Education Technology

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.
    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