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Investigating the Nature of Function Derivatives
Fred Mueller, fmueller@dentonisd.org
Objective(s):
Students develop an appreciation for the graphical representation of the derivative of a function compared with the graph of the function.
Activity Overview:
This activity is intended for the AP Calculus classroom. The supplied GDB files are GDB1 for the Sine function, GDB2 for the Natural Exponential function, and GDB3 for the Cubic function. GDB files for other functions can be saved and also included in the activity. Students are presented with a function on the Graph Tab of Activity Center. The same function is sent to the student calculators as one of the GDB files. Students then investigate the graph of the derivative by returning a set of (x, dy/dx) data points. As students return data points for the derivative, they are plotted on the Activity Center screen and a graphical picture of the derivative evolves. Students are then encouraged to enter into discussions comparing the function and derivative graphs, and among other things the implications of function extrema, zero and undefined slopes, points of inflection, and critical points on the function.
Activity Time: 15 to 30 minutes per function
Before the Activity
Students at this stage are expected to have an appreciation for the value of the derivative at a point on a function being equal to the slope of the tangent line. Derivative data can be developed by having the students manually estimate the slope of tangent lines using the Manual-Fit regression analysis option under the STAT | CALC menu; or by simply using the dy/dx option under the CALC menu. The Manual-Fit approach has the benefit that students grow to better appreciate the value of the derivative as the slope of the tangent line. To speed up the activity and to ensure that the entire domain interval is adequately covered, individual students should be assigned a particular interval to investigate.
TI-Navigator Activity
Load the TI-Navigator program and click on the Activity Center icon. Clear any previous activity data by selecting Edit | Clear Activity Data, then load one of the activity files and choose the Graph tab within Activity Center.
On the TI-Navigator home screen click on the Begin Class button. Have students log into NavNet. Let the students know that they will be receiving several GDB files.
Send the GDB files to the student calculators. Depending on where the files are stored they can be transmitted from either the computer running TI-Navigator, or directly from the Teacher calculator. Although the Activity Center screen is showing only one function, to save time if several different functions are to be investigated, all of the GDB files should be transmitted at the same time.
Once all students have received the GDB files have them exit from NavNet.
Have students load the GDB file corresponding to the function displayed on the Activity Center screen. Students may not be familiar with storing and recalling GDB files. From the Home screen on their calculators, have students select DRAW | STO | RecallGDB. The RecallGDB command will transfer to the Home screen. Type the appropriate GDB number and press ENTER. Students can press GRAPH to make sure they have loaded the function corresponding to the Activity Center display.
Students now collect and record their (x, dy/dx) derivative data. Monitor student understanding and progress using Screen Capture.
After students have had sufficient time to each record about 5 data points, have them log back into NavNet and go to Activity Center. Click on Start Activity.
Students enter their data into the L1 and L2 lists and then send the data to Activity Center and watch the graph of the derivative evolve. The L1 list should have the x values, while the L2 list should have the corresponding dy/dx values.
After the Activity
Students should be encouraged to enter into discussions comparing the function and derivative graphs, and among other things the implications of function extrema, zero and undefined slopes, points of inflection, and critical points on the function.
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