Lagrange Differential Calculus

A first semester calculus is not all engineering students, it is not all physical science majors, it is not all computer science majors, it is not all social science majors, and it is definitely not all math majors. The Calculus I syllabus is designed for a general audience, "just plain folks". If, for example, the computer science concentration requires only one semester of calculus, then it is important that one semester cover all the "core" material from calculus. Calculus I can be taken as a terminal course—there is no postponement of integration or trigonometry or exponential functions to the second semester. It is in the second semester, where the enrollments are often half of first semester calculus, that some branching may take place.
J. Goldstein et al. "Calculus Syllabi, Report of the Content Workshop at the Tulane Conference."

As a result of the Tulane Conference, starting in 1988, the NSF had a Calculus grant program which went on for several years but whose impact has, rather unsurprisingly, been almost nil.

    Indeed, efforts toward a Lean and Lively Calculus ranged from the camouflage of the theory by way of "applications" to the improvement of the "delivery system" by way of "high technology" but the use of the Bolzano-Cauchy-Weirstrass (BCW) theory of limits in introductory texts was never questioned even though the length of the precalculus preparation that it forces deters many "just plain folks" from even considering precalculus and therefore from acceding to careers in science and technology requiring as little as First Semester Calculus.

    As a result, in the search to "revive" calculus, no consideration was ever given to theories other than BCW and no discussion of the comparative merits of alternate approaches took place. In particular, why Leibniz's infinitesimals should have continued to underlay so much mathematics for almost a century after BCW became established did not seem to have intrigued Educologists; neither were the latter troubled in the least by the fact that infinitesimals should have been jettisoned from elementary texts precisely at the time when more and more non-specialists began having to be introduced to calculus, and that infinitesimals should have been replaced, of all things, by limits.

    But that this adherence to BCW should eventually have led to a "watered down, cookbook course in which all (the students) learn are recipes, without even being taught what it is that they are cooking" was predictable if one considers, for instance, the fact that the BCW definition of continuity cannot be made "intuitive" without at the same time withdrawing from the student the means, say, to compute whether or not a given function is continuous. Nobody seems to have noticed that "rigorous" mathematics could not possibly mean today the same thing to professional mathematicians and to "just plain folks" and nobody seems to have wondered why a mathematical correctness that could yesterday bring intellectual satisfaction to Newton, Leibnitz and Euler could not do the same today for "just plain folks". It is as if mathematicians had forgotten that there is something deeply gratifying to doing "naïve" mathematics.

 Elements of Differential Calculus, the text I developed with F. Schremmer-Mattei for an integrated sequence of two four-hour semesters, to parallel Precalculus I, Precalculus II and Calculus I was based on Lagrange's treatment for freeing calculus from "any consideration of infinitesimals, vanishing quantities, limits and fluxions and reduce it to the algebraic study of finite quantities".

    By far the main problem with Lagrange's approach is that it is mistaken for a series approach so that, invariably, one's first reaction is to wonder how students having trouble with a watered down BCW could possibly not have even more trouble with Lagrange. Lagrange, though, was essentially using asymptotic expansions rather than series and he approached functions very much the way we approach real numbers in real life—that is when we use decimal approximations.

    Engineers indeed like to say that the real Real Numbers are the Decimal Numbers. For instance, they view √5 not so much as an irrational number as a symbol standing for any (positive) decimal number two copies of which will multiply approximately to 5. For instance, depending on how many decimals they need, they might write √5 = 2 + (…) or √5 = 2.2 + (…) or √5 = 2.23 + (…), etc, depending on the situation and where (…) is read “a little bit, too small to matter in the current situation”. Similarly, for instance, we write (x0+h)3 = x03 + (...) if we want the output near x0 or (x0+h)3 = x03 + 3x02h + (...) if we want the slope near x0 or (x0+h)3 = x03 + 3x02h + 3x0h2 + (...) if we want the concavity near x0. In both the arithmetic and the algebraic cases, we need not concern ourselves with the convergence of the sequence of approximations and so whether the remainder approaches 0 is another, totally different issue. Indeed, all we need in a first year calculus is the polynomial approximation of f(x0+h) since the nth derivative of f can be defined as the function that for x0 outputs n! times the coefficient of hn in the polynomial approximation of f(x0+h) (Peano). From that, the usual theorems can be proved very simply. Moreover, should we get interested in the convergence of the sequence of approximations, we need only replace (...) by o(hn) and proceed ahead without having to redo everything from the beginning.

    A problem with Lagrange's approach is of course that it does not have all the generality deemed desirable by Educologists. For instance, the function f(x) = cosx + x3sinx is approximated near 0 by 1 – x2/2 even though f ''(0) does not exist. But then, this is rather an advantage if it is the concavity of f at 0 that we are after! Similarly, while we systematically use the fact that f has an extreme at a point x0 when the degree of the principal non-constant term in the approximation is even, which is rather reasonable, there are cases like that of  f(x) = exp(–x–2) at 0 where this does not work (which is not to say that we cannot deal with it). And of course there are the many functions that definitely do not have polynomial approximations. But then, they are also quite unlikely to appear in First Semester Calculus—not to mention on the exams. 

    Conceptually, Lagrange's approach seems especially well suited to "just plain folks" in that it is based on ideas that are already familiar from arithmetic. Computationally, it is also quite appropriate since, just as the processes giving the decimal approximation of a real number depend on the nature of the number, the algebraic processes giving the polynomial approximation of a function also depends on the nature of the function: binomial approximation for polynomial functions, division of polynomials in ascending as well as descending powers for rational functions, solution of narrow band systems of linear equations for transcendental (e.g. exponential and trigonometric) functions. As such, these processes can be introduced on a "just in time" basis, that is they can be delayed until the corresponding type of functions is being dealt with.

After a few years during which the courses were taught by a variety of instructors, my school’s Office of Institutional Research wrote, in a report comparing the integrated 8-hour sequence with the conventional Precalculus I, Precalculus II and Calculus I 10-hour sequence, that

"Of those attempting the first course in each sequence, 12.5% finished the [conventional three semester 10 hour] sequence while 48.3% finished the [integrated two semester 8-hour] sequence, revealing a definite association between the [integrated two semester 8 hour] sequence and completion (chi2(1) = 82.14, p < .001)."

The report also said that the passing rates in Calculus II (integral) for the students coming from the two sequences were almost identical but not significant because most students did not continue into Calculus II.

So, all in all, the choice rests between an approach that can deal with all functions, but that most students cannot deal with, and an approach that cannot deal with all functions but that most students can deal with.<