Intermediate Value TheoremDate: 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. |
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