Classroom capsules

Farmer, Thomas A

A Classroom Capsule is a short article that contains a new insight on a topic taught in the earlier years of undergraduate mathematics. Please submit manuscripts prepared according to the guidelines on the inside front cover to Tom Farmer

Computers and Advanced Mathematics in the Calculus Classroom Kurt Cogswell (mal9@sdsumus.sdstate.edu), South Dakota State University, Brookings, SD 57006

As calculus instructors, we often work with classes consisting largely of nonmathematics majors. As a result, we spend a great deal of time and effort constructing examples and projects that demonstrate the uses of calculus in fields other than mathematics. This is great, and as an ex-physicist I’m all for it. However, now I’m a mathematician, so I’d like to propose an alternative: why not develop projects based on advanced mathematics of the sort typically seen in graduate courses? I can think of at least one good answer to that question: “I have enough trouble getting my students to use the chain rule, and now you want them to prove the Riemann hypothesis?” Well no, not really. However, my experience at the end of the first year of a typical calculus sequence is proof that the idea can work, that students can enjoy it and be motivated by it to study mathematics, and that they can gain significant understanding of topics in calculus in the process. The trick to making it work is to replace the years of “theorem-proof” experience usually prerequisite to studying advanced mathematics with a computer-aided, investigative approach.

Assigning this project was an experiment and, as it turned out, a very successful one. It led to excellent, thoughtful consultations with students and among group members as they worked. With a minimum of hints and nudging on my part, each group produced a written report of their results. The sophistication of their conclusions surprised me, and the enthusiasm with which they discussed them was invigorating. I think the project may have even caused a couple of potential engineers to see the light and consider mathematics as a career, though only time will tell.

Clearly, this experimental approach to investigating advanced concepts and gaining insights into basic concepts has much to recommend it. I can see the same idea being applied in several other freshman and sophomore level courses. For example, in a linear algebra course it could lead students into an experimental investigation of the concepts of functional analysis, or in a multivariable calculus course it could lead to a computer-aided differential geometry project. Such projects could be implemented using any of the existing popular mathematical software packages, or even developed from scratch with computer language compilers. Even the development of such a project could become a project for an advanced student with sufficient computer expertise. I hope others will try these ideas; and if they do, I trust they will experience as much success as I did.

A Natural Proof of the Chain Rule

Stephen Kenton (kentonXecsuc.ctstateu.edu), Eastem Conn. State Univ., Willimantic, CT 06226

The Average Distance of the Earth from the Sun

David L. Deever (ddeever(otterbein.edu) Otterbein College, Westerville OH 43081

In astronomy texts one sometimes finds a description of an Astronomical Unit (AU) as the “average distance of the earth from the sun.” However, consideration of the concept of the average, or mean value, of a function reveals that the parameterization (or, equivalently, the underlying metric) must be specified in order to have a precise definition of that mean value. A more careful definition of AU is as the length of the semi-major axis of the earth’s orbit. In this note we will show how the choice of parameter from among several plausible alternatives can affect the average value of the distance.

Example values. We consider ellipses with eccentricities equal to those of the orbits of the earth, Pluto, and Halley’s Comet. For ease of comparison we use a semi-major axis length of 1 for each case. The following values were calculated using built-in numeric integration on a TI-92 calculator.

Observations and Conclusions

Not surprisingly, the greater the eccentricity the greater the differences arising from the different methods. It may not have been expected that the arc length representation would always produce the length of the semi-major axis, as it did, but this will be demonstrated below. As a consequence it is correct to describe an Astronomical Unit as the average relative to arc length of the earth’s distance from the sun. Invariance over eccentricity of the arc length based average In Figure 1 we offer a proof without words that the average distance using the arc length parameterization is always equal to the length of the semi-major axis.

Shortest Path Solution by Epitrochoid Machine Mark Schwartz (mdschwarEcc.owu.edu), Ohio Wesleyan University, Delaware, OH 43015, and Darryl Adams (daadams$usa.capgemini.com), Cap Gemini America, Cincinnati, OH 45242

Geza Schay (gsWcs.umb.edu), University of Massachusetts at Boston, Boston, MA 02425

We present a new strategy of row reduction to obtain a basis for the left nullspace of a matrix (and, by transposition, one for the nullspace too). The method can also be extended to solve a linear system, and to invert a nonsingular square matrix.

Most introductory linear algebra books contain examples and exercises in which they ask for conditions on the right-hand side of a linear system to ensure consistency. The usual recommendation is to reduce the system Ax = b by elementary row operations to a form Ux = c, where U is an echelon matrix, and set the components of c that correspond to the zero rows of U equal to zero. Since c is obtained from b by elementary row operations, we thus obtain a set of homogeneous linear equations for the components of b. It is then natural to ask questions about this system such as: Are its equations independent and what characterizes its coefficient matrix?

To avoid the components of b being hidden in c and to obtain their coefficient matrix explicitly we write the system as Ax = Ib and reduce the latter or, equivalently, the augmented matrix [ A 1]. Let us look at an example:

the consistency conditions can be written as Mb = 0. Clearly the rows of this M are independent. On the other hand we know that the equation Ax = b is consistent if and only if b is in the column space of A, and so the solutions of Mb = 0 must make up the column space of A. Hence the rows of M must be orthogonal to the column space of A. It is straightforward to check that in this case indeed MA = O.

There seems to be no computational advantage or disadvantage to the proposed methods, since they require just as many operations as the standard ones. What we gain in avoiding back substitution, we lose because of the enlarged matrices. Acknowledgment. The author wishes to thank the referee for suggesting several valuable improvements.

1 We follow the convention of considering the left nu(lspace of A to be a space of column vectors.

References

1. T. Apostol, Mathematical Analysis (2nd ed.), Addison Wesley, 1975.

2. R. Bartle and D. Sherbert, Introduction to Real Analysis (2nd ed.), Wiley, 1992.

3. G. Bilodeau and P. Thie, An Introduction to Analysis, McGraw-Hill, 1997.

4. G. Hardy, A Course in Pure Mathematics (lOth ed.), Cambridge University Press, 1950.

5. W. Kosmala, Introductory Mathematical Analysis, Wm. C. Brown, 1995.

6. S. Lay, Analysis with an Introduction to Proof (2nd ed.), Prentice Hall, 1990.

7. J. Olmsted, Intermediate Analysis, Appleton-Century-Crofts, 1956.

8. W. Rudin, Principles of Mathematical Analysis (3rd ed.), McGraw Hill, 1976.

References

1. J. W. Bruce and P. J. Giblin, Curves and Singularities, 2nd ed., Cambridge University Press, 1992. Leon M. Hall, Trochoids, roses, and thoms-beyond the Spirograph, This JOURNAL 23 (1992) 20-35.

3. I. R. Porteous, Geometric Differentiation for the intelligence of curves and surfaces, Cambridge University Press, 1994.

4. Robert C. Yates, Cunes and Their Properties, reprint of 1952 edition, National Council of Teachers of Mathematics, 1974.

Thomas A. Farmer

Department of Mathematics and Statistics Miami University Oxford, OH 45056-1641

Copyright Mathematical Association Of America May 1999

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