Thomas Calculus 13th [Solutions]

Section 9.4 Graphical Solutions of Autonomous Equations 683
(b) It is different if the population falls below m, for then P 0 as t (extinction). It is probably a more
realistic 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 dP
dt
rP( M P) ( P m ) is negative. Thus the curve is everywhere decreasing.
Moreover, P M is a solution to the differential equation. Since the equation satisfies the existence and
uniqueness conditions, solution trajectories cannot cross. Thus, P M as t .
(d) See the initial discussion above.
(e) See the initial discussion above.
15.
dv
dt
g
k
v 2 , g, k, m 0 and v( t) 0
m
Equilibrium:
Concavity:
(a)
dv 2
mg
g
k
v 0 v
dt m k
2
d y k dv k k
2
dt m dt m m
2 v 2 v g v
2
(b)
(c) V
160 ft
terminal 0.005 s
178.9 122 mph
16. F Fp Fr
ma mg k v
dv
g
k
v, v (0) v
dt m
0
Thus,
dv
mg
2
mg
2
0 implies v
dt
k , the terminal velocity. If v 0 , the object will fall faster and faster,
k
mg
2
approaching the terminal velocity; if v 0 , the object will slow down to the terminal velocity.
k
17. F Fp Fr
ma 50 5 | v|
dv 1 (50
dt m
5| v|)
The maximum velocity occurs when
dv
dt
0 or v 10 ft .
sec
18. (a) The model seems reasonable because the rate of spread of a piece of information, an innovation, or a
cultural fad is proportional to the product of the number of individuals who have it ( X ) and those who do
not ( N X ). When X is small, there are only a few individuals to spread the item so the rate of spread is
slow. On the other hand, when ( N X ) is small the rate of spread will be slow because there are only a
few individuals who can receive it during the interval of time. The rate of spread will be fastest when
both X and ( N X ) are large because then there are a lot of individuals to spread the item and a lot of
individuals to receive it.
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