Local Linearity II

Using Local Linearity to introduce difference quotient and the derivative.

An effective way to introduce difference quotients and derivatives is to write the equation of the “line” you see when you zoom-in on a locally linear function.

First: Ask your class to use their calculator or computer grapher to graph a function, say y = sin(x), or some function they like better.

  1. Ask them to trace over to some point where the graph is “curvy.” (So they will remain on the graph, use the TRACE feature, not the moving cursor.) They do not have to go to, or even be near, the same place.
  2. Then ask them to zoom-in several times until their graph looks like a straight line (locally linear) and save the coordinates of that point as a and b (see the technology hint below).
  3. Then return to the graph and trace one or two clicks left or right to a nearby point on the graph and record the coordinates of that point as c and d.
  4. Write the equation of the line through (a, b) and (c ,d) and enter it in the graphing menu (see technology hint again).
  5. Graph the line. They should see only one “line” because the two graphs are on top of each other.
  6. Re-graph in the standard or Trig window. What do you see now? They should see their original graph with a line that appears tangent to it at the point (a, b).

Next: Discuss what you’ve done, specifically in finding the slope. The value c is a plus a little bit, that is c = a + h. (Or minus a  little bit if h is negative.) So the slope is

\displaystyle \frac{Y1(a+h)-Y1(a)}{(a+h)-a}\text{ or }\frac{f\left( x+h \right)-f\left( x \right)}{h}

and now you are ready to talk about difference quotients and their limit the derivative.

Technology Hints:

When you trace a graph on a calculator the coordinates of the point are written on the bottom of the screen as X and Y, or xc and yc. If you return to the home screen and type X [STO] A and Y [STO] B (or xc [STO] a etc.) the values will be saved to A and B. When you trace to the next point the x and y change, so return to the home screen and save them as C and D.

The line can be written directly in the equation editor in point-slope form by typing Y2 = Y1(A) + (Y1(B)-Y1(A))/(B – A)*(x – A)


A Note on Notation

For quite a while I’ve been writing sin(x), ln(x) and the like with parentheses instead of the usual sin or ln x .

The main reason is that I want to emphasize that sin(x), ln(x), etc. are the same level and type of notation as f(x). The only difference is that sin(x) and ln(x) always represent the same function, while things like f(x) represent different functions from problem to problem. I hope this makes things just a little clearer to the students.

I also favor using (sin(x))² instead of sin²(x), again to make clearer just what is getting squared. Notation can be inconsistent: I don’t think I’ve ever seen ln²(x) or even ²(x).  So this helps in that regard as well.

Of course, when entering functions in calculators or computers you almost always must use the “extra” parentheses in both cases. (Except for the new Casio PRIZM which will understand sin x and ln x, but not sin²(x).)

Now we can use that spot in the notation exclusively for inverse functions, as in {{\sin }^{-1}}\left( x \right) and {{f}^{-1}}\left( x \right). Maybe that will lessen the confusion there.

Another possible inconsistency is trying to write sin′(x)  for the derivative as you do with {f}'\left( x \right)Although, if I saw it I would understand it. (LaTex won’t even parse  sin′(x).)