Calculus

This page is about infinitesmal calculus. For other uses of the word in mathematics and other fields, click here

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.

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 $p(t)$, tells us how far the car is from its starting point after $t$ seconds.

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

$p(t)=vt$ 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. This is a good definition because the slope of the line doesn't depend on point we are looking at. That means that the slope is constant.

But car's don't really move at constant speeds. Cars can accelerate. 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 $f(t)=t^{2}$. What is the rate of change of this function? It doesn't seem like we can find it. In fact, at the moment, this is true. With the tools of algebra and geometry, we cannot the rate of change of this function.

What we can do is find the average rate of change between two points. If we take two points on $f(t)$, call them $t_{0}$ and $t_{0}+{\Delta }t$, we can draw a line between them and take the slope of that line. This is a decent approximation of rate of change in the region between the points. The smaller we make the interval between $t_{0}$ and $t_{0}+{\Delta }t$ the more accurate our approximation gets.

Suppose we wanted to approximate the rate of change of $p(t)$ at $t_{0}$ very accurately. We could keep $t_{0}$ fixed and make ${\Delta }t$ really small. The smaller we make ${\Delta }t$, the more accurate our approximation.

Let $f'(t)$ be the rate of change of the function at $t$. Then the formula for our approximation becomes:

Failed to parse (syntax error): {\displaystyle f'(t) \approx \frac{f(t+{\Delta}t) {–} f(t)}{{\Delta}t}}

This is called the difference quotient. It gives us the slope of the line between $t$ and $t+{\Delta }t$. As we make ${\Delta }t$ smaller and smaller, the difference quotient becomes a better and better approximation of the instantaneous rate of change of $f(t)$ at the point $t$.

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.