The power of being naive

The interactive examples in this post require installing Wolfram CDF player, which is free and works on most desktop computers using Firefox, Safari and Internet Explorer, but not Chrome. The source code is the Mathematica Notebook MM Def Deriv.nb, which is available for free use under a Creative Commons Attribution-ShareAlike 2.5 License. See How to manipulate the diagrams for more information on what you can do with them. The notebook can be read by CDF Player if you cannot make the embedded versions in this post work.

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?

Concept Prob

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.

 

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