Harris S. Shultz

The Function sin x/x.

Harris S. Shultz, Professor of Mathematics at California State University, Fullerton, is the author of more than forty published papers and a textbook, Mathematical Topics for Computer Instruction. He has been Co-Principal Investigator of the South? ern California Mathematics Honors Institute, the Mariana Islands Mathematics Institute, and the C3 Teacher Training Project, all funded by the National Science Foundation. In 1988 he was named Outstanding Professor at California State University, Fullerton, and in 1989 he was the recipient of the California State University Trustees Outstanding Professor Award.

Dependent Probability Spaces.

Ray Shiflett (rcshiflett@roadrunner.com) received his Ph.D. from Oregon State University. He has published in operator, measure, matrix, and number theory, topology, optometry, science fiction, and mathematics education. He has served as chair of mathematics at Wells College, Dean of the College of Science at Cal Poly Pomona, and Executive Director of the National Research Councils Mathematical Sciences Education Board. He enjoys golf, fly fishing, writing songs, and woodworking.

M&m Sequences.

Harris Shultz (hshultz@fullerton.edu), Professor of Mathematics at Cal State Fullerton (Fullerton, CA 92834), earned his Ph.D. from Purdue University. He received the MAA’s Southern California Section Award for Distinguished College or University Teaching in 1992. He has directed numerous institutes for secondary mathematics teachers and currently is co-directing and designing a project providing online professional development for middle school mathematics teachers.

Tugging a Barge with Hyperbolic Functions

For many years now, the problem of determining the arc of a hanging cable has been used in calculus texts as an application of hyperbolic functions. This example is certainly important as it indicates the use of hyperbolic functions in mechanics for engineering students. However, as seen in [1], there is another and perhaps simpler example of the application of hyperbolic functions. This example, based on towing a barge, also makes use of many more properties of the hyperbolic functions. Similar models, such as pulling a trailer, have appeared [6]. The problem can even be traced back to Leibniz, who posed it in terms of pulling a watch on a chain. Whatever the form of the problem, the path of the object has become known as a tractrix (see [2, p. 428]), and one could view the model as an early example in the theory of pursuit problems. In the next section we briefly review a few of the ways in which hyperbolic functions arise. The hanging cable problem is presented, but as a problem in the calculus of variations, rather than using the approach based on balancing forces. However, to show the idea of balancing forces, we develop the model of the hyperbolic arch. Then the barge problem is developed as an exercise in the use of the hyperbolic functions. In the process, we will see another natural and interesting way in which the Gudermannian function appears (see [4]).