# Calculus

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Calculus usually refers to the elementary study of real-valued functions and their applications to the study of quantities. The central tools of calculus are the limit, the derivative, and the integral. The subject can be divided into two major branches: differential calculus and integral calculus, concerned with the study of the derivatives and integrals of functions respectively. The relationship between these two branches of calculus is encapsulated in the Fundamental theorem of calculus. Calculus can be extended to multivariable calculus, which studies the properties and applications of functions in multiple variables. Calculus belongs to the more general field of analysis, which is concerned with the study of functions in a more general setting. The study of real-valued functions is called real analysis and the study of complex-valued functions is called complex analysis.

## Motivation

As was mentioned in the introduction, calculus is considered as two separate, but very interrelated topics. The motivation for the derivative is rather different from that of the integral, yet it turns out that they are very closely related.

### Rate of change

A simple and intuitive way to introduce the derivative is to consider the problem of the rate of change of a function. We will use the concrete example of the position of an automobile on a straight road as an example.

Intuitively, the function describing our car's position should be continuous, meaning it has no holes or jumps in it, and smooth, meaning it has no cusps or sharp turning points. What these assumptions mean in physical terms is that the car always has a position and speed, and its position and speed cannot change instantaneously.

Let's say that the car has a constant speed. What does the function of its position look like? We will assume that the function, which we will denote by ${\displaystyle p(t)}$, tells us how far the car is from its starting point after ${\displaystyle t}$ seconds.

Let ${\displaystyle v}$ be the speed of the car in meters per second. If ${\displaystyle p(0)=0}$, where can we expect the car to be after a second? Well, since the speed of the car is constant, and ${\displaystyle p(0)=0}$, we have:

${\displaystyle p(t)=vt}$ This is just the equation of a line. The slope of the line is equal to the speed of our car. In general, the rate of change of a linear function is equal to its slope.

But in general, the rate of change of a quantity isn't constant. How can we define rate of change for a general smooth and continuous function? Suppose that ${\displaystyle f(t)=t^{2}}$. What is the rate of change of this function? Do we even have a proper definition of rate of change? It doesn't seem like we can find it, or even define it properly! In fact, at the moment, this is true. With the tools of algebra and geometry, we cannot study the rate of change of this function. The derivative is a tool that allows us to define the rate of change of a function. Much of Calculus is devoted to determining when the derivative of a function exists and how to find it.

### Area beneath a curve

Suppose we wanted to find the area underneath a constant function on an interval. We know how to find this area because it is just a rectangle. We can also find the area beneath a linear function on an interval because it is just a trapezoid.

Can we find the area beneath a general function? Just like finding the rate of change of a function, finding the area beneath one on an interval is impossible with just the tools of geometry and algebra. We don't even know how to define area! In fact, the definition of area is a very deep topic, and it turns out that sometimes it cannot even be defined.

In elementary calculus, only functions nice enough to have an area beneath them are studied. The tool used to compute these areas is called the integral. The integral has many interesting properties, and comes in two types: The definite and indefinite integrals. The former is the one we are discussing here. The latter is very interesting as it is tied to the fundamental theorem of calculus.

## Calculus vs. analysis

Strictly speaking, there is virtually no distinction between the topic called calculus and the topic called analysis. The distinction is made on historical and pedagogical grounds. Calculus usually refers to the material taught to first and second year university students. It is usually non-rigorous and more concerned with applications and problem solving than theoretical development. Analysis usually refers to the study of functions in a more technical and rigorous setting, usually starting with a first course in the theoretical foundations of elementary calculus. The elementary treatment of calculus generally follows the historical development pioneered by Isaac Newton and Gottfried Leibniz. The development of introductory Analysis follows the rigorous treatment of the subject that was formulated by mathematicians such as Karl Weierstrass and Augustin Cauchy.