Functions form a fundamental foundation of mathematics, second only in importance to numbers themselves. These are the objects that we model and explore the world with. We have already explored several important families of functions: lines, quadratics, and more general polynomials. In that investigation, we stumbled onto the ideas of calculus. Here we find the foundations to extend calculus to more general functions and then use calculus to define the functions.
This approach is a little unorthodox in that some of the most common functions to play around with in the calculus introductions will actually be developed afterwards. This is not only a logical correction, but I think it helps slow down the steps so that calculus never feels like some kind of magic trick, but rather will become a faithful and well-understood tool.
We start with finishing off what we can do with the arithmetic operations and unknown functions. We explore the algebraic functions, though largely confine our explorations to the explicit types of ratios of polynommials and adding in radical functions. We explore how to do calculus with these functions from a more approximative viewpoint.
Full calculus inevitably deals with notions of infinity and so we explore various notions of infinity, starting with stuff just involving infinite strings of numbers.
We then move on to some infinite notions involving functions which largely goes under the broad foundational notion of limit. The idea of a limit is to look at behavior near the point of interest, but not at it. It is similar to asking a person's neighbors about the person to gather information. Such a tactic does not give you the full information after one asking, but after many asks, a pattern develops and a notion arises. Unlike asking neighbors, computing limits is not considered rude.
With limits undestood, we can then tackle the abstract version of differential calculus and integral calculus. Differential calculus is about gathering slopes and deducing geometric information. Integral calculus is about undoing derivatives as well as computing areas, lengths, volumes, etc. Derivatives give us ways of simplifying problems, including the very important Newton'smethod, while integrals give us a way of assessing the accumulation of errors.
With the tools of calculus firmly set in place, we can finally proceed into some notable transcendental functions. Exponentials arise from notions of population growth, whether it be actual life forms, compound interest, or radioactive decay. To deal with the extreme scales of exponentials, logarithms are useful to have around. They can be derived in a variety of fashions, which we will do, but a somewhat explicit method comes from computing the needed nominal annual interest rate to achieve a desired effective annual rate.
The other major function family is that of the trigonometric functions. While they start off as functions related to right triangles and various geometric computations, they rapidly become far more important as the foundations for periodic functions. If you have repeating behavior, reach for some trigonometric functions. We conclude the trigonometric chapter with a view towards the complex exponential, a convenient idea that merges the ideas and rules of exponentials with that of trigonometric functions.
Our final chapter is an exploration of defining and transforming functions. We start with "polynomials with infinitely many terms", generally known as a power series; convergent ones lead to a Taylor series which can represent a function entirely if everything is nice enough. We then explore functions coming from equations, partiuclarly differential equations. This then leads to whole new families of functions, of which we take a quick look at. Differential equations also lead to some special transformations, such as the Laplace transform which transforms differential equatons into algebraic equations.