Tomorrow I’ll be introducing the intermediate value theorem (IVT) to my calculus class. Recall the statement of the IVT: if is a continuous function on the interval and is between and , then there exists a value such that . In other words, achieves all of the intermediate values between and .
This is a very underappreciated theorem by the students. They think that it is “obvious”—if a function is continuous, then clearly it will hit all of the intermediate values. That’s what continuous means, right?
I remind them that continuity is defined at a point—it is a local definition. I try to convince them that their intuition about continuity is global, and that it is precisely the IVT that supports their intuition.
As for applications of the IVT, I first remind them of the saying “the straw that broke the camel’s back.” That such a straw exists is an application of the IVT. I also have them prove that at every instant there are two antipodal points on earth that have the same temperature. (For each point on a given line of longitude define to be the temperature at minus the temperature at . Since is continuous and , there must be a location on this line of longitude at which . In other words, the temperature at and at are the same.)
My new favorite application of the IVT is the wobbly table theorem: every rectangular table placed on uneven ground can be stabilized by rotating it. This was proved in 2005 by Baritompa, Löwen, Polster, and Ross. Here is an excerpt from their paper.
Balancing a Square Table by Turning–the Intermediate Value Theorem in Action. Consider a wobbling square table. We wobble the table until two opposite vertices of the associated mathematical table are on the ground, and the other two vertices are the same vertical distance above the ground… Let’s call this position of the table its initial position. By pushing down on the table, we can make the hovering vertices touch the ground and, in doing so, we have shoved the “touching” vertices that same vertical distance into the ground. We call this new position of the table its end position… Starting in the initial position, we now rotate the table around the -axis; in doing so, we ensure that at all times the center of the mathematical table is on the -axis, that the same pair of vertices as in the initial position are touching the ground, and that the other two vertices are an equal vertical distance from the ground. Eventually, we will arrive at the end position. So, we started out with two vertices hovering above the ground, and we finished with the same vertices shoved below the ground. Furthermore, the vertical distance of the hovering vertices depends continuously on the rotation angle. Hence, by the Intermediate Value Theorem, somewhere during the rotation these vertices are also touching the ground: that is, the table has been balanced locally.
Polster has a video on his website illustrating the behavior of the table.
They do point out that although it is always possible to stabilize the table, it won’t necessarily be horizontal. So the drinks may slide off and crash to the floor.
It may be obvious but shouldn’t you point out that the table must be square and the legs must be the exact same length? I only mention this because in my years applying mathematics to engineering, errors were often made by not clarifying assumptions. Real world tables usually wobble because two opposing legs are shorter than the other two. I can imagine your students going back to their dorm and rotating a table fruitlessly until finally giving up on theory and folding a paper to go under one leg.
Great blog. Keep it up
Thank you for the comment. You’re right about the need to be precise with our assumptions (see a later post for my thoughts on that!)
In the paper the authors define a mathematical table to be a rectangle with diameter 2. This implies that “the legs” are all the same length, but the table need not be square (although it was in the excerpt that I pasted)—rectangular is good enough. Here’s their theorem.
Theorem 1 (Balancing Mathematical Tables) A mathematical table can always be balanced
locally, as long as the ground function g is continuous.
They do comment on mathematical vs. real tables.
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