I have finally managed to be (almost) done and I have begun to upload the bundle. See
Visitors who remember the previous version of this page will notice that the major organizational difference that was discussed in the Notes From The Mathematical Underground is indeed the one being implemented: instead of introducing a number of concepts on an as-needed basis, the present contents architecture regroups them up-front from a unifying graphical viewpoint. It certainly makes for a very neat, as well as rather standard, architecture, but, as of this day, I am not yet entirely convinced that this is the way to go. It may be that this is too much too soon and possibly too "out of context".
The less important changes, mentioned earlier, corresponding to a more realistic view of what can be accomplished in a fifteen weeks course meeting three hours a week have been retained: the investigation of the Family of Power Functions and that of the Higher Polynomial Functions are relegated to the Epilogue, the Local Analysis Of Cubic Functions and the Global Analysis Of Cubic Functions are in two separate chapters, and the standard "operations on functions" are regrouped in the chapter Operators.
So, here is the table of contents as it has, finally, been implemented. (In the ancillaries as well as in the text.)
I. - TOOLBOX
Relations And Functions. Relations, Quantative Rulers, Finite Numbers And Near-Finite Numbers, Quantitative Screens, Functions, Functions Specified by A Global Input-Output Rule, Functions Specified By A Curve.
Towards Local Analysis. The Fundamental Graphic Problem, Size Of Numbers, Arithmetic Of large, small And medium, Qualitative Rulers, Infinity, Qualitative Analysis.
Graphic Local Analysis. Local Graphs, Local Language, Place of a Local Graph, ∝-Height Inputs and 0-Height Inputs, Shape of a Local Graph, Local Behavior, Feature-Sign Change Inputs, 0-Slope and 0-Concavity Inputs, Extremum Inputs.
From Local Graphs To Global Graph. Smoothness, Interpolation, The Essential Question, The Essential Bounded Graph, Essential Notable Inputs.
II. - GAUGE FUNCTIONS: POWER FUNCTIONS
Regular Positive-Exponent Power Functions. Input-Output Pairs, Normalized Input-Output Rule, Local Graph Near ?, Types of Local Graphs Near ?, The Essential Question, Essential Bounded Graph, Existence of Notable Inputs, Local Graph near 0, Types of Local Graphs Near 0, Essential Global Graph, Types of Global Graphs.
Negative-Exponent Power Functions. Input-Output Pairs, Normalized Input-Output Rule, Local Graph Near ?, Types of Local Graphs Near ?, The Essential Question, Essential Bounded Graph, Existence of Notable Inputs, Local Graph near 0, Types of Local Graphs Near 0, Essential Global Graph, Types of Global Graphs.
Exceptional Power Functions. (Exponent 0 and exponent 1)
Constant Functions. Global Graph, Existence of Notable Inputs, Local Graph Near ∝, Local Graph Near 0.
Linear Functions. Global Graph, Existence of Notable Inputs, Local Graph Near ∝, Local Graph Near 0.
III. - POLYNOMIAL FUNCTIONS
Chains And Operators Constant Functions, Dilation Functions; Chaining Functions, Chains of Constant Functions, Chains of Dilation Functions, Operators On Functions, Dilation Operators, Add-On Operators.
Affine Functions: Local Analysis Input-Output Pairs, Normalized Input-Output Rule, Local Rule Near ?, Local Graph Near ?, Types of Local Graphs Near ?, Local Features Near ?, The Essential Question, Essential Bounded Graph, Existence of Notable Inputs, Local Rule Near a given x0, Approximate Local Rule Near a given x0, Local Graph Near a given x0, Approximate Local Graph Near a given x0, Local Features Near a given x0.
Affine Functions: Global Analysis. Global Theorems, Qualitative Global Graph, Solving Equation Problems (*), Locating Inputs Whose Output is a given y0, Solving Inequation Problem s(*), Locating Inputs Whose Output compares in a given manner with a given y0, Initial Value Problem, Boundary Value Problem.
Quadratic Functions: Local Analysis. Input-Output Pairs, Normalized Input-Output Rule, Local Rule Near ?, Local Graph Near ?, Types of Local Graphs Near ?, Local Features Near ?, The Essential Question, Essential Bounded Graph, Existence of Notable Inputs, Addition formula (*), Local Rule Near a given x0, Approximate Local Rule Near a given x0, Local Graph Near a given x0, Approximate Local Graph Near a given x0, Local Features Near a given x0.
Quadratic Functions: Global Analysis. Global Theorems, Qualitative Global Graph, Locating 0-height Inputs, Locating Inputs Whose Output is a given y0, Locating Inputs Whose Output compares in a given manner with a given y0, Initial Value Problem, Boundary Value Problem.
Cubic Functions: Local Analysis. Local Rule Near ?, Local Graph Near ?, Types of Local Graphs Near ?, Local Features Near ?, The Essential Question, Essential Bounded Graph, Existence of Notable Inputs, Addition formula (*), Local Rule Near a given x0, Approximate Local Rules Near a given x0, Local Graph Near a given x0, Approximate Local Graph Near a given x0, Local Features Near a given x0.
Cubic Functions: Global Analysis. Global Theorems, Qualitative Global Graph, Locating 0-height Inputs, Locating Inputs Whose Output is a given y0, Locating Inputs Whose Output compares in a given manner with a given y0,
IV. - RATIONAL FUNCTIONS
Rational Degree -- Algebra Reviews (*). Rational Degree, Binomial Expansions, Division in Descending Order of Exponents, Default Rules for Division, Comparison With Arithmetic, Division in Ascending Order of Exponents, Graphic Difficulties.
Rational Functions: Local Analysis Near Infinity. Local Input-Output Rule near Near Imp, Local Polynomial Approximations, Local Approximate Input-Output Rule Near Infinity, Height-sign Near Infinity, Slope-sign Near Infinity, Concavity-sign Near Infinity, Local Graph Near Infinity.
Rational Functions: Local Analysis Near x0 Local Input-Output Rule near Near x0, Height-sign Near x0, Slope-sign Near x0, Concavity-sign Near x0, Local Graph Near x0
Rational Functions: Global Analysis. The Essential Question, Locating Infinite-Height inputs (Poles), Offscreen Graph, Feature-sign Change Inputs, Global Graph, Locating 0-Height Inputs (Zeros).
(*) Pulled verbatim from Reasonable Basic Algebra.