Solving an expected value problem without using geometric series

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**Donald J. Albers, An Interview with Tom Apostol**

Apostol's two-volume Calculus and his Mathematical Analysis have had a strong influence on a generation of teachers. In this interview he recalls his early years in Utah, as the first child of a Greek immigrant shoemaker and his mail-order bride, and traces his schooling and early career. His candid observations about his teachers and his colleagues at M. I. T. and Caltech make not only an entertaining contribution to the mathematical folklore; they raise interesting questions about child-rearing and education.

**John S. Robertson, Gudermann and the Simple Pendulum.**

The Gudermannian function gd(x) is a poor relative of the circular and hyperbolic functions, namely, the inverse tangent of sinh(x). It is shown that the motion of a simple pendulum released from its equilibrium position with just enough kinetic energy to reach the inverted position is naturally described by the Gudermannian function. This nonperiodic solution of the pendulum equation can be viewed as completing the family of periodic solutions classically expressed by elliptic functions. The article is accessible to students who have only a brief acquaintance with differential equations.

**Brian J. Loe and Nathaniel Beagley, The Coffee Cup Caustic for Calculus Students**

The coffee cup caustic is the cusped curve one sees in the bottom of a coffee mug when light from a point source shines in obliquely. That this curve is a portion of a certain epicycloid is shown in three ways, each using an interesting mix of ideas from calculus. The first way, viewing the caustic as the envelope of a family of parallel light rays reflected from a semicircular mirror, requires only L'Hospital's rule. Next, the caustic is shown to be the locus of the focal points of the osculating parabolas (second-order Taylor polynomials) for the semicircular mirror. Finally, it is shown that the caustic is the locus where the Jacobian determinant of a certain mapping on the plane vanishes, and this result is generalized to prove that the caustic formed by the reflection of parallel light rays from any smooth curve is the locus of the foci of the osculating parabolas for the curve.

**Classroom Capsules**

Leonard Gillman, Order Relations and a Proof of l'Hospital's Rule.

The epsilon-delta definition of the limit of a function refers to metric properties of the real number system. But in most circumstances in freshman calculus order concepts are sufficient; metric details are an irrelevant distraction. Saying that the limit of f at x=a is a number L means that if A and B are any numbers satisfying AHolly P. Hirst and Wade T. Macey, Bounding the Roots of Polynomials.

Two simple bounds on the roots of a polynomial in terms of its coefficients are derived and compared. Such bounds are useful for determining an appropriate graphing window when the graph of the polynomial is to be found using a computer or graphing calculator. These bounds are well known, but they usually are buried deep inside complex analysis or numerical linear algebra textbooks. Here only the triangle inequality is needed to prove one bound; the other requires as well the formula for the sum of a geometric series.

J. Marshall Ash, Neither a Worst Convergent Series nor a Best Divergent Series Exists.

A simple geometric proof is given for the well-known result that for any convergent series with positive terms there is always a convergent series with much bigger terms, and for any divergent series there is always a divergent series with much smaller terms. The proof just compares the area under an appropriate curve with the sum of the areas of a family of approximating rectangles.

Russell Euler, A Note on Taylor's Series for sin(ax+b) and cos(ax+b).

A simple form is found for the coefficients of the Taylor series expansions indicated, centered at an arbitrary point.

**Student Research Project**

**Irl C. Bivens, Multiple Derivatives of Compositions: Investigating Some Special Cases.**

The student is led to discover that the nth derivative of the composite exp(f(x)) is the product of this function and a certain polynomial in the first n derivatives of f. This polynomial function is shown to have close ties with classical topics in combinatorics. **Computer Corner**

**Paul Vodola, Adventure Games, Permutations, and Spreadsheets.**

Many multimedia adventure games contain puzzles that players must solve in order to proceed, and students enjoy seeing how these can be solved by applying ideas they have learned in their mathematics classes. A common example is a rearrangement puzzle, an inverse problem in which the puzzle designer has applied to a given configuration an unknown permutation and the solver must discover the inverse permutation that unscrambles the configuration. A nontrivial example of this type, the pinball puzzle from the game Shivers, is modeled by a directed graph and solved by using a spreadsheet to systematically generate possible sequences of moves and search for the required unscrambling permutation. Several related games and their associated puzzles are discussed.