This lesson explores the behavior of a function near critical points where the derivative is undefined. The lesson utilizes the T.I. 82 calculator and its ability to both graph functions and to create tables of values. The students will explore the graph of the function and use the calculator to compute the values of the slopes of tangent lines near the critical point. They will then create tables of both x and y coordinates of the function and, next to them for the purpose of comparison, create values for the derivative at these same points.

There are two common situations where a function is continuous and the derivative is undefined. In both of these situations the tangent line is verical. In one case the critical point is an extremum and in the other it is an inflection point. The function Y=(x-1)^1/3*(x+2)^2/3 is an ideal example to explore as it has both situations occur at x=-2 and x=1.

There is an assumption made that the students are familiar with the calculator and its ability to : graph functions, zoom in on specific parts of the graph, use the "trace"facility, calculate derivatives using the "dy/dx"option, create tables, and change the increments of the tables.

When entering both the function itself and its derivative into the calculator it is necessary to use a generous supply of parentheses, as the calculator will not always interpret fractional exponents the way we wish them to. I suggest entering the function above in this format:

and its derivative as :

in order to have the calculator operate on the entire domain. The explanation of this extra detail offers an excuse to review the manipulation of exponents with the students.

The value of the slope of a tangent line near the critical points -2 and 1 can be explored by first using the "trace"option to select a point near the critical point and then having the calculator compute the derivative at that point by selection the "calc"option and then the "dy/dx"(option number 6). Further by "zooming"in to select points closer and closer to the critical point the student can see how the derivative is growing ever larger ( going to infinity ) or approaching vertical. This will also allow students to gain practice with the art of "zooming".

With the function entered as y1 and the derivative entered as y2 the students can examine the values of both near and at the critical points by using the "talble"option. By continuing to use smaller and smaller values for the "change in table"option the students will see that the derivative is growing ever larger as they approach the critical points. At the same time they will see that the derivative is undefined at the critical point ( the "error"message ) while the function itself is defined.

While the calculator makes the use of fractional exponents difficult, it does allow quick computation of very small values of the domain. Such an exploration of both the graph and the values obtained with the table can give students an insight into this unusual situation. If this is followed or preceded with an algebraic analysis of the function and the derivative students should have a much deeper understanding of just how a function can be continuous and yet undifferentiable at the same point.

This lesson relates to the State Goals for Learning numbers: 1,4,5,6,and 7. It also exposes the student to the use of the graphing calculator as a tool for analysis and not just a means to a quick but not necessarily understandable solution to a problem.

A MORE DETAILED LIST OF STUDENT INSTRUCTIONS

1. Pass out the graphing calculators and have the students "clear"all functions on the "y=" screen.
2. Have the students enter the function as y1 and the derivative as y2. Warn them to be careful to include all parentheses.
3. Examine the graph on the "standard"scale. Pay attention to the graph around -2 and 1. Ask the students to make note of the shape of the graph in the neighborhoods of these two points.
4. Using the "Z-box"option on the "zoom"menu, have the students enlarge the area around the point (-2,0). Have them "zoom in"on this graph at least once. Select the "trace"option and move the cursor to a point very near but no on (-2,1). Select "calc"and on that menu select number 6 "dy/dxÓ. The calculator wil first show the x and y coordinates of the chosen point and if "enter"is selected the numerical value of the derivative will appear. If you "zoom in"again and repeate this process you can evaluate the value of the derivative at points very close to our critical point. I suggest trying points on both sides of (-2,0) to get a better feeling for the behavior of the derivative near the critical point. Clearly the students should observe the numerical values of the absolute value of the derivative growing ever larger as the point approaches the critical point.
5. Have the students repeat the process near (1,0) and make note of the difference between the behavior of the derivative near this point and its behavior near (-2,0).
6. Have the students select the "table " option where the calculator will create a chart of values of x, values of the function at x, and values of the derivative at x. By using the "table set"option they can decrease the value of the "change in x"and thus examine values ever closer to those at -2 and AT 1. Have the students at least reduce the "change in x"to .001. They should see that near -2 the derivative is quite large while at -2 they should see an "error"message. This indicates that the derivative is not defined at -2, hence the function is not differentiable there. At the same time they can see that the value of the function is defined, it is 0. Again they should examine this chart near both -2 and 1 and note the differences.
7. Have the students examine the function and derivative and analyze the domain of each to see if their conclusions based upon the graph and the table are consistent with the results of this investigation.