Exercises 3.2
16.
By definition,
p
(
t
) = lim
h
→
0
p
(
t
+
h
)
−
p
(
t
)
h
.
Replacing
h
by
−
h
in the above equation, we obtain
p
(
t
) = lim
h
→
0
p
(
t
−
h
)
−
p
(
t
)
−
h
= lim
h
→
0
p
(
t
)
−
p
(
t
−
h
)
h
.
Adding the previous two equations together yields
2
p
(
t
)
=
lim
h
→
0
p
(
t
+
h
)
−
p
(
t
)
h
+
p
(
t
)
−
p
(
t
−
h
)
h
=
lim
h
→
0
p
(
t
+
h
)
−
p
(
t
−
h
)
h
.
Thus
p
(
t
) = lim
h
→
0
p
(
t
+
h
)
−
p
(
t
−
h
)
2
h
.
19.
This problem can be regarded as a compartmental analysis problem for the population of
fish.
If we let
m
(
t
) denote the mass in million tons of a certain species of fish, then the
mathematical model for this process is given by
dm
dt
= increase rate
−
decrease rate
.
The increase rate of fish is given by 2
m
million tons/yr. The decrease rate of fish is given as
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 Spring '10
 Hald,OH
 Math, Differential Equations, Linear Algebra, Algebra, Equations, Elementary algebra, Constant of integration

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