One of the first challenge problems Paul H. Nahin offers in his new book comes from his experience as a freshman at Stanford. This is a nice yarn.
When I was a freshman at Stanford I did well enough during the first two terms of calculus to be allowed to transfer into the honors section of the course. (That's when I found lots of my fellow freshmen were at least as good at math as I was!) One of the homework problems in that course (Math 53, Spring Quarter 1959) was the following: prove that the shortest path length between any two given points in a plane is that of the straight line segment connecting the two points. This is, of course, "obvious" to anybody with a body temperature above that of an ice cube, but the point of the problem was to construct a proof.
Nahin the freshman hit on the idea of using the arc length integral
He reasoned that if a curve is not a graph of a function it could be made such by choosing a proper system of Cartesian coordinates. The length of a curve does not depend on the coordinate system and thus could be computed in any suitable one.
Now it is clear that to achieve the required minimum the curve needs to have throughout, meaning that it is a straight line.
The homework has been returned with a big red "NO" and a grade 1 (for trying) out of 10.
The problem is to answer the question, "Why? What was wrong with Nahin's solution?"
Curiously, 50 years later Nihin located his old professor - John Lamerti - and turned to him to explain where Nahin the freshman went wrong.
In his gracious reply the professor admitted that a score of 1 out of 10 seem unreasonably negative and found in the submitted homework some justification to raise the score to 7 out of 10 for a good but not quite right try.
He also pointed out that it is not always possible for a curve to be represented in the form and suggested that the problem was given at the time they discussed parametric equations so that a reference to the parametric form
would have been corrected and expected.
As far as I am concerned, my difficulty was not whether to use the parametric form but in the definition of the path. If the problem has implicitly assumed that the paths in question had to be smooth - which may be so due to the course context - then the parametric form was certainly the way to go. But the problem formulation does not say that much and the author does not mention that until a few steps into the yarn. Still, it was a pleasure to read that story.
- P. H. Nahin, Number-Crunching, Princeton University Press, 2011