Activity Overview
In this activity, students learn to solve problems dealing with limits, derivatives, and optimization. They use various features of the TI-86 to study some important ideas covered in a differential calculus course.
Before the Activity
Set up the calculator as explained in the activity
See the attached PDF file for detailed instructions for this activity
Print pages 65 - 78 from the attached PDF file for the class
During the Activity
Distribute the pages to the class.
Follow the Activity procedures:
Graphical and Numerical Investigation of limits:
Set the window and graph y1 = (sin x) / x
Trace the graph near x = 0 and note that y1 is undefined for this point
Turn off the axes, and use the Table feature to recognize the fact that the limit of the expression as x tends to 0 is 1
Understand the reason for using radian measure for calculating limits in calculus
Evaluate the limit by using the Seq function and other alternative methods
A Piecewise Function Limit Problems:
Enter a piecewise function in the graphing editor and observe the graph
Use the Trace tool to decide whether the limit of the function when its variable tends to 1, exists
Verify the observation with the help of the Table feature
Difference Quotients:
Use the difference quotient to approximate the value of the derivative of a function for a given value of the variable
Obtain this derivative using the TANLN option and draw a tangent line at the appropriate point
Compute the first and second derivatives of a function using the der1 and der2 commands
Use these commands to graph functions
Introduction of the Derivative Commands:
Explore the limitations of the nDer and der1 commands, and recognize the fact that an absolute value function is not differentiable at any point
An Optimization Problem:
Enter the function and graph with xRes = 1
Enter Bound limits and find the derivatives
After the Activity
Students complete the exercises on the Activity sheet.
Review student results:
As a class, discuss questions that appeared to be more challenging
Re-teach concepts as necessary