# Applets and Activities for Real Analysis

## Description

This suite of five interactive applets (written with GeoGebra) allows exploration of definitions and theorems commonly presented in first-year analysis courses. Topics include sequence convergence, continuity at a point, the Mean Value theorem, Taylor series, and Riemann sums. Included with each applet is a pair of activities: one for becoming comfortable using the applet, and one for using the applet to explore the associated topic in depth.

Lee Stemkoski and Christopher Storm

GeoGebra is an open source exploration/development tool that allows the creation of applets using a Java-based "player" understood by most browsers. If you have difficulty loading the applet, see www.geogebra.org for more information on the system/browser/java requirements for viewing GeoGebra applications.

## Sequence Convergence

This applet and accompanying activities illustrate the formal definition of a sequence converging to a limit L through a series of interactive exercises.

Open the pdf file and applet in separate windows (or print the pdf file instead) so you can work through the suggested activities.

Click here to download all five LaTeX activity files as a single compressed file. These have the same content as the posted pdf files. They are included in LaTeX form for those who would like to edit them for their own use.

## Continuity

This applet and accompanying activities illustrate the formal definition of continuity of a function at a point through a series of interactive exercises.

Open the pdf file and applet in separate windows (or print the pdf file instead) so you can work through the suggested activities.

Click here to download all five LaTeX activity files as a single compressed file. These have the same content as the posted pdf files. They are included in LaTeX form for those who would like to edit them for their own use.

## Mean Value Theorem

This applet and accompanying activities illustrate the Mean Value Theorem through a series of interactive exercises. The user can, for a given secant line, scroll across values of x until a point is found where the tangent line at x is parallel to the secant line.

Open the pdf file and applet in separate windows (or print the pdf file instead) so you can work through the suggested activities.

Click here to download all five LaTeX activity files as a single compressed file. These have the same content as the posted pdf files. They are included in LaTeX form for those who would like to edit them for their own use.

## Taylor Series

This applet and accompanying activities illustrate the Taylor approximation to a function through a series of interactive exercises. The applet allows the user to see the maximum error that an approximation makes over an interval.  The order of the approximation, the center of the approximation, and the radius of the interval can all be changed by the user.

Open the pdf file and applet in separate windows (or print the pdf file instead) so you can work through the suggested activities.

Click here to download all five LaTeX activity files as a single compressed file. These have the same content as the posted pdf files. They are included in LaTeX form for those who would like to edit them for their own use.

## Riemann Sums

This applet and accompanying activities illustrate Riemann sums through a series of interactive exercises. The applet allows the user to visualize the approximation of an integral with a number of rectangles created by a random partition. The number of subintervals can be varied by the user, and new random partitions can be generated automatically.

Open the pdf file and applet in separate windows (or print the pdf file instead) so you can work through the suggested activities.

Click here to download all five LaTeX activity files as a single compressed file. These have the same content as the posted pdf files. They are included in LaTeX form for those who would like to edit them for their own use.