Setting Within the CourseWhen I brought my class to the computer lab for this activity, my calculus students had already been using the limit definition of a derivative for three days. We had also been discussing the derivative at a point as the function's instantaneous rate of change, and the slope of a line tangent to the function at that point. When we had opportunities to do so, we were taking special notice of times when the derivative was zero and the tangent line was horizontal. We had also spent just a little bit of time more generally discussing the relationships between graphs of functions and the graphs of their derivatives, including a few mentions of concavity. Furthermore, after working with the limit definition for several days, my students had noticed and we had informally discussed the power rule.
The ActivityWith these concepts in mind, we did the following together on GeoGebra:
1) First, we entered a function in the Input bar at the bottom of the window. The function we started with was y = x^3+3x^2-1. In the algebra pane, GeoGebra relabels this as f(x). This function provided opportunities to talk about horizontal tangent lines (critical points) as well as changes in concavity (inflection points). At this point I also showed my students how to use the Move Graphics tool to center their graph in the window, and to adjust the scale on each axis to get a good picture of the function.
2) With our function graphed, we added a new point on the curve by selecting the Point tool and clicking on the curve. GeoGebra automatically labels this point A.
3) We then put a tangent line on the curve at point A. GeoGebra has a tool for this under the fourth box from the left. Once the Tangents tool is selected, click on the curve and click on the point of tangency (point A for us), in no particular order. This is a good time to show students how to use the Move tool if they haven't discovered it already. It's the cursor at the upper left of the tool bar (or pressing the Esc key will select this tool). If students click and hold point A with the move tool, they can slide it along the curve and watch the tangent line change.
4) Now we'll have GeoGebra measure the slope of the tangent line (the value of which is that of the derivative). GeoGebra has a variety of measuring tools. Once the Slope tool is selected, click on the the tangent line and GeoGebra will add a rise/run triangle and display the slope. The value of the slope gets added to the algebra pane as m.
5) We finally add one more point to the sketch, using the Input bar again. Type in (x(A),m) to define this last point, which GeoGebra will call point B. This tells GeoGebra to use the x-value from point A (the point on the curve) and to plot the slope of the curve as the y-value. This point displays the value of the graph's derivative at any x-value. By right-clicking on point B, Trace On can be selected.
With trace turned on, the path of this point will be traced out on the sketch as point A is moved along the curve, revealing the shape of the derivative function. (I also changed the color of point B using the pull-downs at the top of the graphics view.)
The completed sketch opens up a lot of good talking points. Students should be able to see that the local maximum and local minimum of f(x) line up with the zeros of the derivative function. If inflection points have been discussed, it's also nice to see that this point lines up with the vertex of the derivative. As point A is moved back and forth across the inflection point, the tangent line makes a nice rocking motion that shows the slope is beginning to increase/decrease. In general, the tangent line in the sketch provides a good visual connection between the curve of f(x) and the sign of the derivative.
Of course, students can also try to predict the shape of the derivative function, or find the derivative function algebraically and plot it on the sketch as well. Wherever possible, it seems good to keep students' visual and algebraic understandings closely intertwined. (Check out Aspinwall and Shaw's (2002) article, When Visualization is a Barrier to Mathematical Understanding, in Mathematics Teacher vol. 95, no. 9)
This GeoGebra sketch is also a good tool for students to explore derivatives for functions that they don't yet know how to differentiate algebraically. Helping my students work through this activity took less time than I had anticipated, so they had about ten minutes of class time in the computer lab after we were done with the first function. After I explained that they could change f(x) by double-clicking it in the algebra pane without altering any of the other features of the sketch, most of them started exploring other functions. Students were discovering on their own that cosine is the derivative of sine, and one of them was looking at the graph of tangent, and noticing similarities (but important differences) between the sketched derivative and the secant function. Having a tool available to explore derivatives increased their curiosity. I may pull up the same sketch again in class as we learn about the derivatives of specific types of functions.
If you would like to go to the completed sketch, here is the GeoGebra Tube link.