### Technology for Math 192

Mathematical experiments
Almost every aspect of the practice of mathematics, both pure and applied, has been improved and amplified in recent decades by the widespread availability of "computer algebra systems", CAS. This technology is much more than just algebra, of course. It is systematic and powerful programs that permit algebraic manipulation
• What is the coefficient of x6y4z2 in (x+y+z)12?
13860
and numerical approximation
• What is an approximation to the only root of 3x+cos(2x2)=0?
-.3258460227
and graphing
• How often do the curves y=x2 and y=2x intersect?

The freedom to work with "exact" symbolic computation, with numerical approximation (with specified accuracy) and with visual display of data (human beings learn much more from pictures than from lists of numbers!) is very useful.

Almost all of my teaching and research is now improved by access to powerful programs which allow me to experiment. I can get examples which are useful for instruction. I've also used these programs to try to understand complicated phenomena which I could not easily explain (further explanation is available!).

Computer help
Many students have graphing calculators. These are useful, but are limited by speed and memory size. Simple errors may occur. There are large computer programs with powerful numerical, symbolic, and graphical capabilities. These still may have the potential for errors, but much effort has gone into their programming. The most widely distributed programs are Maple, Mathematica, and Derive. In this course Maple will be favored, since almost every large computer system at Rutgers has Maple installed. These programs are not infallible but they can be very helpful. Other programs available with special capabilities. For example, Matlab, a program originally directed at problems of linear algebra, is installed on systems of the Engineering School.

Also, I usually use Maple because I have a copy at home. Students should learn how to use such programs.

Maple in Math 192
I think it would be useful to become familiar with Maple. It will help in Math 192 (you can check your homework!). Maple is not required, but ... it likely will also be useful in other science and engineering courses. Even if you later work with, say, Matlab, I think that familiarity with a variety of programs is an asset.

The Maple exercises
If you work through these pages, I think you will have done enough with Maple to at least understand when it is likely to be helpful.

 General introduction 8/30/2005 Playing with arithmetic 8/30/2005 Playing with algebra 8/30/2005 Playing with calculus 8/30/2005 Playing with graphs 8/30/2005

The answers to the questions above were obtained with the following Maple instructions. Please: this is not an effort to impress you, but rather to show you how easy is is to get the answers.

• coeff(coeff(coeff((x+y+z)^12,x^6),y^4),z^2);
The command coeff(P,monomial) finds the coefficient of the monomial in the expression P. Layering three repetitions of coeff finds the desired coefficient.

• fsolve(3*x+cos(2*x^2)=0,x);
fsolve is a general "floating point" approximate equation solver. Care must be used if there's more than one root. There are also symbolic solvers, useful when there is a nice formula for the solution.

• plot({2^x,x^2},x=-2..5,thickness=2)
The plotting programs are wonderful. In several variable calculus, the three-dimensional plots will be very helpful.

Programming in Maple
Maple is also a programming environment. Maple programs are known as procedures. The language has many statements supporting program flow such as if ...then and while and do etc., and also has a variety of data types. There's no time in this course to teach this material, but students should know that programmng is possible.

I have several books on Maple programming. My current favorite is Maple A comprehensive introduction by Roy Nicolaides and Noel Walkington, Cambridge University Press (\$65, 484 pages, available for less in places on the web). There are also many web pages which discuss programming in Maple. For example, here's one online tutorial. Warning: such pages are only for the enthusiast!