Category Archives: Writing and Reading

Theorems

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Theorems are statements that summarize the results that are true in mathematics. Theorems are statements that have been proved true; but the emphasis in AP Calculus is not on proof. Rather, it is on what the theorems mean and how to use them.

Theorems have two parts: the “if …” clause called the hypothesis and the “then …” clause called the conclusion. Students need to know both parts. In many theorems the conclusion is some sort of formula. The students need to know this, but also need to know when they can use it (the hypothesis tells them that).

An early important theorem is the Intermediate Value Theorem (IVT). Take some time with this theorem. “Play” with it. The hypothesis requires that the function be continuous on a closed interval. Use graphs (sketches, no equation needed) to show cases where the conclusion is both true and false when the function is not continuous. Can the function take on values not between f(a) and f(b)? Can you find a case where the hypothesis is met, but the conclusion is false? (Let’s hope not!)

Consider the theorem (p\to q), its converse (q\to p), its inverse (\sim p\to \sim q) and its contrapositive (\sim q\to \sim p) by looking at graphs of each case. (For the IVT the converse and inverse are false. The contrapositive of any true theorem is also true.)

Finally, for this and for all the important theorems that you use this year, express them in words, “play” with them by making change to the hypothesis, and look at graphs. Don’t just state the theorem and expect students to understand it, remember it and use it correctly.

The next post will be about definitions, which are similar to theorems in lots of ways.

For Any – For Every – For All

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The universal quantifier \forall  –  for any – for every – for all

Many theorems and definitions in mathematics use the phrases “for any”, “for every” or “for all.” The upside-down A is the symbol. The three phrases all mean the same thing!

For example, we have the definition “A function is increasing on an interval if, and only if, for all pairs of numbers x1 and x2 in the interval, if x1 < x2 then f(x1) < f(x2).” Whenever you have a theorem or definition with one, restating it with the other two will help students understand it better: “for all pairs of numbers,” “for any pair of numbers” and “for every pair of numbers.”

Increasing and Decreasing Functions

The symbols in the definition above tell the whole story – sure they do. As with any theorem or definition, use the Rule of Four. The definition above is the analytic part. The graphical part is the obvious – the graph goes up to the right. The numerical part is that as the x-values increase in a table, so do the y-values. The verbal part is the two preceding sentences and all the talking you’re going to have to do to explain this.

The function y=\sin \left( x \right) increases on the closed interval \left[ -\tfrac{\pi }{2},\tfrac{\pi }{2} \right] and the function decreases on the closed interval  \left[ \tfrac{\pi }{2},\tfrac{3\pi }{2} \right]. The fact that  \tfrac{\pi }{2} is in both intervals is not a problem since it is in the intervals, not at the point, that the function increases or decreases.  This is because \sin \left( \tfrac{\pi }{2} \right)  is larger than all (every, any) values in \left[ -\tfrac{\pi }{2},\tfrac{\pi }{2} \right]  , and also larger than all (any, every) of the values in \left[ \tfrac{\pi }{2},\tfrac{3\pi }{2} \right] .

“Playing” with theorems: You will soon have a theorem that says, “If the derivative of a function is positive on an interval, then the function is increasing on the interval.” Nothing in the paragraph above contradicts this, because the hypothesis says nothing about what is true if the derivative is zero. For this you have to go back to the definition. The converse of this theorem is false. Counterexample: f\left( x \right)={{x}^{3}} is increasing on any (all, every) interval containing the origin, yet f'\left( 0 \right)=0 . The AP exams do not make a big deal of this; they accept either open or closed intervals for increasing or decreasing.

A Note on Notation

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For quite a while I’ve been writing sin(x), ln(x) and the like with parentheses instead of the usual sin x or ln x .

The main reason is that I want to emphasize that sin(x), ln(x), etc. are the same level and type of notation as f(x). The only difference is that sin(x) and ln(x) always represent the same function, while things like f(x) represent different functions from problem to problem. I hope this makes things just a little clearer to the students.

I also favor using (sin(x))² instead of sin²(x), again to make clearer just what is getting squared. Notation can be inconsistent: I don’t think I’ve ever seen ln²(x) or even f ²(x).  So this helps in that regard as well.

Of course, when entering functions in calculators or computers you almost always must use the “extra” parentheses in both cases. (Except for the new Casio PRIZM which will understand sin x and ln x, but not sin²(x).)

Now we can use that spot in the notation exclusively for inverse functions, as in {{\sin }^{-1}}\left( x \right) and {{f}^{-1}}\left( x \right). Maybe that will lessen the confusion there.

Another possible inconsistency is trying to write sin′(x)  for the derivative as you do with {f}'\left( x \right). Although, if I saw it I would understand it. (LaTex won’t even parse  sin′(x).)