Math021, week 7
Definition 7.1
When there is is a quantity changing over time and
f
(
t
)
is that
quantity at time
t
.
f
0
(
t
)
is called the rate of change of the given quantity at time
t
.
Example 7.2
In a chemical reaction chamber, the concentration of a certain
reactant
A
after
t
seconds is
[
A
](
t
) =
1
k
(
t
+
c
)
M. Here
k
and
c
are two fixed
positive numbers.
Show that the rate of change of the concentration of
A
is
proportional to the square of the concentration of
A
.
solution:
The rate of change of the concentration of
A
(rate of reaction) is
d
dt
[
A
]
=
d
dt
1
k
(
t
+
c
)
=

1
k
(
t
+
c
)
2
=

k
[
A
]
2
.
Thus, the assertion is proved.
Example 7.3
Water is pumped into a inverted conical vessel with semivertical
angle
π/
6
at a rate of
R m
3
s

1
. Evaluate the rate at which water level in the
vessel is raising at the moment when the water level is at
H
meters.
solution:
Let
V
(
t
) be the volume of water in the vessel after
t
seconds.
h
(
t
) is the water
level in the vessel after
t
seconds. Then,
V
(
t
) =
1
3
h
(
t
)
π
(
h
(
t
) tan
π
6
)
2
=
π
9
h
(
t
)
3
for all
t.
Taking derivatives on both sides of the equation yields
V
0
(
t
) =
π
3
h
(
t
)
2
h
0
(
t
)
.
Suppose that after
t
0
seconds, the water level in the vessel is
H
meters. That
is,
h
(
t
0
) =
H
. It is given that
V
0
(
t
) =
R
for all
t
. In particular,
V
0
(
t
0
) =
R
.
We evaluate the previous equation at
t
0
and it yields
R
=
π
3
H
2
h
0
(
t
0
)
1
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or
h
0
(
t
0
) =
3
R
πH
2
.
The water level is raising at a rate of
3
R
πH
2
ms

1
at the moment when the water
level is
H
meters.
Example 7.4
One is keeping an eye on a car which is moving in a straight
line. At the moment when the car is cloest to the guy, the car is
D
meters away
from the guys and it is moving at a speed of
V
meters per second. Evaluate the
rate at which the guy turns his head at that moment.
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 Fall '10
 LUXNU
 Rate Of Change, lim, relative maximum

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