Derivative Example (Calculus Intro)

This is a classroom activity in which the "slope of a curve" is calculated and graphed at many points along the curve. The concept of a "derivative function" that passes through all those points is developed, and an equation for that function is sought. ~ ~ ~ Each student in a class is assigned an x value. Student names are entered in the GeoGebra spreadsheet accordingly. For each student the task will become: Find the "slope of the curve" at that x value (or more specifically, the slope of the line that is tangent to the curve at that point). Teacher chooses one x value to use as an example before asking students to perform the steps individually. On a Texas Instruments TI-83 or TI-84 graphing calculators, start by pressing:
  • Zoom
  • 4: ZDecimal
Then press
  • Trace
  • (enter the assigned x value)
Then repeatedly press the following sequence of keys:
  • Zoom
  • 2: ZoomIn
  • Enter
  • Trace
Repeat that last sequence of four keystrokes until you've zoomed in far enough for the curve to appear to be a straight line. This is a good time to discuss the concept of "local linearity." Record both coordinates for the assigned x-value as (x1, y1). Still in the Trace feature of the calculator, arrow either to the left or right a bit, landing on a nearby point. Record the new coordinates as (x2, y2). The slope of the line tangent to the curve may be approximated by calculating the rise/run between these two points, right? (Discuss as a class.) Discuss how many decimal places would be appropriate for this calculation. Discuss whether to (1) write the coordinates on paper and then type in the calculation or (2) exclusively use calculator techniques to perform such calculations without having to write anything on paper. Calculate that slope and enter into the "tangent" column of the GeoGebra spreadsheet. A point will automatically plot in the graphics region, graphing the slope as a function (y-value) for each x-value. Notice that the point (x,y)=(0,2) has already been entered into the defaults spreadsheet, so its point is already plotted. Discuss to reinforce student understanding of how the y-values of this plot are giving the slope of the original curve. The teacher may choose to enter the data as students call out their results aloud. Alternatively, a copy of this Google Sheet may be shared with the class, and students may enter their results themselves. At various points the teacher may then simply copy/paste from the Google Sheet into the GeoGebra spreadsheet. As a majority of the points get plotted, the class may start to speculate about what the "derivative curve" will look like. Try to identify and correct outliers that don't seem to fit an overall pattern. Enter an equation for the "derivative curve" and transform as needed until you find an equation that appears to fit the collected data. Repeat the activity for other functions. Rejoice, go learn some more wondrous Calculus!