 2 − x2 p 3.5.7. 2x3 / 1 + x4 1 √ 3.5.8. √ 4 x(5 − x)3/2 3.5.9. 6 + 18x 2x + 1 x2 + x + 1 3.5.10. + 1−x (1 − x)2 p p 3.5.11. −1/ 25 − x2 − 25 − x2 /x2   .r 169 1 −169 − 1 −x 3.5.12. 2 x2 x 3.5.14. 4.3.6. 7 3.5.22. 5 + 5/x2 3.5.1. 4x3 − 9x2 + x + 7 √ 3.5.2. 3x2 − 4x + 2/ x 3.2.2. −20x4 + 6x + 10/x3 √ 4.3.5. − 2/2 3.5.21. 24x2 (1 − 4x3 )−3 3.4.9. 13/18 3.2.1. 15x2 + 24x 4.3.4. 1 3.5.20. −12x(6 − 2x2 )2 3.4.8. y = 19/169 − 5x/338 −16/7 Appendix A Selected Answers 3.5.19. 6x(x2 + 5)2 3.4.6. y = 11x/16 − 15/16 −1/4 498 ! . 4.1.2. nπ ± π/6, any integer n √ √ 4.1.3. ( 2 + 6)/4 √ √ √ 4.1.4. −(1 + 3)/(1 − 3) = 2 + 3 4.1.11. t = π/2 4.3.1. 5 4.3.2. 7/2 4.3.3. 3/4 4.5.5. − csc2 x 4.5.7. 3x2 sin(23x2 ) + 46x4 cos(23x2 ) 4.5.8. 0 4.5.9. −6 cos(cos(6x)) sin(6x) 4.5.10. sec θ tan θ sin θ = (1 + sec θ)2 (cos θ + 1)2 4.5.11. 5t4 cos(6t) − 6t5 sin(6t) 4.5.12. 3t2 (sin(3t) + t cos(3t))/ cos(2t) + 2t3 sin(3t) sin(2t)/ cos2 (2t) 4.5.13. nπ/2, any integer n 4.5.14. π/2 + nπ, any integer n √ √ 4.5.15. y = 3x/2 + 3/4 − 3π/6 √ √ 4.5.16. y = 8 3x + 4 − 8 3π/3 √ √ 4.5.17. y = 3 3x/2 − 3π/4 Appendix A Selected Answers 4.5.18. π/6 + 2nπ, 5π/6 + 2nπ, any integer n 4.6.1. x/y 4.6.2. −(2x + y)/(x + 2y) 4.6.3. (2xy − 3x2 − y 2 )/(2xy − 3y 2 − x2 ) 4.6.4. sin(x) sin(y)/(cos(x) cos(y)) √ √ 4.6.5. − y/ x 4.6.6. (y sec2 (x/y)−y 2 )/(x sec2 (x/y)+y 2 ) 4.6.7. (y − cos(x + y))/(cos(x + y) − x) 4.6.8. −y 2 /x2 4.6.9. 1 4.7.15. ∞ 4.7.16. ∞ 4.6.13. y = x/2 ± 3 √ √ √ √ 4.6.14. ( 3, 2 3), (− 3, −2 3), √ √ √ √ (2 3, 3), (−2 3, − 3) √ √ 4.6.15. y = 7x/ 3 − 8/ 3 1/3 1/3 1/3 1/3 (−y1 x+y1 x1 +x1 y1 )/x1 (x1 − 2x31 − 2x1 y12 ) (x − 4.6.17. (y − y1 ) = (y1 + 2y13 + 2y1 x21 ) x1 ) 4.7.1. 1 4.7.2. 1/6 4.7.3. −∞ 500 Appendix A Selected Answers 5.1.7. none 5.1.8. min at x = 7π/12 + kπ, max at x = −π/12 + kπ, for integer k. 5.1.9. none 4.7.17. 2/7 5.1.10. local max at x = 5 4.7.18. 2 5.1.11. local min at x = 49 4.7.19. −∞ 5.1.12. local min at x = 0 4.7.20. 0 4.7.21. ∞ 4.7.22. 1/2 4.7.23. 0 4.6.12. y = 2x ± 6 4.6.16. y = 4.7.13. −1/2 4.7.14. 5 499 4.7.24. 1/2 4.7.25. 5 √ 4.7.26. 2 2 4.7.27. −1/2 4.7.28. 2 4.7.29. 0 4.7.30. ∞ 4.7.31. 0 4.7.32. 3/2 4.7.33. ∞ 4.7.34. 5 4.7.35. −1/2 5.1.13. Local min of 1 at every point of [0, 1], local max of 1 at every point of (0, 1). 5.1.16. one 4.7.9. −3 4.7.10. 1/2 5.1.3. max at x = 2, min at x = 4 5.1.4. min at x = ±1, max at x = 0. 5.3.15. none 5.3.16. min at nπ 5.2.3. max at x = 2, min at x = 4 5.4.1. concave up everywhere 5.4.2. concave up when x < 0, concave down when x > 0 5.2.6. none 5.4.3. concave down when x < 3, concave up when x > 3 √ 5.4.4. concave up when x < −1/ 3 or √ x > 1/ 3, concave down when √ √ −1/ 3 < x < 1/ 3 5.2.5. min at x = 1 5.2.7. none 5.2.8. min at x = 7π/12 + kπ, max at x = −π/12 + kπ, for integer k. 5.2.9. none 5.2.10. max at x = 0, min at x = ±11 5.4.5. concave up when x < 0 or x > 2/3, concave down when 0 < x < 2/3 5.2.13. min at nπ, max at π/2 + nπ 5.4.6. concave up when x < 0, concave down when x > 0 5.2.11. min at x = −3/2, neither at x = 0 5.2.15. min at π/2+2nπ, max at 3π/2+2nπ 5.1.2. min at x = −1, max at x = 1 5.3.14. min at 2−1/3 5.2.4. min at x = ±1, max at x = 0. 5.2.14. min at 2nπ, max at (2n + 1)π 4.7.8. −1/4 5.3.13. max at −1, min at 1 5.2.2. min at x = −1, max at x = 1 4.7.37. ∞ 5.1.1. min at x = 1/2 5.3.12. none 5.3.17. max at nπ, min at π/2 + nπ 4.7.36. does not exist 4.7.38. y = 1 and y = −1 5.3.11. max at −5−1/4 , min at 5−1/4 5.3.18. max at π/2+2nπ, min at 3π/2+2nπ 4.7.5. 1/3 4.7.7. 3/2 5.3.9. max at x = 63/64 5.3.10. max at x = 7 5.2.1. min at x = 1/2 4.7.4. 1/16 4.7.6. 0 5.3.8. min at x = 7π/12 + nπ, max at x = −π/12 + nπ, for integer n. 5.3.1. min at x = 1/2 5.3.2. min at x = −1, max at x = 1 5.3.3. max at x = 2, min at x = 4 5.3.4. min at x = ±1, max at x = 0. 5.3.5. min at x = 1 4.7.11. 0 5.1.5. min at x = 1 5.3.6. none 4.7.12. −1 5.1.6. none 5.3.7. none 5.4.7. concave up when x < −1 or x > 1, concave down when −1 < x < 0 or 0 b: 2πb2 + p 2πa2 b √ arcsin( a2 − b2 /a), a 2 − b2 2 sin θ − 3 sin3 θ , 12.2.6. 3 cos3 θ − 2 cos θ 3 cos4 θ − 3 cos2 θ + 2 2 cos3 θ(3 cos2 θ − 2)3 507 508 Appendix A Selected Answers 12.4.7. x = 4 cos t − cos(4t), y = 4 sin t − sin(4t) 12.4.8. x = 2 cos t + cos(2t), y = 2 sin t − sin(2t) 12.4.9. x = cos t + t sin t, y = sin t − t cos t 12.5.1. There is a horizontal tangent at all multiples of π. 13.2.3. 1 n=1 n P∞ diverges, so verges 13.2.4. −3/2 13.2.5. 11 13.2.6. 20 13.2.7. 3/4 13.2.8. 3/2 12.1.4. r 3 cos θ sin2 θ = 1 √ 12.1.5. r = 5 12.3.8. π/12 12.5.2. 9π/4 Z 2π 1√ 12.5.3. 5 − 4 cos t dt 2 0 12.5.4. Four points:  s  √ √ 5 5 − 5 −3 − 3 ,  , ±5 4 8   s √ √  −3 + 3 5 , ±5 5 + 5  4 8 12.5.5. 11π/3 12.1.6. r 2 = sin θ sec3 θ 13.3.8. converges 12.3.9. 3π/16 12.5.6. 32/3 13.3.9. N = 5 a < b: 2πb2 + 2πa2 b √ ln b2 − a 2 b + a √ b2 − a a2 12.1.2. θ = arctan(3) 12.1.8. r = 2/(sin θ − 5 cos θ) 12.1.9. r = 2 sec θ 12.1.10. 0 = r 2 cos2 θ − r sin θ + 1 12.1.11. 0 = 3r 2 cos2 θ − 2r cos θ − r sin θ 12.1.12. r = sin θ 2 2 2 2 2 12.1.21. (x + y ) = 4x y − (x + y )y 12.1.22. (x2 + y 2 )3/2 = y 2 12.1.23. x2 + y 2 = x2 y 2 2 2 12.1.24. x + x y = y 12.3.4. π/12 + 12.3.5. πa √ 3/16 2 12.3.7. 2 − π/2 12.1.7. r sin θ = sin(r cos θ) 4 12.3.2. 9π/2 √ 12.3.3. 3/3 12.3.6. 41π/2 12.1.3. r = −4 csc θ 2 ! 12.3.1. 1 2 12.2.1. (θ cos θ + sin θ)/(−θ sin θ + cos θ), (θ 2 + 2)/(−θ sin θ + cos θ)3 cos θ + 2 sin θ cos θ 12.2.2. , cos2 θ − sin2 θ − sin θ 3(1 + sin θ) (cos2 θ − sin2 θ − sin θ)3 √ 12.3.10. π/4 − 3 3/8 √ 12.3.11. π/2 + 3 3/8 12.5.8. 16/3 12.3.12. 1/2 12.5.9. (π/2, 1) 12.3.13. 3/2 − π/4 √ 12.3.14. π/3 + 3/2 √ 12.3.15. π/3 − 3/4 3 12.3.16. 4π /3 12.3.17. π 12.5.10. 5π 3 /6 12.5.11. 2π 2 p 12.5.12. (2π 4π 2 + 1 + ln(2π + p 4π 2 + 1))/2 13.1.1. 1 2 12.3.18. 5π/24 − √ 13.1.3. 0 3/4 √ 12.3.19. 7π/12 − 3 √ 12.3.20. 4π − 15/2 − 7 arccos(1/4) 12.3.21. 3π 3 12.4.6. x = t − 12.5.7. 2π 13.1.4. 1 13.1.5. 1 13.1.6. 0 13.2.1. lim n2 /(2n2 + 1) = 1/2 n→∞ sin(t) cos(t) , y =1− 2 2 13.2.2. lim 5/(21/n + 14) = 1/3 n→∞ ∞ X 1 3 din n=1 13.2.9. 3/10 13.3.1. diverges 13.3.2. diverges 13.3.3. converges 13.3.4. converges 13.3.5. converges 13.3.6. converges 13.3.7. diverges 13.3.10. N = 10 13.3.11. N = 1687 13.3.12. any integer greater than e200 13.4.1. converges 13.4.2. converges 13.4.3. diverges 13.4.4. converges 13.4.5. 0.90 13.4.6. 0.95 13.5.1. converges 13.5.2. converges 13.5.3. converges 13.5.4. diverges 13.5.5. diverges Appendix A Selected Answers 13.5.6. diverges 13.10.1. 13.5.7. converges 13.5.8. diverges n=0 13.10.2. 13.5.9. converges 13.10.3. 13.10.4. 13.10.6. ∞ X n=0 13.7.7. converges 1+ 13.7.8. diverges 13.8.2. R = ∞, I = (−∞, ∞) 13.10.9. 13.8.3. R = e, I = (−e, e) 13.8.4. R = e, I = (2 − e, 2 + e) 13.8.5. R = 0, converges only when x = 2 13.8.6. R = 1, I = [−6, −4] 13.9.1. the alternating harmonic series ∞ X 13.9.2. (n + 1)xn n=0 13.9.3. ∞ X (n + 1)(n + 2)xn ∞ X (2n − 1)! xn , R = 1 2n−1 (n − 1)! n! 2 n=1 (−1)n x4n+1 /(2n)! n=0 13.10.10. ∞ X 13.12.13. converges 13.12.14. converges 13.12.16. 13.12.17. 13.12.18. 13.12.20. 13.12.21. 13.12.22. 13.12.23. 13.12.24. 13.12.25. (−1)n xn+1 /n! n=0 x4 x6 x12 x2 + − +···+ 2 24 720 12! 13.11.2. 1000; 8 13.12.26. 13.11.1. 1 − x3 2x5 13.11.3. x + + , error ±1.32. 3 15 13.12.1. diverges 13.12.3. converges −1 xn+2 13.9.5. C + (n + 1)(n + 2) n=0 13.12.5. diverges 13.12.4. diverges 13.12.6. diverges 13.12.31. π = ∞ X n=0 (−1)n 4 2n + 1 √ √ 14.1.6. 3, 26, 29 √ √ √ 14.1.7. 14, 2 14, 3 14. 14.1.8. (x − 1)2 + (y − 1)2 + (z − 1)2 = 4. 14.1.9. (x − 2)2 + (y + 1)2 + (z − 3)2 = 25. 14.1.10. (x − 3)2 + (y + 2)2 + (z − 1)2 = 33. 14.1.11. (x − 2)2 + (y − 1)2 + (z + 1)2 = 16, (y − 1)2 + (z + 1)2 = 12 √ √ converges 14.2.6. 10, h0, −2i, h2, 8i, 2, 2 17, diverges h−2, −6i √ √ (−∞, ∞) 14.2.7. 14, h0, 4, 0i, h2, 0, 6i, 4, 2 10, (−3, 3) h−2, −4, −6i √ √ (−3, 3) 14.2.8. 2, h0, −2, 3i, h2, 2, −1i, 13, 3, (−1, 1) h−2, 0, −2i √ √ radius is 0—it converges only when 14.2.9. 3, h1, −1, 4i, h1, −1, −2i, 3 2, √ x=0 6, h−2, 2, −2i √ √ √ √ √ (− 3, 3) 14.2.10. 14, h2, 1, 0i, h4, 3, 2i, 5, 29, (−∞, ∞) h−6, −4, −2i ∞ X (ln(2))n n 14.2.11. h−3, −3, −11i, x √ √ √ n! 139, −3/ 139, −11/ 139i h−3/ n=0 √ √ √ ∞ h−12/ 139, −12/ 139, −44/ 139i X (−1)n n+1 x n+1 14.2.12. h0, 0, 0i n=0 √ ∞ 14.2.13. 0; h−r 3/2, r/2i; h0, −12ri; where X 2 2n+1 x r is the radius of the clock 2n + 1 13.12.15. converges 13.12.19. ∞ X (n + 1)(n + 2) n x ,R=1 13.9.4. 2 n=0 ∞ X (x − 2) ,R=2 n2n ∞ X 1 · 3 · 5 · · · (2n − 1) n x = n!2n n=1 13.12.2. converges n=0 (−1)n−1 n 13.10.8. x + x3 /3 ∞ X 13.12.8. converges 13.12.12. converges n (−1)n (n + 1)(x − 1)n , R = 1 13.10.7. 1 + 13.7.6. converges 13.8.1. R = 1, I = (−1, 1) ∞ X 13.12.7. converges 13.12.11. converges (x − 1) ,R=1 n n=1 13.6.8. converges conditionally 13.7.5. converges (−1)n−1 13.10.5. ln(2) + 13.6.6. converges absolutely 13.6.7. diverges ∞ X Appendix A Selected Answers 13.12.10. converges (x − 5)n ,R=5 5n+1 n=1 13.6.4. converges absolutely 13.6.5. converges conditionally (−1)n 510 13.12.9. converges xn /n!, R = ∞ n=0 13.6.2. diverges 13.6.3. converges conditionally ∞ X ∞ X 509 (−1)n x2n /(2n)!, R = ∞ n=0 13.5.10. diverges 13.6.1. converges absolutely ∞ X 13.12.27. n=0 13.12.28. 1 + x/2 + ∞ X 1 · 3 · 5 · · · (2n − 3) n (−1)n+1 x 2n n! n=2 13.12.29. ∞ X (−1)n x2n n=0 ∞ X (−1)n 2n+1 13.12.30. x 2n +1 n=0 14.3.1. 3 14.3.2. 0 14.3.3. 2 14.3.4. −6 14.3.5. 42 √ √ 14.3.6. 6/ 7, ≈ 0.39 √ √ 14.3.7. −11 14 29/406, ≈ 2.15 Appendix A Selected Answers 14.3.8. 0, π/2 14.3.9. 1/2, π/3 √ 14.3.10. −1/ 3, ≈ 2.19 √ 14.3.11. arccos(1/ 3) ≈ 0.96 √ 14.3.12. 5, h1, 2, 0i. √ 14.3.13. 3 14/7, h9/7, 6/7, 3/7i. √ 14.3.14. 10, 10 3 √ √ 14.3.15. 10 2, 10 2 √ 14.3.16. h0, 5i, h5 3, 0i √ √ 14.3.17. h0, 15 2/2i,h15 2/2, 0i √ 14.3.18. 10/ cos(π/6) = 20/ 3. 14.3.19. Any vector of the form ha, −7a/2, −2ai √ √ √ 14.3.20. h1/ 3, −1/ 3, 1/ 3i 14.3.21. No. 14.3.22. Yes. 14.4.1. h1, −2, 1i 14.4.2. h4, −6, −2i 14.4.3. h−7, 13, −9i 14.4.4. h0, −1, 0i 14.4.5. 3 √ 14.4.6. 21 2/2 14.4.7. 1 14.4.8. 16 √ 14.4.9. 443/2 √ 14.4.10. 3 166/2 14.5.1. (x − 6) + (y − 2) + (z − 1) = 0 14.5.2. 4(x + 1) + 5(y − 2) − (z + 3) = 0 14.5.7. −68x − 153y + 85z = −1462/7 14.5.8. 3x − 19y − 10z = −10 14.5.9. h1, 0, 3i + th0, 2, 1i 14.5.10. h1, 0, 3i + th1, 2, −1i 14.5.11. th1, 1, −1i 14.5.12. −2/5, 13/5 14.5.14. neither 14.5.15. parallel 14.5.16. intersect at (3, 6, 5) 14.5.17. same line √ 14.5.21. 7/ 3 √ 14.5.22. 4/ 14 √ √ 14.5.23. 131/ 14 √ 14.5.24. 68/3 √ 14.5.25. 25/ 198 √ 14.5.26. 50/ 237 √ √ 14.5.27. 3 26/ 14 √ 14.5.28. 3 √ 14.5.29. 42/7 √ 14.5.30. 21/6 √ 14.6.1. ( 2, π/4, 1), √ √ ( 3, π/4, arccos(1/ 3)); √ (7 2, 7π/4, 5), √ √ ( 123, 7π/4, arccos(5/ 123); √ (1, 1, 1), ( 2, 1, π/4); (0, 0, −π), (π, 0, π) 2 2 14.6.2. r + z = 4 14.5.3. (x − 1) − (y − 2) = 0 14.6.3. r cos θ = 0 14.5.5. 4(x − 1) − 6y = 0 14.6.5. z = e−r 14.5.4. −2(x − 1) + 3y − 2z = 0 14.5.6. 7x − y + 11z = 11 14.6.4. r 2 + 2z 2 + 2z − 5 = 0 14.6.6. z = r 2 511 512 Appendix A Selected Answers 14.6.7. sin θ = 0 14.6.8. 1 = ρ cos φ 14.6.9. ρ = 2 sin θ sin φ. 14.6.10. ρ sin φ = 3 14.6.11. φ = π/4 15.2.15. ht2 /2, t3 /3, sin ti 15.2.18. (1, 1, 1) when t = 1 and s = 0; √ θ = arccos(3/ 14); no 15.2.19. −6x + (y − π) = 0 √ √ 15.2.20. −x/ 2 + y/ 2 + 6z = 0 14.6.13. z = mr; cot φ = m if m 6= 0, φ = 0 15.2.21. if m = 0 15.2.22. 14.6.14. A sphere with radius 1/2, center at 15.3.1. (0, 1/2, 0) 15.3.2. 14.6.15. 0 < θ < π/2, 0 < φ < π/2, ρ > 0; 15.3.3. 0 < θ < π/2, r > 0, z > 0 15.1.5. h3 cos t, 3 sin t, 2 − 3 sin ti 15.1.6. h0, t cos t, t sin ti √ 15.2.1. h2t, 0, 1i, r′ / 1 + 4t2 15.2.2. h− sin t, 2 cos 2t, 2ti, q r′ / sin2 t + 4 cos2 (2t) + 4t2 15.2.3. h−et sin(et ), et cos(et ), cos ti, √ r′ / e2t + cos2 t √ √ 15.2.4. h 2/2, 2/2, π/4i + √ √ th− 2/2, 2/2, 1i √ 15.2.5. h1/2, 3/2, −1/2i + √ √ th− 3/2, 1/2, 2 3i √ √ 15.2.6. 2/ 5/ 4 + π 2 √ √ √ √ 15.2.7. 7 5 17/85, −9 5 17/85 15.2.9. h0, t cos t, t sin ti, h0, cos t − √ t sin t, sin t + t cos ti, r′ / 1 + t2 , √ 1 + t2 15.2.10. hsin t, 1 − cos t, t2 /2i 15.2.11. h1 + t2 /2, 2 + t3 /3, 3 − t2 /2i 15.2.12. h−4/3 + t3 /3, 1 − cos t + cos 1, 2 + sin t − sin 1i 15.2.13. t = 4 √ 15.2.14. 37, 1 15.3.4. 15.3.5. 15.3.6. 15.3.7. 15.3.8. 15.3.9. (−1, −3, 1) √ √ √ h1/ 2, 1/ 2, 0i + th−1, 1, 6 2i √ 2π 13 √ (−8 + 13 13)/27 √ √ 5/2 + ln( 5 + 2)/4 √ √ (85 85 − 13 13)/27 R5√ 1 + e2t dt 0 √ 2 2/(2 + 4t2 )3/2 √ 2 2/(1 + 8t2 )3/2 √ 2 1 + 9t2 + 9t4 /(1 + 4t2 + 9t4 )3/2 √ 12 17/289 15.4.1. h− sin t, cos t, 1i, h− cos t, − sin t, 0i, 0, 1 15.4.2. h− sin t, cos t, 2ti, h− cos t, − sin t, 2i, √ √ √ 4t/ 4t2 + 1, 4t2 + 5/ 4t2 + 1 15.4.3. h− sin t, cos t, et i, √ h− cos t, − sin t, et i, e2t / e2t + 1, √ √ 2e2t + 1/ e2t + 1 15.4.4. h2t, 2, 6t − 3i, h2, 0, 6i, √ (40t − 18)/ 40t2 − 36t + 13, √ 14/ 40t2 − 36t + 13 15.4.5. het , cos t, et i, het , − sin t, et i, √ (2e2t − cos t sin t)/ 2e2t + cos2 t, √ √ t 2e | cos t + sin t|/ 2e2t + cos2 t 15.4.6. h−3 sin t, 2 cos t, 0i, h3 cos t, 2 sin t, 0i 15.4.7. h−3 sin t, 2 cos t + 0.1, 0i, h3 cos t, 2 sin t + t/10, 0i Appendix A Selected Answers 15.4.8. h−3 sin t, 2 cos t, 1i, h3 cos t, 2 sin t, ti 15.4.9. h−3 sin t, 2 cos t + 1/10, 1i, h3 cos t, 2 sin t + t/10, ti 16.3.7. −1/(x2 y), −1/(xy 2 ) 16.3.8. z = −2(x − 1) − 3(y − 1) − 1 16.1.6. (a) {(x, y) | |x| ≤ 3 and |y| ≥ 2} (b) {(x, y) | 1 ≤ x2 + y 2 ≤ 3} (c) {(x, y) | x2 + 4y 2 ≤ 16} 16.2.1. No limit; use x = 0 and y = 0. 16.5.12. 6(x − 1) + 3(y − 2) + 2(z − 3) = 0 z = (x − 2) + 4(y − 1/2) 16.5.13. h2 + 4t, −3 − 12t, −1 − 8ti r(t) = h2, 1, 4i + th2, 4, −1i 16.5.14. h4 + 8t, 2 + 4t, −2 − 36ti −0.42 (a) f ′ (x)g(y), f (x)g ′ (y); (b) f ′ (xy)y, f ′ (xy)x; (c) f ′ (x/y)/y, −xf ′ (x/y)/y 2 2 2 2 16.4.3. 2xyt cos(st) + 2x s, 2xys cos(st) + 2x2 t 2 2 2 16.2.3. No limit; use x = 0 and x = y. 16.4.5. x/z, 3y/(2z) 16.4.6. −2x/z, −y/z 16.2.5. Limit is 1. 16.4.7. −y 2 /(2z), −xy/z 16.2.6. Limit is zero. 16.4.8. z/x, z/y ′ 16.2.8. Limit is zero. 16.2.9. No limit; use x = 0 and y = 0. 16.2.10. Limit is zero. 16.2.11. Limit is −1. 16.2.12. Limit is zero. 16.3.1. −2xy sin(x2 y), −x2 sin(x2 y) + 3y 2 16.3.2. (y 2 − x2 y)/(x2 + y)2 , x3 /(x2 + y)2 2 16.3.3. 2xex +y 2 2 , 2yex +y 2 16.3.4. y ln(xy) + y, x ln(xy) + x p 16.3.5. −x/ 1 − x2 − y 2 , p −y/ 1 − x2 − y 2 16.3.6. tan y, x sec2 y 16.5.19. in the direction of h8, 1i 16.5.20. ≈ 5/8 knots/mi 2 16.4.4. 2xy t − 4yx s, 2xy s + 4yx t 16.2.7. Limit is −1. 16.5.18. (a) h4, 9i (b) h−81, 2i or h81, −2i 2 16.4.1. 4xt cos(x + y ) + 6yt cos(x + y ) 16.2.2. No limit; use x = 0 and x = y. 16.2.4. Limit is zero. 16.5.15. h4 + 8t, 2 + 20t, 6 − 12ti 16.5.16. h0, 1i, h4/5, −3/5i 16.4.2. 2xy cos t + 2x2 t 2 Appendix A Selected Answers 16.5.11. 2(x − 1) − 6(y − 1) + 6(z − 3) = 0 z = 6(x − 3) + 3(y − 1) + 10 2 514 16.5.10. Any direction perpendicular to ∇T = h1, 1, 1i, for example, h−1, 1, 0i 16.3.9. z = 1 16.1.1. z = y 2 , z = x2 , z = 0, lines of slope 16.3.10. 1 16.3.11. 16.1.2. z = |y|, z = |x|, z = 2|x|, diamonds 16.3.12. 2 2 16.1.3. z = e−y sin(y 2 ), z = e−x sin(x2 ), 16.3.15. −2x2 2 z=e sin(2x ), circles 16.3.17. 16.1.4. z = − sin(y), z = sin(x), z = 0, lines of slope 1 16.1.5. z = y 4 , z = x4 , z = 0, hyperbolas 513 16.4.9. (a) V = (nR − 0.2V )/P (b) P ′ = (nR + 0.6P )/2V (c) T ′ = (3P − 0.4V )/(nR) √ 16.5.1. 9 5/5 √ 16.5.2. 2 cos 3 √ √ 16.5.3. e 2( 3 − 1)/4 √ 16.5.4. 3 + 5 √ √ 16.5.5. − 6(2 + 3)/72 16.5.6. −1/5, 0 16.5.7. 4(x − 2) + 8(y − 1) = 0 16.5.8. 2(x − 3) + 3(y − 2) = 0 16.5.9. h−1, −1 − cos 1, − cos 1i, √ − 2 + 2 cos 1 + 2 cos2 1 16.5.21. ∇g(−1, 3) = h2, 1i 16.6.1. fx = (y 3 − yx2 )/(x2 + y 2 )2 , fy = (x3 − xy 2 )/(x2 + y 2 )2 , fxx = (2x3 y − 6xy 3 )/(x2 + y 2 )3 , fyy = (2xy 3 − 6x3 y)/(x2 + y 2 )3 , fyx = (6x2 y 2 − x4 − y 4 )/(x2 + y 2 )3 16.6.2. fx = 3x2 y 2 , fy = 2x3 y + 5y 4 , fxx = 6xy 2 , fyy = 2x3 + 20y 3 , fxy = 6x2 y 2 2 16.6.3. fx = 12x + y , fy = 2xy, fxx = 24x, fyy = 2x, fxy = 2y 16.6.4. fx = sin y, fy = x cos y, fxx = 0, fyy = −x sin y, fxy = cos y 16.6.5. fx = 3 cos(3x) cos(2y), fy = −2 sin(3x) sin(2y), fxy = −6 cos(3x) sin(2y), fyy = −4 sin(3x) cos(2y), fxx = −9 sin(3x) cos(2y) 2 2 16.6.6. fx = ex+y , fy = 2yex+y , 2 fxx = ex+y , 2 2 fyy = 4y 2 ex+y + 2ex+y , fxy = 2yex+y 2 3x2 2y 3 , fy = 3 , 4 +y ) x + y4 9x4 3x − , fxx = 3 x + y4 2(x3 + y 4 )2 2 6 6y 8y fyy = 3 − , x + y 4 (x3 + y 4 )2 2 3 −6x y fxy = 3 (x + y 4 )2 −y −x , zy = , 16.6.8. zx = 16z 4z 16z 2 + x2 zxx = − , 162 z 3 2 2 4z + y , zyy = − 16z 3 −xy zxy = 64z 3 y+z x+z 16.6.9. zx = − , zy = − , x+y x+y x+z y+z , zyy = 2 , zxx = 2 (x + y)2 (x + y)2 2z zxy = (x + y)2 16.6.7. fx = 2(x3 16.7.1. minimum at (1, −1) 16.7.2. none 16.7.3. none 16.7.4. maximum at (1, −1/6) 16.7.5. none 16.7.6. minimum at (2, −1) 16.7.7. f (2, 2) = −2, f (2, 0) = 4 √ 16.7.8. a cube 1/ 3 2 on a side 16.7.9. 65/3 · 65/3 · 130/3 = 2 · 653 /27 ≈ 20343 cubic inches. 16.7.10. It has a square base, and is one and one half times as tall as wide. If the volume is V the dimensions are p p p 3 2V /3 × 3 2V /3 × 3 9V /4. p 16.7.11. 100/3 √ 16.7.12. |ax0 + by0 + cz0 − d|/ a2 + b2 + c2 Appendix A Selected Answers 16.7.13. The sides and bottom should all be 2/3 meter, and the sides should be bent up at angle π/3. p p b − 1/4 16.7.17. |b| if b ≤ 1/2, otherwise b − 1/4 √ 16.7.19. 256/ 3 p p p 16.8.1. a cube, 3 1/2 × 3 1/2 × 3 1/2 3 16.8.2. 65/3 · 65/3 · 130/3 = 2 · 65 /27 Appendix A Selected Answers Z 17.1.32. (a) 17.1.7. 1/2 17.1.9. (2/9)23/2 − (2/9) 17.1.10. (1 − cos(1))/4 √ 17.1.11. (2 2 − 1)/6 3 0 Z (b) (c) Z Z e2 17.2.7. π/2 17.1.20. 5/3 17.2.11. π/12 17.2.13. −a5 /15 17.1.23. 4π 17.2.14. 12π 16.8.16. 3, 3, 3 √ 16.8.17. a cube of side length 2/ 3 16.8.18. An equilateral triangle with side 4. 17.1.1. 16 17.1.2. 4 17.1.3. 15/8 17.1.26. 128/15 17.1.27. 1800π m3 √ (e2 + 8e + 16) √ 5 5 17.1.28. − e+4 − 15 3 5/2 e 1 + 15 15 √ 17.1.30. 16 − 8 2 √ 17.2.8. π/2 − 1 √ 17.2.9. 3/4 + π/6 17.1.22. 2a3 /3 16.8.15. 5, −9/2 f (x, y) dy dx 0 17.2.5. 5π/3 17.2.10. 8 + π 17.1.25. 31/8 f (x, y) dx dy 17.2.6. π/6 17.1.19. 2 17.1.24. π/32 f (x, y) dy dx 0 y/4 17.2.4. 4/9 16.8.11. standard $65, deluxe $75 16.8.14. maximum e4 , no minimum f (x, y) dx dy 17.2.2. 32π/3 − 4 3π √ 17.2.3. (2 − 2)π/3 17.1.21. 81/2 16.8.13. 35, −35 2 0 Z √9−x2 16.8.9. (1, 2, 2) √ √ 16.8.10. ( 5, 0, 0), (− 5, 0, 0) 16.8.12. x = y = 2/3, φ = π/3 Z 4Z 0 Z 3 0 17.2.1. 4π f (x, y) dy dx 0 ey Z π/2 Z sin x (d) (e) 9−x2 0 0 17.1.8. π/64 16.8.3. It has a square base, and is one and 17.1.12. π − 2 one half times as tall as wide. If 17.1.13. (e9 − 1)/6 the volume is V the dimensions are p p p π 4 3 2V /3 × 3 2V /3 × 3 9V /4. − 17.1.14. 15 4 √ 16.8.4. |ax0 + by0 + cz0 − d|/ a2 + b2 + c2 17.1.15. 1/3 16.8.5. (0, 0, 1), (0, 0, −1) 17.1.16. 448 p √ √ 16.8.6. 3 4V × 3 4V × 3 V /16 17.1.17. 4/5 √ √ √ 16.8.7. Farthest: (− 2, 2, 2 + 2 2); 17.1.18. 8π closest: (2, 0, 0), (0, −2, 0) 16.8.8. x = y = z = 16 516 17.1.5. 5/6 17.1.6. 12 − 65/(2e). 16.7.14. (3, 4/3) 16.7.16. |b| if b ≤ 1/2, otherwise 17.1.4. 1/2 515 17.2.12. (1 − cos(9))π/2 17.2.15. π 17.2.16. 16/3 17.2.17. 21π 17.3.6. x̄ = 6/5, ȳ = 12/5 17.3.7. x̄ = 14/27, ȳ = 28/55 17.3.8. (3/4, 2/5) ! √ 81 3 ,0 17.3.9. 80π 17.3.10. x̄ = π/2, ȳ = π/8 Z 2π Z 1+cos θ 17.3.11. M = (2 + 0 0 sin θ(2 + 0 cos θ)r dr dθ, Z 2π Z My = 0 1+cos θ cos θ(2 + 0 cos θ)r 2 dr dθ. Z π/2 Z cos θ 17.3.12. M = (r + 1)r dr dθ, Mx = −π/2 0 Z π/2 Z cos θ −π/2 0 −π/2 0 1)r 2 dr dθ, Z π/2 Z My = 2 1)r dr dθ. Z π/2 Z 17.3.13. M = Z 3π/2 Z π/2 Mx = Z sin θ(r + cos θ cos θ(r + 1+cos θ r dr dθ + −π/2 cos θ 1+cos θ r dr dθ, 0 π/2 Z 1+cos θ r 2 sin θ dr dθ + −π/2 cos θ 3π/2 Z 1+cos θ 2 17.3.2. x̄ = 4/5, ȳ = 8/15 Z 17.3.3. x̄ = 0, ȳ = 3π/16 My = 17.3.4. x̄ = 0, ȳ = 16/(15π) Z 17.3.5. x̄ = 3/2, ȳ = 9/4 1+cos θ 2 17.2.19. 2π 17.3.1. x̄ = ȳ = 2/3 0 cos θ)r dr dθ, Z 2π Z Mx = r sin θ dr dθ, π/2 0 Z π/2 Z 1+cos θ r 2 cos θ dr dθ+ −π/2 cos θ 3π/2 Z 1+cos θ 2 π/2 r cos θ dr dθ. 0 Appendix A Selected Answers 17.4.1. 17.4.2. 17.4.3. 17.4.4. 17.4.5. √ πa h2 + a2 √ πa2 m2 + 1 √ 3/2 √ π 2 √ π 2/8 17.4.6. π/2 − 1 √ d2 a2 + b2 + c2 17.4.7. 2abc √ 17.4.8. 8 3π/3 17.5.1. 11/24 517 17.5.3. −3e2 /4 + 2e − 3/4 17.5.4. 1/20 18.2.17. 1243/3 17.6.11. πkha3 /6 18.2.18. ln 2 + 11/3 17.6.12. π 2 /4 18.2.19. 3 cos(1) − cos(2) − cos(4) − cos(8) 17.6.13. 4π/5 17.6.14. 15π √ √ 17.6.15. 9kπ(5 2 − 2 5)/20 √ 17.7.1. 4π 3/3 17.5.8. π 3π 17.5.10. 16 17.5.11. 32 18.3.2. x4 /4 − y 5 /5 18.3.3. no f 17.7.3. 2/3 18.3.5. y sin x e2 − 1 2e2 17.7.5. 36 √ √ 17.7.6. 32( 2 + ln(1 + 2))/3 18.3.6. no f 17.7.10. (4/3)πabc √ 18.2.1. 13 11/4 17.5.7. 151/60 18.3.1. no f 18.3.4. no f 17.7.8. π(1 − cos(1))/24 17.5.6. 11/84 18.2.2. 0 18.2.3. 3 sin(4)/2 18.2.4. 0 18.5.5. −2π, 0 18.2.20. −10/3 17.7.2. 0 17.7.7. 3 cos(1) − 3 cos(4) 17.5.5. π/48 Appendix A Selected Answers 17.6.10. πkh2 a2 /12 17.7.4. 17.5.2. 623/60 518 18.5.6. 0, 2π √ 18.6.3. 25 21/4 √ 18.6.4. π 21 √ 18.6.5. π(5 5 − 1)/6 √ 18.6.6. 4π 2 18.6.7. πa2 /2 18.6.8. 2πa(a − √ 18.6.9. π((1 + 4a ) 18.3.7. xyz 18.6.11. πa2 − 2a2 √ 18.6.12. πa2 1 + k 2 /4 √ 18.6.13. A 1 + a2 + b2 √ 18.6.14. A k 2 + 1 18.3.8. -144 18.3.9. 6 18.3.10. 1/e − sin 3 √ √ 18.3.11. 1/ 77 − 1/ 3 18.6.15. 8a2 18.4.1. 1 18.7.1. (0, 0, 3/8) 18.4.2. 0 7 7 18.4.3. 1/(2e) − 1/(2e ) + e/2 − e /2 18.7.2. (11/20, 11/20, 3/10) 18.4.4. 1/2 18.7.3. (0, 0, 2275/682) 18.7.5. 16 18.7.4. on center axis, h/3 above the base 18.2.5. 2e3 17.5.13. x̄ = ȳ = 0, z̄ = 16/15 18.2.6. 128 18.4.5. −1/6 √ √ √ √ 18.4.6. (2 3−10 5+8 6)/3−2 2/5+1/5 17.5.14. x̄ = ȳ = 0, z̄ = 1/3 18.2.7. (9e − 3)/2 18.4.7. 11/2 − ln(2) 18.7.6. 7 18.4.9. −17/12 18.7.8. −137/120 17.6.2. π(1 − √ 2/2) 17.6.3. 5π/4 17.6.4. 0 17.6.5. 5π/4 17.6.6. 4/5 17.6.7. 256π/15 17.6.8. 4π 2 3π 17.6.9. 16 18.2.8. ee+1 − ee − e1/e−1 + e1/e + e4 /4 − e−4 /4 18.2.9. 1 + sin(1) − cos(1) 18.2.10. 3 ln 3 − 2 ln 2 18.4.8. 2 − π/2 18.4.10. 0 18.4.11. −π/2 18.7.7. −π 18.7.9. −2/e 18.7.10. πb2 (−4b4 − 3b2 + 6a2 b2 + 6a2 )/6 18.2.11. 3/20 + 10 ln(2)/7 18.4.12. 12π 18.7.11. 9280 kg/s 18.2.12. 2 ln 5 − 2 ln 2 + 15/32 18.4.13. 2 cos(1) − 2 sin(1) − 1 18.7.12. 24ǫ0 18.2.13. 1 18.2.14. 21 + cos(1) − cos(8) 18.2.15. (ln 29 − ln 2)/2 18.2.16. 2 ln 2 + π/4 − 2 − 1)/6 18.6.10. 2π((1 + a2 )3/2 − 1)/3 17.5.12. 64/3 17.6.1. π/12 a 2 − b2 ) 2 3/2 18.5.1. −1, 0 18.5.2. 0, a + b 18.5.3. (2b − a)/3, 0 18.5.4. 0, 1 18.8.1. −3π 18.8.2. 0 18.8.3. −4π 18.8.4. 3π Appendix A Selected Answers 19.2.1. y = Ae−5t 18.8.5. A(p(c − b) + q(a − c) + a − b) 18.9.2. a2 bc + ab2 c + abc2 19.2.3. y = Ae− arctan t 18.9.3. e2 − 2e + 7/2 19.2.4. y = Ae−t 3 19.2.5. y = 4e−t 18.9.5. 384π/5 19.2.6. y = −2e3t−3 18.9.6. π/3 19.2.8. y = e2 e−e 18.9.8. π/2 19.5.12. e−6t (4 cos t + 24 sin t) 19.2.13. y = e1−t −2 19.2.15. k = ln 5, y = 100e−t ln 5 2t 2t 19.1.8. y = ±1, y = (1 + Ae )/(1 − Ae ) 19.1.9. y 4 /4 − 5y = t2 /2 + C 19.2.17. y = 10 e 19.2.18. y = 10e−t ln(2)/6 19.1.11. y = M + Ae−kt 10 ln(15/2) − 10 ≈ 2.52 minutes ln 5 M 19.1.13. y = 1 + Ae−M kt 19.1.12. 19.1.14. y = 2e3t/2 ln 2 19.1.15. t = − k 41 days 19.3.1. y = Ae−4t + 2 19.3.2. y = Ae2t − 3 2 19.3.3. y = Ae−(1/2)t + 5 5 ln 300 ≈ ln 2 19.5.14. 2 cos(2t − π/6) √ 19.5.15. 5 2 cos(10t − π/4) √ −2t 19.5.16. 2e cos(3t − π/4) 19.3.6. y = Ae−t/2 + t − 2 1 19.3.7. y = At2 − 3t c 2√ t 19.3.8. y = + t 3 19.3.9. y = A cos t + sin t t A +1− sec t + tan t sec t + tan t 19.4.1. y(1) ≈ 1.355 19.3.10. y = 19.7.1. A sin(t) + B cos(t) − cos t ln | sec t + tan t| 19.5.17. 5e4t cos(3t + arcsin(4/5)) 19.5.18. (2 cos(5t) + sin(5t))e−2t 19.5.19. −(1/2)e−2t sin(2t) 19.6.1. Ae5t + Bte5t + (6/169) cos t − (5/338) sin t √ t 19.3.4. y = Ae−e − 2 19.3.5. y = Aet − t2 − 2t − 2 19.1.17. 100e−200 ln 2/191 ≈ 48 mg; 5730 ln 50 ≈ 32339 years ln 2 19.1.18. y = y0 et ln 2 19.1.19. 500e−5 ln 2/4 ≈ 210 g ) 6 t ln(3/2) 3/2 19.1.16. 600e−6 ln 2/5 ≈ 261 mg; 19.2.16. k = −12/13, y = exp(−13t 1/13 19.6.2. Ae− 2t √ 2t + Bte− +5 19.6.3. A cos(4t) + B sin(4t) + (1/2)t2 + (3/16)t − 5/16 √ √ 19.6.4. A cos( 2t)+B sin( 2t)−(cos(5t)+ sin(5t))/23 19.6.5. et (A cos t + B sin t) + e2t /2 √ √ 6t + Be− −3t 2t 19.6.6. Ae 19.6.7. Ae t 6t + 2 − t/3 − e−t /5 −3t + Be − (1/5)te 19.6.8. Ae + Be3t + (1/2)te3t 19.6.9. A cos(4t) + B sin(4t) + (1/8)t sin(4t) 19.6.10. A cos(3t) + B sin(3t) − (1/2)t cos(3t) 19.6.11. Ae−6t + Bte−6t + 3t2 e−6t 19.6.12. Ae4t + Bte4t − t2 e4t 19.6.19. e−6t (2 cos t + 20 sin t) + 2e−4t   592 23 cos(3t) + sin(3t) e−3t + 19.6.20. − 325 975 23 11 cos t − sin t 325 325 19.6.21. e−2t (A sin(5t) + B cos(5t)) + 8 sin(2t) + 25 cos(2t) 19.6.22. e−2t (A sin(2t) + B cos(2t)) + (14/195) sin t − (8/195) cos t 19.5.13. 2e−3t sin(3t) 19.2.14. y = 0 p 19.1.7. y = ± t2 + C 19.6.18. −(4/27) sin(3t) + (4/9)t 19.5.9. (16t − 3)e4t √ √ √ 19.5.10. −2 cos( 5t) + 5 sin( 5t) √ √ 19.5.11. − 2 cos t + 2 sin t 19.2.12. y = −2e(1/t)−1 19.1.6. none 19.6.17. 4et + e−t − 3t − 5 19.5.8. 5e−6t + 20te−6t 19.2.11. y = 4t2 19.1.5. y = nπ, for any integer n. 19.6.16. Ae2t + Be−2t + te2t 19.5.7. −2e−3t + 2e4t t 19.2.10. y = 0 tn+1 19.1.3. y = +1 n+1 19.1.4. y = t ln t − t + C 19.6.14. Ae4t + Be−3t + (1/144) − (t/12) √ √ 19.6.15. A cos( 5t) + B sin( 5t) + 8 sin(2t) 19.5.6. −(1/4)e−5t + (5/4)e−t 19.2.9. y = 0 19.1.2. y = arctan t + C 19.6.13. Ae−t + Be−5t + (4/5) 19.4.4. y(1) ≈ 2.30 ω + 1 ωt ω − 1 −ωt e + e 19.5.4. 2ω 2ω 19.5.5. 2 cos(3t) + 5 sin(3t) 19.2.7. y = e1+cos t 18.9.7. 10π Appendix A Selected Answers 19.4.3. y(1) ≈ 1.05 /3 18.9.4. 3 520 19.4.2. y(1) ≈ 40.31 19.2.2. y = Ae2t 18.9.1. both are −45π/4 19.1.10. y = (2t/3) 519 1 19.7.2. A sin(t) + B cos(t) + e2t 5 19.7.3. A sin(2t) + B cos(2t) + cos t − sin t cos t ln | sec t + tan t| 19.7.4. A sin(2t) + B cos(2t) + 1 sin(2t) sin2 (t) + 2 1 t sin(2t) ln | cos t| − cos(2t) + 2 2 1 sin(2t) cos(2t) 4 t3 2t 19.7.5. Ae2t + Be−3t + e −  2  152t t 2t 2 e − + 5 25 125 5 19.7.6. Aet sin t + Bet cos t − et cos t ln | sec t + tan t| 1 cos t(cos3 t+ 10 3 sin3 t − 2 cos t − sin t) + 1 sin t(sin3 t − 3 cos3 t − 2 sin t + 10 1 1 cos(2t) − sin(2t) cos t) = 10 20 19.7.7. Aet sin t+Bet cos t−