When logic and calculus collide

[Richard Wallace]

Originally Posted by Lil' Lavery

"Two runners start a race at the same time and finish in a tie. Prove that at some time during the race they have the same velocity."
Well, if you consider the position equation f(t)=g(t)-h(t) (g being the position of one runner and h the other). Then the velocity equation would be f'(t)=g'(t)-h'(t), so if the velocities are the same, f'(t)=0. When t=0 and when the race end (another value t), and possibly other ponts, f(t)=0.

Everything about your argument is fine up to this point. f'(t) must be zero for some t between the start and finish of the race. This can be proved using the Mean Value Theorem; the result is called Rolle's Theorem.

And because the derivative of a constant is 0, then f'(t) would also be 0.

This does not follow from the problem statement. In practice a runner's velocity at the finish line is rarely zero, and in some kinds of racing (relays, sailing, etc.) the starting velocity is not zero, either. But on a purely mathematical level, the statement "the derivative of a constant is zero" is neither precise nor pertinent here.

Precisely, the derivative of the function f(t) = C is zero for all t. However, in this example you have assumed that the function f(t) = 0 at the beginning and the end of the race, but not for times before the beginning or after the end. What the two runners did before t=0 and after crossing the finish line is not defined, nor is that information needed to prove the result.