### Articles

**Prelude to Musical Geometry**

Brian J. McCartin

When notes that differ by one or more octaves are considered equivalent, the twelve tones corresponding to the keys in an octave on a piano can be pictured as equally spaced points on a circle, like the numbers on the face of a clock. Exploration of this simple geometrical model yields many insights into the structure of classical tonal music. Rotations or reflections of the clock points representing a set of tones correspond to the musical operations of transposition or inversion of the associated chord. A special feature of the major scales is exposed, indicating a possible reason for the wide appeal of diatonic music (music that employs these seven-tone scales). This introduction to fundamental ideas in the mathematical foundations of music theory will appeal to students of both mathematics and music, arts whose practitioners search for and explore pleasing patterns.

**A Simple Decision Rule for a Guessing Game**

Luiz Felipe Martins

A contestant receives an initial amount of money, say *r* dollars, and then tries to guess a number chosen at random among the integers from 1 to *n*, paying a fee of 1 dollar before each guess. This game is favorable to the player if *V*(*n*), the expected number of guesses needed to guess the mystery number, is less than the inital stake *r*. A closed formula for *V*(*n*) is found, by clever use of recursion. Then the values of *n* for which *V*(*n*) is a positive integer are determined, which provides a rule for quickly determining whom the game favors, for any integer stake *r*.

**The Long Arm of Calculus**

Ethan Berkhove and Rich Marchand

A simplified model treats the space shuttle's robot arm as two hinged segments constrained to move in a plane, the free end being used to grasp a satellite. The plane region from which the robot arm can retrieve a payload is found and the corresponding torque on the joint where the arm is attached to the shuttle is determined. Finally, the optimal procedure for delivering a captured satellite to the shuttle bay is found. This apparently straightforward student project makes use of a surprising variety of ideas from multivariable calculus: cross products, contour plots, the mapping associated with a function, Lagrange multipliers, even setting up and solving a differential equation. The project helps students develop modeling skills, for at every turn they must carefully consider both physical and mathematical aspects of their results.

**Differential Forms for Constrained Max-Min Problems: Eliminating Lagrange Multipliers**

Frank Zizza

The Lagrange multiplier method for multivariable optimization introduces a new variable for each constraint. But constraints reduce the degrees of freedom, so in principle they should not increase the number of variables in a problem. Lagrange's condition for an extremum is first rephrased by replacing the gradients of functions by differentials. Using the wedge product of vectors, it is shown that the critical points for a function of n variables subject to a system of constraint equations satisfy a system of n equations in n unknowns that is as easy to write down as the Lagrange equations. Specific examples are used to compare the methods. This concrete application provides a natural way to introduce differential forms in the introductory calculus sequence, making full use of their invariance under change of coordinates. The discussion also clarifies why the Lagrange multiplier method works.

**Computation of Planetary Orbits**

Donald A. Teets and Karen Whitehead

Many calculus textbooks derive Kepler's laws of planetary motion from Newton's law of motion and the universal inverse square law of gravitation. In this article a variety of techniques from the typical second semester calculus course follow this up to show how the orbit of a planet can be determined from just two observations. Specifically, given the position vectors from the sun to a planet at two different times, the authors show how to determine the orbit parameters that describe the shape of the elliptical orbit, its orientation in space, and the position of the planet along the orbit at any time. At a key step the trapezoidal rule is used to replace an integral by a numerical approximation, but astronomical data is used to check that the method gives useful results.

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Classroom Capsules

**J. B. Thoo, A Picture Is Worth a Thousand Words**

A simplified version of the diagrams used by G. Strang to illustrate the four fundamental subspaces associated with a matrix is introduced. Six examples show how these diagrams can illustrate important results in linear algebra such as the rank-nullity theorem, least squares approximation, and the pseudoinverse of a matrix.

**Bennett Eisenberg, How Much Should You Pay for a Derivative? **

A simple model for the behavior of a stock price is used to introduce the analysis of financial derivatives such as stock options. How much should one be willing to pay today for an option to buy a stock on some future date at a set price? Should the price of the option be the expected value of the stock on the expiration date of the option? Taking into account the possibility of buying the stock at todayÕs price, and allowing use of borrowed money to create packages of stock and debt that will duplicate the performance of a stock option, the unique price of an option is determined. This simplified example introduces the important idea of financial arbitrage and alerts students to practical situations when the use of expected values for decision making is inappropriate.

**Saad M. Adnan, Candies and Dollars**

A boy with a jar of candies consumed one candy the first day and gave away 10% of the remainder. On the second day he ate two candies and gave away exactly 10% of the remaining number. If he continued in this fashion until the jar was empty, how many candies did the jar originally contain? A generalized version of this puzzle problem is solved by a recursive analysis that involves summing a finite geometric series and then differentiating the resulting rational function! A variant in which the jar of candies is replaced by dollars in an account and dollars are added each day, rather than being consumed, is posed for readers to solve.

**Dan Kalman and James White, A Simple Solution of the Cubic**

In trying to solve an unrelated problem the authors came upon an identity that they realized could be applied to solve any reduced cubic equation. Their method is described and shown to be closely related to one published 75 years ago but not widely known. The note ends with a suggestion for an easy-to-remember procedure for solving cubics.

**Marc D. Sanders and Barry A. Tesman, MATH and Other Four-Letter Words**

An amusing student project asks students to work in groups to choose a four-letter word and find parametric equations which when graphed will spell out this word. The difficulty of the project can be adjusted by specifying in greater or less detail various conditions the parametric equations or their graphs are required satisfy.

**Michael W. Ecker, A Novel Approach to Geometric Series**

Imagine that you are to receive a gift that is subject to a tax at the rate r, where 0 a, how much should her bequest be? From one point of view the total bequest $*S* should satisfy *S* = *a* + *ar* + where each term is the tax owed on the preceding term. On the other hand, since the amount received is the bequest minus the total tax, we have *a* = *S* - *rS* . Solving this equation for *S* then gives the formula for the sum of a geometric series, for 0 r

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Computer Corner

**Milton P. Eisner, The Probability of Passing a Multiple-Choice Test**

If a student knows a certain percentage of the material to be tested on a multiple-choice test and guesses at random among the possible answers to questions on the unknown material, what is the probability he will achieve a passing score on the test? The test can be modeled as a binomial experiment with the number of questions, the number of possible answers to each question, and the minimum passing score as parameters. Using a spreadsheet to compute the probabilities, the effects of varying each of the parameters are investigated by examining graphs of the probability of passing as a function of the student's knowledge level. Variants and extensions of the model, such as penalties for guessing, are then considered.

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Software Review

*IDEA: Internet Differential Equations Activities*, is a website at Washington State University that serves as a repository for resources and teaching materials in differential equations and dynamical systems. This review discusses several *IDEA* modeling projects that are ready for classroom use, and *DynaSys*, a Microsoft Windows package for investigating systems of differential or difference equations that can be downloaded from the *IDEA* site at no charge.