It is often said, by way of intuitive explanation, that the derivative of a function at a point is the slope of the line that “best fits” the function through that point. Can this be pressed into a mathematically-rigorous definition, somewhat along the lines of the prisoners’ dilemma?
That is, two parties, who cannot communicate with each other, are given the same function and the same point, and told that if they both come up with the same line through that point that is best relative to the function, they will be set free, otherwise they will both have to serve time.
Would it then be some kind of theorem in Probability/Logic/Game Theory, that they would, if acting rationally and intelligently, come up with the tangent line “almost surely”?
Regards,
Mike Jones
2.Jun.2011 (Beijing time)