The CDF files in G&G posts no longer work. I have been unable to find out why.I expect to produce another document on abstractmath.org that will include this example and others. A link willl be posted here when it is done.

This is my first experiment at posting an active Mathematica CDF document on my blog. To manipulate the graph below, you must have Wolfram CDF Player installed on your computer. It is available free from their website.

This is a new presentation of old work. It is a graph of a certain fifth degree polynomial and its first four derivatives.

The buttons allow you to choose how many derivatives to show and the slider allows you to show the graphs from up to a certain point.

How graphs like this could be used for teaching purposes

You could show this in class, but the best way to learn from it would be to make it part of a discussion in which each student had access to a private copy of the graph.  (But you may have other ideas about how to use a graph like this.  Share them!)

Some possible discussion questions:

1. Click button 1. Now you see the function and the derivative. Move the slider all the way to the left and then slowly move it to the right.  When the function goes up the derivative is positive.  What other things do you notice when you do this?
2. If you were told only that one of the functions is the derivative of the other, how would you rule out the wrong possibility?
3. What can you tell about the zeroes of the function by looking at the derivative?
4. Look at the interval between and .  Does the function have one or two zeroes in that interval?  On my screen it looks as if the curve just barely  gets above the axis in that interval.  What does that say about it having one or two zeroes?  How could you verify your answer?
5. Click button 2.  Now you have the function and first and second derivatives.  What can you say about maxima, minima and concavity of the function?
6. Find relationships between the first and second derivatives.
7. Now click button 4.  Evidently the 4th derivative is a straight line with positive slope.  Assume that it is.  What does that tell you about the graph of the third derivative?
8. What characteristics of the graph of the function can you tell from knowing that the fourth derivative is a straight line of positive slope?
9. What can you say about the formula for the function knowing that the fourth derivative is a straight line of positive slope?
10. Suppose you were given this graph and told that it was a graph of a function and its first four derivatives and nothing else.  Specifically, you do not know that the fourth derivative is a straight line.  Give a detailed explanation of how to tell which curve is the function and which curve is each specific derivative.

Making this manipulable graph

I posted this graph and a lot of others several years ago on abstractmath.org.  (It is the ninth graph down).  I fiddled with this polynomial until I got the function and all four derivatives to be separated from each other.  All the roots of the function and all its derivatives are real and all are shown.  Isn’t this gorgeous?

To get it to show up properly on the abmath site I had to thicken the graph line.  Otherwise it still showed up on the screen but when I printed it on my inkjet printer the curves disappeared. That seems to be unnecessary now.

Mathematica 8.0 has default colors for graphs, but I kept the old colors because they are easier to distinguish, for me anyway (and I am not color blind).

Inserting CDF documents into html

A Wolfram document explains how to do this.  I used the CDF plugin for WordPress.  WordPress requires that, to use the plugin, you operate your blog from your own server, not from WordPress.com.  That is the main reason for the recent change of site.

The Mathematica files are New5thDegreePolynomial.nb and New5thDegreePolynomial.cdf on my public folder of Mathematica files.  You may download the .cdf file directly and view it using CDF player if you have trouble with the embedded version. To see the code you need to download the .nb file and open all cells.

Here are some notes and questions on the process.  When I find learn more about any of these points I will post the information.

1. At the moment I don’t know how to get rid of the extra space at the top of the graph.
2. I was surprised that I could not click on the picture and shrink or expand it.
3. It might be annoying for a student to read the questions above and have to go up and down the screen to see the graph.  I had envisioned that the teacher would ask the questions and have the students play with the graph and erupt with questions and opinions.  But you could open two copies of the .cdf file (or this blog) and keep one window showing the graph while the other window showed the questions.
4. Which raises a question:  Could it be possible to program the graph with a button that when pushed would make the graph (only) appear in another window?

Other approaches

1. I have experimented with Khan Academy type videos using CDF files.  I made a screen shot and at a certain point I pressed a button and the graph appropriately changed.   I expect to produce an example video which I can make appear on this blog (which supposedly can show videos, but I haven’t tried that yet.)
2. It should be possible to have a CDF in which the student saw the graph with instructional text underneath it equipped with next and back buttons.  The next button would trigger changes in the picture and replace the text with another sentence or two.  This could be instead of spoken stuff or additional to it (which would be a lot of work).  Has anyone tried this?

Note

My reaction to Khan Academy was mostly positive.  One thing that struck me that no one seems to have commented on is that the lectures are short. They cover one aspect (one definition or one example or what one theorem says) in what felt to me like ten or fifteen minutes.  This means that you can watch it and easily go back and forth using the controls on the video display.  If it were a 50-minute lecture it would be much harder to find your way around.

I think most students are grasshoppers:  When reading text, they jump back and forth, getting the gist of some idea, looking ahead to see where it goes, looking back to read something again, and so on.  Short videos allow you to do this with spoken lectures. That seems to me remarkably useful.