The derivative of eax is aeax and its second derivative is a2 eax, so its 3/2 derivative clearly should be a(3/2) eax. But what are we to do with the 3/2th derivative of x2 ? Unexpected complications arise when you attempt to make a sensible definition of fractional derivatives, but the authors make everything clear. Fractional derivatives, by the way, have a surprisingly large number of applications.
The Pop-up Cuboctahedron
Binomial coefficients arrange themselves in two dimensions in Pascal's triangle, and trinomial coefficients array themselves on the vertices of a cuboctahedron in three dimensions. So, it would be nice is we could make cuboctahedrons. The author shows how a model can be constructed with cardboard, rubber bands, and some skill.
Modeling the Gait of a Running Animal
No matter how many legs you have, when you run you go forward and up for a while and then forward and down. A simple model of running, more quantitative than that but using only basic ideas of first-semester calculus, gives results that agree with observations of running animals.
Recursions that Produce Pythagorean Triangles
Peter W. Wade and William R. Wade
Everyone knows a few Pythagorean triangles, at least the 3, 4, 5 and 5, 12, 13 triangles (though the 8, 15, 17 triangle suffers from unwarranted neglect), but not everyone knows how to find all of them, especially not in a systematic manner. The authors show how to find all Pythagorean triangles of a given height, where "height" is defined to be the difference between the length of the hypotenuse and the longest leg.
is the Minimum Value for Pi
C. L. Adler and James Tanton
A circle consists of all points at the same distance from a center, and [pi] is the ratio of the circle's circumference to twice that distance. "So what else is new?" you may think, but you may not have thought that different ways of measuring distance give different values of [pi]. The authors show that 3.14159265358979... is special.
The Eigenvalues of an Infinite Matrix
Finite-dimensional matrices have finitely many eigenvalues, and it should be no surprise that infinite-dimensional matrices have infinitely many eigenvalues. But where are they, and what do they look like? Sometimes they can look like a cardioid, and the author shows what else can happen.
General Arithmetic Triangles and Bhaskara's Equation
Raymond Beauregard and E. R. Suryanarayan
There are triangles other than right triangles that have sides with integer lengths and integer area. The 13, 14, 15 triangle has area 84, and the 8, 29, 35 triangle has an area whose calculation is left as an exercise for the reader. The authors describe the relation between such triangles and Bhaskara equations x2 - M y2 = d2.
Related Rates Collide with Vectors
The problem about Entity A moving along a straight line at a constant velocity and Entity B moving along another straight line at another constant velocity, how fast is the distance between them changing? appears in every calculus text. There are two ways of looking at the problem (not counting the all too common deer-in-the-headlights stare of incomprehension) that both seem reasonable, but which lead to different answers. The author explains why one really isn't reasonable.
A Rational Solution to Cootie
Arthur Benjamin and Matthew Fluet
A previous Classroom Capsule on the game of Cootie (29 (1998), 222-224) found the expected length of a game to eight significant figures. Now, three authors show in two different ways that the expected length of a game is, to infinitely many significant figures, 584684533/11943936, a satisfying result.
An "Extremely" Cautionary Tale
Find the least distance from a surface to a point. The author did, once using Lagrange multipliers and once using substitution, and got two different answers. I will not spoil the paper by telling you which was right and why the other was wrong. What does your intuition tell you is best? Substitution or Lagrange multiplier?
A Quick Construction of Tangents to the Ellipse
Drawing tangents freehand can lead to terrible errors. If you ever need a good tangent to an ellipse, here is a very quick straightedge-and-compass construction that will do the job.
Student Research Project
Linear Functions and Rounding
Jack E. Graver and Lawrence J. Lardy
A U. S. cents to German pfennigs exchange table, one that appeared in print,
can be produced by no exchange rate. How did it get constructed? The authors answer that question and suggest several others for investigation.