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