From Arithmetic To Differential Calculus

FreeMathTexts RBA Text In Preparation A2DC Development About Acknowlegments CopyLeft LaTeX Notes

Based on my experience teaching Arithmetic and Basic Algebra courses, and since the only prerequisite for Lagrange Differential Calculus had been polynomial algebra, I thought that it should be possible to design a prequel to Lagrange Differential Calculus that "developmental" students could reasonably do in one four/five-hour semester. It would thus be the first in a three four-hour semesters sequence, From Arithmetic to Differential Calculus, whose second and third semester would be based on a rewrite of Lagrange Differential Calculus.

Given the size of the undertaking, though, I thought that I would just write a "proto-text", that is only a proof of concept, and leave the actual implementation to whomever might be interested. The first few chapters appeared in the AMATYC Review, in which I had had a column for about ten years previously. However, I soon bowed to the wisdom of the Hestenes dictum and started to work on a Text for A2DC.

But then, I started using parts of the prequel in my classes and thus needed Homeworks, Reviews and Exams. So, it occurred to me that standalones could be extracted from A2DC and that, in fact, it would probably be better to begin with them but within an overall framework in which A2DC should still be relatively simple to produce once the standalones were done. (This, by the way, is the reason for the discontinous numbering of the contents files in the source of RBA.)

The prequel was specified through a kind of reverse engineering, namely as what was strictly necessary for Lagrange Differential Calculus. Here are a few examples:

In order to define, say, the exponential function as solution of the Initial Value Problem f '(x) = f(x), f(0) = 1, it is necessary to see an equation as an a priori specification of something that may or may not exist—as opposed to being a "question" to be answered—and the idea needs to be introduced a long time before this point so as to give students the time to get completely used to it. So the idea is developed in the middle part of RBA as the investigation of Basic Problems, Affine Problems, etc.
In order to feel comfortable with the idea of (locally) approximating a function, it is necessary to see approximation as a natural thing to do—as opposed to finding "the" (exact) answer to the given question. The idea is thus introduced in RDA where, for instance, 1/3 = 0.3 + (...) or 1/3 = 0.33 + (...) or 1/3 = 0.334 + (...), etc with (...) standing for "a little bit of no significance in the present situation".
In order to realize that the + in 3x⁵ + 4x² is not the symbol for addition but for a (linear) combination—and thus we really ought to use &, it helps to have already seen that this is the exact same situation as in 3 Apples + 4 Bananas, 3 Hundreds + 4 Tens, 3 Halves + 4 Thirds, 3f + 4g, etc—And of course as in 3e₁ + 4e₂ = (3, 4) in linear algebra.

Hence the three parts of A2DC:

Part I. Decimal-Metric Arithmetic, Arithmetic Functions, Comparisons and Operations, Equations/Inequations Problems, Laurent Polynomial Algebra.
Part II. Algebraic Functions: Power Functions, Polynomial Functions, Rational Functions.
Part III. Transcendental Functions as solution of Initial Value Problems: Exponential Functions, Logarithmic Functions, Circular Functions, Hyperbolic Functions.

and the corresponding standalones:

Reasonable Decimal Arithmetic. (RDA) Essentially the arithmetic part of Part I of A2DC.

Decimal-Metric Number-phrases
Approximation
Arithmetic Functions
Comparisons and Operations

Reasonable Basic Algebra. (RBA) A reduction of RDA with the algebra necessary for Lagrange Differential Calculus.

Decimal Arithmetic

Equations/Inequations

Polynomial Algebra

Reasonable Arithmetic and Algebra. (RAA) An assembly of RDA and RBA that corresponds to a reduction of Part I of A2DC.
- Decimal-Metric Number-phrases
- Comparisons and Operations
- Equations/Inequations
- Polynomial Algebra

Reasonable Algebraic Functions. (RAF) A reduction of Part II of A2DC that de-emphasizes derivatives.

Power Functions
Polynomial Functions
Rational Functions

Reasonable Transcendental Functions. (RTF) A reduction of Part III of A2DC that de-emphasizes derivatives.

Exponential Functions
Logarithmic Functions
Box Functions
Circular Functions
Hyperbolic Functions

FreeMathTexts RBA Text In Preparation A2DC Development About Acknowlegments CopyLeft LaTeX Notes

Page Updated January 23, 2008