Intermediate Value Theorem

Date: 09/19/2002 at 15:41:24
From: Kedar Vin
Subject: Prove by intermediate value theorem

Question: A hiker starts walking from the bottom of a mountain at 
6:00 a.m., following a path, and arrives at the top of the mountain at 
6:00 p.m. The next day he starts from the top at 6:00 a.m. and takes 
the same path to the bottom of the mountain, arriving at 6:00 p.m. 

How can we prove by the intermediate value theorem that there is a 
point on the path that the hiker will cross at exactly the same time 
of the day on both days?

Date: 09/19/2002 at 18:13:21
From: Doctor Shawn
Subject: Re: Prove by intermediate value theorem

Kedar,

This is a really neat problem! In order to solve it, take the 
following thought experiment: assume that the hiker has a twin
sister, and the sister starts at the top of the mountain when the
hiker starts at the bottom. The sister starts down at 6:00 a.m., and 
the hiker starts up at the same time. They're both walking along the 
same path, and although they can stop, they can't walk backward, and 
they can't deviate from the path. The hiker arives at the top of the
mountain at 6:00 p.m., and the sister gets to the bottom at the same
time. No matter how much they stop, or run, they're going to have to 
pass each other SOMEWHERE along the path. There's your problem in a 
nutshell, only the hiker is in a sense meeting himself.

To be a little more rigorous, the Intermediate Value Theorem states
that if you have a continuous function in a range with two output
values (call them v and w), then the function must hit every other
value between v and w.  Why?  Otherwise the function couldn't be
continuous. In this case, the hiker's path is a function with output
values distance v. time, at the bottom of the mountain at 6:00 a.m. 
and at the top of the mountain at 6:00 p.m.

           |
Distance   |   c                 b                
           |
           |
           |
           |
           |
           |   a                 d
           ___________________________
             6 a.m.            6 p.m.
                   TIME

On day 1, the hiker goes from point a to b. On day 2, the hiker goes
from point c to point d. Can you see that when drawing continuous
lines between those points, the lines will have to cross at some
point? Otherwise, there would be a discontinuity in the function (the
hiker falls off the path and dies).

Does that help?  Feel free to write back if you'd like to talk about
this some more, or if you have another question.

- Doctor Shawn, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 09/19/2002 at 19:02:50
From: Kedar vin
Subject: Thank you (prove by intermediate value theorem)

I know you get lots of thanks e-mail but really thanks a lot.