Introduction

Calculus is the study of the following concepts.

1. Limits
2. Continuity
3. Derivatives
4. Integrals

If you understand these four core concepts then you understand calculus. The remaining parts of calculus are just theorems and applications of these four concepts. In this article we are going to focus on these core concepts. The goal of the article is to give a basic description of these core concepts and how they fit together.

Numbers

There are different types of numbers. A number system is a set of a certain kind of number. Examples of number systems are natural numbers, integers, rational numbers, real numbers and complex numbers.

In introductory calculus we work with real numbers. Real numbers is just a fancy name for the number system that you are used to working with. It is possible to have a calculus that works with complex numbers. Therefore we usually specify that we are working with real numbers.

Functions

You are probably already familiar with the concept of a function. A function is a rule that associates an element of a set with elements of another set. Usually the sets are the sets of real numbers but they don't have to be. A real function is a function that associates a real number with a real number. A complex function is a function that associates a complex number with a complex number. When it comes to introductory calculus we are only concerned with real functions. Therefore from this point the terms function and real function will be used interchangeably.

A function is usually denoted by f(x). Here f(x) is the value that the function associates with the number x. A function does not have to be defined for all values of x. For example the function 1/x is not defined when x is 0.

A graph of a function is the set of all points (x, f(x)). You have probably already seen a graph of a function.

Absolute value

A number can be positive or negative. The absolute value of a number |x| is equal to the numerical value of x disregarding the sign of the number. Perhaps an example will shed more light. The absolute value of 42 or |42| is equal to 42. The absolute value of -13 or |-13| is equal to 13.

The absolute value of a number is the distance between the number and 0. The absolute value of a number can be thought of as the magnitude of the number. The absolute value is used when formally defining the concept of a limit.

Limit

A limit is one of the core concepts of analysis. To understand and in fact to be able to define the concepts of continuity, derivatives and integrals you have to understand limits. Before we can define limits we first need to introduce two new symbols. Traditionally these are the lower case greek letters delta (δ) and epsilon (ε).

Intuitively a limit of a function f(x) at a point x0 is the value that f(x) approaches if x gets arbitrarily close to x0 without x actually landing on x0. Here is the formal definition of a limit.

The function f(x) approaches the value L at x0 if for every ε > 0 there exists a number δ > 0 such that |f(x0 + δ) - L| < ε.

The above definition can be a bit tricky to wrap your head around. I suggest finding a math book that goes over the concept of limits and that has examples that you can do yourself. Remember that to be able to understand the rest of calculus you have to understand limits.

Limits have a special notation. The limit of the function f(x) as x approaches c is written in the following way.

Continuity

Continuity is a core concept in calculus. Intuitively it is easy to see if a function is continuous or not. A function is continuous if it is possible to draw the graph of a function without lifting the pencil. To be able to mathematically define the concept of continuity you have to understand the concept of a limit.

We can formally define the concept of continuity in the following way.

A function f(x) is continuous at a point c if the limit of f(x) as x approaches c exists and that the limit is equal to f(x). Or using mathematical notation the function f(x) is continuous at the point c if the following holds.

Slope of a line

Before we look at derivatives it is good to review how the slope of a line is calculated. Imagine that we have a line between the points (x1, y1) and (x2,y2). Let k be the slope of the line between the two points. Then the slope of the line is calculated in the following manner.

Next assume that we have a function f(x). Let x1 and x2 be two points on the x axis. Then (x1, f(x1)) and (x2, f(x2)) are two points that lie on the graph of f(x). It is now possible to calculate the slope of the line between the points f(x1)) and (x2, f(x2)). The slope of the line is calculated in the following manner.

Let x0 be a point on the x axis. Let h be a number that is not equal to 0. Then x0 and (x0 + h) are two different numbers. We can now again calculate the slope of the line between the points (x0, f(x0)) and (x0 + h, f(x0 + h)).

This last equation is important because it shows up in the definition of the derivative.

Derivaties

The derivative is one of the core concepts of calculus. Consider the following limit.

Simplifying the denominator we get the following limit.

If the limit exists then the function f(x) is said to be differentiable at the point x0 and the derivative of the function f(x) at the point x0 is equal to the value of the limit.

Since the derivative is one of the core concepts in it has it's own notation. The derivative of the function f(x) at the point x can be written as f'(x), Df(x) or df/dx.

It is possible to show that a function that is differentiable at a point is continuous at that same point. Therefore a function that is differentiable will also be continuous.

The derivative f'(x) of the function f(x) is also a function. Sometimes it is possible to take the derivative of f'(x). This is called the second derivative of f(x) and is denoted by f''(x). The nth derivative of f(x) is denoted by f(n)(x).

The derivative has a geometric interpretation. The derivative of a function f(x) at a point is the slope of the tangent line to the function f(x) at that point.In fact the tangent line of the function f(x) at the point x is defined as the line that goes through the point (x, f(x)) and whose slope is equal to f'(x).

Primitive functions

A function F(x) is said to be a primitive function of f(x) if the derivative of F(x) is equal to f(x). If F(x) is a primitive function to f(x) then F(x) + C will also be a primitive function to f(x). Primitive functions are important due to the fundamental theorem of analysis which we will come to later.

Step functions

Before we can introduce the concept of an integral we need to introduce step functions. A step function is a function where the x-axis can be divided into a finite number of pieces. On each piece the step function is constant. In other words a step function is piecewise constant.

Intervals

Let a and b be two numbers such that a < b. Then the interval between a and b is defined as follows.

[a,b] = x such that a ≤ x ≤ b.

Integrals

Another core concept in calculus are integrals. First we define the integral of a step function. To calculate the integral of a step function on an interval on the x axis we first divide the interval into subintervals in such a way that the step function is constant on each subinterval. Assume there are n subintervals and that the subintervals have the following x coordinates.

[x0,x1],[x1,x2], ..., [xn-1,xn].

Then the integral of the step function is defined in the following manner. The integral is equal to the sum of the following term for each subinterval. Take the width of the subinterval times the value of the function in the sub interval.

Therefore the integral can be viewed as the area under the curve of the function. However negative parts of the function subtract area from the curve. If the step function is never negative then the integral is in fact equal to the area under the curve.

Let the integral of a step function f(x) be denoted by I(f).

We now turn to the problem of defining an integral for a function that is not a step function. Assume that we have a function f(x) that is continuous on an interval [a,b]. Then it is possible to find a step function L(x) such that L(x) ≤ f(x) for every x in [a,b]. L(x) is a lower bound for f(x). It is also possible to find a step function U(x) such that f(x) ≤ U(x) for every x in [a,b].

Let f(x) be a function that is defined on the interval [a,b]. If for any ε there exists step functions L(x) and U(x) such that

|I(U) - I(L)| < ε

then the function f(x) is said to be integratable on the interval [a,b]. Furthermore the integral of f(x) is equal to the value that lies between I(L) and I(U) when ε approaches 0.

More informally you can think of the function f(x) being approximated with a step function. The approximation gets better and better as ε gets smaller. The integral of the function f(x) is then equal to the integral of the step function that is the approximation of f(x). And we already have a way of defining the integral of a step function so we are done.

Since integrals are so important in calculus they have their own notation. The integral of the function f(x) from a to b is denoted as follows.

Principal theorem of analysis

Let S(x) be the integral of the function f(t) on the interval [a,x]. The S(x) is differentiable and S'(x) = f(x). This is known as the principal theorem of analysis. Another way of stating this is that S(x) is a primitive function to f(x).

Differential equations

A differential equation is an equation where the solution is a real function. The following is an example of a differential equation.

y'(x) + k * y(x) + 3 = 0

Here y(x) is the unknown function. Once you find a solution to a differential equation it is easy to check that it is correct. Just take the derivative of the function and plug it into the equation. If the left hand side of the equation equals the right hand side than you have found a solution.

Factorial

The factorial of a number n is defined is defined as follows.

n! = n*(n-1)*(n-2)*...*3*2*1

The factorial is a useful shorthand mathematical notation. The factorial is used when defining Maclaurin polynomials and Taylor polynomials.

Maclaurin and Taylor approximations

Assume that you have a continuous function f(x). It is possible to approximate f(x) using a polynomial.

p(x) = p0 + p1x + p2x2 + ... + pnxn

The issue now is to find the values of the coefficients in the polynomial. If we want to approximate a function around the origin then we can use the Maclaurin polynomial. The Maclaurin polynomial of order n of the function f(x) is defined as follows.

If we want to approximate the function around the point a then we can use a Taylor polynomial instead. The Taylor polynomial of order n is defined as follows.

Conclusion

We have now gone through and defined limits, continuity, derivatives and integrals which are the core concepts of calculus. Furthermore we have looked at differential equations, Maclaurin polynomials and Taylor polynomials. This is what is usually covered in your first calculus course.

If you are interested in learning calculus I highly recommend getting a introductory book on calculus. The purpose of this article is to give an overview of how the core concepts of calculus fit together. Most of the definitions have been simplified. Another reason to get a book is that typesetting of mathematics in books is much better than what can be achieved on the web. Buy or borrow a calculus book. You won't regret it.