Functions arise whenever one quantity depends on another. Mathematically speaking, a function is a rule that assigns to each element x in a set D (called the domain) exactly one element, called f(x), in a set called the range. Because we continually make theories about dependencies between quantities in nature and society, functions are important tools in the construction of mathematical models. In this module, we will learn the theory of functions, see many examples and their graphs, as well as apply these functions. We will learn how to implement these functions in Python as well.

Calculus is the science of measuring change. Early in its history, its tools were developed to solve problems involving the position, velocity, and acceleration of moving objects. Prior to the development of calculus, there was no way to express this change in a variable. In this section, we introduce the notion of limits to develop the derivative of a function. The derivative, commonly denoted as f'(x), will measure the instantaneous rate of change of a function at a certain point x = a. This number f'(a), when defined, will be graphically represented as the slope of the tangent line to a curve. We will see in this module how to find limits and derivatives both analytically and using Python.

The derivative is defined as a limit of the difference quotient. Computing this limit symbolically is very challenging for complicated functions. In this section, we develop rules that find the derivative without having to fall back on the limit definition each time. These rules are purely algebraic in nature and help us gain intuition into the behavior of a derivative function. More importantly, these rules help to demystify the Derivative() function and show the steps to produce the functions output. Understanding the process allows for mastery, adaptation, and more complicated applications of these concepts.

One major topic in calculus is what is called "integral calculus," which involves finding areas or volumes of regions by adding up small slices. We start to think about areas or volumes as an accumulation of the smaller slices that make them and from that we can apply the theory of integral calculus to measure net change and total accumulations. Then, by the Fundamental Theorem of Calculus, this is then related back to where we started: derivatives. This module introduces some of the most beautiful and useful applications of calculus. Algebraic techniques will be shown alongside of numerical computations using Python.