Discussion about math, puzzles, games and fun. Useful symbols: ÷ × ½ √ ∞ ≠ ≤ ≥ ≈ ⇒ ± ∈ Δ θ ∴ ∑ ∫ • π ƒ ¹ ² ³ °
 

You are not logged in. #1 20091029 10:17:36
Intermediate Value TheoremJust rattled off a page about the Intermediate Value Theorem "The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  Leon M. Lederman #2 20091105 11:12:15
Re: Intermediate Value TheoremAnyone care to tear it to pieces? "The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  Leon M. Lederman #3 20091105 20:15:22
Re: Intermediate Value TheoremHi MathsisFun;
Rather different tone than normal. Almost a challenge. You are proud of this one. You should be. Last edited by bobbym (20091105 21:04:59) In mathematics, you don't understand things. You just get used to them. I have the result, but I do not yet know how to get it. All physicists, and a good many quite respectable mathematicians are contemptuous about proof. #4 20091105 23:11:59
Re: Intermediate Value TheoremWow, thanks! "The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  Leon M. Lederman #5 20091106 00:47:14
Re: Intermediate Value TheoremI'd say the round trip better demonstrates Rollé's Theorem instead of the IVT. Why did the vector cross the road? It wanted to be normal. #6 20091106 04:13:10
Re: Intermediate Value TheoremRolle’s theorem requires the curve to be differentiable (i.e. smooth) not just continuous. MathsIsFun’s example is about continuous routes, not smooth ones. You could e.g. walk up the side of a pyramid to the apex and down the another side – in which case no part of your route is horizontal. Last edited by JaneFairfax (20091106 04:36:43) #7 20091106 04:31:47
Re: Intermediate Value TheoremOr your route could be like this: In which case the IVT applies but Rolle’s theorem does not! Last edited by JaneFairfax (20091106 04:37:02) #8 20091106 13:51:06
Re: Intermediate Value TheoremThe IVT says a lot more than "you will be at the same height". If you walk in a circle (this works for any parametrized continuous path, but circle is easiest to see), let your height at time t be described by h(t). Here we are walking around the circle in 1 unit time of time. Then there will exist at least one point which will be exactly the same height as the opposite side of the circle. "In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..." #9 20091106 14:27:38
Re: Intermediate Value TheoremIt would be nice to include that! "The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  Leon M. Lederman #10 20091107 10:46:41
Re: Intermediate Value TheoremHi all;
We could argue for awhile about why it is "likely to be wrong" just because it is your own invention. True, that example does have some kinks in it, but does that mean there was a greater than 50% chance that it was an error from the start. In mathematics, you don't understand things. You just get used to them. I have the result, but I do not yet know how to get it. All physicists, and a good many quite respectable mathematicians are contemptuous about proof. #11 20091108 14:32:33
Re: Intermediate Value Theorem
Kinks? What Kinks?
I have attempted to illustrate that ... please have a look at the revised page (at bottom). "The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  Leon M. Lederman #12 20091108 15:43:44
Re: Intermediate Value Theorem
Looks good, I suppose an attempt at proof would be just a little too much? It is rather surprising how elementary the proof is. "In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..." #13 20091108 17:20:48
Re: Intermediate Value TheoremHi Mathsisfun;
Correction, No kinks. Only, I haven't seen the IVT for only a 2D graph and not a 3D surface. Following up on it I was led to the Borsuk  Ulam theorem which may or may not apply. In mathematics, you don't understand things. You just get used to them. I have the result, but I do not yet know how to get it. All physicists, and a good many quite respectable mathematicians are contemptuous about proof. #14 20091126 05:15:53
Re: Intermediate Value TheoremThe intermediate value theorem sounds easy! 