Timeline for Why do we teach calculus students the derivative as a limit?
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| Aug 12, 2021 at 22:34 | comment | added | Qfwfq | (...) In my own experience, in high school (second year) we defined real numbers by Dedekind cuts (a construction easily forgotten and immediately replaced by intuitive use of the usual properties! But we were told exactly what $\sqrt{2}$ is in that framework). Then in the fifth year of high school real analysis was done pretty rigorously including epsilon-deltas, continuity, derivatives, fundamental theorem of calculus (but with Riemann integral done with no proofs and with only semi-rigorous definition; and no mention of Cauchy sequences). | |
| Aug 12, 2021 at 22:34 | comment | added | Qfwfq | @Steven Landsburg : we're probably using different definitions of "freshman calculus". In some countries this not-yet-rigorous version of elementary real analysis simply doesn't exist: people learn to deal with all the rigorous definitions and all the proofs from the get go. There, every freshman is supposed to get that knowledge of epsilon-deltas etc during "calculus" because "calculus" is done that way. So the question is probably location-dependent. (...) | |
| Aug 10, 2021 at 22:02 | comment | added | Steven Landsburg | @Qfwfq : "the he nonstandard analysis definition would require a knowledge of logic that no freshman is supposed to have". One could just as easily say that the epsilon-delta definition would require a knowledge of Cauchy sequences or Dedekind cuts that no freshman is supposed to have. In calculus, we simply postulate and work with the basic properties of the real numbers, without actually proving that anything has those properties. I don't see where there's any difference in principle between that and simply postulating the properties of the non-standard reals. | |
| Aug 9, 2021 at 12:30 | comment | added | Gerald Edgar | Because if we don't teach the derivative as a limit, we end up with questions like this: matheducators.stackexchange.com/q/21203/127 | |
| Aug 9, 2021 at 8:47 | answer | added | Alessandro Della Corte | timeline score: -1 | |
| May 3, 2021 at 14:35 | comment | added | François Brunault | I find it important when teaching an important notion, to present different points of view and their interrelation, rather than a unilateral approach. The $h \to 0$ thing may be the first time students are exposed to manipulating quantifiers in a "non-trivial" way, which has to be done at some point. Finding the derivative of $x^2$ can make the link with the symbolic approach. Here is a provocative analogue: why would we teach addition and multiplication in elementary school while calculators can do it? One reason could be they learn what is behind the scene (and a first example of algorithm). | |
| Sep 8, 2020 at 19:56 | answer | added | Michael Renardy | timeline score: 0 | |
| Jun 21, 2019 at 21:51 | comment | added | sfmiller940 | More generally what's the point of teaching real analysis if it's mangled into a commodity called "calculus"? | |
| Dec 4, 2017 at 8:21 | review | Close votes | |||
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| Oct 7, 2017 at 22:20 | review | Close votes | |||
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| Aug 15, 2017 at 2:56 | review | Close votes | |||
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| Sep 14, 2015 at 20:34 | comment | added | isomorphismes |
You can do derivatives over ℤ without travelling off onto philosophical side roads. With a lagged difference operator (setting $h=1$) you can show that diff(1,4,9,16,25) = 3,5,7,9. This is simple enough that even non-university students could understand. One can't treat sinc this way, but maybe you could introduce h↓0 second (talk about 1/.00000003 and 1/−.000000003), after they understand symbolic differentiation of polynomials.
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| Mar 3, 2015 at 0:36 | answer | added | Michael Hardy | timeline score: 3 | |
| Dec 26, 2014 at 9:50 | comment | added | Alexey Muranov | "What benefit is there in introducing to calculus students the $h→0$ definition of a derivative?" -- you have to define it anyway, don't you? I agree that calculating a derivative of something like $x\mapsto 3x^2$ by definition can be boring, it is better to ask a student to derive the product or the quotient rule for the derivative, for example. | |
| Dec 22, 2014 at 11:45 | answer | added | Mikhail Katz | timeline score: 2 | |
| Dec 15, 2014 at 6:49 | history | edited | user9072 |
edited tags; edited tags
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| Dec 14, 2014 at 23:29 | history | protected | S. Carnahan♦ | ||
| Dec 14, 2014 at 23:25 | answer | added | Steven Landsburg | timeline score: 16 | |
| Dec 15, 2013 at 13:40 | comment | added | LSpice | I'm going to stuff my favourite calculus pet peeve in a comment here: the claim (which I have seen!) that $\lim_{x \to 0} \sin(x)/x$ is computed using L'H\^opital's rule. (Why is it a pet peeve? Well, aside from mere pedagogical preference, what limit do you compute to find $\sin'(0)$? In a word---I can't resist the pun---it's circular.) | |
| May 19, 2012 at 6:02 | answer | added | David Feldman | timeline score: 8 | |
| May 19, 2012 at 3:54 | answer | added | William N | timeline score: 1 | |
| Sep 28, 2011 at 20:58 | answer | added | Adam Harris | timeline score: 4 | |
| Apr 3, 2011 at 23:23 | answer | added | Toby Bartels | timeline score: 1 | |
| Apr 3, 2011 at 22:17 | comment | added | Toby Bartels | I'm starting to teach calculus for business majors now, and the book (contrary to some remarks above) does use the limit definition of derivative. But it does not give any definition of limit! I don't really see the point of this. If we're going to hand-wave limits (and completely fudge the definite integral, which this book also does), then why not hand-wave derivatives? (And if we leave limits until later, then we get to useful applications faster.) | |
| Jan 31, 2011 at 4:43 | answer | added | Jason | timeline score: 2 | |
| Jan 29, 2011 at 19:25 | answer | added | Anna Varvak | timeline score: 20 | |
| Jan 26, 2011 at 6:28 | answer | added | Patrick I-Z | timeline score: 5 | |
| Jan 23, 2011 at 15:32 | answer | added | Misha | timeline score: -4 | |
| Nov 7, 2010 at 6:31 | answer | added | Jeremy West | timeline score: 10 | |
| Nov 6, 2010 at 23:22 | answer | added | Michael Hardy | timeline score: 12 | |
| Nov 6, 2010 at 22:30 | answer | added | Bob Pego | timeline score: 8 | |
| Nov 2, 2010 at 15:37 | answer | added | Terry Tao | timeline score: 27 | |
| Oct 9, 2010 at 4:08 | comment | added | Thierry Zell | The edit has clarified the issue, but I still see two questions rolled into one -- which may explain why the discussion is sometimes at cross purpose: 1. Why do we define the derivative as a limit. 2. Should we expect the students to compute derivatives using limits. It's in 2 that the bigger controversy appears to reside. And I would agree that any problem that hints at jumping through hoops arbitrarily is not desirable; fortunately, we have here examples that show how the limit definition be needed even at the calculus level. | |
| Sep 28, 2010 at 21:54 | answer | added | Alexander Woo | timeline score: 6 | |
| Sep 28, 2010 at 11:10 | comment | added | darij grinberg | Short answer: because a derivative IS a limit. Even in algebra, the shortest definition of the derivative (of a polynomial or rational function over any ring) at a point is to taking the limit of the differential quotient at this point. "Limit", of course, means that we cancel as much as we can from numerator and denominator, and then evaluate at the point. If you want to make stuff simpler, you can try eliminating the precise definition of the analytic notion of "limit", but that again comes at the cost of functions such as $e^x$ and $\sin x$. | |
| Sep 28, 2010 at 4:04 | answer | added | roy smith | timeline score: 6 | |
| Sep 28, 2010 at 3:06 | comment | added | J. M. isn't a mathematician | @Thierry: I'm a chemist by training; as I recall the treatment we had only gave a passing definition of limits (so no $\epsilon-\delta$) and the derivative as a limit, and much of the remainder was taught as algorithms handed down from a mountain (though I appreciate they took the time to define the natural logarithm as an integral and the exponential as the inverse, and then deduce the requisite properties from those). | |
| Sep 28, 2010 at 1:26 | answer | added | Victor Protsak | timeline score: 21 | |
| Sep 27, 2010 at 22:49 | comment | added | Nick S | Sorry for repeating, I included this in another comment below. But if any student asks why the definition is needed, ask him/her to derivate $f(x)= x^2 \sin(\frac{1}{x})$ for $x \neq 0$ and $f(0)=0$. | |
| Sep 27, 2010 at 20:15 | comment | added | Thierry Zell | @Gerald: This is one of the biggest challenges for me when teaching calc 1: half of the class (if I'm lucky) will go on to take at least 3 more semesters of the stuff (to diff eq), and they must know the calc 1 material forward and backwards. Then there's always one half that only needs a calc 1 on their transcript (e.g. for med school). A few might be somewhere in between (Chem majors maybe?). These guys have to work hard to get that credit, but then again, that might be the rationale behind requiring it. (We do have a separate calc for business.) | |
| Sep 27, 2010 at 17:48 | history | edited | JBL | CC BY-SA 2.5 |
minor typos corrected
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| Sep 27, 2010 at 16:41 | vote | accept | Steven Sam | ||
| Sep 27, 2010 at 16:41 | history | edited | Steven Sam | CC BY-SA 2.5 |
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| Sep 27, 2010 at 15:33 | answer | added | Igor Belegradek | timeline score: 21 | |
| Sep 27, 2010 at 15:32 | comment | added | Gerald Edgar | There is such a course at many universities. Called "Calculus for Business" or "Calculus for Life Sciences" or such. Only calculus for engineers, physical scientists and mathematicians emphasizes the limit. But then at small colleges, where they have a "one size fits all" course, what to do? | |
| Sep 27, 2010 at 14:28 | history | made wiki | Post Made Community Wiki by Steven Sam | ||
| Sep 27, 2010 at 14:07 | comment | added | JBL | In the process, they would discover that computing derivatives from scratch (as a special case of computing limits of formulas) is quite difficult; this would allow them to appreciate the rules of differentiation, and where they come from. (I think there is a similar problem in the way the Fundamental Theorem of Calculus is taught -- if antidifferentiation and definite integration are taught simultaneously, one loses appreciation for the miracle that we can compute definite integrals without resorting to Reimann summation or the like.) | |
| Sep 27, 2010 at 14:04 | comment | added | JBL | 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) | |
| Sep 27, 2010 at 13:58 | answer | added | Jeff Strom | timeline score: 99 | |
| Sep 27, 2010 at 12:37 | answer | added | Thierry Zell | timeline score: 13 | |
| Sep 27, 2010 at 12:20 | answer | added | Deane Yang | timeline score: 144 | |
| Sep 27, 2010 at 12:14 | comment | added | Qfwfq | -1, as I don't see the point in asking such a question. That's simply the most effective definition of derivative (the nonstandard analysis one would require a knowledge of logic that no freshman is supposed to have!). By the way, in Italy it is perfectly normal to learn and use the epsilon-delta definition of limit/continuity/derivative at the last year of high school... | |
| Sep 27, 2010 at 12:10 | answer | added | Pietro Majer | timeline score: 42 | |
| Sep 27, 2010 at 11:35 | comment | added | Holger Partsch | 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". | |
| Sep 27, 2010 at 10:08 | comment | added | KConrad | Each time you meet a new "basic" function, you need to compute its derivative before you can apply your standard toolbox (product rule, chain rule) to derivatives of that function composed with other functions. So with polynomial, exponential, trigonometric, etc. functions their derivatives have to come from somewhere. On the one hand you can refer to a uniform approach to start finding their derivatives (which is the h--> 0 limit definition) or you can just tell students the answers and then everything seems like more of a black box. And in economics, derivatives use h = 1 as "small"! | |
| Sep 27, 2010 at 10:03 | answer | added | Pablo Lessa | timeline score: 17 | |
| Sep 27, 2010 at 9:20 | comment | added | Michael Greinecker | One can do calculus based on infinitesimals, which are probably somewhat easier to manipulate than limits: math.wisc.edu/~keisler/calc.html | |
| Sep 27, 2010 at 8:07 | answer | added | Alex | timeline score: 6 | |
| Sep 27, 2010 at 7:42 | answer | added | Andrew Stacey | timeline score: 53 | |
| Sep 27, 2010 at 7:21 | answer | added | Aleksei Averchenko | timeline score: 6 | |
| Sep 27, 2010 at 7:15 | answer | added | Denis Serre | timeline score: 19 | |
| Sep 27, 2010 at 6:29 | answer | added | Theo Johnson-Freyd | timeline score: 12 | |
| Sep 27, 2010 at 6:27 | answer | added | Tommaso Centeleghe | timeline score: 10 | |
| Sep 27, 2010 at 6:08 | comment | added | Steven Sam | I guess it's hard to get across what I'm trying to ask (maybe I don't even understand), but I am not trying to say that we should get rid of definitions in the first place (maybe the headline is misleading). Maybe in line with Austin's comment that students already have a black-box view of mathematics would be the question of how to get students to care (or why should they care) about the definitions in the first place (in the case of derivatives it seems particularly easy for students to not care once the symbolic rules are in place). | |
| Sep 27, 2010 at 6:05 | answer | added | Pete L. Clark | timeline score: 37 | |
| Sep 27, 2010 at 5:53 | answer | added | Gjergji Zaimi | timeline score: 6 | |
| Sep 27, 2010 at 5:42 | comment | added | Austin Mohr | 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. | |
| Sep 27, 2010 at 5:41 | comment | added | Alex B. | 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? | |
| Sep 27, 2010 at 5:29 | history | asked | Steven Sam | CC BY-SA 2.5 |