# Intermediate Value Theorem

I wish we could do “baby posts” that were formatted different. Little asides that don’t warrant full posts.  Like this one.

Anyway, today, the AP Calculus BC teacher (and all around awesome person) asked me if I had any good ways to introduce the intermediate value theorem.

That’s the most boring theorem ever. Saying that if you have a continuous function $f(x)$ on $[a,b]$, and $u$ is between $f(a)$ and $f(b)$, then there exists a $c$ in $[a,b]$ such that $f(c)=u$.

In other words, if you have a continuous curve that goes from point $(x_1,y_1)$ to $(x_2,y_2)$, then at some point along the curve’s journey from the first point to the second point, it’s going to pass through every $y$ value between $y_1$ and $y_2$.

If you still don’t see it, just draw two points on a coordinate plane and try to connect them with a continuous function. You’ll see it then.

Anyway, it’s boring. So she was right to ask for ideas. I searched and found none.

So I suggested a warm-up for the class — before they know anything about this theorem. I asked her to throw this up on the board:

INDIVIDUAL CHALLENGE: I am so wise. I have drawn a function $f(x)$ on $[1,5]$, with $10$ between $f(1)$ and $f(5)$, such that then there does not exist a $c$ on $[1,5]$ such that $f(c)=10$. Are you as clever?

And then I wrapped up some Jolly Ranchers for her to give to the first student who could do it.

She said it went really well. And it took a few minutes (read that: minutes) before the first student got it. Perfect warm-up.

The reason I really liked this idea, and wanted to share it, is because: (a) kids were motivated by it, (b) kids were forced to grapple with complex mathematical language, (c) kids got to play around (by drawing different graphs — a puzzle-y thing), and (d) kids discovered the Intermediate Value Theorem on their own.

Let’s think about the last point. The first 5 or so graphs students would draw would not satisfy the challenge. And they’d see the problem: that the graphs they were drawing were continuous. So the only way to satisfy the challenge would be to make their function discontinuous. So not only would they learn the IVT, but they’d really remember the restriction: you need a continuous function for the IVT to hold.

I’m sure many of you probably introduce the IVT this way. It’s certainly not new or revolutionary. But I am now excited to when I get to teach the IVT.

PS. I also am really impressed by this consequence (click link to see proof). The consequence of total boringness happens not to be boring at all!:

The theorem implies that on any great circle around the world, the temperature, pressure, elevation, carbon dioxide concentration, or any other similar quantity which varies continuously, there will always exist two antipodal points that share the same value for that variable.

1. It’s also useful for estimating roots of an equation that’s near impossible to solve explicitly.

1. DavidC says:

Or even to know there are roots! Once when I was talking about the IVT, I tried to get students to think about this:

‘We talk about ‘the (positive) square root of two’, but how do we know there is such a thing?’

I think it worked well for some but not others.

2. Here is a (contrived) story that I’ve done in the past.

“Exactly at sunrise one morning, a Buddhist monk set out to climb a tall mountain. The narrow path was not more than a foot or two wide, and it wound around the mountain to a beautiful, glittering temple at the mountain peak.
The monk climbed the path at varying rates of speed. He stopped many times along the way to rest and to eat the fruit he carried with him. He reached the temple just at sunset.

At the temple, he fasted and meditated for several days. Then he began his journey back along the same path, starting exactly at sunrise and walking, as before, at variable speeds with many stops along the way.

Will the monk ever be at the same position on the mountain at the same time of day?”

One year we went out to a big staircase we had at the school and I had kids be monks climbing the mountain and descending the mountain. Each ascent/descent had to be 30 seconds, and each stair had to be touched at least once other kids plotted height as a function of time (ala Dan’s graphing videos). Then I started one kid at the top and one kid at the bottom and told the bottom kid to never occupy the same stair as the top kid.

1. I forgot to finish that story… hijinks ensued as the kid ran up then realized he couldn’t pass the other guy and kept running up and down the shrinking space that he had.

1. Love it! Especially them going and being the monks climbing. It shows how one kid is “forced” to be on the same stair, no matter how they try not to be.

3. Matt E. sent me some good IVT problems that I wanted to have here so I wouldn’t lose them:

***

The Intermediate Value Theorem

All of these problems can be solved using the Intermediate Value Theorem… but it’s not always obvious how to use it!

1) Show that for any closed shape in the plane, there exists a line that divides the area of the shape exactly in half.

2) Show that for any two closed shapes in the plane, there exists a line that divides the area of each shape exactly in half.

3) Show that for any closed shape in the plane, there exists a line that divides both the area and the perimeter exactly in half.

4) Show that at any instant there exist two points directly opposite one another (called antipodal points) on the Earth’s equator with exactly the same air temperature.

5) Last summer, I climbed a mountain, starting at an elevation of 1000 ft at 7AM, and ending (at the top) at an elevation of 5000 ft at 7 PM. The next day I took a different path down the mountain, but I started at 7 AM and returned to my starting point at 7 PM. Prove that there was some time between 7 AM and 7 PM for which I was at exactly the same elevation on each day at that moment.

6) Show that for any closed shape in the plane there exists a pair of perpendicular lines that divide the area of the shape into equal quarters.

4. chris says:

Hello! I just taught this theorem yesterday as well. I opened with “True or False: You were once 3 feet tall.”

This tends to bring out a great intuitive explanation, and then we can see how to formalize the thinking with the IVT. After that, I launch into many of the examples samjshah brought up.

1. A similar starter:

Was there ever a time in your life where your weight in pounds was exactly the same as your height in inches?

This isn’t the Intermediate Value Theorem specifically, but the same style of reasoning by continuity is necessary.

Sam, you mentioned “same stair” and I would avoid that language: it actually is possible for two people never to be on the “same stair” going up and down, since stairs are discrete and not continuous. A and B can switch stairs at the same moment. (The graphs of y = [x] and y = [9-x] do not intersect, where [x] is the greatest integer function.)

1. DavidC says:

It seems to me like that is the intermediate value theorem, just with a little bit of extra work… (inches minus pounds starts out positive, ends up negative, so passes through zero).

Unless the possible values of weights and heights are only a dense but not complete (e.g., the rationals) subset of the real line. And how would you know the difference? :)

5. Oops! Should have refreshed the page first!

6. Tom says:

I find the way to get students to really appreciate the IVT is the application mentioned
here. Straightening a wobbly table is actually a useful technique (I have used this many times since I learned about it).

7. DavidC says:

The applications are definitely exciting, but my favorite thing about this theorem is how it sheds light on what continuity (completeness) really mean.

Completeness is never taught formally in a first calculus course (appropriately! though I have heard of one disastrous exception…), but students can start to have a sense of what it would mean for the number line to have ‘holes’ and the importance of the fact that it doesn’t.

1. DavidC says:

P.S. This is one reason I really like Sam’s introduction to the theorem.

(Another reason I like it: Thinking about ‘functions’ is hard too, and I think students sometimes forget that something can be a function without being one of the functions we commonly talk about, most of which are continuous.)