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I'm not teaching calculus right now, but I talk to someone who does, and the question that came up is why emphasize the $h \to 0$ definition of a derivative to calculus students?

Something a teacher might do is ask students to calculate the derivative of a function like $3x^2$ using this definition on an exam, but it makes me wonder what the point of doing something like that is. Once one sees the definition and learns the basic rules, you can basically calculate the derivative of a lot of reasonable functions quickly. I tried to turn that around and ask myself if there are good examples of a function (that calculus students would understand) where there isn't already a well-established rule for taking the derivative. The best I could come up with is a piecewise defined function, but that's no good at all.

More practically, this question came up because when trying to get students to do this, they seemed rather impatient (and maybe angry?) at why they couldn't use the "shortcut" (that they learned from friends or whatever).

So here's an actual question:

What benefit is there in emphasizing (or even introducing) to calculus students the $h \to 0$ definition of a derivative (presuming there is a better way to do this?) and secondly, does anyone out there actually use this definition to calculate a derivative that couldn't be obtained by a known symbolic rule? I'd prefer a function whose definition could be understood by a student studying first-year calculus.

I'm not trying to say that this is bad (or good), I just couldn't come up with any good reasons one way or the other myself.

EDIT: I appreciate all of the responses, but I think my question as posed is too vague. I was worried about being too specific, so let me just tell you the context and apologize for misleading the discussion. This is about teaching first-semester calculus to students straight out of high school in the US, most of whom have already taken a calculus course in high school (and didn't do well or retake it for whatever reason). These are mostly students who have no interest in mathematics (the cause for this is a different discussion I guess) and usually are only taking calculus to fulfill some university requirement. So their view of the instructor trying to get them to learn how to calculate derivatives from the definition on an assignment or on an exam is that they are just making them learn some long, arbitrary way of something that they already have better tools for.

I apologize but I don't really accept the answer of "we teach the limit definition because we need a definition and that's how we do mathematics". I know I am being unfair in my paraphrasing, and I am NOT trying to say that we should not teach definitions. I was trying to understand how one answers the students' common question: "Why can't we just do this the easy way?" (and this was an overwhelming response on a recent mini-evaluation given to them). I like the answer of $\exp(-1/x^2)$ for the purpose of this question though.

It's hard to get students to take you seriously when they think that you're only interested in making them jump through hoops. As a more extreme example, I recall that as an undergraduate, some of my friends who took first year calculus (depending on the instructor) were given an oral exam at the end of the semester in which they would have to give a proof of one of 10 preselected theorems from the class. This seemed completely pointless to me and would only further isolate students from being interested in math, so why are things like this done?

Anyway, sorry for wasting a lot of your time with my poorly-phrased question. I know MathOverflow is not a place for discussions, and I don't want this to degenerate into one, so sorry again and I'll accept an answer (though there were many good ones addressing different points).

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What competing definition do you have in mind? –  Pete L. Clark Sep 27 '10 at 5:34
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Maybe I misunderstand your question. But what would be the point of teaching students the symbolic rules as axioms without explaining to them how they are derived? Would you advocate teaching maths undergraduates the combinatorial properties satisfied by character tables of finite groups, so that they can work out the tables in most cases, without proving any of the properties or maybe even without explaining what a character is? –  Alex B. Sep 27 '10 at 5:41
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I think your only alternative is to present the "magic" differentiation rules with no justification. It is already common for students to have a black-box view of mathematics; I don't think you want to encourage it. Perhaps you want to begin with the definition via limits and then derive the rules from there. Emphasize to your students that "Why didn't we just use the rule from the start?" is not a valid question. The rule is a consequence of the definition, not a self-evident truth. –  Austin Mohr Sep 27 '10 at 5:42
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there is an article by Solomon Friedberg entitled "Teaching mathematic graduate students how to teach" in the Notices of the AMS (52) 2005, where the question you ask and its didactical implications is part of a "case study". –  Holger Partsch Sep 27 '10 at 11:35
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If calculus class were devoted to the project of getting students to learn to appreciate mathematics by a process that resembles mathematics (which they aren't, perhaps with good reason), then one could do this by simply holding off on the introduction of the power, product, chain and quotient rules. The geometric problem of computing tangent lines is natural and easy to motivate; the limit definition is reasonably easy to motivate from the geometric problem; and then students could spend reasonable amount of time flailing around trying to compute derivatives of different functions. (Cot'd) –  JBL Sep 27 '10 at 14:04

31 Answers 31

Dear friends (and foes;-), limits are not needed to understand and do calculus. You can read my article at http://www.mathfoolery.com/Article/simpcalc-v1.pdf and my recent translation of a 1981 talk by V.A. Rokhlin at http://mathfoolery.wordpress.com/2011/01/01/a-lecture-about-teaching-mathematics-to-non-mathematicians/ I hope it will get you thinking.

The following is my response to fedja, it may be of interest to those who haven't read my article. First, let me indicate briefly my suggestions on how to approach calculus. The most important principle (of V.I. Arnold, explicitly stated by him in his recent Lectures on Partial Differential Equations) is to concentrate on examples, calculations, and applications, staying away from the generalities before they become necessary and the ideas behind them are well understood in concrete situations.

I will mostly discuss differentiation, see my article and my talk slides at http://www.mathfoolery.com/talk-2010.pdf for more details. In high school they teach kids how to factor algebraic expressions, long and synthetic division of polynomials, the fact that if $f(a)=0$ then $x-a$ divides $f(x)$ (for a polynomial $f$). Why not use it and develop differentiation of polynomials? Indeed, if you ask a high school student to make sense of $\frac{x^2-a^2}{x-a}$ for $x=a$, (s)he is very likely just to factor the numerator, cancel $x-a$ and stick in $x=a$ to get $2a$ (or $2x$). This stuff is purely algebraic and all the differentiation rules are immediate.

Parenthetically, once they are established for polynomials, they are forced upon us for any reasonable understanding of differentiation, at least for uniformly differentiable because of the Weierstrass approximation theorem, for example. Also differentiation as an aspect of factoring becomes apparent.

We also can similarly develop differentiation of rational functions and roots, and use implicit differentiation to do other algebraic functions. This gives us already a lot to play with and to apply. The sine function can be treated geometrically, as an aspect of kinematics of the uniform rotation. This broadens the range of potential applications.

To get to the geometric and intuitive meaning of differentiation, we can notice that $x^n-a^n-na^{n-1}(x-a)$ has a double root at $x=a$, or we can look at the expression $f(x+h)$, $f$ being a polynomial, as a polynomial in $h$ with coefficients depending on $x$. It has a constant term $f(x)$, the linear term $f'(x)h$ and all the higher order terms, so $f(x+h)-f(x)-f'(x)h=h^2r(x,h)$ where $r$ is a polynomial. This way, if we restrict $x$ and $h$ to some finite interval, we arrive at our basic estimate, uniform in $x$ and $h$:

$$|f(x+h)-f(x)-f'(x)h| \le Kh^2$$

that indicates how close is a polynomial to its affine approximation using its differential.

This inequality allows us to explain why polynomials with positive derivatives are increasing. We simply notice first that if $f' \gt C$ and $0 \lt h \lt C/K$, then $f(x) \lt f(x+h)$, and therefore $f(A) \lt f(B)$ when $A \lt B$. Then, by applying this fact to $f(x)+Cx$, we see that $f(B)-f(A) \gt C(A-B)$ for any $C \gt 0$ when $f' \ge 0$, and therefore $f(A) \le f(B)$. This is called the monotonicity principle, and it is the most complicated theorem in this approach to calculus. Everything else follows from it.

Now, to broaden our scope, we promote our basic estimate to the definition status and call the functions that satisfy this definition (uniformly) Lipschitz differentiable (LD). Derivatives of polynomials are polynomials, and differentiation of polynomials is related to their factoring. Likewise, derivatives of LD functions are Lipschitz. Indeed, we can rewrite our basic estimate as $|\frac{f(x)-f(a)}{x-a}-f'(a)|\le K|x-a|$, then notice that $ \frac{f(x)-f(a)}{x-a} = \frac{f(a)-f(x)}{a-x}$ and conclude that $|f'(x)-f'(a)| \le 2K|x-a|$. Moreover, $f$ is LD if and only if $f(x)-f(a)$ factors through $x-a$ in the class of Lipschitz functions of 2 variables, $x$ and $a$. Differentiation rules are straight forward.

I suggest to develop integration in parallel with differentiation (since they work and are understood better together) starting with simple examples of Newton-Leibniz theorem, and working our way to approximating definite integrals by approximating the integrands by the functions that are easy to integrate, say, piecewise-linear functions showing integrability of, say, Lipschitz function, positivity of integral and proving Newton-Leibniz.

Now I can get to fedja's objections. The Lipschitz theory takes care of all the piecewise-analytic functions, and that's almost everything that we deal with in elementary calculus. When we run into functions that don't fit into the Lipschitz theory ($x^{3/2}$, for example), we broaden our definitions by replacing $h^2$ in our basic estimate (with $|h|^{3/2}$ for our example). Then we observe that the theory still holds for the weaker estimates. The Holder estimates i.e. the ones we get by replacing $h^2$ with $|h|^{1+\gamma}$, $0 \lt \gamma \lt 1$ gives us much more room to play. All moduli of continuity are not needed for any problem involving only a finite number of functions, and it covers the vast majority of problems we encounter in calculus.

Now, in the classical treatment the extreme value theorem is used to prove the Lagrange theorem that is used to prove that a function with positive derivative is increasing. But in our approach we have a direct proof of this fact, so we don't need it. And we don't need the intermediate value theorem to prove the Newton-Leibniz since it can be proven directly using positivity of the integral. By the way, both of these theorems are non-constructive.

One may ask about minima and maxima within this approach. The monotonicity principle takes care of this topic, since it assures us that the point where the derivative changes its sign form plus to minus will be a local maximum; the similar obvious result is true for a local minimum.

Fedja also said that the inverse function theorem fails miserably within Lipschitz functions. My guess is that he was talking about the theorem that says that the inverse of a monotonic continuous function on a closed interval is continuous. This is true, of course, but it may be a good thing, since it raises our awareness of the fact that the inverses of very nice functions can be computationally horrendous. It also can be a motivation to consider some other moduli of continuity. As for the inverse function theorem about local invertibility of the differentiable functions, its treatment within the Lipschitz class is not much different from the standard, and is somewhat simpler.

In any case, fedja and I should probably take our dialogue elsewhere. Thanks for all the comments.

I also want to mention that similar approaches to calculus and introductory analysis have been tried with a good measure of success by Hermann Karcher at Bonn University and Mark Bridger of Notheastern University. See Karcher's lecture notes with an English summary at http://www.math.uni-bonn.de/people/karcher/MatheI_WS/ShellSkript.pdf and 2007 book "Real Analysis: a Constructive Approach" by Mark Bridger, where he defines differentiation via factoring of $f(x)-f(a)$ into $x-a$ in the class of continuous functions. Karcher said (in a recent e-mail to Dick Palais): "I taught my last Calculus course by first using only Lipschitz continuity. At the end of the first semester I reached uniform convergence of functions and continuity. From the second semester on the procedure was the standard one. The students liked it a lot, I still meet one or the other and they still smile." There is also a nice book by Peter Lax "Calculus with Applications," where he deals with uniform instead of the pointwise notions.

Added on 1/28/2011. I have just got an e-mail from Hermann Karcher. It says: "it was nice to hear from you again. I read Rokhlin's talk. Maybe one has to go even farther: Sometimes I think, everybody needs his own explanation, and a successful teacher is good in guessing what each individual child needs.I also read your paper. You won't be surprised that I am familiar with your arguments. I am now retired for 7 years. My last three semester beginners course was my most successful one. Today I attended the PhD colloquium of the student of a younger colleague. That student (and some of the younger people in the audience) had been in that last course of mine. They were happy to see me again and say how much fun those three semesters had been. For various reasons I was then in a situation where I could completely ignore, how the standard course in analysis proceeds. During the first semester I did my own stuff,ending up at continuous functions and their uniform convergence at the END of the first term. The next two semesters proceeded as usual - but all the fun we had came from building the foundations differently and the fun stayed with us. I wish you similar experiences. Don't try to convert too many grown ups, just enjoy teaching. Best regards --Hermann Karcher."

By the way, an English translation of Karcher's lecture notes is in progress. I also heard today via facebook form Ursula Weiss who is a math professor in Germany (we were both graduate students at Brandeis in the late 1980s) that she has just finished translating the first chapter.

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"As small as you want" is exactly the expression you tried to eliminate. Once you start saying "there exists $a$ such that $f'(a)$ is as small as you want", I see no difference with "for every $\varepsilon>0$..." and you defeat the whole purpose of the alternative exposition. Moreover, in place of one standard "for every ..., there exists...", you introduce several nonstandard ones. The main aim of the whole exercise is to kill this construction, not to multiply it. –  fedja Jan 26 '11 at 6:21
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@Misha: If you interpret "What do you mean when you write X?" as picking on you, we cannot have a conversation. It's also distressing that you don't find the fact that a professor of mathematics finds your writings on calculus hard to follow as being anything other than criticism. As I said, I came to this answer with some interest in your point of view. After receiving your reaction, this is no longer the case. –  Pete L. Clark Jan 26 '11 at 8:02
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@Misha: please don't say that I don't want to understand the ideas. I have said multiple times that I do want to do so, so suggesting otherwise is essentially accusing me of intellectual dishonesty and/or bad faith. That's pretty insulting given the demonstrable amount of time I have put into asking and answering questions on this website. If you want to claim I am not competent to follow your arguments: that's fine; go ahead. Perhaps you could clarify what your audience is, though, if you do not intend your calculus writings to be easily understandable by a PhD mathematician. –  Pete L. Clark Jan 26 '11 at 11:34
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As to your second question, I believe that yes, this is being read, and, moreover, you, probably, have about as much attention now from wide mathematical audience to your teaching ideas as you are ever going to get. You certainly turned some people away (Pete, say) by the moment and what happens next depends on what you say, but I find public discussions more useful than private e-mails (full public record of speeches and independent arbitrage are big pluses. Besides, you can really win some people you don't know to your side). Of course, if you do not care, we can stop here. –  fedja Jan 27 '11 at 19:21
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«I probably understood him a bit better than he cared for...» Well, that's surely going to attract good will! –  Mariano Suárez-Alvarez Jan 27 '11 at 23:57

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