Something is working

This fall I'm teaching Calculus I for the third time here at Coastal.  Perhaps because I just taught it at the end of the summer, I am finding that it is working particularly well.  I'm not entirely sure why.  Student interaction is good, and I'm sure that is helping.  But beyond that, the lectures seem to be flowing in a way they have not previously.

One possibility is the way I just happen to be presenting the material.  I thought I'd take a moment and record a couple of examples here. 

1. To introduce limits, I started by asking how we might graph $f(x) = \frac{sin(x)}{x}$.  We had previously talked about the graph of $f(x) = \frac{1}{x}$, and what happens at $x = 0$.  We said that while you cannot plug in 0, you can ask what happens near 0.  So we tried that again.  What happens to $f(x)$ as $x$ gets closer and closer to 0.  We made a table, everyone agreed that the $y$-values were getting closer to 1.  Only then did I start using the language "limit."  I gave them the notation as a way to quickly write down what we just did.  This seemed much more natural than giving them a definition of a limit (out of thin air) and then showing them a bunch of examples.
2. For left/right-hand limits, I only introduced them after we discovered the problem in finding limits if the $y$-values approach different values from each side.  I gave them a graph and asked them to find limits in a variety of cases.  We all agreed that at the jump discontinuity, the limit did not exist.  But of course, we can be more descriptive than this.  Coming from one side, the limit does exist.  Then I introduced that notation.
3. For continuity, I drew two graphs, one continuous, and one with a jump discontinuity.  I asked the class what was different about the graphs.  We agreed that the continuous one was continuous and the other not.  I asked what other ways a graph might not be continuous.  We came up with a graph with a removable discontinuity and an infinite discontinuity.  Now, how might we say something about these in terms of limits?  
4. Continuing with continuity, to introduce the difference between continuous at a point vs interval, I covered up the discontinuity and asked if the rest of the graph was continuous.  We agreed it was, so decided that we needed to express continuity at specific points.  This led us to the limit definition of continuity.  This was so much more natural than giving the definition, and then figuring out what it means.

That's all we have done so far.   I'm not sure if this luck will continue.  The common theme seems to be that instead of giving the definition and trying to apply it (which of course has it's place and value), I'm asking them to solve problems, then coming up with the math-way of saying what we are doing. 

I don't know if this is helping the students (we have not had any exams yet) but it definitely feels better to me.   I plan to make an effort to continue using this "technique" for the rest of the class and see how it goes.

Product rule vs. trig functions

What is the best first example of the product rue?  Often, books like to use $f(x) = x e^x$.  But this is a horrible first example.  Look at the derivative: $\frac{df}{dx} = e^x + xe^x$.  Can you see the format of the product rule there?  Not in the least.  It would help to use $x^2e^x$, but the real problem is that you do not see the difference between $e^x$ and its derivative.

A much better first example might be $f(x) = x^2\sin x$.  Now $\frac{df}{dx} = 2x\sin x + x^2 \cos x$. Very nice.  You can see exactly how the product rule is used.  But there is a problem: most textbooks do not cover the derivatives of trig functions until the section after the product rule.  The reason for this appears to be the desire to keep all the trigonometric function derivatives in one place.  To get the derivative of tangent, you need the quotient rule, which should definitely be in the same section as the product rule.  What to do?

Whether textbooks like it or not, I think it is worth it to teach the derivatives of sine and cosine first, then the product and quotient rule, and then as an application of the quotient rule, do tangent, and the other three basic trig functions.  This takes a little more forethought, but the benefits clearly outweigh the costs.

The "Zeno's Paradoxes and Calculus" Paradox

One of my fondest memories of taking freshman calculus was the brief discussion of Zeno's paradoxes. For anyone unfamiliar, the particular one I remember is the Dichotomy paradox:

That which is in locomotion must arrive at the half-way stage before it arrives at the goal.

--Aristotle, Physics VI:9, 239b10

That is, if you are walking towards the wall, first you must travel half way there.  Then you must travel half way from there to the wall, then half way again, and so on.  Thus you will never reach the wall!  (Actually, this is backwards from the traditional reading of the paradox: before you travel half way there, you must first travel half way to that half way point, and before that, half way to there, so in fact you never start moving at all!)

This paradox is often used as an example of a great mystery that calculus can help us solve.  I so wish that were the case.

I have given my students this example in my own calculus courses.  It always goes over very well.  When first describing the situation, I ask given all this whether I will ever reach the wall, to which most students say that I will not.  "Great," I say, and proceed to walk straight into the wall.  It is a fun activity that engages students, and is related to mathematics.  Except that it is not related to mathematics.

The usual explanation of the paradoxes using calculus is to show that the geometric series with ration 1/2 converges.  But this is not what is perplexing about Zeno's paradox.  In fact, doing the math behind this series is much more complicated than just looking at a picture of a line divided first in half, then the next part in half again, and so on.  Clearly the sum of 1/2, 1/4, 1/8, ... is 1.  The fact that mathematicians have been able to develop a the notions of limit and infinite series to a level of precision which agrees with our intuition is remarkable, yes, but the result is not surprising.  This explanation acts as if the perplexing thing about Zeno's paradox is that the result of traveling these half distances is just the whole distance, and not an infinite distance.  After all, at first glance, adding up an infinite number of things should not give you something finite.

Perhaps better would be to use the geometric series to represent time.  Say you walk one mile at one mile per hour.  In half an hour, you have walked half a mile.  Then you need 1/4 of an hour to get through the next 1/4 of a mile.  Then you need 1/8 of an hour to go the next bit, then 1/16 of an hour, and so on.  You add up all these times, you get 1 hour, and you have traveled 1 mile (again by adding up all those distances).  Alright, so this definitely is convincing.  I am now sure that I will reach my goal in a finite amount of time.  Of course, I knew that already: I walked right into the wall.  Anyway, Aristotle even gave that explanation, and he didn't know any calculus.

The reason Zeno's paradox is compelling is that it requires you accomplish an infinite number of steps.  Not an infinite number of steps in 1 hour, but an infinite number of steps at all.  I can see that this particular infinite sum is a finite number, but what I cannot see is that I would be able to ever arrive at that number by physically entering infinitely many terms into my calculator (even if I could do so at an ever increasing rate).  This seems like a problem for physics, not mathematics. 

And yet, Zeno's paradox is such a great teaching tool.  If only there were a way to use it that did it justice.

Random examples

Some examples are better than others - most of the time.  For example, when first teaching the product rule, it is not a good idea to use $xe^x$: since the derivative of $e^x$ is $e^x$, students don't see the form of the product rule explicitly.  That said, there are times when the technique being taught are so general, that they would work equally well with any example.  In cases like these, I like to construct a random example, with the class's help.

Suppose I wanted to show my students that the Taylor series for any polynomial is simply the polynomial back again.  If students have not thought about this yet, the result can be rather surprising (after all, the Taylor series for the other standard examples look nothing like the starting function).  What I don't want to do is look down in my notes, carefully copy down a polynomial and start from there.  While this would definitely be an example worth sharing, I fear that students would not be impressed.  Of course I, the powerful math professor, could come up with an example of a function which is identical to it's Taylor series.  Just another example, they would think.

Instead, I write on the board "Find the Taylor series for $f(x) =$" and then dramatically point at a student and demand, "What's your favorite number!?"  After regaining his or her composure, the student will say, perhaps, 7.  I write $7 x^3 +$ and then repeat with another student.  Continuing in this fashion, the class and I together come up with a random polynomial.  And wouldn't it be amazing if this random polynomial happened be it's own Taylor polynomial?  Surely that cannot be a coincidence.

Random examples like these can be used all over the place, although it is important not to use them as a substitute for a well written lecture.  As with any interaction with individual students in class, this technique will keep students alert.  Most find it humorous (especially when a student can't remember their favorite number).  And most importantly, when used correctly, the random example can drive home the fact that the mathematical technique can be used in any case, not just the special cases the professor has prepared.