## The power of being naive

To manipulate the demos in this post, you must have Wolfram CDF Player installed on your computer. It is available free from the Wolfram website. The code for the demos is in the Mathematica notebook MM Def Deriv.nb. See How to manipulate the diagrams for more information on what you can do with them.

# Learning about the derivative as a concept

The derivative $f'(x)$ of $f(x)$ is the function whose value at $a$ is the slope of the line tangent to the graph $y=f(x)$ at the point $(a,f(a))$.

To gain understanding of the concept of derivative the student need to see and play with the pictures that illustrate the definition. This can be done in stages:

• Give an intuitive, pictorial explanation of the tangent line.
• Show in pictures what the slope of a line is.
• Show in pictures how you can approximate the tangent line with secant lines.

Of course, many teachers and textbooks do this. I propose that:

The student will benefit in the long run by spending a whole class session on the intuitive ideas I just described and doing a set homework based only on intuition. Then you can start doing the algebraic stuff.

This post provides some ideas about manipulable diagrams that students can play with to gain intuition about derivatives. Others are possible. There are many on the Mathematica Demonstrations website. There are others written in Java and other languages, but I don't know of a site that tries to collect them in one place.

My claim that the student will benefit in the long run is not something I can verify, since I no longer teach.

## Present the tangent line conceptually

The tangent line to a curve

• is a straight line that touches the curve at a point on the curve,
• and it goes in the same direction that the curve is going, like the red line in the picture below. (See How to manipulate the diagrams.)

My recommendation is that you let the students bring up some of the fine points.

• The graph of $y=x^3-x$ has places where the tangent line cuts the curve at another point without being parallel to the curve there. Move the slider to find these places.
• The graph of $y=\cos(\pi x)$ has places where the same line is tangent at more than one point on the curve. (This may requre stepping the slider using the incrementers.)
• Instigate a conversation about the tangent line to a given straight line.
• My post Tangents has other demos intended to bother the students.
• Show the unit circle with some tangent lines and make them stare at it until they notice something peculiar.
• "This graph shows the tangent line but how do you calculate it?" You can point out that if you draw the curve carefully and then slide a ruler around it so that it is tangent at the point you are interested in, then you can draw the tangent carefully and measure the rise and run with the ruler. This is a perfectly legitimate way to estimate the value of the slope there.

## Slope of the tangent line conceptually

This diagram shows the slope of the tangent line as height over width.

• Slide the $x$ slider back and forth. The width does not change. The height is measured from the tangent line to the corner, so the height does change; in particular, it changes sign appropriately.
• This shows that the standard formula for the derivative of the curve gives the same value as the calculated slope of the tangent. (If you are careful you can find a place where the last decimal places differ.) You may want to omit the "derivative value" info line, but most students in college calculus already know how to calculate the formulas for the derivative of a polynomial– or you can just tell them what it is in this case and promise to show how to calculate the formula later.
• Changing the width while leaving $x$ fixed does not change the slope of the tangent line (up to roundoff error).
• In fact I could add another parameter that allows you to calculate height over width at other places on the tangent line. But that is probably excessive. (You could do that in a separate demo that shows that basic property that the slope of a straight line does not change depending on where you measure it — that is what a curve being a straight line means.)
• This graph provides a way to estimate the slope, but does not suggest a way to come up with a formula for the slope, in other words, a formula for the derivative.

## Conceptual calculation of the slope

This diagram shows how to calculate the value of the slope at a point using secant lines to approximate the tangent line. If you have a formula for the function, you can calculate the limit of the slope of the secant line and get a formula for the derivative.

• The function $f(x)=x^3-x$.
• The secant points are $(x-h,f(x-h))$ and $(x+h, f(x+h))$. $h$ is called "width" in the diagram.
• Moving $x$ with the slider shows how the tangent line and secant line have similar slopes.
• Moving the width to the left, to $0$ (almost), makes the secant line coincide with the tangent line. So intuitively the limit of the slope of the secant line is the slope of the tangent line.
• The distance between the secant points is the Euclidean distance. (It may be that including this information does not help, so maybe it should be left out.)
• The slope of the secant line is $\frac{f(x+h)-f(x-h)}{(x+h)-(x-h)}$ when $h\neq0$. This simplifies to $3x^2+h^2-1$, so the limit when $h\to0$ is $3x^2-1$, which is therefore a formula for the derivative function.

## Testing intuitive concepts

Most of the work students do when studying derivatives is to solve some word problems (rate of change, maximization) in which the student is expected to come up with an appropriate function $f(x)$ and then know or find out the formula for $f'(x)$ in the process of solving the problem. In other words there is a heavy emphasis on computation and much less on concept.

The student in the past has had to do very few homework problems that test for understanding the concept. Lately some texts do have problems that test the concept, for example:

This is the graph of a function and its derivative. Which one is the function and which is its derivative?

Note that the problem does not give you the formula for the function, nor does it have to.

Many variations are possible, all involving calculating parameters directly from the graph:

• "These are the first and second derivatives of a function. Where (within the bounds of the graph) is the function concave up?"
• "These are the first and second derivatives of a function. Where (within the bounds of the graph) are its maxima and minima?"
• "This straight line is the derivative of a function. Show that the function is a quadratic function and measure the slope of the line in order to estimate some of the coefficients of the quadratic."

### How to manipulate the diagrams

• You can move the sliders back and forth to to move to different points on the curve.
• In the first diagram, you can click on one of the four buttons to see how it works for various curves.
• The arrow at the upper right makes it run automatically in a not very useful sort of way.
• The little plus sign below the arrow opens up some other controls and a box showing the value of $a$, including step by step operation (plus and minus signs).
• If you are using Mathematica, you can enter values into the box, but if you are using CDF Player, you can only manipulate the number using the slider or the plus and minus incrementers.

## Tangents

This is an experiment in exposition of the mathematical concepts of tangent.  It follows the same pattern as my previous post on secant, although that post has explanations of my motivation for this kind of presentation that are not repeated here.

To manipulate the diagrams in this post, you must have Wolfram CDF Player installed on your computer. It is available free from the Wolfram website. The Mathematica notebooks used here are listed in the references below.

### Tangent line

A line is tangent to a curve (in the plane) at a given point if all the following conditions hold (Wikipedia has more detail.):

1. The line is a straight line through the point.
2. The curve goes through that point.
3. The curve is differentiable in a neighborhood of the point.
4. The slope of the straight line is the same as the derivative of the curve at that point.

In this picture the curve is $y=x^3-x$ and the tangent is shown in red.  You can click on the + signs for additional controls and information.

Etymology and metaphor

The word “tangent” comes from the Latin word for “touching”.  (See Note below.) The early scholars who talked about “tangent” all read Latin and knew that the word meant touching, so the metaphor was alive to them.

The mathematical meaning of “tangent” requires that the tangent line have slope equal to the derivative of the curve at the point of contact. All of the red lines in the picture below touch the curve at the point (0, 1.5). None of them are tangent to the curve there because the curve has no derivative at the point:

The curve in this picture is defined by

The mathematical meaning restricts the metaphor. The red lines you can generate in the graph all touch the curve at one point, in fact at exactly at one point (because I made the limits on the slider -1 and 1), but there are not tangent to the curve.

Tangents can hug!

On the other hand, “touching” in English usage includes maintaining contact on an interval (hugging!) as well as just one point, like this:

The blue curve in this graph is given by

The green curve is the derivative dy/dx. Notice that it has corners at the endpoints of the unit interval, so the blue curve has no second derivative there. (See my post Curvature).

Tangent lines in calculus usually touch at the point of tangency and not nearby (although it can cross the curve somewhere else).  But the red line above is nevertheless tangent to the curve at every point on the curve defined on the unit interval, according to the definition of tangent. It hugs the curve at the straight part.

The calculus-book behavior of tangent line touching at only one point comes about because functions in calculus books are always analytic, and two analytic curves cannot agree on an open set without being the same curve.

The blue curve above is not analytic; it is not even smooth, because its second derivative is broken at $x=0$ and $x=1$.  With bump functions you can get pictures like that with a smooth function, but I am too lazy to do it.

### Tangent on the unit circle

In trigonometry, the value of the tangent function at an angle $\theta$ erected on the x-axis is the length of the segment of the tangent at (1,0) to the unit circle (in the sense defined above) measured from the x-axis to the tangent’s intersection with the secant line given by the angle. The tangent line segment is the red line in this picture:

This defines the tangent function for $-\frac{\pi}{2} < x < \frac{\pi}{2}$.

The tangent function in calculus

That is not the way the tangent function is usually defined in calculus. It is given by $latex \tan\theta=\frac{\sin\theta}{\cos\theta}$, which is easily seen by similar triangles to be the same on $latex -\frac{\pi}{2} < x < \frac{\pi}{2}$.

We can now see the relationship between the geometric and the $\frac{\sin\theta}{\cos\theta}$ definition of the tangent function using this graph:

The red segment and the green segment are always the same length.
It might make sense to extend the geometric definition to $\frac{\pi}{2} < x < \frac{3\pi}{2}$ by constructing the tangent line to the unit circle at (-1,0), but then the definition would not agree with the $\frac{\sin\theta}{\cos\theta}$ definition.

The root is “tangent-” but the nominative is “tangens”. This is normal behavior for Latin participles, but it causes angst among people who write about etymology of trig functions. Ignore “tangens”, it is the root that matters.

The Latin root comes from the Indo-European root “tag”. Don’t worry about how the n got into “tangent”; this is typical Indo-European behavior. The English word “tag” does not come from the IE root, but “thwack” may.

## Picturing derivatives

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 $latex x=-4$ 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 $latex x=1.5$ and $latex x=1.75$.  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 $latex x$ 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.