# Odd Answers - i i i main page 763 i APPENDIX E Answers to...

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APPENDIX E Answers to Odd-Numbered Exercises Chapter 1 Section 1.1 True-False Review 1. False 3. True 5. False 7. False 9. False 11. False Problems 1. y(t) = gt 2 / 2 ; t = q 200 g 4 . 52 sec. 3. (a) 180 g 42 . 0 m/sec. (b) t = q 180 g 4 . 28 sec. 5. = 2 / 2 2 t ; h 470 m. 7. A a ; φ = ( n an integer). 13. After 4 p.m. 15. y = kx 4 . 17. x 2 + 2 y 2 = k . 19. y = ke x . 21. y =− 1 m x + k . 23. y m = . 25. y = 2 . Section 1.2 True-False Review 1. False 3. True 5. False Problems 1. 2, nonlinear. 3. 2, nonlinear. 5. 4, linear. 7. ( −∞ , ) . 9. ( −∞ , 4 ) or ( 4 , ) . 11. ( −∞ , ) . 13. ( −∞ , 0 ) or ( 0 , ) . 15. ( −∞ , ) . 17. ( −∞ , ) . 19. r 3 , 1. 21. r 1. 763

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764 APPENDIX E Answers to Odd-Numbered Exercises 27. y(x) = ln x x ,x > 0. 29. = 1 + sin x x > 0. 31. = 2 x + c for all x> 0. 33. If n 6=− 1 , 2, = x n + 2 (n + 1 )(n + 2 ) + c 1 x + c 2 . If n =− 1, = x ln | x |+ c 1 x + c 2 .I f n 2, = c 1 x + c 2 ln | x | . Solution is valid on ( −∞ , ) for n 0 and on ( −∞ , 0 ) or ( 0 , ) for n< 0. 35. = 3 + x cos x . 37. = xe x 2 e x + 5 x + 5. 39. = e x 1 e x . 49(a) = 1 1 4 x 2 + 1 64 x 4 +··· . Section 1.3 True-False Review 1. True 3. False 5. True 7. True Problems 1. y 0 y x . 3. y 0 = y 2 x 2 2 xy . 5. y 0 = x y ± p x 2 + y 2 . 7. y 0 x 2 + 2 xy y 2 y 2 + 2 xy x 2 . 15. (c) (i) = 1 x 2 + 1 on ( −∞ , ) , (ii) = 1 x 2 on ( 0 , ) , (iii) = 1 x 2 1 on ( 1 , 1 ) . (d) = 0. Section 1.4 True-False Review 1. True 3. True 5. False 7. True 9. True Problems 1. = ce x 2 . 3. = ln (c e x ) . 5. = c(x 2 ) . 7. = cx 3 2 x 1 . 9. = (x 1 ) + c(x 2 ) 2 (x 1 ) c(x 2 ) 2 and 1. 11. = c + c 1 ± x a x b ² 1 /(a b) . 13. = a( 1 + p 1 x 2 ) . 15. = 0. 17. (a) v(t) = a e gt/a e e + e = a tanh (gt/a) , where a = mg/k . (b) No. (c) y(t) = a 2 g ln [ cosh (gt/a) ] . 19. = ln (e x e 3 + e) . 21. (a) No. (b) 1 2 ln ( 1 + v 2 0 ) . 25. T( 40 ) = 225 F; t 96 . 4 minutes. 27. (a) 500 F. (b) t 6 : 07 p.m. Section 1.5 True-False Review 1. True 3. True 5. True 7. False 9. False Problems 1. 2560. 3. t 35 . 86 hours. 5. 1091. 7. (a) P 1 > 2 P 0 P 2 P 0 + P 2 (b) No. 9. (a) Equilibrium solutions: P(t) = 0 ,P(t) = T . Isoclines: P = 1 2 ± T ± q rT 2 + 4 k r ² . Slope: positive for P>T , negative for 0 <T <P . Con- cave up for P>T/ 2, concave down for 0 <P <T/ 2. (c) For 0 <P 0 <T , population dies out; For P 0 >T , popula- tion grows. 11. Equilibrium solutions: = 0 = T,P(t) = C . Slope: positive for T<P<C , negative for 0 <P < T and for P>C . Changes in concavity occur when P = 1 3 ³ C + T ± p C 2 CT + T 2 ´ .Fo r0 0 , population dies out; for T<P 0 <C , population grows asymptotically toward C ; for P 0 >C , population declines asymptotically toward C .
APPENDIX E Answers to Odd-Numbered Exercises 765 13. P(t) = Ce ln (P 0 /k)e rt . 15. (a) More. (b) 31.06 min. (c) 85.02 min. 19. Approx. 52 years. Section 1.6 True-False Review 1. False 3. True 5. True Problems 1. y(x) = e x (e x + c) . 3. = x 2 1 + ce x 2 . 5. = 1 1 + x 2 ( 4tan 1 x + c) . 7. = x 3 ( 3ln x 1 ) + c ln x . 9. x(t) = 4 e t (t 1 ) + c t 2 . 11. = ( tan x + c) cos x . 13. = 1 α + β e βx + ce αx + β 6= 0 , e (x + c), α + β = 0 . 15. = x 2 (x 4 + 1 ) . 17. = ( 4 t)( 1 + t) . 19. = ( 2 e x + 1 , if x 1 , e x (e + 2 ), if x> 1 . 29. = x 1 ( cos x + x sin x + c) .

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Odd Answers - i i i main page 763 i APPENDIX E Answers to...

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