# Activities

• • • ##### Subject Area

• Math: Calculus: Derivatives
• Math: Calculus: Limits of Functions

• ##### Author College

60 Minutes

• TI-86
• ##### Other Materials
This is Activity 5 from the EXPLORATIONS Book: Using the TI-86 in Collegiate Mathematics: A Tutorial
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Differential Calculus

#### 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.

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