Microworld Title Page:
Derivatives and Graphs of Functions
Individual and Institutional Members may sign in. Click here to join the Library

Requires the free Java MathwrightWeb ActiveX Control to read in your Browser.
Download free MathwrightWeb to view Microworlds in your browser, then press

Browser problems? No Problem! Download the free Mathwright32 Reader, then press

Once you download our free Mathwright32 Reader above, then simply click Get This Microworld, and it will be downloaded to your machine and installed in a directory there. You may find it whenever you want to view it, by going to the Start, Programs, Mathwright32 Reader menu.

• "Run ActiveX Controls and Plugins" set either to enable or prompt.
• "Initialize and Script ActiveX Controls not marked as safe" set either to enable or prompt.

Microworld: Derivatives and the Graphs of Functions: (All in One)
Click the Hyperlink above to visit the Microworld.
Author: James White

The aim of this Microworld is to give students the opportunity to experiment with some of the general applications of the derivative to the study of the behavior of functions. This includes their critical behavior (local extrema, inflections) The reader may investigate built-in examples or supply her own functions.

This Microworld may also serve as an all-purpose function grapher, graphing also the first and second derivatives of any (differentiable) user-defined functions.

When you click the hyperlink at the top of the page, you will be taken to the Microworld Page, which contains the portal to the Microworld itself, as well as the Story. To give an idea of how this goes in general, we copy the full story for this page below. The portal is not represented here. It will be a sky-blue rectangle which appears at the end of the story below.

---

Derivatives and the Graphs of Functions
(Just the Story. To see the Microworld, click link at top of page)

by James E. White

You may open this 1-page Microworld at the bottom of this page by pressing Start. Before you open it, you might like to press F11 to view the Microworld in Full Screen mode. If you do not check: Disk Version, then it will behave as an applet. If you would like to write programs and play with them in this Microworld, then check Disk Version. In that case, things happen more quickly because you will get a much smaller self-extracting executable rather than all files in unzipped form.

If you elect Disk Version, you will be asked to find or create a Resource Directory for the Microworld in your Documents directory. Just select the "New Folder" icon, give it a name, and then open it. When you click OK, you will be notified that the self-extracting executable is being downloaded. It's about 90 KB and, once it arrives, you will be told to select UNZIP. Then close the dialog, and, voila, you have the Microworld, and never have to download it in the future. When you open the Microworld in the future, check: Disk Version, go to the Resource Directory, and it will open immediately.

In general, we tell the Mathematical Story and/or place the Table of Contents above the Microworld, and the Interaction Instructions below it.

The Story:

The purpose of this WorkBook is to experiment with some of the general applications of the derivative to the study of the behavior of functions. This includes their critical behavior, their local extrema, regions of monotonicity (increasing or decreasing), and their inflection points.

Each of those will tell us something useful about the graph of a function. On page 3 of the Microworld, we will work with 15 built-in examples to illustrate this theme. And when you are comfortable with the ideas, then you can try your own functions. We will show you how to do that, too.

As you supply your own functions, you will see how the graph looks, and what relation the graph has to its critical points and inflections. A function is a convenient way to represent a relationship between two variables. When two variables, say x and y stand for measurements of some pair of quantities that change together in some process, and when for each value that x takes only one y value corresponds to it, then we say that y depends on x. This is because we only need to know the value of x to determine the pair.

When x and y are bound together in this way, we often say also that y is a function of x. It means the same thing. The idea is that somehow each value that x takes determines the value that y takes in the measurement process. When, in some process, y is a function of x, we may write: y = f(x). And we think of f (the function) as a rule that assigns to each value of x, the y that corresponds to it in the process.

Now, processes are most often thought of as temporal things, but they may be more abstract than that. When the process is temporal, then one natural choice for the independent variable, say x, is the measurement of elapsed time itself, as for example, when we let an object fall and bind the time to the height above the ground.

But we may, for example imagine an experiment in which we construct circles with various diameters, and bind together the measurement of diameter to the measurement of circumference. In that case, the "process" is not really temporal, but it is a conceptual and geometrical process in which, for each construction, given a unit of measurement, a pair of numbers is bound together: We may call the diameter of the circle D, and the circumference C. Then we might say that C was a function of D, say C = f(D). But now something interesting happens. For such an abstract process, we are permitted to write: C = pi * D. And the functional relationship between D and C is now expressed in symbolic terms. We have, in fact, an algebraic procedure for producing C whenever we know the value of D. Simply multiply D by pi. So we say: C = f(D) = pi * D. For "abstract" relationships such as this, or as those proposed by our models, the functional dependence is reduced to an algebraic procedure. And we describe the functional relationship with the formula that is used to produce the value of the dependent variable (y) when the value of the independent variable (x) is known.

This is fine. But we must understand that the algebraic procedure is a property of our model. The functional dependence between the variables measured in the process expresses a richer and a deeper thing. In a sense, it expresses a belief (that the model attempts to articulate) that a certain process of measurement will yield the results predicted by the formula.

In this Microworld, we will see a few things that can be learned about functions - as snapshots of processes that bind measured variables - from their pictures. A graph of a function is indeed a "picture" of it. It is just the set of bound pairs (x,y) where y = f(x) drawn in a plane. The independent variable (often t or x) is represented on the abscissa (or horizontal axis), and the dependent variable is represented on the ordinate (or vertical axis). Much can be learned about the behavior of the variables from a picture like this. The information is qualitative, that is, "fuzzy," but that is usually the way it is with real data before it is modeled by formulas.

The examples that we will use will in fact come from formulas. But you should keep in mind that many of the conclusions you draw from these examples will still be meaningful even when the data is not produced by formula, but by some measurement process. Let us start with a simple thing we can observe about a function.

1. Monotonicity

We say that a function is monotonic on an interval (a,b) of values of the independent variable if either: It is increasing as the independent variable increases in (a,b) (monotonic increasing) or it is decreasing as the independent variable increases (monotonic decreasing). For example, if the independent variable is amount spent on advertising, and the dependent variable is net profit, then it is very useful to know if the "function" is monotonic on an interval in time, and which way! That ought to affect corporate strategy.

Let's see what this looks like. In the upper right corner of the screen is a button: Presently, example #1 is installed. There are 15 examples, and to get one, just write the number in the field and press the button. The current example function definition will appear in the box below:

It will always be a function of x (The independent variable will always be x). You may type numbers into the Get Example field, or just click for the next example. When you tire of our examples, create your own. Just write their definition in the " f(x) = " box and the system will use them. Be sure the independent variable is x in your formula. And you may set the Abscissa, Ordinate, and Domain intervals (for example. to get a close-up view, or to look at the graph on some restricted domain) by typing the interval in the appropriate box below the graph screen, and then pressing the button. To see the graph, press the graph function button in the cluster of buttons:

You will see the first example.

The 3 highlighted points correspond to the "zeros" of the function, the values of x for which the function takes the value 0. Now, what are the regions of monotonicity? To see the intervals on which y increases with x, press the increasing button in that cluster. You will see:

The section of the graph highlighted in black is the part of the single interval on which f is increasing. As you move from left to right, the graph rises there. Next, to see where f is decreasing, press the decreasing button. You see:

And the regions highlighted in white are over the 2 intervals on which f is decreasing. On each of those intervals, y gets smaller as x gets larger. What can we say about the points on the borders between increasing and decreasing f ? For example when f stops decreasing (with increasing x) and begins to increase, we say that f has attained a local minimum.

This happens at the point whose coordinates are approximately ( -(3/2) , -1) We shall see what the exact coordinates are shortly. Again, at the point whose coordinates are roughly ( 3/2 , 1) the function attains a local maximum. Neither of these values is an absolute (global) minimum or maximum, as you can easily see from the graph. They are called critical points.

Roughly, a critical point is a point at which the monotonicity can change. It may not change, but it can change at such a point. Your Calculus text gives a more specific definition, and we will come to that later. We include in this WorkBook as critical points, the endpoints of the interval on which the function is defined, the local maxima and minima (extrema), the points at which the function fails to be defined (vertical asymptotes, say) and certain other points where the tangent line becomes horizontal. To understand the global behavior of the function on its domain, and especially to learn where it attains its local and global maxima and minima (extrema), it is necessary to be able to identify these points.

This is what the exploration page sets out to help you do. Now to see the critical points of the function, press the critical points button. You see:

The highlighted points are just the local maxima and minima we described above. Try changing the Domain of the function in this example to [-2, 2]. That is, type -2, 2 in the Domain field (no brackets) and press the Domain button. You see the picture:

2. Generating your own examples

If all of this is crystal clear, then continue. If it is not, try this with some of our other examples, or (especially) make a few examples up of your own, and see what is going on. To create your own example, type the function of x into the field:

It will always be a function of x (The independent variable will always be x).

You may also like to set the window parameters: Abscissa (horizontal extent), Ordinate (vertical extent) and Domain (interval in which the function is defined):

Just type the appropriate numbers next to the button, then press the button. For exercise, describe explicity (and exactly; don't just read them from the screen) the intervals of monotonicity of examples: 1, 3, 4, 7. 9, and 10. If it is not clear how to do that now, read on, and we will explain.

3. Critical points

Determining the intervals of monotonicity means finding the places where function stops increasing and starts decreasing or vice-versa. Now this can happen in a number of ways. For example if

We see 5 critical points:

The monotonicity changes at the vertical asymptote x = 0. The system indicates that 0 is a critical point in this case by drawing a short vertical grey line through the x-axis at the asymptote. However, if the function has a unique tangent line at each point, this sort of pathology cannot occur. In fact, it is plausible that:

Observation: If a function has a unique tangent line at each point in an interval, then monotonicity can only change at points where that tangent line is horizontal.

For exercise, justify this observation in your own words. You needn't give a formal proof, but try to find a convincing argument. Or, if you don't believe it, look for a counterexample. You might like to look through the example functions 1-15 to help you decide.

Now this observation, assuming it is true, gives us a powerful tool for discovering the local maxima and minima of a function defined on an interval. Under the conditions of the observation, such maxima and minima can only occur either at the endpoints of the interval, or else at points x that satisfy the equation: f ' (x) = 0. In your Calculus text, the points that satisfy the latter equation are usually called critical points. We may explore this fact by graphing the derivative of a function together with the function. To do this with the second example, go to the second cluster of buttons:

and, after you have graphed the function, press graph derivative. You will see:

The red graph is the graph of the derivative of

The expression for the derivative is printed in the field:

The system will do this for your examples also. Notice that f has a single local minimum at approximately x = 1. This is reflected in the fact that f ' has a zero at that point (the red dot). In fact, that is the only point on this interval where the derivative vanishes. Now it doesn't follow from these remarks that all solutions of f ' (x) = 0 yield local extrema.

But if there is a local extremum, then we can expect that if the derivative exists at that point, it must be 0.

4. Concavity and the second derivative

Now a natural next question to ask is this: If the derivative vanishes at a point, and if it does correspond to an extremum there, what sort of extremum is it? That question has a natural answer in applying the same monotonicity analysis to the derivative that we applied to the function. Consider this. Suppose the derivative is increasing on an interval. That means that on that interval, as you move from left to right, the slopes of tangent lines are increasing. In particular, if at the critical point the slope of the tangent line is 0, then to the left of the critical point, the slope is negative, and to the right, it is positive.

The picture is something like this:

This implies that the critical point is a local minimum. We have the two rules of thumb:

1) If is increasing in the vicinity of a point where = 0, then the critical point is a local minimum, and the function's graph is concave up near that point.

2) If is decreasing in the vicinity of a point where = 0, then the critical point is a local maximum, and the function's graph is concave down near that point. In fact, when the derivative is strongly increasing, the function is concave up, and when strongly decreasing, it is concave down. To visualize this, press the buttons: increasing and decreasing on the graph derivative cluster of buttons. You see:

In the yellow region, the derivative is decreasing. The white section of the graph of f over that interval is concave downward. The two red regions correspond to sections where the derivative is increasing, and you can see that the graph of f is concave upward on the part x > 0. It is also concave upward on the negative red part, right up to the place where the derivative begins decreasing (the yellow part) but this is not so easy to see. You can see this more clearly with example 3:

The original graph (black and white) is concave down over the yellow (decreasing) intervals of the derivative, and is concave up over the red (increasing) intervals of the derivative. And what about the points between the red and yellow regions on the derivative graph?

These correspond to local maxima and minima of the derivative (under some restrictive conditions). At these points, the second derivative must vanish: f ' ' (x) = 0. The local extrema of the derivative of f are the points where the concavity of f changes (because they are points where the monotonicity of the derivative changes).

These points are called inflection points. Under suitable differentiability conditions on f, we may say that inflection points can only occur where f ' ' (x) = 0 but that condition is not sufficient. For example f(x) = x^4 has 0 second derivative at x = 0, but the concavity of f does not change there. For exercise, determine the inflection points (where concavity changes) for example functions 1 through 10. Again, give exact answers, not decimal approximations.

Experiment with your own examples. That is important for your understanding of these things.

---

Return to the listing of MathwrightWeb Microworlds

---

		- James E. White, Ph.D. , Library Director,
		author of this website, Mathwright 2000, MindScapes,
		MathwrightWeb, and Mathwright32