But despite these concessions to modern times, my proposed syllabus most closely resembles what would have been taught in Calculus courses before any of us were students. The attached Table of Contents from the book, Practical Analysis in One Variable, gives an idea of what could be taught in the first two semesters. The material passes from the analysis of numbers, with the highlight of the construction of the real numbers as solutions of nonlinear equations, to the analysis of transcendental functions, as solutions of differential equations. Sequences and limits are discussed primarily from the point of view of Cauchy sequences, which are very practical. For example, the standard engineering way to test convergence is to test the Cauchy condition on a couple of elements of a sequence.
Lipschitz continuity is emphasized over the standard weaker definition.
This fits the idea of trying to understand nonlinear functions by replacing them by linear approximations and it removes several technical difficulties. The derivative is defined through the linearization of a function, again fitting the notion of approximating a nonlinear function by a linear function. This also serves as an introduction to asymptotic analysis. Integration is defined as a procedure for computing an approximate solution of a differential equation. The mathematical highpoint is the proof that numerical solution produced by the rectangle rule generates a Cauchy sequence that converges to the solution of the differential equation.