# What is the derivative of tangent

*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.

**(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 .

**(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 .

**(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.

Source: tutorial.math.lamar.eduCategory: Bank

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