It’s the limit of distance over time, as time approaches 0.
The problem which the limit formulation resolves is:
To calculate instantaneous velocity, you can’t just divide distance by time, because how much time is there in an instant?
I beg to differ. When I took first year calc, it was all “here are a whole bunch of concepts that you learned informally in high school, but now we’re doing them formally”.
So “limit” when from “value it gets close to as the value gets in gets close to whatever” to the formal epsilon-delta definition. “Derivative” went from “slope of the tangent” to the limit of a Newton quotient. “Continuous” went from “can draw the graph without lifting your pencil” to “has a limit at every point and the value is equal to the limit”. And so on.
In that class, if you did an integral and “cancelled the dx’es” instead of applying the CoV theorem, it was marked wrong. Ironically, that loosened up in higher math classes where the policy was, “yeah, yeah, we know you got it, just use the short cut”. But first year calc was all about the formalism.
I took “first year calc” to mean “the first time you ever learnt about calculus”. And I stand by my position that, though everyone is different, for most students, they shouldn’t be worrying about formality when first being introduced to calculus.
The main benefit of f’ vs df/dx or derivative(f(x),x) is that you can express the former without making reference to the name of the variable. You havent done anything fundamentally interesting if you change the name of your dependent variable so f(x) becomes f(u), but with the Leibniz notation you have to change the name of the derivative function to df/du. Not to mention what all happens if the function only appears composed with another expression, like f(3x-7y).
I wonder how you feel about the level of formality I described a few posts above, where I noted that we learned the epsilon-delta definition of limits and did a few simple proofs to “get the idea”. But we didn’t prove things like the Chain Rule or anything like that.
Examples we did were like: Use the epsilon-delta definition to prove that
lim[sub]x→4[/sub] (x[sup]2[/sup] - 2x + 3) = 11
Is that commonly still taught in first-semester calculus today? Is that about the right level of formality for a first semester course?
In my experience (which is not too different from what the end of your post describes), there’s some lip-service given to such epsilontics at the beginning of class, a few homework problems, everyone moves on quickly, and not very much of it sticks for most students. (And why should it? The formality is addressing concerns of a sort they generally have no reason to care about. Of course, some do.).
The implicit contract soon struck between student and teachers is “With rare and insignificant exception, everything that comes up will be continuous where defined, so limit calculation is simply function evaluation”, and indeed, why not? Might as well introduce calculus using a category of smooth functions rather than arbitrarily pathological ones.
I should probably keep our of this discussion, but here goes.
Newton used f’(x) for the derivative, except he used a dot atop the f for time derivative. I am not sure how he conceived of it. Leibniz, by contrast, called it dy/dx (with y = f(x)) and doubtless thought of it as the actual ratio of the infinitesimal change of y resulting from an infinitesimal change in x. So it was an actual fraction, except what is an infinitesimal. Not for nothing that Bishop Berkeley ridiculed the derivative as the ghost of departed quantities.
Recognizing that the foundations of calculus, some 19th century mathematicians, notably Cauchy, put calculus on a rigorous foundation using limits defined via the familiar epsilon/delta definition. This state lasted well over a century. But the epsilontics (I will call it), while rigorous, was not the way mathematicians thought about these things, for the most part. They still thought in terms of infinitesimals and only after converted their arguments into epsilontics. Finally, around 1960, the logician Abraham Robinson, figured out how to introduce infinitesimals rigorously. For the record, the usual construction uses ultrafilters. This is rather complicated and certainly not for college freshmen. Or sophomores or juniors for the most part. This is not an insuperable problem. After all, we don’t ask our freshmen to actually understand real numbers, nor should we. And yet the mean value theorem depends crucially on the completeness of the reals.
Let me write a ~ b to mean that a - b is infinitesimal. Every “extended real” is the sum of an “ordinary” number and an infinitesimal. Then f is continuous function at x if and only if whenever h is infinitesimal (that is h ~ 0), then f(x+h) ~ f(x). That’s all. The function f has a
derivative at x if the ordinary part of (f(x+h) - f(x))/h does not depend on h. The infinitesimal part generally does. But the ordinary part of that fraction (which can well be written dy/dx) is the derivative.
Let’s see how this works with a simple function f(x) = x^2. Then f(x+h) = x^2 +2xh +h^2 and (f(x+h) - f(x))/h = 2x +h ~ 2x. The computation is not that different from the one using limits, but there are no epsilons.
Here is a kindergarten model for infinitesimals. Let R* denote the set of all convergent sequences of real numbers, except that we consider x = y if the sequences differ in only finitely many terms. Also x ~ y if the sequence x - y converges to 0. So an infinitesimal is just a sequence that converges to 0. You can carry out the same computation with the squaring function (apply it term-wise) and derive the same conclusion. So what is wrong with this definition? Well, basically, an ordinary function–think sin(x)–can be defined on this model if and only if it is, in the usual sense, continuous. Not only can’t you deal with discontinuous functions but you need epsilontics to describe which functions you can deal with. Extended reals defined by ultrafilters do not have this problem. Every function automatically extend.
Actually, the above applies to finite extended reals. By inverting infinitesimals (always non-zero, you get infinite numbers and they are useful in defining integrals. And yes you can multiply dy/dx by dx and get dy. It really does all work.