I could use "Educator Ideas" now

[Bicho the Inhaler]

1) How in the world does one teach the epsilon-delta definition of a limit? I tried to explain it to two classes of college freshmen, and I don't think I got through at all to 80-90% of them.

2) Is it possible to make calculus fun for a mostly non-math major undergraduate audience?

[MatthewV]

2) tell them there are many chances for partial credit. It won't help them be interested, but it might make them try a wee bit more.

[Samadhi]

Why are non-math major undergraduates taking calculus?

[Samadhi]

The only one I can think of might be a computer science major. But, seriously, if you can't get calculus and it's required for your major, maybe you should look for another major.

[Chuck]

Start a 0.99999... = 1 debate. Everyone loves those.

[Courk]

Calculus not fun? But it is fun! Every problem is like a little mystery.

Umm... maybe explain practical applications for calculus they might run into? Ask them all to write down their majors, and come up with problems that they might see once they get a job. (I assume this is a class of no more than 30-40 people.)

[Courk]

And epsilon delta... I think I remember doing that, but I can't remember exactly what it was. Would it be possible to explain it here, and I'll see if I can think of another way to explain it?

[extropalopakettle]

It's just the definition of what it means when we say that the limit of f(x) as x appraoches x0 equals L. The limit equals L means that f(x) gets as close to L as you choose, and stays at least that close - that is, for any tiny epsilon you care to choose, f(x) will be within epsilon of L, i.e. differ from L by no more than epsilon, for all values of x that are close enough to x0, close enough being within delta of x0, for a delta that depends on the chose epsilon ... and there will always be such a delta, no matter what epsilon you choose. That's what it means to say the limit of f(x) as x appraoches x0 equals L.

[extropalopakettle]

any non-zero epsilon, that is

[MatthewV]

So they need to understand that it is basically how much you could be off by.

I think 100 level Calculus is almost a core class now.

[extropalopakettle]

And I agree that 0.999... = 1 is a perfect example. Not 0.999... approaches 1. It equals 1. Because 0.999... represents an infinite sum, and an infinite sum, if it means anything, is taken to mean the limit of the finite sum of some terms, for more and more terms. This limit equals 1.

[extropalopakettle]

[Leptonn]

Bicho, I really feel for you. The best explanation for epsilon-delta proofs has been a comparison to two very boring people playing a game. One gives a small number (epsilon); if the other can give an appropriate delta, then the second player wins the round. If the second player can *always* win, then it is a proof. I'm sorry... I didn't explain that very well.

As regards question II, I would suggest that university Calculus is just one of those courses that you need to plow through. Those who have a reason to try to understand it will do so; those that don't want to be there will have a good excuse to leave, and those that actually have trouble with it but are taking a mostly non-mathematical programme will keep trying at it. I know a few comp. sci. people who fit into the last category.

You could, at the best, maximize interest levels by showing some cute proofs (sqrt(2) being irrational, and so on).

[Bicho the Inhaler]

Wow...thanks for all your responses.

Samadhi - I think pretty much everyone has to take some kind of calculus nowadays. This is sort of the "medium-difficulty" track for science-oriented people who didn't necessarily master calculus in high school.

Among the hard things about the epsilon-delta definition are that (a) the definition is hard to understand, having two quantifiers; (b) it introduces two Greek letters you've probably never seen before; (c) it requires you to argue that the limit is something, instead of calculating it in a linear fashion. Also, it is the only way to make limits mathematically precise. (Actually, it isn't, but good luck teaching nonstandard analysis to the freshmen!)

Intuitively, the "limit of f(x) as x approaches a" is the answer to the question "what can we say about f(x) if x is really close to, but not equal to, a?" I think they all pretty much understand that. If the limit is equal to L, then that means that "if x is really close to a, then f(x) is really close to L." Then I said that this isn't a mathematical statement, since "really close to" can't be given a precise meaning here. I fumbled about trying to say that "really close" doesn't mean "within 1/1,0000,000" or "within 1/1,000,000,000" or "within 1/1,000,000,000,000," but it has to be able to mean any of these. This is already starting to be too abstract; maybe I shouldn't have tried to derive the definition from intuition and just done some examples to convince them that it is the right definition. But anyway, then I restated it as "we can make f(x) as close as we want to L by restricting x to be within a certain distance of a." That can be translated directly into the epsilon-delta definiton:

[Courk]

I'm always a fan of word problems (espcially picture problems, but that's more trig). I also liked it when we had to find things like the smallest surface area of a cylinder that had this-and-that volume.

But as for normal people... I don't know.

[Courk]

Or was it maximizing the volume with using as little material as possible? I'm sure you know what kind of problems I'm talking about.

[Samadhi]

[mith]

Courk: both, but usually the latter.

In my experience, a lot of students have problems with word problems... or for that matter anything that involves more than plugging numbers into some memorized formula. They just don't get prepared early on for "creative" thinking in math.

[Chuck]

Maybe a better introduction. Something like:

"Remember when you were told by your grade school arithmetic teacher that you can't divide by zero? Well, you CAN divide by zero if you sneak up on it. The procedure is called taking limits. This course will teach you how."

[old grey mare]

If I had the job of making epsilon-delta proofs interesting I would put them in a historical/philosophical context.

In the enlightenment days of the 18th century, religious ideas were under attack by the ‘rationalists’. In defense of his religion, Bishop George Berkeley launched a counter-attack on the non-religious mathematicians claiming they were being hypocritical. He argued that many non-religious mathematicians readily objected to the inconsistencies of popular religions while remaining blind to their own inconsistencies. He wrote a book called “The Analyst” that attacked the foundations of calculus. In particular he observed that mathematicians would say you cannot divide by zero, while simultaneously saying you should compute a tangent to a curve by dividing (f (x + h)-f (x))/h as h became zero. How could ‘h’ be both zero and non-zero?? Can you divide by zero or not? It’s kind of an oxymoron to use the phrase “slope of a point on a curve” when a slope is really an attribute of a line, and a line is defined by at least two distinct points.

This (apparent) inconsistency in mathematics needed an answer, or at least an explanation. In the early days of calculus things remained vague and non-rigorous. In the 19th century the likes of Cauchy and Weierstrass answered the call for rigor with the epsilon-delta approach to calculus.

In this approach, h never really needed to go to zero. It is only necessary to make h (i.e. delta) sufficiently small so that (f (x + h)-f (x))/h is as close to the real tangent as you need (i.e. epsilon). Then when you extend into integral calculus you can compute areas under curves as accurately as you need by making h small enough, and then this extends into physics, mechanics, etc.

In a sense, the epsilon-delta is the practical engineering approach to abstract mathematics. It shows that with a sufficient starting precision you have a way of ensuring you can get close enough to where you need to be without having to be perfect. This way, users of calculus are spared the hubris of having to deal with infinities and infinitesimals, and so they need not answer for the paradoxes of the infinite.

Your philosophy students may appreciate the metaphysical controversy. The history students may appreciate the history. The engineering students may appreciate how epsilon-delta brings calculus to realistic practical terms. And who doesn’t like a good religion argument?

If your students are still bored after that then I give up.

[Antrax]

Bicho, one method I've seen here is to explain what he'd like to describe (ie, what a limit is), then take definitions from the students (or offer some wrong definitions), and show why they fail. I don't remember the exact examples the professor used, but after he showed what's wrong with each of the tweaked definitions, the real definition made a lot more sense, because all the parts sort of fit in.

[doormouse]

I memorized how to do epsilon-delta proofs and managed to get them right. Epsilon-delta proofs almost made me change my major.

[jadesmar]

My suggestion is to simply explain the concept without naming the variables. Once the concept has been suitably understood by the majority, variable/greek letters can be added.

It is really too easy to lose a group of people having the mindset that "mathematics is difficult" by substituting variable names for description.

The letters epsilon, delta, L and x should ultimately be the last things mentioned.

[doormouse11]

haha! That's a good idea - I laughed because it reminds me of something my professor told me.

There was this professor from Germany at the UTSA a while ago. She was a little confusing and hard to understand - some students actually filed a complaint that they couldn't understand her because she was using German letters in all of her proofs and stuff. A little investigation showed she was using Greek letters.

[Bicho the Inhaler]

jadesmar, that's actually kind of what I did. I started with a verbal explanation and then brought in epsilon and delta. Wouldn't it be almost impossible to apply the definition to an example without having names for everything?

ogm, that's an interesting story. Sadly, I don't know if I'll have time to digress into the history. I'll keep it in mind, though...maybe if I bomb again on Monday, I'll keep it as a backup

MatthewV, I do give partial credit for progress. (That will be clear after I hand back quiz #2.)

[Bicho the Inhaler]

It went better today! What I decided to do was to state the definition and show how to use it for a linear function, i.e., exactly what I would want to see on, say, a quiz. That seemed to do a lot more good than trying to base it on intuition. (It isn't intuitive, as ogm's tale illustrates.) In discussing one of the homework problems for today, I also got to describe the relation between epsilon and delta in an engineering context (epsilon=tolerable error in final product; delta=tolerable error in instruments). In one of my sections, I even had time to relate the George Berkeley story, which seemed to go over reasonably well.

So thanks for the help, guys. It's nice to see that The Grey Labyrinth can still be a resource for educators despite the absence of "Educator Ideas."

As for question 2, or rather the revised

2') Is it possible to make calculus interesting for a mostly non-math major undergraduate audience?

I still have a long way to go. I think the historical/engineering context was a step in the right direction, though. It seems obvious now that non-math people would appreciate seeing connections to outside mathematics. (I'm a math guy, myself, so I don't always think of these things!)

[MatthewV]

2) print off a bunch of my fractals and say "look, you can use calculus to make pretty pictures"

[worm]

wow, old gray mare, thanks for teaching me something today. guess i can pretty much go home now

[old grey mare]

Glad to contribute. I'm always working on good reasons to go home.

[wordcross]

Practical applications are always a good way to go.

Personally, calc I seemed to make more sense to me when the problems were phrased in a physics related manner. Projectile motion with speed, velocity, and acceleration, were probably the mainstay of my understanding of vectors, derivatives, and Integrals.

limits are bloody annoying. But i finally got them. I have never heard of them called epsilon delta anythings. well, not that i recall.

I always did better in classes where they focused more on the doing than the naming. Yes, you do need some way to indicate what is what, but greek letters are so foreign a concept to most of these people that they get distracted and confused. You *can* use english letter equivalents, since I assume they've had algebra and understand the concept of substituting constants with variables. They've just never seen greek variables and think they have some sort of added significance that they don't.

also, number crunching. give them simple number crunching work. not even for a whole limit equation, just part of it.

http://whyslopes.com/etc/CalculusAndBeyond/ch14PS.html
I was looking at this page to see if i could figure out what epsilon deltas are. extro's explanation left me scratching my head. Anyway, the part where it says:

[wordcross]

anyway, my two cents. it may be an impractical way to teach it, i dunno. I always seemed to learn differently from most people.
_________________
Has anyone really been far even as decided to use even go want to do look more like?

[Jack_Ian]

Nothing to do with "epsilon-delta", but it fits in with the forum title.

[GH]

I'm really sorry I missed this thread the first time around. I can still remember the way I learned it my senior year in high school, which is just over 20 years ago!

My teacher used to talk about creating an "epsilon attack" and a "delta defense." His point was that if you claim that L is the limit of F(x), then for every epsilon, you need to be able to define the delta on the X-axis to make all values F(x) land inside L-epsilon..L+epsilon.

To further illustrate his point, he used to use colored chalk, and he regularly said the phrase, "If X is red, then Y is orange." Orange was the +/- epsilon "attack" range drawn around limit L, and X was the range on the X-axis defined by the specified delta "defense" that we created.

If this is difficult to picture, I'll try again to explain, but this method was (obviously) very effective for me. I can still hear him talking about "epsilon attack" and "delta defense," and I'm fairly certain that I'll go to my grave remembering that the definition of a limit is, "If X is red, Y is orange."