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## Monday, November 21, 2011

### IVT the sensible way

I heard of a lovely activity for pseudo-discovering the Intermediate Value Theorem at a recent compulsory workshop for the calculus course I teach. It has everything i like in a thing. I do not have a record of the name of the teacher who presented it (and even if I did, I don't know if he wants to be famous on the Internet) so if you are he, please email me if you want credit.

(begin basic text of student handout/activity:)

The intermediate value theorem states: If a function y = f(x) is continuous on a closed interval [a,b], then f(x) takes on every value between f(a) and f(b).

Think about what you remember of conditional statements (from your geometry course:)

1) State the hypothesis of the IVT.

2) State the conclusion of the IVT.

3) In the following, be sure to use the endpoints (a, f(a)) and (b, f(b)).

A. Sketch a diagram where both the hypothesis and the conclusion hold true.

B. Sketch a diagram where the hypothesis is false, but the conclusion is true.

C. Sketch a diagram where the hypothesis and the conclusion are false.

D. Sketch a diagram where the hypothesis is true, but the conclusion is false.

(on to the back of the page)

4) Which one is impossible to do? Explain why.

5) Compare your diagrams with a partner. How are they similar? Different? If they are different, are they both valid?

6) Is any real number exactly 1 less than its cube?

A. Create a function whose roots satisfy the equation.

B. Find f(1) and f(2). How do you know there is a point (c, 0)? What do you know about c?

1. I wish you luck. Not to rain on anyone's parade, but you'll be lucky if even one or two of your students get the IVT.

Sure, they should mostly be able to use it for the usual toy problems like #6 that get trotted out for it, but it's really one of the few purely structural aspects of the Calculus whose only real purpose is to make the later parts (like the DMVT) work. And even the DMVT is mostly there for structural purposes as well, as is any theorem at this point that concludes "there exists..."

To a large extent, these don't matter nearly as much as their practical implications like the fact that a function with a positive derivative on a region is increasing there, or Fermat's theorem, which is the basic tool for optimization problems.

But the IVT is worse than the DMVT; that one at least doesn't seem entirely obvious, while the IVT does. Properly understood, the IVT is the thing that makes the Calculus work, and it's incredibly deep and subtle for something that almost all of your class will gloss right over.

The reaction of a student -- of any age -- realizing the true meaning of the IVT is sort of like Calvin's after his father discussed the record player (http://www.freewebs.com/calvin-hobbes-org/dadandcalvinsrecordplayer.jpg)

2. 1. grid paper

2. staircase

3. Live graphing stories: student performers go up the stairs, varying styles. Rest of class must plot elevation v. time.

4. Discuss if elevation is continuous. Decide what point we're measuring (left foot? head? belt?)

5. Tell next performer to go up the stairs, but they are not allowed to have (foot/head/belt) pass the 7th stair.

6. Watch performer squirm below the barrier as classmates shout warnings.

3. I just saw a commercial for Tempurpedic mattresses and I thought of the IVT. It was late so I couldn't call anyone, and not anyone I could call would have cared anyway. But the setup is something like this "And we have something for everyone! Soft (show picture of person sinking into mattress holding a sign with 'soft' on it), firm (on top of mattress with barely a dent, with 'firm' sign), and everything in between (person with 'medium' sign)." So everything between the two values of Firm and Soft is "medium"? Maybe Tempurpedic is not continuous. Discreet Mattress-matics?