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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

29 Answers 29

up vote 84 down vote accepted

This is a good question, given the way calculus is currently taught, which for me says more about the sad state of math education, rather than the material itself. All calculus textbooks and teachers claim that they are trying to teach what calculus is and how to use it. However, in the end most exams test mostly for the students' ability to turn a word problem into a formula and find the symbolic derivative for that formula. So it is not surprising that virtually all students and not a few teachers believe that calculus means symbolic differentiation and integration.

My view is almost exactly the opposite. I would like to see symbolic manipulation banished from, say, the first semester of calculus. Instead, I would like to see the first semester focused purely on what the derivative and definite integral (not the indefinite integral) are and what they are useful for. If you're not sure how this is possible without all the rules of differentiation and antidifferentiation, I suggest you take a look at the infamous "Harvard Calculus" textbook by Hughes-Hallett et al. This for me and despite all the furor it created is by far the best modern calculus textbook out there, because it actually tries to teach students calculus as a useful tool rather than a set of mysterious rules that miraculously solve a canned set of problems.

I also dislike introducing the definition of a derivative using standard mathematical terminology such as "limit" and notation such as $h\rightarrow 0$. Another achievement of the Harvard Calculus book was to write a math textbook in plain English. Of course, this led to severe criticism that it was too "warm and fuzzy", but I totally disagree.

Perhaps the most important insight that the Harvard Calculus team had was that the key reason students don't understand calculus is because they don't really know what a function is. Most students believe a function is a formula and nothing more. I now tell my students to forget everything they were ever told about functions and tell them just to remember that a function is a box, where if you feed it an input (in calculus it will be a single number), it will spit out an output (in calculus it will be a single number).

Finally, (I could write on this topic for a long time. If for some reason you want to read me, just google my name with "calculus") I dislike the word "derivative", which provides no hint of what a derivative is. My suggested replacement name is "sensitivity". The derivative measures the sensitivity of a function. In particular, it measures how sensitive the output is to small changes in the input. It is given by the ratio, where the denominator is the change in the input and the numerator is the induced change in the output. With this definition, it is not hard to show students why knowing the derivative can be very useful in many different contexts.

Defining the definite integral is even easier. With these definitions, explaining what the Fundamental Theorem of Calculus is and why you need it is also easy.

Only after I have made sure that students really understand what functions, derivatives, and definite integrals are would I broach the subject of symbolic computation. What everybody should try to remember is that symbolic computation is only one and not necessarily the most important tool in the discipline of calculus, which itself is also merely a useful mathematical tool.

ADDED: What I think most mathematicians overlook is how large a conceptual leap it is to start studying functions (which is really a process) as mathematical objects, rather than just numbers. Until you give this its due respect and take the time to guide your students carefully through this conceptual leap, your students will never really appreciate how powerful calculus really is.

ADDED: I see that the function $\theta\mapsto \sin\theta$ is being mentioned. I would like to point out a simple question that very few calculus students and even teachers can answer correctly: Is the derivative of the sine function, where the angle is measured in degrees, the same as the derivative of the sine function, where the angle is measured in radians. In my department we audition all candidates for teaching calculus and often ask this question. So many people, including some with Ph.D.'s from good schools, couldn't answer this properly that I even tried it on a few really famous mathematicians. Again, the difficulty we all have with this question is for me a sign of how badly we ourselves learn calculus. Note, however, that if you use the definitions of function and derivative I give above, the answer is rather easy.

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I emphatically agree that students don't know what a function is. But then again, it is a deceptively deep concept. As for modern treatments that emphasize other things than the standard, did you ever look at "Calculus in Context", the five colleges calculus? Available at math.smith.edu/Local/cicintro/cicintro.html My problem with this approach though is that even if you can convince me easily that it's the right thing to do mathematically, how will it mesh with the courses in other disciplines that students take, which will expect much more traditional material. –  Thierry Zell Sep 27 '10 at 12:55
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Any changes to math courses should of course be done only in close collaboration with other departments who rely on the math courses. But I think you'll find that, with the computational tools available, many of them will quite sympathetic to a "concept first, hand computation second" approach. Besides, I don't argue against teaching symbolic computation, just delaying it. And I do also like the Calculus in Context book but have not had experience using it. I suspect it works best with students with a stronger background than the ones I teach. –  Deane Yang Sep 27 '10 at 13:39
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Harry, that is exactly how any pure mathematician, including me, would do it. But that's the hard way. For an engineer or physicists, who thinks in units and dimensional analysis and views the derivative as a "sensitivity" as I've described above, the answer is dead obvious. –  Deane Yang Sep 27 '10 at 17:51
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Alexander and Pietro, unfortunately I said "I would like to see...", which means I don't really get to banish symbolic methods for a whole semester. In fact, I advise being pragmatic and teaching in a fashion that will not alienate you from your department or school administration. That said, if you want to slip in more understanding (which I claim actually helps students learn the symbolic methods better), I recommend taking problems from the Harvard calculus textbook, as well as their precalculus text ("Functions Modeling Change"). Especially those where no formula is given for the function. –  Deane Yang Sep 28 '10 at 2:07
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Your description of a function as a box seems to miss the most important part: that whenever you put a given number in, you always get the same output. That is, the box behavior should be single-valued. (Otherwise, we might imagine a black-box that accepts a given input and outputs a random number, perhaps different every time, and although this accords with your description, it is not a function.) –  Joel David Hamkins Jan 24 '11 at 11:36

I'm teaching Calc 1 this semester, and I've stumbled onto something that I like very much.

First of all, I start (always) by having my students draw bunches of tangent lines to graphs, compute slopes and draw the "slope graphs" (they also do "area graphs", but that's not relevant to this answer). They build up a bit of intuition about slope and slope graphs.

Then (after a few days of this) I ask them to give me unambiguous instructions about how to draw a tangent line. They find, of course, that they are stumped.

In the past, I went from this to saying "we can't get a tangent line, but maybe we can get an approximately tangent line" and develop the limit formula.

This semester, I said, "we have an intuitive notion of tangency; suppose someone offered a definition of tangency -- what properties would it satisfy?" We had a discussion with the following result: tangency at point $x = a$ should satisfy:

  1. tangency (of one function with another) should be an equivalence relation
  2. if two linear functions are tangent at $x= a$, they are equal.
  3. a quadratic has a horizontal tangent line at its vertex.
  4. if $f$ and $g$ are tangent at $x = a$, then $f(a) = g(a)$.
  5. if $f_1$ is tangent to $f_2$ at $x = a$ and $g_1$ is tangent to $g_2$ at $x = a$ then $f_1 + g_1$ is tangent to $f_2 + g_2$ at $x = a$ and similarly for the products.
  6. the evident rule for composition.

Using these rules, we showed that if $f$ has a tangent line at $x = a$, it has only one. So we can define $f'(a)$ to be the slope of the tangent line at $x = a$, if it exists!

The axioms are enough to prove the product rule, the sum rule and the chain rule. So we get derivatives of all polynomials, etc., assuming only that tangency can be defined.

Then (limits having presented themselves in the computation of area) I defined $f$ to be tangent to $g$ if $\lim_{x\to a} {f(x) - g(x) \over x-a} = 0$. We derive the limit formula for the derivative, and check the axioms.

EDIT: Here's some more detail, in case you're wondering about implementing this yourself. I had the initial discussion about tangency in class, writing on the board. A day or so later, I handed out group projects in which the axioms were clearly stated and numbered, and the basic properties (as outlined above) given as problems.

The students' initial impulse is to argue from common sense, but I insisted on argument directly from the axioms. There was one day that was kind of uncomfortable, because that is very unfamiliar thinking. I had them work in class several days, and eventually they really took to it.

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This is very nice. –  Deane Yang Sep 27 '10 at 14:20
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I second Deane's comment. –  Mark Meckes Sep 27 '10 at 17:12
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Is there really a unique equivalence relation satisfying these rules? I do not see how these rules could ever access a function which is not a polynomial. If not, saying that you can define f'(a) to be the slope of the tangent line at x=a presupposes that you have chosen one of the many equivalence relations which satisfy these properties. –  Steven Gubkin Nov 3 '10 at 15:41
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See this MO question: mathoverflow.net/questions/44774/… –  Steven Gubkin Nov 4 '10 at 14:23
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@Jeff: The MO question I linked to shows that for C^\infty functions, these axioms do characterize differentiation. –  Steven Gubkin Nov 11 '10 at 15:21

I'm going to answer this part:

does anyone out there actually use this definition to calculate a derivative that couldn't be obtained by a known symbolic rule?

Yes. $sin(x)$.

My point is that of course we can just learn the derivative of this function, but then we could just learn the derivative of any function. So looking for a "complicated function" that needs the limit definition is pointless: we could just extend our list of examples to include this function. It's a bit like the complaint that there's no closed form for a generic elliptic integral: all we really mean is that we haven't given it a name yet.

In fact, one could do $x^2$ like this, or even $x$, but I think that $sin(x)$ has a good pedagogical value. If you can get them first to ponder the question, "What is $sin(x)$?" then it might work. I'm teaching a course at the moment where I'm trying to get my students out of the "black box" mentality and start thinking about how one builds those black boxes in the first place. One of my starting points was "What is $sin(x)$?". Or more precisely, "What is $sin(1)$?". If you take that question, it can lead you to all sorts of interesting places: polynomial approximation of continuous functions, for example, and thence to Weierstrass' approximation theorem.

Many students will just want the rules. But if the students refuse to learn, that's their problem. My job is to provide them with an environment in which they can learn. Of course, I should ensure that what they are trying to learn is within their grasp, but they have to choose to grasp it. So I'm not going to give them a full exposition on the deep issues involving the ZF axioms if all I want is for them to have a vague idea of a "set" and a "function", but I am going to ensure that what I say is true (or at the least is clearly flagged as a convenient lie).

Here's a quote from Picasso (of all people) on teaching:

So how do you go about teaching them something new? By mixing what they know with what they don't know. Then, when they see vaguely in their fog something they recognise, they think, "Ah, I know that." And it's just one more step to, "Ah, I know the whole thing.". And their mind thrusts forward into the unknown and they begin to recognise what they didn't know before and they increase their powers of understanding.

We all remember professors who forgot to mix the new in with the old and presented the new as completely new. We must also avoid the other extreme: that of not mixing in any new things and simply presenting the old with a new gloss of paint.

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+1 for "if the students refuse to learn, that's their problem. My job is to provide them with an environment in which they can learn." –  Mark Meckes Sep 27 '10 at 14:44
    
+1 for the same –  Sean Rostami Sep 28 '10 at 4:42
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@Mark and Sean, I have to admit that I'm a little put off by the cheerleading for this particular phrase -- stripped from the context Andrew provided it, it comes across rather as, "I don't do a bad job of teaching, my students do a bad job of learning." I think this is an attitude those of us who teach should be careful to avoid, in general. (Of course, everyone who has ever taught has come across specific cases where it might be applied.) –  JBL Sep 28 '10 at 13:12
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@JBL: point taken, although I disagree with your restatement. "My job is to provide them with an environment in which they can learn" is the sentiment of someone who takes doing that job well seriously. I really liked the line because of several recent conversations about students who don't take notes, skip class, rush through homework, and don't ask questions. Such students are the exception rather than the rule, but they can get under one's skin. For times like that, I thought Andrew's line would be a good substitute for the glib old saw about a horse and water. –  Mark Meckes Sep 28 '10 at 13:57
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@Mark: Yes, I didn't think that either you or Sean agreed with my rephrasing, just that it seems to me that this statement (in isolation) has a little of that ring to it. Students who behave as you describe are extremely irritating, but I think teachers would do well to avoid sounding like we think this is the norm :) –  JBL Sep 28 '10 at 15:06

I agree with the above comments.

The point of my comment-question "What competing definition do you have in mind?" was to emphasize something that seems to be under-emphasized in the question itself: the reason we speak of derivatives as limits is because that's the definition of the derivative, and we want to give a definition of the concept that is going to be discussed for much of the semester.

[It is possible to give other definitions of a derivative, but they are all variations on the same theme and, in particular, all use either the concept of limit or the (equivalent!) concept of continuity. For instance, Caratheodory has a nice definition of the derivative in terms of functions vanishing to first order, but this is not going to be any more palatable to the freshman calculus student.]

[Added: I admit that I forgot about nonstandard analysis when I wrote the above paragraph. That indeed has a somewhat different feel from the usual limits and continuity. One the one hand, although I have never taught calculus this way, I rather doubt that doing so would suddenly make the difficult concepts of continuity and differentiability go over easily. On the other hand, I certainly couldn't decide to teach a nonstandard approach to calculus because it would be...nonstandard. The curriculum among different sections, different classes and different departments has to have a certain minimal level of coherence, and at the moment the majority of the grad students and faculty in every math department I have ever seen are not familiar enough with nonstandard analysis to field questions from students who have learned calculus by this approach.]

If we don't give a definition of the most important concept in the course, then we lose all pretense of developing things in a logical sequence. In particular, it's hard to see how to discuss the derivations of any of the basic rules the students will actually be using to compute derivatives, and thus we would be forced to reduce calculus to a (long!) list of algorithms based on certain unexplained rules.

Nevertheless I take your question seriously, since I have taught a fair amount of freshman calculus in recent years. It is absolutely correct that a lot of students get impatient, angry and/or confused at the limit definition of the derivative (or really, at anything having to do with limits and/or continuity). I do derivations of things like the product rule and the power rule rather quickly in class, because I know that something like half the class isn't following and doesn't care to follow. And yet I do them anyway (not all of them, but more than half) because, to me, not to do them makes the course something I could not bring myself to teach (and, by the way, would put it well below the level of the AP calculus class I had in high school: I feel honorbound to give to my calculus students not too much less than was given to me). Thus there is a real disconnect between the calculus class that I want to teach and the calculus class that something like half of the students want to take. It's discouraging.

I would be happy to hear that I am making a false dichotomy between giving the limit definition of the derivative and just giving algorithms to solve problems. I definitely experiment with different kinds of explanation beyond (and instead of!) just a formal proof. Here are some things I have tried:

1) Take the definition of continuity as primary, and define the limit of a function at a point as the value at which one can (re)define the function to make it continuous. I think this should be helpful, since I think most people have an intuitive idea of a "continuous, unbroken curve" and much less of the limit of a function at a point.

2) Emphasize physical reasoning. The last time I taught freshman calculus, I spent the entire first day talking about velocities: first average velocity, then instantaneous velocity. If a differentiation rule has a plausible physical interpretation -- e.g. the chain rule says that rates of change should multiply -- then I often give it.

3) Emphasize "chemical reasoning", i.e., dimensional analysis:. I often give the independent variable and the dependent variable units and emphasize that the units of the derivative are different from the units of the original function. In this way one can see that the conjectured product rule $(fg)' = f'g'$ is dimensionally wrong and thus nonsense. (And again, the chain rule is "obvious" from a unit conversion perspective.) Similarly dimensional analysis should stop you from saying that the volume of a cylinder is $\pi rh$.

Unfortunately none of these things have worked with the portion of the class that doesn't want to hear anything but how to solve the problems.

Added: To more directly address your specific question: yes, there are problems one can ask of freshman calculus students which require them to use the limit definition of the derivative rather than (just) the differentiation rules, but I do not recommend asking many of these questions, since the students find them very difficult. A personal example: when I was teaching Math 1A (first semester calculus) as a graduate student at Harvard, we had communal exams but the course head (who was a tenured professor of mathematics, hence a very brilliant person) had the final say. On the first exam, we decided that one of the questions was too hard, so at the last minute the course head replaced it with the following one (which he did not show to us):

Consider the function $f(x)$ defined as $x^a \sin(\frac{1}{x^2})$ for $x \neq 0$ and $f(0) = 0$. What is the smallest integer value of $a$ such that $f$ is (i) continuous, (ii) differentiable, (iii) twice differentiable?

I had the good fortune to grade this problem. Out of $200$ or so exams, the median score was $0.5$ out of $12$. About three students wrote down the right numerical answer for part (iii), but this was not supported by any work or reasoning whatsoever.

Added: by the way, it's not as though the above question is "bad" in the sense that it's not testing mathematical competence and depth of understanding of calculus. I think it absolutely is, just at a level way above that which one should be testing in a freshman class for non math majors. For the next few years, when the story came up in a social setting involving mathematical hotshots, after telling it I would press them for an answer to part c) on the spot. Most people I asked did not get it. (Note that I would not of course give them pen and paper and a quiet spot to think about the problem for some period of time. I generally required an answer after a minute or so. Let's hold PhD mathematicians to higher standards than freshman non-majors after all!) For instance, I watched a cloud pass over one Fields Medalist's face as he got very confused. After a while though I stopped using this as a pop quiz in addition to a story: I can't explicitly remember why, but I'd like to think it dawned me how obnoxious it was to put people on the spot like that...

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+1. Differentiation can be done without limits too, en.wikipedia.org/wiki/Formal_derivative I interpreted the question as distinguishing between derivatives in analysis and "generating functions". Even most trigonometric functions have combinatorial meaning and so their derivatives can be computed formally. But as you say that misses the point of calculus (continuity, physical reasoning etc.). –  Gjergji Zaimi Sep 27 '10 at 8:37
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The freshman-level example is especially biased because students tend to believe that any problem with two letters in it is very hard. But I've had a reasonable degree of success with a problem of this type, using a value for a (disclaimer: at a good school, though no Harvard). But it's also because I'd spent some time on this in class; you can't spring this on students out of the blue like that professor did and expect they'll do well. –  Thierry Zell Sep 27 '10 at 11:41
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Teaching undergraduate calculus using nonstandard analysis is not out of the question. I haven't done it but I know others who have, using for example Henle and Kleinberg's Infinitesimal Calculus. –  Timothy Chow Sep 27 '10 at 14:50
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@Mariano: no, I was dead serious. The point was that this person was far too bright to realize that this was a ridiculously hard question for freshman calculus. –  Pete L. Clark Sep 27 '10 at 19:33
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A much simpler version of your problem is the following (I actually used this problem in the past): Consider $f(x)= x^2 \sin( \frac{1}{x})$ for $x \neq 0$ and $f(0)=0$. Find $f'(0)$. This function is differentiable at zero but $f'$ is not eve continuous at 0 (so no power series representation), so I doubt that any other approach than the limit would work. I think this is one of the simplest examples which explains why the limit is needed. Also, what if one discovers a completelly new great function $f$, how does one find $f'$? –  Nick S Sep 27 '10 at 22:45

While I think that ideally, even in a freshman course of calculus, students should receive some historical notions about the development of the ideas of infinitesimal calculus, I think that, even in a freshman course of calculus, the true definition of derivative of a function should be given, that is, via the first order approximation. A function $f:(a,b)\to\mathbb{R}$ is differentiable at $x$ if there exists $m$ such that $$f(x+h)=f(x)+mh+o(h)\quad \mathrm{as}\\ \\ h\to0. $$

The fact that the coefficient $m$ (the derivative) can be characterized, and sometimes efficiently computed, as a limit of a quotient, has certainly to be observed, and should be applied immediately to treat some elementary functions like $x^2$, $1/x$ or $e^x$, as usual. But I would never give it as a definition.

I think there is a philosophical issue here. It may seem simpler to define something as the result of a procedure for getting it, compared with defining it via a characteristic property. But the latter way is superior, and on a long distance, simpler. And in the case of students who will stop there their mathematical education, then, I prefer they at least see the true idea behind, rather that being able to compute the derivative of $\cos(e^x)$ : when will that be of use for them?

The definition via first order expansion is very natural, and more understandable to the freshman students. It has a more direct geometrical meaning. It reflects the physical idea of linearity of small increments (like in Hooke's law of elasticity, etc). It is much closer to the practical use of derivatives in approximations. It makes easier all the elementary theorems of calculus (consider how needlessly complicated becomes the proof of the theorem for the derivative of a composition by introducing a useless quotient). Finally, it is closer to the generalization to Fréchet differential, which is a good thing for those students that will continue their study in maths.

A funny remark, from my experience. Ask students that received the definition of derivative as limit of incremental quotient, to compute $\lim_{x\to 0 }\sin(x)/x$. Will anybody say, it's the derivative of $\sin(x)$ at $0$, that is $\cos(0)=1$? No, they will try and use the "rule of de L'Hopital"!

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From my prof back in the days: "Some of you might have heard of a thing called L'Hopital's rule. It has a lot of hypotheses that no-one ever checks, and students always apply it when the quotient is in the wrong form, so I won't teach it and you'd better not use it." And now I do the same... (I didn't mind when he said that because I was one of the ones who'd never heard of it.) –  Thierry Zell Sep 27 '10 at 12:42
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Yes! I think that introducing the differential of a function of several variables would be much easier for students if they had this point of vue on derivative. –  Benoît Kloeckner Sep 27 '10 at 15:57
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@Andrew: and if I was the chairman in another maths department, I'd immediately engage him with double salary. De'LHopital himself would be embarassed to know somebody's still wasting time with such an awkward theorem like that thing that brings his name. Theorems, like cakes, don't always come out well; that thing came out very badly, and left a mess in the oven. Today, it may be at most of some historical interest. Teach the Landau notation instead! (Btw, as you probably know, Edmund Landau was fired from Göttingen in 1933, with the pretext of his way of teaching a calculus course.) –  Pietro Majer Sep 27 '10 at 19:13
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@Pietro: what you give is essentially Caratheodory's definition, as alluded to in my answer. It's so close to the usual definition that I don't really believe that students have a significantly easier time with it. However, I believe that when you teach calculus, this definition inspires you and you do a very good job teaching it, more so than you would with the standard definition. I suspect that most "the students find it easier when..." statements are like this, but that's fine -- finding the version that you can get behind enthusiastically and explain well is part of good teaching. –  Pete L. Clark Sep 27 '10 at 19:49
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About that funny remark: even saying $\lim_{x\to 0} \sin(x)/x$ is the derivative of $\sin(x)$ at 0 may be viewed as cheating, since the typical textbook approach is to use a geometric argument to prove $\lim_{x\to 0} \sin(x)/x = 1$ and then use that limit to prove that $\frac{d}{dx} \sin(x) = \cos(x)$. –  Mark Meckes Sep 28 '10 at 14:27

I wanted to add one further point to the many good answers already given here: "black box" symbolic computation, in the absence of understanding the formal definitions, can work when everything goes right, but is very unstable with respect to student errors (which are sadly all too common). Knowledge of definitions provides a crucial extra layer of defence against such errors. (Of course, it is not the only such layer; for instance, good mathematical or physical intuition and conceptual understanding are also very important layers of defence, as is knowledge of key examples. But it is a key layer in situations which are too foreign, complicated, or subtle for intuition or experience to be a good guide.)

For instance, without knowing the formal definition of the derivative, a student could very easily start with a true formula such as

$$ (x^2)' = 2x$$

and do something like "substitute x=3" to obtain the false formula

$$ (9)' = 6.$$

(An example that I have actually seen: someone attempted to prove Fermat's last theorem by starting with the equation

$$ a^n + b^n = c^n$$

and then differentiating with respect to $n$. Ironically, a variant of this type of trick actually works when solving FLT over polynomial rings, but that's another story...)

Now, without bringing in the definition of a derivative (and of a function), how could you explain to the student what went wrong here in a way that the student will actually remember? Saying that one can use the law of substitution or the trick of differentiating both sides in some situations, but not in others, is likely to be recalled inaccurately, if at all (and may have the side effect that the student may view such basic moves as substitution as somehow being "suspect", thus avoiding it in the future).

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People say I'm mean for asking for the derivative of $\pi^2$, but I think it's a memorable example for the students. Another place the blind symbol manipulation goes wrong is on $\sin^{-1} x = \arcsin x$. Many students are willing to assume that if $f(x)=g(x)$ then $f'(x)=g'(x)$, which is only a consequence for some meanings of the first equation. –  Douglas Zare Nov 6 '10 at 16:43
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@Terry: I agree that the black-box use is an issue, but some might argue that it's possible to fix without necessarily going all the way to a limit definition, because the root problem is conceptual understanding of functions, and more precisely of the derivative as a function. A simple aphorism would suffice: "chug then plug, don't plug and chug!" More seriously, one could discuss the difference between the derivative as a function and the derived number at a point (slope) purely graphically, with no references to limits. –  Thierry Zell Nov 7 '10 at 0:01
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@Douglas: I didn't think the derivative of $\pi^2$ was a mean thing to ask! Who thinks so? students? colleagues? Never mind, I'll be sure to borrow it for next time. (Even meaner would be to ask for the derivative of $e^2$, btw.) As for the blind symbol manipulation, I cannot believe that it took me all these years before seeing for the first time (in an exam) that the derivative of $\arctan x$ was $\mathrm{arcsec}^2 x$. In retrospect, I should have been expecting this for a long time! –  Thierry Zell Nov 7 '10 at 0:09
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I think this type of error is avoidable if you introduce $x$ not as a variable, but as a special symbol for identity function. Even more, you can use bold $\mathbf{x}$ for the identity function and normal $x$ for a value. –  Anixx Jan 2 '11 at 18:35
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When my father was a judge at a high school math fair, a student gave a presentation on calculus. During the question period after the presentation, he asked the student "If f(x) = 3^2, what is the derivative of f(x)?" The student said "6". My father then asked "If f(x) = 9, what is the derivative of f(x)?" The student said "0". He asked the first question again and the student still said "6". Of course this student did not go on to the next round of the math fair. –  KConrad Jan 23 '11 at 20:31

The derivative of $x|x|$ is best computed via the "limit" definition. A more general example is $xf(x)$ where $f$ is any continuous function, and we are computing the derivative at $x=0$.

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This is a good example -- Steven, take note :). (Though I would tweak it to get something with nonzero derivative, if possible.) –  JBL Sep 27 '10 at 16:08
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If you want example with nonzero derivative consider $x|x|+x$. –  Igor Belegradek Sep 27 '10 at 17:12
    
For the more general example, one should also ask that $f$ is not differentiable at $0$. (By varying such $f$, we can get any derivative at all, or none. Taking $f(x) = x \sin(1/x)$, extended continuously, gives a very interesting example that's come up elsewhere on this page.) –  Toby Bartels Apr 3 '11 at 23:08
    
This example can be done straightforwardly without limits, using infinitesimals (either informally or in NSA). –  Ben Crowell Oct 6 '12 at 2:10

I am surprised that no answer has explicitly mentioned the fundamental theorem calculus yet: that is a classic, and important, instance of calculating the derivative using the limit definition. So, for example, the integral sine function

$$\int_0^x \frac{\sin t}{t} dt $$

has important applications in signal processing and the cumulative distribution function of the normal distribution $N(a,\sigma^2)$

$$\frac{1}{\sqrt{2\pi\sigma^2}}\int_{-\infty}^x e^{-\frac{(t-a)^2}{2\sigma^2}}dt$$

is the bread and butter of probability and statistics. Both functions are not elementary and their derivatives, while significant, would be impossible to calculate by other means.

I also disagree with the comment that piecewise defined functions "are not good at all" for illustrating the definition of the derivative based on limits. In fact, piecewise polynomial functions, in the form of splines, are used in mechanical engineering (e.g. to design the shape of the car body), and provide a neat opportunity to relate conceptual and computational aspects of derivatives.

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+1, the guys who deal in splines always make a big deal out of left and right continuity for making the approximant $c^n$ (for whatever value of $n$ is needed by the application). I too feel that the error function and the sine integral are too important not to at least be given a passing mention in the context of the Fundamental Theorem. –  J. M. Sep 28 '10 at 1:55
    
I agree with both points here. –  Deane Yang Sep 28 '10 at 2:08

The way that Calculus is traditionally taught gives a false impression that every function worth looking at can be differentiated using the rules of differentiation. This comes from a misconception that any function worth looking at can be described by an algebraic formula, or using trigonometric or logarithmic functions.

That's just not the case: the most common everyday functions don't have any formulas. Some examples:

  1. Price of a company stock over several decades.
  2. Volume of water in a water tower over the course of a week
  3. Median price of a house in your area (adjusted for inflation), over the course of 100 years.
  4. US National Debt over the last two hundred years.
  5. US Deficit

For such functions, rate of change has a very real meaning. I find that students who had Calculus in high-school are stumped if I give them an example like that and ask them to graph the rate at which, say, the US national debt has changed throughout US history, and how that relates to the deficit.

Understanding the derivative as both rate of change and the slope of the tangent line helps, and the only good way to tie those concepts is with using limits.

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@Anna: I agree heartily. To take it a step further: I have often thought that the traditional calculus sequence lacks an applied component which severely limits its usefulness to those who are not going on to physics and math. As you say, when given a real world function of interest, you are generally not given an algebraic expression for it. Rather, in order to apply the methods of calculus in a quantitative way, there needs to be a step where you create a mathematical model of the function. I was never taught how to do this step myself, and it seems not to be at all trivial... –  Pete L. Clark Jan 29 '11 at 22:38
    
Once or twice I tried to remedy this by asking exam questions like: "Write down an explicit function which has local minima at $\pm 2$ and approaches infinity as $x$ approaches $\pm infinity$." I was hoping that a student would see that a simple function satisfying these conditions is a fourth degree polynomial with double roots at $\pm 2$, thus $f(x) = (x-2)^2 (x+2)^2$. But they had a lot of trouble with this, and the answer to my question "Does teaching the standard curriculum give them the tools to answer this question?" was "No." But wouldn't it be great if students could do this? –  Pete L. Clark Jan 29 '11 at 22:44
    
@Pete: I have been thinking a lot about how the calculus we teach is too neat and not applied enough these days, and I've been thinking about it not because of my calculus classes, but because of my higher-level courses where I've had to use "messy" or "exotic" stuff like Taylor expansions pretty often to motivate some avenues of investigation. I wish my students were more comfortable with this, and expected less of the neat closed-form formula results. Though of course, we should teach them to appreciate when closed-form stuff happens too! –  Thierry Zell Mar 12 '11 at 1:25
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I appreciate your point, but your examples are all discontinuous functions, so in fact the notion of a limit doesn't quite work. For example, the US deficit can only change in steps of one cent, the amount of water only in steps of one molecule. These could all be used as examples to show that the Cauchy-Weierstrass limit does not actually connect to reality. Perhaps a better point to make with these examples would be that students could benefit by understanding discrete/numerical calculus as well as the classical calculus of continuous real functions. –  Ben Crowell Oct 6 '12 at 2:47

The definition of derivatives is useful in exercises about functional equations. Ever solve $f(x+y)=f(x)f(y)$ ? A more elaborate one is $[f(x):f(y):f(z):f(t)]=[x:y:z:t]$, functions preserving the cross-ratio (= anharmonic ratio).

However, we should not neglect the interest of the black-box side of mathematics. We should remember that it is this aspect which has made mathematics so much unavoidable in Science. Somehow, it participates to the ``unreasonable effectiveness of Mathematics in the Natural Sciences'' (E. Wigner's famous statement). After all, the definition of derivatives has the same status as the constructions of ${\mathbb Z},{\mathbb Q},{\mathbb R},{\mathbb C}$. One can spend a year without thinking about them, while using these fundamental objects every hour, by applying rules. Do you remember the construction of the polynomial algebra $k[X]$ ? How would you define $\pi$ ? In a more advanced situation, chemists have efficient rules to deal with characters of representations of finite groups, and they do not need to read a justification, or to remember it, even though the first Chapter of J.-P. Serre's book was intended to be read by his chemist wife. Mathematics is the tool box of Science. It is even a tool box for itself, in the sense that new topics use the older ones. To go further, we must accept older truths. Of course, it is way better to accept them for good reasons, that is, because we have completely understood the definitions. But if the half of a classroom, who does not intend to do mathematical research, neglects the definition and prefer focussing on the rules, there is no problem at all, provided they apply the rules correctly. There are many ways to learn rules, one of them being solving a lot of exercises.

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+1. But, I think a point some other people have been making is that students should at least see the formal definition. But I agree that you shouldn't beat it over the head of students who aren't happy... –  Matthew Daws Sep 27 '10 at 12:07
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If scientists want their students to learn a sequence of formal rules, I think they should be the ones to teach them. Personally I want to teach mathematics, and mindless manipulation of symbols is not math. –  Steven Gubkin Nov 11 '10 at 15:31

An example I like is $\exp(-\frac{1}{x^2})$ and the "bump functions" one can construct with it.

First of all, this example is important in differential geometry (e.g. Whitney's embedding theorem) and complex analysis (as an example of a real $C^\infty$ function which isn't holomorphic).

In second place, even in first year calculus it's an important illustration of the concept of derivative and of Taylor's theorem. It's important in my opinion to understand why all derivatives at zero are zero (i.e. because it goes to zero faster then any polinomial) but even so the function is changing values.

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The example is well-chosen, but your parenthesis sounds misleading to me :there exist functions that go to zero faster than any polynomial at zero, while they are not even twice derivable. –  Benoît Kloeckner Sep 27 '10 at 15:53
    
You're right! Thank you. To be honest, I didn't consider this possibility at the time of writing. I'll leave the parenthesis as is, since it still has some content and you're comment is right below. –  Pablo Lessa Sep 27 '10 at 16:57
    
@ Benoi: Are there examples of such functions which are continuous in a neighborhood of 0? I ask because the only examples I can find are along the lines of the characteristic function of the rationals time exp(-1/x^2). –  Steven Gubkin Nov 2 '10 at 22:59
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Take $f(x) = \exp(-1/x^2)$ and $g$ a continuous nowhere differentiable function. The function $h(x) = f(x)g(x)$ is continuous, goes to zero faster then any polynomial when $x \to 0$ but isn't differentiable at any point other than $0$. Hence $h$ isn't twice differentiable. –  Pablo Lessa Nov 9 '10 at 12:10

This is a question that I also struggle with sometimes. On the one hand, I understand the value of sweeping things under the carpet when students are not ready for them yet. When I learned Calculus in High School, we talked about -- but never properly defined -- limits (I'm can't recall if we did the limit derivatives). Yet, we managed to go pretty far into the material, e.g. establishing recurrence relations for integrals of the type $\int_a^b e^{-x}\sin(n x)$. This lack of definition was a very frustrating point for me, and when I finally learned about $(\epsilon,\delta)$ two years later, a wave of relief washed over me. Yet, I'm pretty sure that my cohorts did not feel the same way, hence my sympathy for teachers who want to keep things simple by hiding the definition.

At the same time, I don't want my Calc course to be a series of magic tricks, so I always insist on the logical construction of the course: we want to investigate slopes of tangents. We want to work exactly, not approximately. This is why we'll get into limits in the first place (not very historical, but a logical development). So what do I do?

  • I briefly cover $(\epsilon,\delta)$ without really applying it. Just to show the difference between a "wordy" definition and a mathematical one.
  • I insist on the fact that the limit laws are derived fro this rigorous definition. (You can sketch the proof for the sum of limits for instance).
  • This sets up for students how the mathematical edifice is built: abstract definitions to formalize intuition, big gun theorems proved rigorously from these definition (limit laws, derivative laws...). Add a few examples to the mix and then you're set up for practical, mechanical computations (the stuff that computers do).
  • I am upfront about the fact that I don't expect my students to use the $(\epsilon,\delta)$ definition, though I like them to memorize it. The payoff will be later.
  • I also stress that, historically, calculus was done without this definition for a long time: so it can be done, they will be able to do it, but it also has its limitations when dealing with more abstract material.

In a course that is set up in this way, it is quite natural to cover the limit definition of derivatives. There are a lot of good reasons why one should do that anyway, some of which have already been addressed. Functions which are defined piecewise do require this, and that includes important examples like $\exp(-1/x^2)$ and fun ones like $x^2\sin(1/x)$. The rigorous derivation of the derivative of $\sin x$ is another good example.

There are also wrong ways of doing this. In the comments, Holger pointed to the case study in the Notices article Teaching mathematics graduate students how to teach. Here, the problem asked to use the definition of derivative to compute the slope of a certain cubic at a point. By the time the exam rolls around, you have easier ways of doing this, so of course the students would feel that this is an arbitrary and confusing question.

[Actually, I took so long to write this that I've been ninja'd by Pietro on this example.] One example that I have yet to see though is Taylor series: defining the derivative in this way makes it obvious that $$ f(a+h) \approx f(a)+f'(a)h+o(h)$$ and sets you up for the higher order ones. Yes, you can see that from the graph too, but at that level most of my students have a terrible time reasoning from an abstract graph.

Given how fundamental these ideas are, especially in Physics, I can never stress enough these kinds of relationship in my course.

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So far no one's mentioned (or did I miss it?) that if you make students compute $$ \lim_{w\to 5} \frac{w^6 - 5^6}{w - 5}, $$ then some of them will use L'Hopital's rule to do that, if you don't tell them not to.

Here's an example of something I have students do with the limit definition of the derivative: http://wnk.hamline.edu/~mjhardy/1170/notes/quiz.10.19.pdf

They find all sorts of creative ways of getting things wrong when doing this. Here's another: http://wnk.hamline.edu/~mjhardy/1170/homework/13th.pdf

I think after they've done several like this, they actually do learn what this is for, and that it's not being used as a way to avoid quick and efficient ways of computing derivatives.

But I have them thinking about instantaneous rates of change without using limits on the first day of the course: http://wnk.hamline.edu/~mjhardy/1170/handouts/September.8.pdf

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Fantastic handouts! I think you would get a lot more upvotes on this answer if people clicked on the links. –  Steven Gubkin Jan 28 '11 at 15:20
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Sorry about the broken links. Those were for the duration of the semester. I've reinstated the first one. To be continued....... –  Michael Hardy Mar 12 '11 at 21:36
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OK, for now the one that ends with "13th.pdf" can be considered superseded by this: wnk.hamline.edu/~mjhardy/1170/homework/5th.pdf –  Michael Hardy Mar 12 '11 at 21:38
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THe "handout" referred to there is here: wnk.hamline.edu/~mjhardy/1170/handouts/March.2nd.2011.pdf –  Michael Hardy Mar 12 '11 at 21:38
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All links give 404 errors. –  Julien Puydt May 19 '12 at 13:06

I hope my answer is read as a response to the question asked, rather than as either a defense of or disagreement with the choices the pedagogists (is that a word?) make.

I think one of the main reasons to teach derivatives in terms of the $h\to 0$ limit is that it captures the dual notions of "instantaneous velocity" and "slope", which are respectively physical and geometric.

(Ok, now I will mention some personal opinions about teaching calculus. I love physics, and sometimes pretend to be a physicist, so for me the geometric/physical meanings of calculus are very important. So I would love if they were emphasized more. Unfortunately, we do not do enough in introductory calculus classes in that direction, and it is very hard to present functions and ask students to find the slopes of their graphs without essentially teaching them these black-box techniques. So I don't know whether it's worth it: maybe we should just do the algebraic part of calculus --- it's the only thing we tend to test anyway. I also don't really think that MO is the best place to get into that discussion, though, and I don't think that OP intended as such.)

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Interesting answer. I must stress the connections with Physics than I realized, since a student asked me just last week if I taught Physics also. You mention that we tend to only test the algebraic part of calculus; clearly, we should put our money where our mouth is. Because I emphasize the theory, I make sure there are at least some non-algebraic questions on my tests (you can make up some easy ones). If a prof thinks that the audience can only handle the algebraic stuff, I'm ok with it, but then you should be teaching only that. –  Thierry Zell Sep 27 '10 at 12:00
    
I agree that there should be at least somewhat more physics in calculus courses than their currently is. To do it otherwise seems like another example of the decortication of calculus: the subject takes place in a certain intellectual context and as a response to certain scientific problems. A lot of students get really worried when you talk physics though, since they think you're expecting them to have some outside knowledge that they in fact don't have... –  Pete L. Clark Jan 29 '11 at 22:18
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...The irony of this for me personally is that I haven't taken a physics class since high school, so all the physics I now know is that which comes up when explaining closely related topics in mathematics. For instance, if I am supposed to talk about first order differential equations, I feel compelled to talk at least a little about second order differential equations, especially those of the form F(x) = c x'', because this is the best motivation I know as to what solving differential equations means and why it's important:... –  Pete L. Clark Jan 29 '11 at 22:21
    
...the equation gives a law governing the behavior of some object over time, and the solution to the equation tells you what the consequences of that law are for the behavior of the object. In particular, my favorite example is m x'' = - kx. To solve this equation without writing down Hooke's Law, a picture of a spring, carefully explaining the physical significance of the minus sign...what a disservice that would be! By making it physical, you give the student a chance to use her intuition: "Well, if I were forced to satisfy this differential equation, what would I do?" –  Pete L. Clark Jan 29 '11 at 22:27

Well, the definition of derivative is probably one of the best application of the notion of limit, from a didactical point of view. If you define the derivative as a limit process then students who understand it will not miss the geometric flavour: the slope of the tangent line is the limit as $h\rightarrow 0$ of the slope of the line through $(x, f(x))$ and $(x+h, f(x+h))$. I think this is beautiful and relatively simple, once you get the students to think about it for a minute. Plus, it answers the question "When do we agree that the graph of $f$ admit the existence of a tangent line at $(x, f(x))$"?

Of course one has to keep in mind that for most students the useful thing to learn is how to compute practically a derivative without using the definition but rather applying a collection of rules. Nevertheless I think it is important to give them an idea of where all these rules come from. Think about those students who want to get a a math major? No?

In Italy in the so called "scientific high school", the schools that provide you with the widest and most basic education (you learn a bit of everything) with a focus in math, physics, chemistry perhaps, ecc.. we are taught the limit using the $\epsilon-\delta$ definition, and the derivative from its definition. This is to say that I think it is possible to have students learn this theoretical aspects of calculus, if high school kids do.

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As you probably know, the Italian government is now planning to gradually change the high school teaching programs into three main topics: "Religion", "Use of guns", "Commercials". So the content of the maths programs of the Italian high school is becoming soon a historical topic. If this is the trend, I guess (I hope) Italy itself will become soon a historical topic too :-( –  Pietro Majer Sep 27 '10 at 12:22
    
I also learnt calculus in high school using the $\epsilon$-$\delta$ definition. This was an ordinary public (state-run) high school in the United States. –  Toby Bartels Apr 3 '11 at 21:11
    
It can create issues for students (due to their typical lack of logic skills in first year) to think about derivatives giving a criterion for the existence of tangent lines. This is because the first thing they think of after a line is a circle, and the tangent to a circle is hard to describe in this way. –  Glen Wheeler Oct 21 at 4:03

The standard definitions of limit, continuity and derivative are things of beauty mathematically - flexible and well-honed like fine woodworking tools. But to get calculus students to care, and appreciate their meaning and significance, takes some motivation.

A pretty good way to motivate $\epsilon$-$\delta$ is that it has to do with determining what control on input error ($\delta$) is needed to guarantee meeting a given tolerance for output error ($\epsilon$). How accurately do you have to aim a spacecraft to ensure it enters Martian orbit without burning up the way Beagle 2 did, costing hundreds of millions? Students can appreciate this is a serious question, and that it is fair to insist they be able to handle simple examples like $f(x)=-100x+50$, $\epsilon=10^{-2}$. (In large lectures for freshmen, I wouldn't do much more than Lipschitz examples or something carefully designed so $\delta$ is easy to find without cases. Many calculus students are adults but, ahem, need practice with inequalities.)

One can tell engineering students who just want the formulas that they'll be surprised to find that in a couple of years they'll be estimating ``sensitivity coefficients'' numerically from black-box software or experiment. Gee, sensitivity coefficients are just derivatives, and they'll be estimated from the definition, not symbol-pushing.

Speaking of which, it's nice to express the error in the definition explicitly: $$ \frac{f(x)-f(a)}{x-a} = f'(a)+E_a(x), $$ and do the algebra that occurs to few to write $$ f(x)=f(a)+ f'(a)(x-a)+ E_a(x)(x-a) $$ This makes the nature of linear approximation a bit more apparent.

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It's funny that actually many students believe that the symbiosis is always the other way around, i.e. derivatives are used to compute limits (l'Hopital etc.). My favorite example of an elegant calculation of a derivative using the limit definition comes from basic physics. Ask the students why the acceleration of an object performing uniform circular motion is always perpendicular to the velocity. One could come up with a non-elegant solution by writing the equations of motion and using a derivatives table, or one could observe the nice geometric proof of considering an infinitesimal isosceles triangle formed by the two velocity vectors that are a few seconds apart and notice that $\Delta \vec{v}$ is the base of this triangle and points toward the center of the circle.

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If you use the derivative table for a scalar product (applied to the square product), from v.v=cte, you get 2v.a=0 -- that doesn't look that non-elegant. In fact, I use this very example to show my students (which are more into physics) that this rule gets results they're already well-acquainted with. That convinces them more than the proof... –  Julien Puydt May 19 '12 at 13:16

I believe it gives a good conceptual or practical reasoning to why we would want to study calculus in the first place. As it was first introduced to me, we can always talk about the average speed a car has traveled over a certain distance. Even very small distances, but apart from a speedometer, how can we say 'I'm travelling XX km/h right now'. There enters the limit definition, where we want the instantaneous rate of change!

If we only presented the formal rules for differentiation, we run in to the same problem as high school students who dislike math present "But my calculator can just do it! Why do I need to learn this?!". If the fundamentals are not taught, one day they will be forgotten.

There are certainly other rigorous approaches to the derivative out there. The delta-epsilon method, which most students in their first year simply struggle to grasp as easily as the $h \rightarrow 0$ definition. This approach is typically reserved for the math majors who go on to take a course in analysis, not the general first calculus course for all science majors.

While I do not use this definition in practice, I am primarily not calculating derivatives, so take that for what it's worth I suppose.

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A problem I like to give students to solve shortly after introducing the derivative is to evaluate $f'(2)$ for $f(x) = x^x$. Of course, this function can be rewritten as $f(x) = e^{x\ln x}$ but in my experience students don't think of this. In fact, students who have seen Calculus before almost universally reach the solution $f'(2) = 4$ which they get from the mistaken idea that $f'(x) = x\cdot x^{x-1} = x^x$. The only students that usually get this problem correct are those that haven't yet learned any of the computational methods and only know the definition.

I teach the limit definition and emphasize the physical and geometric interpretations, and then move from that to the concept of the tangent line and linear approximation. I think these concepts encapsulate most of what is significant (intuitively) about the definition. I dislike exam questions that require students to compute derivatives using the limit definition when they know a "better" way to do it. It isn't too hard to write a problem where no formula for the function is given and ask students questions about the sign or approximate magnitude of the derivative or whether or not the function should even have a derivative. For students who to do not intend to pursue mathematics, this seems appropriate to me. Even those who become mathematicians will almost surely see these ideas again in complete detail in an elementary analysis course.

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How do you differentiate x^x using only the limit definition? This seems like a tall order to me. –  Steven Gubkin Nov 11 '10 at 15:35
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I only meant that they would numerically estimate it at x = 2 using the limit definition. It is easy to estimate using the definition, but if they try to differentiate and plug in 2 they will probably get the wrong answer. –  Jeremy West Nov 11 '10 at 18:13

It is worth noting that there is a lot of historical precedent for teaching it as a limit, which occurs already in Euclid. I.e. Euclid characterizes the tangent to a circle as the unique line such that between it and any other line through the same point, one can interpose a secant (Prop. 16, Book III). (Strictly, he says equivalently that one cannot interpose another line between the tangent and the circle itself, i.e. every other line through the point is a secant.) Thus the tangent is the limit of those secants. Thus I believe one can easily say that the limiting point of view is the original one of Euclid. From this point of view, the idea of limit is the one used so fruitfully by the Greeks, and the contribution of the mathematicians of later times is to make that notion more precise.

On the other hand, if you want to avoid the conceptual difficulty students have with limits, you can follow Descartes instead, at least for derivatives of polynomials, and characterize the tangent line as the unique line such that subtracting its equation from the original function gives a polynomial with a double root at the given point. This leads to motivating the Zariski cotangent space, as M/M^2.

Both points of view also have a nice dynamic interpretation as realizing the tangent as the unique line intersecting the curve doubly at the point, understood as the limit of the two secant intersections,and measured by the presence of a double root.

But if you want a defense of limits, I suggest Euclid Prop. 16, Book III as ample precedent.

If you want a defense of making students practice using the limit definition, I propose that as noted above, this is the only way to get them to appreciate the fundamental theorem of calculus. That theorem cannot be appreciated by memorizing rules for derivatives, One must understand the definition and apply it to an abstractly defined area function. I suggest that one reason many students do not understand why the fundamental theorem of calculus is true, is that (again as noted above) they have not grasped either what an abstractly defined function is, nor what a derivative truly means.

So if you want them to understand the relation between the derivative and the integral, then I agree with others that they need to know what a function is and derivative is. The reasoning here is that once someone understands something, he can use it in more settings than could possibly be covered by any set of rules.

Another practical benefit of testing the use of the h-->0 definition to obtain derivatives of simple functions, is that it forces practice in algebra, trig identities, and exponentials, skills which most of my students are almost completely lacking.

However, I recommend you teach it any way that makes sense to you. after all you understand it, so whatever you say based on that understanding will be useful. Make up your mind what seems important to you, and go for it!

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If it is just a question of definition but not a question of computation, I have heard when I was a student the following definition:

Let $f$ be a real continuous function, class $C^0$. Let $$ \Delta f(x,x') = \frac{f(x') - f(x)}{x'-x} \quad\mbox{defined on}\quad {\bf R}^2 -\{x=x'\} $$ If $\Delta f$ admits a continuous extension on the diagonal $\{x=x'\}$ then it is unique, and $f$ is said to be of class $C^1$. The function $f'x) = \Delta f(x,x)$ is then called the derivative of $f$

Of course this is the standard definition, nothing new under the sun, but the $\epsilon-\delta$ calculus if hidden, and of course, not for long :-)

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It's an interesting alternative to the usual, but ultimately I'm afraid it's even more challenging than the usual one. Of course, here I'm not talking about using the definition to perform computations, but even to grasp the intuitive meaning, resorting to functions of two variables strikes me as more than a student can comfortably handle. –  Thierry Zell Jan 27 '11 at 16:14
    
This also has the technical problem of not covering the case when a function is differentiable but not continuously so. (Of course, this might appear in a context where one is uninterested in such functions.) We can fix this by making it slightly more complicated: ask whether $\Delta{f}(a,-)$, the restriction of $\Delta{f}$ to a given vertical line $\{x = a\}$, admits a continuous extension; if so, then this is unique, and we define $f'(a)$ to be the new value $\Delta{f}(a,a)$. –  Toby Bartels Apr 3 '11 at 21:22

The answer I give my students is that mathematicians want to know what a word (in this case 'derivative') means in all cases, and the definition of the derivative is a communal agreement about what to say in strange cases such as the absolute value function. (Well, since I banish symbolic stuff from the first two weeks, I say 'function whose graph has a sharp corner like this one (draws on board)'.)

If students press further, I point out that in a literature class they are expected to learn the communal agreement on the difference between a 'simile' and a 'metaphor'. It helps that I am at a liberal arts institution and not a technical one.

Let me also use this opportunity to share a pedagogical trick:

I find it helpful (third time I've tried it) to break up the definition of $f^\prime(2)$ into two parts:

1) Define a new function $E_2$ by the formula $$E_2(x)=\frac{f(x)-f(2)}{x-2}.$$ 2) Take the limit of $E_2$ at 2.

To pull this off, you do need to take the function $E_2$ somewhat seriously; graph it, write formulas for it, et c.

Rationale:

1) It always helps to break up complicated definitions into smaller pieces.

2) It emphasizes that you take limits of functions (in the sense of machines that accept a single number as input and gives a single number of output) rather than of symbolic expressions.

3) Students get to really understand why a discontinuous function or something like the absolute value function is not differentiable (at the relevant point).

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This semester I am teaching Calculus I following Rogawski's textbook. The chapter on limits spends some serious effort on getting comfortable with average rates of change, going as far as creating a table of values (easy to do with a graphing calculator) and numerically estimating the limit, before going on to derivatives in the next chapter devoted to derivatives. By the way, I don't see how this approach could accomplish a worthy goal stated in Rationale 3. –  Victor Protsak Sep 29 '10 at 7:12

One way to avoid limits without losing too much is to teach the calculus of finite differences. Conceptually, the move from numbers to lists-of-numbers as first-class mathematical objects is easier than the move from numbers to real-valued-functions-of-a-real-variable, and the easier move also forms a good stepping stone to the harder one. One can develop the calculus of finite differences mutatis mutandis and thereby make the transition to infinitesimal calculus essentially painless. (So, for example, one should work not with polynomials per se, but with linear combinations involving rising or falling powers).
All the black box rules have their analogues, and all are reasonably easy to see and/or prove. Passing the limit, when it happens, comes as a welcome simplification.

Aside from the conceptual challenge of functions themselves, students find limits difficult because of their quantifier complexity. I have never understood why standard algebra pedagogy suppresses quantifiers, thus, for example, leaving many students unable to distinguish between unknowns (literals bound by existential quantifiers), variables (literals bound by universal quantifiers) and constants (literals that belong to the language itself). Students who miscalculate the derivative of $\pi^2$, mentioned elsewhere, don't get this distinction. People who become mathematicians usually "got it" without anyone spelling all this out, and then they learned about quantifiers studying logic in college, so they regard quantifiers as sophisticated and advanced. But most students don't "get it," and I think this accounts for the huge attitude downturn when they get to algebra.

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$\frac{\sin x}{x}$ at $x = 0$ should be a good example.

P.S.: Talking about esoteric definitions, if you can introduce stationary point without derivatives, you can then introduce derivatives using sheaves, like you would introduce vectors on a smooth manifold. It would broaden the consciousness of your freshmen, he-he ^_^

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There was a recent article in the American Math Monthly, Analysis with Ultrasmall Numbers, that might be of interest. For a summary of its implementation in a high school classroom, see http://maths.york.ac.uk/www/sites/default/files/odonovan-slides.pdf.

A quick skim of its implementation seems to suggest that it provides a groundwork for some of the informal manipulations used in calculus-based physics classes.

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Another alternative way of teaching calculus is via infinitesimals (for example the book Elementary Calculus, An Infintesimal Approach by Keisler). The way of thinking about calculus via infinitesimals is obviously very natural, and mathematicians (e.g. the pioneers of calculus, Euler etc...) have used arguments using infinitesimals long before they should have been really allowed to do so. Keisler's book (and in general the area of `Non-standard analysis') makes rigorous our intuition regarding infinitesimals, and is a set of rules that teach us how to formally reason with them. In my opinion this system is intuitive, but the student can never really have a proper understanding of what they are doing "from the ground up" with out some basic knowledge of model theory. The limit approach is less intuitive, but at least a student doesn't have to just accept some rules without truly understanding what's behind them. Possibly this infinitesimal approach is a half way house between teaching it properly with limits and just teaching rules of differentiation to people who aren't interested.

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I am a STUDENT in 11th grade who has just finished BC Calculus.

I don't have a PhD or even a high school diploma, obviously.

But I think that beginning with h->0 is essential. Otherwise, we don't have any definition of a derivative.

But more to the point, I don't think it's strictly necessary to learn the concepts completely before you do symbolic calculations.

How would the teacher make sure that the students eventually learn them before it's too late, then: Leibniz's lovely notation.

Newton's notation is essentially a meaningless shorthand. Prime is arbitrarily chosen to mean a derivative. I don't like that. (I like it as a shorthand, but there is no real meaning behind it).

But Leibniz's has actual meaning: dy/dx is analogous to delta-y/delta-x.

If we are always used to writing dy/dx=... or df(x)/dx=..., then it is no great stretch to write things like df(x)=...dx. And this leads us nicely into differential equations by separation of variables, and concepts such as substituting variables when you integrate.

In my humble opinion, using Newton's notation should be avoided as much as possible, because it doesn't make it clear what you are doing and turns people into robots, mindlessly following the rules of differentiation.

I don't think that Newton's method id all bad. I think it may be good when you are taking derivatives of higher orders, because once you have the concept of derivatives down, it's more important to see that you are taking the derivative of another derivative. (The "exponents" in Leibniz's notation make it a bit confusing).

If I were a Calculus teacher (and I very well may become one someday), I would all but scrap Newton's notation.

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What puzzled me as a calculus student decades ago was the meaning of the dx and the dy in dy/dx. I thought I'd solved the problem by understanding these components not to have independent meaning outside the "quotient" dy/dx . But then I met differentials in the context of multi-variable calculus and felt despair. Even worse, I saw differentials presented very formally - as "expressions" that "varied" in particular ways under coordinate changes. Only years later, in graduate school, did I finally understand dy and dx as linear variables living in cotangent spaces. –  David Feldman May 19 '12 at 6:49
    
Which is all to say Leibniz's notation has its pedagogical dangers too! –  David Feldman May 19 '12 at 6:49
    
What's wrong with viewing a dx as "a small change in x" and a dy as "a small change in y (resulting from the small change in x)"? –  Steven Gubkin May 19 '12 at 7:22
    
The primes for derivatives are not Newton's notation. That is Lagrange's notation. Newton used a dot for derivatives, which is absent from math books but still used by physicists. See en.wikipedia.org/wiki/Notation_for_differentiation –  KConrad Jun 30 '13 at 16:38

I'm interested in the differentials-based approach advocated by Dray and Manogue at Bridging the Vector Calculus Gap. This is for multivariable calculus, but they do discuss the one-variable version (pdf). As they mention, people have reviewed calculus (especially for science courses) in these terms, but has anybody lately taught it this way?

(Also, the theory behind this approach is a little unclear beyond the first derivative, which is what led me to this question.)

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I suppose that I should mention now that I have taught it this way, twice, in a terminal mathematics course intended for non-STEM majors (especially business): tobybartels.name/MATH-1400. –  Toby Bartels Dec 24 '11 at 6:44

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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