When logic and calculus collide

[Richard Wallace]

Originally Posted by Lil' Lavery

f'(t) was the difference of the two derivatives, not the derivative of either of the runners positons, and so could (and would have to be) zero. g'(t) and h'(t) were the velocities of the runners, and niether could have been zero (unless it was a race with absolutely no distance or where niether runner moved). The problem did state that both runners started at the same time, but you're right in that it may not mean they both had a velocity of 0 at the beginning.
I get the Calculus behind it (including the mean value theorem, which was the lesson where we were given this problem), I was trying to justify it logically, and I guess that it must be logically true. I've thought about it a little more and at one point they must both have been travelling the same speed.

Sean, I was pretty sure that you understood the calculus assignment, and the MVT. And I'm pretty sure you will be going much further into engineering math during the next few years. That's why I thought it worthwhile to quibble.

My quibble was with the assumption of any knowledge concerning f'(t) or g'(t) or h'(t) before the start and after the finish. Part of the quibble is practical: nothing about these derivatives is given in the problem statement. The other part is mathematical: Rolle's Theorem does not depend on any particular constraints on f'(t) at the endpoints of the interval.

I only jumped in here because I've seen this kind of error in setting up engineering math problems a few times before. The assumption that I'm quibbling over makes no difference to the proof sought in this problem, but there are many engineering problems for which incorrect assumptions about starting and/or ending conditions will lead to the wrong conclusion. And you are going to see them.

When I am a doddering old retiree and you are overseeing the design of infrastructure for the Martian colony, I want your mathematical reasoning to be perfect.