Section 1 What Is Calculus
In almost all universities, the core and starting point of academic mathematics is calculus. Why is this so? What is calculus and why is it so important? In this introductory section, I want to give you a brief idea of what calculus is about, as well as provide some motivation for why it ought to be studied.
Calculus is the study of the dynamic behaviour of functions. Okay, that’s a definition, but why does that help? What is a function and why is the study of the behaviour of functions so central? Why `dynamic’?
I am going to spend a whole week in Chapter 2 to review functions. For now, in brief, a function is a mathematical way of encoding a dependance of one quantity on another. Mathematics is a descriptive language that seeks to capture the quantitative aspects of the world. In the world, many quantites depend on other quantaties. In an environment, humidity may depend on temperature. In a vehicle, fuel efficieny may depend on speed. In an economy, productivity may depend on interest rates. A function is the mathematical thing that measures these relationship.
This is why a function is, arguably, the most important mathematical tool for describing the quantitative world. The (quantitative) world is not collection of completely seperated and independent measurements; it is a collection of deeply interconnected systems. It is full of dependencies. The function is the tool that describes the reality of the (quantitative) world.
In particular, most quantities in the world depend on time and change over time. Very little in the world is completely static. A function can describe how something changes and evolves over time.
In this way, mathematicians use functions to model the world. To be able to do so, they need a theory of functions and their behaviour. That theory is calculus.
But some of you might now ask: why do we need a new theory? We saw functions in high school and the algebraic techniques of high school mathematics were pretty good in describing functions. This is true: algebra goes a long way to understanding functions. However, algebra has one very important and significant weakness: it is static. Algebra can calculate one unknown quantity from other known quantities, but only it a snapshot, a single fixed moment in time. Once quantities start changing and shifting, algebra can not longer keep track.
This is why we need the new techniques of calculus. Functions are dynamic: they are constantly change, constantly in flux. They represented systems where nothing is static. The new tools in calculus (the derivative, the integral, the infinite series) are all ways to understand dynamic systems, where the change is happening all around.
In Section 3.1, I’ll give more specific motivating problems for the course. For now, I hope this introduction gives a good concept and motivation. We want to describe the (quantitative) world. We observe that in the world, quantities are interrelated and interdependent. They are also dynamic, constant in flux. To make a mathematical description of the world, the description must also be dynamic and interdependant. That requires functions, and calculus is the study of their dynamic behaviour.
To get to the tools of calculus, I’m going to spend two weeks on material that may be review for some of you. In Chapter 1, I’m going to cover analytic geometry. This is the set of geometric tools we use to understand the shape of functions, as well as other shapes in two dimensional space. It is necessary for understanding and, in particular, visualizing functions. In Chapter 2, I’m going to talk about the concept of a function and go over the main properties and operation. These two chapters are preparation for the idea of a calculus, which will properly start in Chapter 3.
