Vol. 38, No. 5, pp. 330-408
Pursuit Curves for the Man in the Moone
Andrew J. Simoson
This article considers an old version of the classical pursuit problem as posed by Francis Godwin in 1599, who imagines a wedge of swans flying from the earth to the moon in twelve days, always flying at constant speed toward the moon. The return trip, during which the swans always fly toward the earth at the same speed, takes eight days. A little calculus is used to analyze the consistency of these flight times using both an earth-moon model and a sun-earth-moon model.
More Mathematics in the Bedroom: A Paradoxical Probability
Paul K. Stockmeyer
A standard mattress can be positioned on a bed frame in any of four orientations. Suppose that four times a year the mattress is rotated into one of the three possible new orientations, chosen at random. According to an article in The American Scientist, a computer simulation suggests the rather surprising result that over a period of 10 years, the most frequently occurring orientation occurs 31 percent of the time, while the least frequently occurring orientation occurs just 19 percent of the time. This paper contains an investigation of this phenomenon, which supports the claim of the simulation. Along the way, it considers a similar but more easily explored problem in coin flipping, deriving the distribution functions for the more frequent and less frequent outcomes of heads and tails for a sequence of n independent flips of a fair coin.
Everyone knows what makes a 3-4-5 triangle special, but how many know what makes a 4-5-6 triangle special? It is an integer-sided triangle in which one angle is twice another. It is enjoyable to search for these things, but for those who are impatient, this article derives explicit polynomial formulas that generate all of the basic examples of triangles with integer sides and one angle a rational multiple of another.
Do Dogs Know Bifurcations?
Roland Minton and Timothy J. Pennings
When a dog (in this case, Tim Pennings' dog Elvis) is in the water and a ball is thrown downshore, it must choose to swim directly to the ball or first swim to shore. The mathematical analysis of this problem leads to the computation of bifurcation points at which the optimal strategy changes.
Partial Fractions in Calculus, Number Theory, and Algebra
C. A. Yackel and J. K. Denny
This paper explores the development of the method of partial fraction decomposition from elementary number theory through calculus to its abstraction in modern algebra. This unusual perspective makes the topic accessible and relevant to readers from high school through seasoned calculus instructors.
An area Approach to the Second Derivative
In this note we prove an alternate limit formula for the second derivative. The formula is based on a geometric construction, and as a corollary an unusual characterization of parabolas is obtained.
Saddle Points and Inflection Points
Félix Martinez de la Rosa
In the study of surfaces in multivariate calculus, we notice some similarities between saddle points on surfaces and inflection points on curves. In this note, we make a direct connection between the two concepts.
Conic Sections from the Plane Point of View
Sidney H. Kung
With an appropriately chosen o-xyz system and property of the cone, the author has obtained an equation of the curve of intersection of a right circular cone and an inclined plane in the form x2 + Ay2 + By2 + Cy + D = 0, where the coefficients involve the vertex angle of the cone and the angle of the cutting plane. This form is the same as that derived from the locus definition of conic sections.
The Convergence Behavior of
Cong X. Kang and Eungeong Yi
We classify the convergence behavior of the one-parameter family in the title using only what is taught in introductory calculus courses.