Experiments in Mathematics Using Mathematica and Maple

C.T.J. Dodson Homepage and E.A. Gonzalez Homepage


Introduction

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We believe that to understand and be comfortable with mathematical concepts and methods, it is necessary to do mathematics and, traditionally, doing meant with a pencil and paper. Now, quite a modest home computer can provide a platform for a powerful computer algebra package like Mathematica, which can perform all of the operations encountered in high school mathematics and beyond, and provide graphical representations of functions, including animations.

The capability of rendering very accurate graphics for mathematical functions greatly enhances the learning experience, and helps intuition work in new situations, before beginning to do the algebra and calculus needed to solve a problem. For example, if you can see that a function has a clear minimum from its graph, then you are more likely to be able to identify its location precisely by analysis. The materials described in the various sections accessible through this location were developed as part of the Ontario EDNET distance learning initiative at the University of Toronto for high school High School Pre-Calculus and Calculus syllabuses, and as part of the mathematical courseware development program for the Faculty of Applied Science and Engineering at the University of Toronto for the calculus and vector calculus syllabuses: University Calculus and Vector Calculus.


Availability of Interactive Notebooks

The interactive Mathematica notebooks that we have developed for the two senior high school years are available at

    http://www.ma.umist.ac.uk/kd/ednet/maths/mathematica/

in the directories PreCalc and Calculus. The following table provides direct hypertext links to the different sections:

   Precalculus Workbooks:          Calculus Workbooks:

      Problem Answers               Problem Answers
      Working Material              Working Material

Note that each file is available in three formats: normal and two compressed (Z and gz). In general compressed files are smaller (gz is the smallest) and will download faster; however, you must be able to uncompress the file once you have your own copy. Select the file of your choice using your browser's method of downloading files to your local disk. For example, in Netscape button 3 can be used to select a file for downloading.


Outline and Organization

They take you from basic algebra, functions and sequences, to calculus and its applications. The notebooks are live documents and allow addition of your own examples and explorations---just like a traditional exercise book---so you can expand the study of any particular themes that interest you, adding more topics, graphs and animations. Each of our notebooks contains open-ended experiments with some hints for project work. In fact, all mathematics on a computer is experimental mathematics and we hope that you get in the habit of trying to check results by analytic methods!

In a number of the notebooks we have included animations of families of functions to illustrate the role of parameters, you will easily find ways to develop these animations for your own interests.

The software Mathematica is provided in many schools, colleges and universities, and student editions are available at low cost. Moreover, Mathematica is one of the standard mathematical packages used by working scientists, engineers and teachers.


Related Maple Packages

Parallel materials have also been written using Maple and are available at

    http://www.ma.umist.ac.uk/kd/ednet/maths/maple/

in the directories PreCalc and Calculus. The following table provides direct hypertext links to the different sections:

   Precalculus Workbooks:          Calculus Workbooks:

      Problem Answers               Problem Answers
      Working Material              Working Material

Note that each file is available in three formats: normal and two compressed (Z and gz). In general compressed files are smaller (gz is the smallest) and will download faster; however, you must be able to uncompress the file once you have your own copy. Select the file of your choice using your browser's method of downloading files to your local disk. For example, in Netscape button 3 can be used to select a file for downloading.


Advanced Mathematica Notebooks

More advanced Mathematica notebooks, aimed at first and second year college and university students were developed by D.C.M. Burbulla and C.T.J. Dodson and are available from http://mathsource.wri.com and from the location http://www.ma.umist.ac.uk/kd/stmath/stmath.html



Comments welcomed at dodson@umist.ac.uk