Patrick R. Coulton

Department of Mathematics
Eastern Illinois University
Charleston, IL 61920-3099
Office: 322
Phone: (217)-581-6276
e-mail : cfprc@eiu.edu

Maple 6 Labs

Use the download capability of your browser to download the links. The links are *.mws (version 6) files and not html files.

Introductiuon

This set of Maple labs has been developed over a number of years. I have put these labs into the Maple 6 format. I am not sure what this means in terms of compatibility with earlier versions of Maple. Most of the labs were originally written for earlier versions. This merely points to a problem that seems to be endemic with commercial software. That is, the constant revision of commercial software without any real advance in the technology. It is my hope that a consortium of educators in technical areas will some day set standards for updating software that will provide some stability to the communities they serve. Until that day, I must apologize to those who do not yet have access to Maple 6.

There are, of course, many possible approaches to doing computer laboratories in calculus. I have been using Maple and Mathematica in the classroom for more than 10 years. My basic approach is seasoned by this. In these labs I have emphasized an empirical method of understanding calculus. It is my belief that the empirical approach was important in the development of calculus. By empirical I mean an approach to calculus which provides intuition concerning the theory which is gotten from experience in calculations rather than by a memorized set of manipulations. As an example, it is worthwhile to prove the power rule for a few low dimensional cases in class. One can go through the same essential proofs simplifying the differential quotient for a large number of cases in a very short time using Maple. The pattern should become obvious. More importantly, working at a computer is less passive than hearing a lecture. But note that the emphasis here is on the student forming a conjecture based on experience with a large number of cases. Surely this is how a great deal of mathematics is done. At the same time it should be easier to follow a general proof of the power rule in the lecture if the student already has some intuition.

A second example of this approach is to demonstrate each new integral method by means of an actual Riemann sum. Repeated use of the converging Riemann sum ,to prove that integration works, should give the student a better understanding of the theory of integration and the practical aspect of solving non-trivial integrals numerically.

It is true that the calculus of Newton and Leibniz does not exhibit this notion of empirical calculus, but we must remember that Newton wrote his calculus from a geometric point of view emulating Euclid and the Greek geometers. Moreover, Newton wrote his calculus long after he had made his important contributions. The development of calculus by Barrow and Newton was symbolic rather than geometric. Liebniz on the other hand was interested primarily in the universal, symbolic, and manipulative character of calculus. Leibniz's methods dominated Continental mathematics until contradictions occurred which cast into doubt this naive approach. These contradictions led to the modern theory of limits, differentiability and Riemann sums.

Calculus I

  1. The Definition of Limits
    This interactive lab asks the student to make a conjecture concerning the relation between epsilon and delta in the definition of limits.

  2. The Definition of Continuity
    This interactive lab guides the student through some simple examples of discontinuous functions and discusses right and left hand limits in Maple.

  3. Power Rule
    This lab uses Maple's expand and simplify commands to develop an understanding of the patterns inherent in the power rule for differentiation.

  4. Newton's Method
    This lab attemps to emphasize the relation between the linear approximation to a curve (or tangent line) and Newton's Method.

  5. Critical Points of Polynomials
    This short lab uses Maple's plot command to compare a polynomial function with its derivative. The student should observe the critical points from the roots of its derivative, then use Newton's Method to approximate the roots.

  6. Inflection Points of Rational Functions
    This lab uses maple's plot command to compare a polynomial function with its higher order derivatives. Using Newton's Method the student can approximate the roots of the second derivative thereby finding the inflection points.

  7. Symmetry of Even and Odd Functions
    This lab investigates the symmetry properties of even and odd functions using a variety of examples. The student is encouraged to make conjectures regarding products and quotietns

  8. Riemann Sums
    We calculate the Riemann sum using the left-hand, right-hand and mid-point of the intervals. The student compares the methods and observes the relation between actual error and the number of terms in the sum.

  9. The Fundamental Lemma of Calculus
    The lab demonstrates that the Fundamental Lemma of Calculus can be obtained using Maple and that the proof using limits of integral functions or Riemann sums can be obtained in a somewhat empirical way.

  10. Integrals on Symmetric Intervals
    The lab investigates the integrals of even and odd functions on symmetric intervals using Riemann sums and actual integrals

  11. The Disk Method
    This lab investigates the Reimann sum applied to a disk method problems. The student observes that the volume formulas for the cone and hemisphere can be obtained.

  12. The Shell Method
    This lab uses Riemann sums and the shell method for the cone and the hemisphere.

  13. Work
    This lab investigates the Riemann sum approximation for the work integral using a standard water tank problem.