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Example 1 Suppose
that the amount of water in a holding tank at t minutes is given by  . Determine each of the following.
(a) Is
the volume of water in the tank increasing or decreasing at  minute?
[Solution]
(b) Is
the volume of water in the tank increasing or decreasing at  minutes?
[Solution]
(c) Is
the volume of water in the tank changing faster at  or  minutes?
[Solution]
(d) Is
the volume of water in the tank ever not changing? If so, when? [Solution]
Solution
In the solution to this example we will use both notations
for the derivative just to get you familiar with the different notations.
We are going to need the rate of change of the volume to
answer these questions. This means
that we will need the derivative of this function since that will give us a
formula for the rate of change at any time t. Now, notice that the
function giving the volume of water in the tank is the same function that we
saw in Example 1 in the last section
except the letters have changed. The
change in letters between the function in this example versus the function in
the example from the last section won’t affect the work and so we can just
use the answer from that example with an appropriate change in letters.
The derivative is.

Recall from our work in the first limits section that we
determined that if the rate of change was positive then the quantity was
increasing and if the rate of change was negative then the quantity was
decreasing.
We can now work the problem.
(a) Is the volume of water
in the tank increasing or decreasing at  minute?
In this case all that we need is the rate of change of the
volume at  or,

So, at  the rate of change is negative and so the
volume must be decreasing at this time.
[Return to Problems]
(b) Is the volume of water
in the tank increasing or decreasing at  minutes?
Again, we will need the rate of change at  .

In this case the rate of change is positive and so the
volume must be increasing at  .
[Return to Problems]
(c) Is the volume of water
in the tank changing faster at  or
 minutes?
To answer this question all that we look at is the size of
the rate of change and we don’t worry about the sign of the rate of
change. All that we need to know here
is that the larger the number the faster the rate of change. So, in this case the volume is changing
faster at  than
at  .
[Return to Problems]
(d) Is the volume of water
in the tank ever not changing? If so,
when?
The volume will not be changing if it has a rate of change
of zero. In order to have a rate of
change of zero this means that the derivative must be zero. So, to answer this question we will then
need to solve

This is easy enough to do.

So at  the volume isn’t changing. Note that all this is saying is that for a
brief instant the volume isn’t changing.
It doesn’t say that at this point the volume will quit changing
permanently.
If we go back to our answers from parts (a) and (b) we can
get an idea about what is going on. At
 the volume is decreasing and at  the volume is increasing. So at some point in time the volume needs
to switch from decreasing to increasing.
That time is  .
This is the time in which the volume goes from decreasing
to increasing and so for the briefest instant in time the volume will quit
changing as it changes from decreasing to increasing.
[Return to Problems]
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