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Before moving on to the section where we learn how to compute derivatives by avoiding the limits we were evaluating in the previous section we need to take a quick look at some of the interpretations of the derivative. All of these interpretations arise from recalling how our definition of the derivative came about. The definition came about by noticing that all the problems that we worked in the first section in the Limits chapter required us to evaluate the same limit.**Rate of Change**

The first interpretation of a derivative is rate of change. This was not the first problem that we looked at in the Limits chapter, but it is the most important interpretation of the derivative. If f(x) represents a quantity at any x then the derivative f′(a) represents the instantaneous rate of change of f(x) at x=a.

**Example 1 **Suppose that the amount of water in a holding tank at t minutes is given by V(t) = 2t^{2 }− 16t + 35. Determine each of the following.

(a) Is the volume of water in the tank increasing or decreasing at t=1 minute?

(b) Is the volume of water in the tank increasing or decreasing at t=5 minutes?

(c) Is the volume of water in the tank changing faster at t=1 or t=5 minutes?

(d) Is the volume of water in the tank ever not changing? If so, when?**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 t=1 minute?**

In this case all that we need is the rate of change of the volume at t=1 or,

So, at t=1 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 t = 5 minutes?**

Again, we will need the rate of change at t=5.

In this case the rate of change is positive and so the volume must be increasing at t=5.

**(c) Is the volume of water in the tank changing faster at t=1 or t=5 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 t=1 than at t=5.

**(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.

4t − 16 = 0 ⇒ t = 4

So at t=4 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 t=1 the volume is decreasing and at t=5 the volume is increasing. So, at some point in time the volume needs to switch from decreasing to increasing. That time is t=4.

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.

Note that one of the more common mistakes that students make in these kinds of problems is to try and determine increasing/decreasing from the function values rather than the derivatives. In this case if we took the function values at t=0, t=1 and t=5 we would get,

V(0)=35 V(1)=21 V(5)=5

Clearly as we go from t=0 to t=1 the volume has decreased. This might lead us to decide that AT t=1 the volume is decreasing. However, we just can’t say that. All we can say is that between t=0 and t=1 the volume has decreased at some point in time. The only way to know what is happening right at t=1 is to compute V′(1) and look at its sign to determine increasing/decreasing. In this case V′(1) is negative and so the volume really is decreasing at t=1.

Now, if we’d plugged into the function rather than the derivative we would have gotten the correct answer for t=1 even though our reasoning would have been wrong. It’s important to not let this give you the idea that this will always be the case. It just happened to work out in the case of t=1.

To see that this won’t always work let’s now look at t=5. If we plug t=1 and t=5 into the volume we can see that again as we go from t=1 to t=5 the volume has decreased. Again, however all this says is that the volume HAS decreased somewhere between t=1 and t=5. It does NOT say that the volume is decreasing at t=5. The only way to know what is going on right at t=5 is to compute V′(5) and in this case V′(5) is positive and so the volume is actually increasing at t=5.

So, be careful. When asked to determine if a function is increasing or decreasing at a point make sure and look at the derivative. It is the only sure way to get the correct answer. We are not looking to determine is the function has increased/decreased by the time we reach a particular point. We are looking to determine if the function is increasing/decreasing at that point in question.**Slope of Tangent Line**

This is the next major interpretation of the derivative. The slope of the tangent line to f(x) at x=a is f′(a). The tangent line then is given by,

y = f(a) + f′(a)(x − a)

**Example 2** Find the tangent line to the following function at z=3.**Solution:**

We first need the derivative of the function and we found that in Example 3 in the last section. The derivative is,

Now all that we need is the function value and derivative (for the slope) at z=3.

The tangent line is then,**Velocity**

Recall that this can be thought of as a special case of the rate of change interpretation. If the position of an object is given by f(t) after t units of time the velocity of the object at t=a is given by f′(a).

**Example 3 **Suppose that the position of an object after t hours is given by,

Answer both of the following about this object.

(a) Is the object moving to the right or the left at t=10 hours

(b) Does the object ever stop moving?

Solution:**Solution:**

Once again, we need the derivative and we found that in Example 2 in the last section. The derivative is,

**(a) Is the object moving to the right or the left at t=10 hours?**

To determine if the object is moving to the right (velocity is positive) or left (velocity is negative) we need the derivative at 10

So, the velocity at t=10 is positive and so the object is moving to the right at t=10.

**(b) Does the object ever stop moving?**

The object will stop moving if the velocity is ever zero. However, note that the only way a rational expression will ever be zero is if the numerator is zero. Since the numerator of the derivative (and hence the speed) is a constant it can’t be zero.

Therefore, the object will never stop moving.

In fact, we can say a little more here. The object will always be moving to the right since the velocity is always positive.

We’ve seen three major interpretations of the derivative here. You will need to remember these, especially the rate of change, as they will show up continually throughout this course.

Before we leave this section let’s work one more example that encompasses some of the ideas discussed here and is just a nice example to work.

**Example 4** Below is the sketch of a function f(x). Sketch the graph of the derivative of this function, f′(x).**Solution:**

At first glance this seems to an all but impossible task. However, if you have some basic knowledge of the interpretations of the derivative you can get a sketch of the derivative. It will not be a perfect sketch for the most part, but you should be able to get most of the basic features of the derivative in the sketch.

Let’s start off with the following sketch of the function with a couple of additions.

Notice that at x=−3, x=−1, x=2 and x=4 the tangent line to the function is horizontal. This means that the slope of the tangent line must be zero. Now, we know that the slope of the tangent line at a particular point is also the value of the derivative of the function at that point. Therefore, we now know that,

f′(−3)=0 f′(−1)=0 f′(2)=0 f′(4)=0

This is a good starting point for us. It gives us a few points on the graph of the derivative. It also breaks the domain of the function up into regions where the function is increasing and decreasing. We know, from our discussions above, that if the function is increasing at a point then the derivative must be positive at that point. Likewise, we know that if the function is decreasing at a point then the derivative must be negative at that point.

We can now give the following information about the derivative.

Remember that we are giving the signs of the derivatives here and these are solely a function of whether the function is increasing or decreasing. The sign of the function itself is completely immaterial here and will not in any way effect the sign of the derivative.

This may still seem like we don’t have enough information to get a sketch, but we can get a little bit more information about the derivative from the graph of the function. In the range x<−3 we know that the derivative must be negative, however we can also see that the derivative needs to be increasing in this range. It is negative here until we reach x=−3 and at this point the derivative must be zero. The only way for the derivative to be negative to the left of x=−3 and zero at x=−3 is for the derivative to increase as we increase x towards x=−3.

Now, in the range −3 < x < − 1 we know that the derivative must be zero at the endpoints and positive in between the two endpoints. Directly to the right of x = −3 the derivative must also be increasing (because it starts at zero and then goes positive – therefore it must be increasing). So, the derivative in this range must start out increasing and must eventually get back to zero at x=−1. So, at some point in this interval the derivative must start decreasing before it reaches x=−1. Now, we have to be careful here because this is just general behavior here at the two endpoints. We won’t know where the derivative goes from increasing to decreasing and it may well change between increasing and decreasing several times before we reach x = −1. All we can really say is that immediately to the right of x = −3 the derivative will be increasing and immediately to the left of x=−1 the derivative will be decreasing.

Next, for the ranges −1 < x < 2 and 2 < x < 4 we know the derivative will be zero at the endpoints and negative in between. Also, following the type of reasoning given above we can see in each of these ranges that the derivative will be decreasing just to the right of the left-hand endpoint and increasing just to the left of the right hand endpoint.

Finally, in the last region x > 4 we know that the derivative is zero at x=4 and positive to the right of x=4. Once again, following the reasoning above, the derivative must also be increasing in this range.

Putting all of this material together (and always taking the simplest choices for increasing and/or decreasing information) gives us the following sketch for the derivative.

Note that this was done with the actual derivative and so is in fact accurate. Any sketch you do will probably not look quite the same. The “humps” in each of the regions may be at different places and/or different heights for example. Also, note that we left off the vertical scale because given the information that we’ve got at this point there was no real way to know this information.

That doesn’t mean however that we can’t get some ideas of specific points on the derivative other than where we know the derivative to be zero. To see this let’s check out the following graph of the function (not the derivative, but the function).

At x=−2 and x=3 we’ve sketched in a couple of tangent lines. We can use the basic rise/run slope concept to estimate the value of the derivative at these points.

Let’s start at x=3. We’ve got two points on the line here. We can see that each seem to be about one-quarter of the way off the grid line. So, taking that into account and the fact that we go through one complete grid we can see that the slope of the tangent line, and hence the derivative, is approximately -1.5.

At x=−2 it looks like (with some heavy estimation) that the second point is about 6.5 grids above the first point and so the slope of the tangent line here, and hence the derivative, is approximately 6.5.

Here is the sketch of the derivative with the vertical scale included and from this we can see that in fact our estimates are pretty close to reality.

Note that this idea of estimating values of derivatives can be a tricky process and does require a fair amount of (possible bad) approximations so while it can be used, you need to be careful with it.

We’ll close out this section by noting that while we’re not going to include an example here we could also use the graph of the derivative to give us a sketch of the function itself. In fact, in the next chapter where we discuss some applications of the derivative we will be looking using information the derivative gives us to sketch the graph of a function.

**Practice Problems: Interpretation Of The Derivative**

**For problems 1 and 2 use the graph of the function, f(x), estimate the value of f′(a) for the given values of a.**

**1. Use the graph of the function, f(x) , estimate the value of f′(a) for****(a) a = −2****(b) a = 3****Solution:****(a) a= −2****Step 1**

Given that one of the interpretations of the derivative is that it is the slope of the tangent line to the function at a particular point let’s first sketch in a tangent line at the point on the graph.**Step 2**

The function is clearly decreasing here and so we know that the derivative at this point will be negative. Now, from this sketch of the tangent line it looks like if we run over 1 we go down 4 and so we can estimate that,

f′(−2) = −4

**(b) a = 3****Step 1**

Given that one of the interpretations of the derivative is that it is the slope of the tangent line to the function at a particular point. Let’s first sketch in a tangent line at the point.

**Step 2**

The function is clearly increasing here and so we know that the derivative at this point will be positive. Now, from this sketch of the tangent line it looks like if we run over 1 we go up 2 and so we can estimate that,

f′(3)=2

**2. Use the graph of the function, f(x), estimate the value of f′(a) for****(a) a = 1(b) a = 4**

Given that one of the interpretations of the derivative is that it is the slope of the tangent line to the function at a particular point let’s first sketch in a tangent line at the point on the graph.

**Step 2**

The function is clearly decreasing here and so we know that the derivative at this point will be positive. Now, from this sketch of the tangent line it looks like if we run over 1 we go up 1 and so we can estimate that,

f′(1)=1

**(b) a = 4****Step 1**

Given that one of the interpretations of the derivative is that it is the slope of the tangent line to the function at a particular point. Let’s first sketch in a tangent line at the point.

**Step 2**

The function is clearly decreasing here and so we know that the derivative at this point will be positive. Now, from this sketch of the tangent line it looks like if we run over 1 we go up 5 and so we can estimate that,

f′(4)=5

**For problems 3 and 4 sketch the graph of a function that satisfies the given conditions.**

**3. Sketch the graph of a function that satisfies f(1) = 3, f′(1) = 1, f(4) = 5, f′(4) = −2.****Solution:****Step 1**

First, recall that one of the interpretations of the derivative is that it is the slope of the tangent line to the function at a particular point. So, let’s start off with a graph that has the given points on it and a sketch of a tangent line at the points whose slope is the value of the derivative at the points.

**Step 2**

Now, all that we need to do is sketch in a graph that goes through the indicated points and at the same time it must be parallel to the tangents that we sketched. There are many possible sketches that we can make here and so don’t worry if your sketch is not the same as the one here. This is just one possible sketch that meets the given conditions.

While, it’s not really needed here is a sketch of the function without all the extra bits that we put in to help with the sketch.

**4. Sketch the graph of a function that satisfies f(−3) = 5, f′(−3) = −2, f(1) = 2, f′(1) = 0, f(4) = −2, f′(4) = −3.****Soluton:****Step 1**

First, recall that one of the interpretations of the derivative is that it is the slope of the tangent line to the function at a particular point. So, let’s start off with a graph that has the given points on it and a sketch of a tangent line at the points whose slope is the value of the derivative at the points.

**Step 2**

Now, all that we need to do is sketch in a graph that goes through the indicated points and at the same time it must be parallel to the tangents that we sketched. There are many possible sketches that we can make here and so don’t worry if your sketch is not the same as the one here. This is just one possible sketch that meets the given conditions.

While, it’s not really needed here is a sketch of the function without all the extra bits that we put in to help with the sketch.

**For problems 5 and 6 the graph of a function, f(x), is given. Use this to sketch the graph of the derivative, f′(x).**

**5. Below is the graph of some function, f(x). Use this to sketch the graph of the derivative, f′(x).****Solution:****Step 1**

From the graph of the function it is pretty clear that we will have horizontal tangent lines at x = −2, x = 1 and x = 5. Because we will have horizontal tangents here we also know that the derivative at these points must be zero. Therefore, we know the following derivative evaluations.

f′(−2) = 0 f′(1) = 0 f′(5) = 0

**Step 2**

The points we found above break the x-axis up into regions where the function is increasing and decreasing. Recall that if the derivative is positive then the function is increasing and likewise if the derivative is negative then the function is decreasing. Using these ideas we can easily identify the sign of the derivative on each of the regions. Doing this gives,

**Step 3**

In the range x < −2 we know the derivative must be positive and that it must be zero at x = −2 so it makes sense that just to the left of x = −2 the derivative must be decreasing.

In the range −2 < x < 1 we know that the derivative will be negative and that it will be zero at the endpoints of the range. So, to the right of x = −2 the derivative will have to be decreasing (goes from zero to a negative number). Likewise, to the left of x = 1 the derivative will have to be increasing (goes from a negative number to zero).

Note that we don’t really know just how the derivative will behave everywhere in this range, but we can use the general behavior near the endpoints and go with the simplest way to connect the two up to get an idea of what the derivative should look like.

Following similar reasoning we can see that the derivative should be increasing just to the right of x = 1 (goes from zero to a positive number), decreasing just to the left of x = 5 (goes from a positive number to zero) and decreasing just to the right of x = 5 (goes from zero to a negative number).

**Step 4**

So, putting all of this together here is a sketch of the derivative. Note that we included the a scale on the vertical axis if you would like to try and estimate some specific values of the derivative as we did in Example 4 of this section.

**6. Below is the graph of some function, f(x). Use this to sketch the graph of the derivative, f′(x).****Solution:****Step 1**

Because the three portions of the function are actually lines and the tangent line to a line would just be the line itself we can easily compute the derivative on each portion of the curve.

On each of the portions we can use the grid included on the graph to compute the slope of each part. Knowing the slope of the graph on each portion will in turn tell us the slope of the tangent line for each portion. This in turn tells us that the derivative on each of the three portions is then,

**Step 2**

Recall Example 4 from the previous section. In that example we showed that the derivative of the absolute value function does not exist at x=0. The limit on the left side of x=0 (which gives the slope of the line on the left) and the limit on the right side of x=0 (which gives the slope of the line on the right) were different and so the overall limit did not exist. This in turn tells us that the derivative doesn’t exist at that point.

Here we have the same problem. We’ll leave it to you to verify that the right and left-handed limits at x=−1 are not the same and so the derivative does not exist at x=−1. Likewise, the derivative does not exist at x=2.

There will therefore be open dots on the graph at these two points.

**Step 3**

Here is the sketch of the derivative of this function.

**7. Answer the following questions about the function W(z) = 4z ^{2} − 9z .**

We know that the derivative of a function gives us the rate of change of the function and so we’ll first need the derivative of this function. We computed this derivative in Problem 5 from the previous section and so we won’t show the work here. If you need the practice you should go back and redo that problem before proceeding.

So, from our previous work we know that the derivative is,

W′(z) =8z − 9

Now all that we need to do is to compute: W′(−1) = −17 . This is negative and so we know that the function must be decreasing at z = −1.

**(b) Is the function increasing or decreasing at z = 2?**

Again, all we need to do is compute a derivative and since we’ve got the derivative written down in the first part there’s no reason to redo that here.

The evaluation is : W′(2) = 7 . This is positive and so we know that the function must be increasing at z = 2.

**(c) Does the function ever stop changing? If yes, at what value(s) of z does the ****function stop changing?**

Here all that we’re really asking is if the derivative is ever zero. So we need to solve,

So, the function will stop changing at

**8. What is the equation of the tangent line to f(x) = 3 − 14x at x = 8.****Solution:**

We know that the derivative of a function gives us the slope of the tangent line and so we’ll first need the derivative of this function. We computed this derivative in Problem 2 from the previous section and so we won’t show the work here. If you need the practice you should go back and redo that problem before proceeding.

Note that we did use a different set of letters in the previous problem, but the work is identical. So, from our previous work (with a corresponding change of variables) we know that the derivative is,

f′(x) = −14

This tells us that the slope of the tangent line at x = 8 is then : m = f′(8) = −14 . We also know that a point on the tangent line is: (8, f(8)) = (8, − 109).

The tangent line is then,

y = −109 − 14(x − 8) = 3 − 14x

Note that, in this case the tangent is the same as the function. This should not be surprising however as the function is a line and so any tangent line (i.e. parallel line) will in fact be the same as the line itself.

**9. The position of an object at any time t is given by (a) Determine the velocity of the object at any time t.(b) Does the object ever stop moving? If yes, at what time(s) does the object stop moving?**

We know that the derivative of a function gives is the velocity of the object and so we’ll first need the derivative of this function. We computed this derivative in Problem 9 from the previous section and so we won’t show the work here. If you need the practice you should go back and redo that problem before proceeding.

Note that we did use a different letter for the function in the previous problem, but the work is identical. So, from our previous work we know that the derivative is,

**(b) Does the object ever stop moving? If yes, at what time(s) does the object stop moving?**

We know that the object will stop moving if the velocity (i.e. the derivative) is zero. In this case the derivative is a rational expression and clearly the numerator will never be zero. Therefore, the derivative will not be zero and therefore the object never stops moving.

**10. What is the equation of the tangent line to ****Solution**

We know that the derivative of a function gives us the slope of the tangent line and so we’ll first need the derivative of this function. We computed this derivative in Problem 8 from the previous section and so we won’t show the work here. If you need the practice you should go back and redo that problem before proceeding.

Note that we did use a different set of letters in the previous problem, but the work is identical. So, from our previous work (with a corresponding change of variables) we know that the derivative is,

This tells us that the slope of the tangent line at is then : .We also know that a point on the tangent line is:

The tangent line is then,

**11. Determine where, if anywhere, the function g(x) = x ^{3} −2x^{2} + x − 1 stops changing.**

We know that the derivative of a function gives us the rate of change of the function and so we’ll first need the derivative of this function. We computed this derivative in Problem 7 from the previous section and so we won’t show the work here. If you need the practice you should go back and redo that problem before proceeding.

From our previous work (with a corresponding change of variables) we know that the derivative is,

g′(x) = 3x

If the function stops changing at a point then the derivative will be zero at that point. So, to determine if we function stops changing we will need to solve,

So, the function will stop changing at and x = 1.

**12. Determine if the function increasing or decreasing at the given points.****Solution:****(a) t = 5**

We know that the derivative of a function gives us the rate of change of the function and so we’ll first need the derivative of this function. We computed this derivative in Problem 10 from the previous section and so we won’t show the work here. If you need the practice you should go back and redo that problem before proceeding.

So, from our previous work we know that the derivative is,

Now all that we need to do is to compute:

This is positive and so we know that the function must be increasing at t = 5.

**(b) t = 10**

Again, all we need to do is compute a derivative and since we’ve got the derivative written down in the first part there’s no reason to redo that here.

The evaluation is:

This is positive and so we know that the function must be increasing at t=10.

**(c) t = 300**

Again, all we need to do is compute a derivative and since we’ve got the derivative written down in the first part there’s no reason to redo that here.

The evaluation is:

This is positive and so we know that the function must be increasing at t=300.

Final Note

As a final note to all the parts of this problem let’s notice that we did not really need to do any evaluations. Because we know that square roots will always be positive it is clear that the derivative will always be positive regardless of the value of t we plug in.

**13. Suppose that the volume of water in a tank for 0 ≤ t ≤ 6 is given by Q(t) = 10 + 5t − t ^{2}.**

We know that the derivative of a function gives us the rate of change of the function and so we’ll first need the derivative of this function. We computed this derivative in Problem 4 from the previous section and so we won’t show the work here. If you need the practice you should go back and redo that problem before proceeding.

So, from our previous work we know that the derivative is,

Q′(t)=5−2t

Now all that we need to do is to compute: Q′(0) = 5. This is positive and so we know that the volume of water in the tank must be increasing at t = 0.

**(b) Is the volume of water increasing or decreasing at t=6?**

Again, all we need to do is compute a derivative and since we’ve got the derivative written down in the first part there’s no reason to redo that here.

The evaluation is: Q′(6) = −7 . This is negative and so we know that the volume of water in the tank must be decreasing at t = 6.

**(c) Does the volume of water ever stop changing? If yes, at what times(s) does the volume stop changing?**

Here all that we’re really asking is if the derivative is ever zero. So we need to solve,

So, the volume of water will stop changing at 5/2.

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