rubber band stretch problem
The question is, is it true that if you stretch a rubber band by moving one end to the right and the other to the left, some point of the band will end up in its original position?
What I wrote:
Yes. You may stretch it left and right, but the middle will remain in place assuming the two forces are equal.
Pretend the rubber band is a graph. It may be stretched horizontally, but the y-intercept will not change unless the graph is shifted (the y-intercept is equivalent to the center of the rubber band)
So the middle of the rubber band will remain in place as long as the rubber band itself is not moved or shifted other than the stretch, equal on both sides.
It was graded and I was told the reasoning was not exactly correct, I need to connect it to the intermediate value theorum. Can anyone explain how I would do so? Thanks!
CaptainBlack beat me to it... well well I post it anyway
There was no specification of how far left or right the rubber band ends where moved so I figure the teacher didn't like your special case explanation with equal lengths(or forces)to the quite general problem statement.
Originally Posted by mistykz
The intermediate value theorem states that(from wolfram mathworld)...
"If f is continuous on a closed interval [a,b], and c is any number between f(a) and f(b) inclusive, then there is at least one number x in the closed interval such that f(x)=c."
Let the rubber band itself constitute the x-axis. Then let f(x) be the distance point x have moved from it's original position when the rubber band is stretched. Now, the left endpoint a will move to the left and the right endpoint b to the right so that f(a)<0 and f(b)>0.
As f is continuous and f(a)<0<f(b) "there is at least one number x in the closed interval such that" f(x)=0."