ISM_T11_C09_B - 602 Chapter 9 Further Applications of...

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602 Chapter 9 Further Applications of Integration 8. y y y y y 1 w œ±œ ± 32 2 ab (a) y 0 and y 1 is an unstable equilibrium. œœ (b) y 3y 2y y y y 3y 2 y 1 ww œ± ± œ ±± a b a b a b 23 2 3 (c) 9. 1 2P has a stable equilibrium at P . 2 2 1 2P dP d P dP dt dt dt œ œ± œ± ± " # 2 2 10. P 1 2P has an unstable equilibrium at P 0 and a stable equilibrium at P . dP dt œ œ " # 14 P P P12 P dP dt dt 2 2 œ ± ± abab
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Section 9.4 Graphical Solutions of Autonomous Differential Equations 603 11. 2P P 3 has a stable equilibrium at P 0 and an unstable equilibrium at P 3. dP dt œ± œ œ ab 2 2P 3 4P 2P 3 P 3 dP dt dt 2 2 œ±œ ±± a b 12. 3P 1 P P has a stable equilibria at P 0 and P 1 an unstable equilibrium at P . dP dt œ±± œ œ œ ˆ‰ " " # # 6P 6P+1 P P P P P 1 3 3 dt dt 6 6 2 33 2 2 ± œ ± ± ± ± ## # ±² " a b Š‹ ÈÈ
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604 Chapter 9 Further Applications of Integration 13. Before the catastrophe, the population exhibits logistic growth and P t M , the stable equilibrium. After the ab Ä 0 catastrophe, the population declines logistically and P t M , the new stable equilibrium. Ä 1 14. rP M P P m , r, M, m 0 dP dt œ± ± ² The model has 3 equilibrium oints. The rest oint P 0, P M are asymptotically stable while P m is unstable. For œœ œ initial populations greater than m, the model predicts P approaches M for large t. For initial populations less than m, the model predicts extinction. Points of inflection occur at P a and P b where a and Mm M m œ ³± ± ³ " 3 22 ±‘ È b. œ ³³ " 3 È (a) The model is reasonable in the sense that if P m, then P 0 as t ; if m P M, then P M as t ; if ´Ä Ä _ ´ Ä _ P M, then P M as t . ²Ä Ä _ (b) It is different if the population falls below m, for then P 0 as t (extinction). If is probably a more realistic ÄÄ _ model for that reason because we know some populations have become extinct after the population level became too low. (c) For P M we see that rP M P P m is negative. Thus the curve is everywhere decreasing. Moreover, ²œ ± ± dP dt P M is a solution to the differential equation. Since the equation satisfies the existence and uniqueness conditions, µ solution trajectories cannot cross. Thus, P M as t . _ (d) See the initial discussion above. (e) See the initial discussion above. 15. g v , g, k, m 0 and v t 0 dv k dt m 2 ²   Equilibrium: g v 0 v dv k dt m k 2 mg œ± œÊœ É Concavity: 2 v 2 v g v dv k k k dt m dt m m 2 2 2 ± ˆ‰ ˆ (a) (b) (c) v 178.9 122 mph terminal 160 ft 0.005 s œ É
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Section 9.4 Graphical Solutions of Autonomous Differential Equations 605 16. F F F œ± pr ma mg k v È g v, v 0 v dv k dt m 0 œ È ab Thus, 0 implies v , the terminal velocity. If v , the object will fall faster and faster, approaching the dv dt k k mg mg 22 0 œœ ² ˆ‰ terminal velocity; if v , the object will slow down to the terminal velocity. 0 mg k 2 ³ 17. F F F ma 50 5 v kk 50 5 v dv 1 dt m The maximum velocity occurs when 0 or v 10 . dv ft dt sec 18. (a) The model seems reasonable because the rate of spread of a piece of information, an innovation, or a cultural fad is
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This note was uploaded on 10/13/2011 for the course MATHEMATIC 103 taught by Professor Thommas during the Spring '11 term at LCC Intl University.

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ISM_T11_C09_B - 602 Chapter 9 Further Applications of...

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