Before we can go into a discussion of calculus itself, it is important to set up some of the underlying concepts that calculus uses. You might view this as something analogous to learning the letters of the alphabet before you can start learning to read words and sentences. Without letters, you aren’t going to be able to grasp words. But once you start getting the words down, you don’t have to think about the letters very often any more. In the same way, there are certain mathematical ideas that are important for understanding how calculus works and what it can do, and you have to understand those first. But as you grow in understanding of calculus, you will have had enough practice with these ideas that you don’t think much about them anymore because they begin to come naturally.
My goal here is to lay out some of the key concepts that will come up at various times in the discussion of what calculus is, what it means, and what it does. You don’t have to be fluent in these ideas to understand the underlying concepts, but it certainly doesn’t hurt.
One of the most important uses of mathematics is in tracking how things are related to each other in the world around us. As an example, you might ask “if I throw a ball this hard, how far will it go?” The underlying relationship is between the force you put into your throw and how far the ball travels. In any situation like this one, you’re going through in your mind a mathematical concept called a function.
In mathematics, a function is a rule for transforming one kind of object into some other kind of object. Usually, a function turns one number into another number. The central idea of a function is that if you know the input, then there is only one way to get an output out of that. As in our example, if you know how hard you threw your ball, the the laws of physics determine exactly how far that ball will go (of course there are other factors like wind and angles, but you get the point).
Certain shorthand notations are helpful in talking about functions. Very often, we use the letter to symbolize a function. If we need to use a few different functions at the same time, we often use the letters and next (just because those are next in the alphabet). You can use any symbol you want to represent your function, the choice of , , and is merely a matter of convenience. We will also often write to denote the function with the input value . We then write and in the realm of the dots, we write the rule we are supposed to use. For example, the function is a rule according to which you take your number , multiply by that 2, then add 3. If you want to use a specific number in your function, you replace all the ‘s with the number you intend to use. For example, and .
There is one other slightly less common but equally important notation we should introduce. Sometimes, it is important to know the specific input values we are about. For example, you can’t throw a ball with negative force, so if you write a function describing the relationship in our example, you won’t care about whether or not you “can” plug in negative numbers. You might also care what sorts of numbers your output looks like – you can’t throw the ball a negative distance. In these situations, we use the notation (read “ is a function from to ). When we write , what we mean is that symbolizes all the inputs allowed for the function , and all of the outputs of will be somewhere in .
When we express and , we usually use what is called interval notation. Interval notation is a way to express a range of values by writing down the endpoints of the ranges. When we want to write all numbers, we write . You can basically just read this as “all numbers.” We can also use interval notation to express a limited range of numbers. For example, the interval represents all numbers between 1 and 3, not including 1 and 3 themselves. The interval represents all numbers between 1 and 3, including 1 and 3. In these examples, the round/open brackets ( ) translate to “not including” and the square/closed brackets [ ] translate to “including.” As another example, the interval represents all numbers between 1 and 3, including 3 but not including 1. In intervals, is used to mean that the interval never ends one some side. For example, represents all numbers larger than 1. Finally, you can combine intervals using the symbol (read this as “union”). So, represents all numbers that are either between and or between and .
Types of Functions
Functions can be just about anything you want them to be, but there are some special and helpful examples that are often used as examples in calculus and other areas of math. So, we will go through a few of those examples here.
A polynomial is any function built by adding/multiplying variables with numbers or other variables. Normally this takes the form of multiplying by itself a certain number of times, multiplying the result by some constant number, and adding together other terms constructed in the same way. Here are a few examples of polynomials:
An exponential is formed by multiplying a constant to itself a variable number of times. One example of an exponential function is . To calculate a few values of this exponential function, and . An exponential function can use any number as its input, although the way to compute an exponential with a fraction (like ) or with even weirded exponents uses some different methods… and very often you can’t really “simplify” an exponential expression.
If you use the definition of an exponential as repeated multiplication, you can deduce a few rules about how exponentials work. For example, and will always be true. These rules form the basics of exponentials. Using these, we can also figure out a few other rules. What if, for instance, we want to know ? Well, the first of our two rules would tell us that . This quick equation leads us to conclude that . We can do something similar with negative exponents to see that a negative exponent is really just division. That is, (see if you can see why – as a hint, you can use the first of the two rules along with the fact that ). We can use the second rule to decipher the meaning of . Whatever this means, it should satisfy . Since , this leads us to conclude that . Similar ideas can be used for other fractional exponents.
The need to do calculations with exponentials is actually relatively rare, but it is important to know that there are these rules that can help us simplify expressions that have exponentials in various ways.
Graphs of Functions
Graphs are universally taught in school because they are a good way of visualizing information about mathematics. The idea is to use a two-dimensional grid to show how a function relates numbers to other numbers. This two-dimensional plane is often called the -plane. The reason is that the “horizontal” dimension is usually represented by an variable, and the “vertical” dimension is usually represented by a variable. Points on a graph are normally written as , which means “ units horizontally and units vertically from the starting point.” This “starting point” is often called the origin (as it is the point from which the others ‘originate’ in a sense).
All equations, and in particular functions can be represented in the form of a graph. The way to do this is to plot all the points with coordinates . In other words, the -coordinate is the input, and the -coordinate is the output. Below is an example, a graph of the function :
Circles and Distance
The concepts of distance and circles are tightly intertwined, and so we discuss them at the same time. Before doing so, it is useful to discuss a third concept – the Pythagorean Theorem – that is fundamental to understanding each. The Pythagorean Theorem tells us how to calculate the side lengths of a right-angled triangle. More specifically, if are the side lengths of a right-angled triangle with the longest side, then the formula will always be true.
How does this apply to the notion of distances? Imagine drawing two points in the plane (i.e. the -plane). How far are these from each other? The answer is the length of a line segment that connects them. If you’re having trouble visualizing this, draw two points on a piece of paper and connect them with a straight line. Now, draw a line in the east-west direction starting at one point and ending “beneath” or “above” the other point. Then connect this new line to the second point with a line running in the north-south direction. If the north-south line has length , the east-west line has length , and the distance between the two points is , then the Pythagorean Theorem tells us that . By taking square roots, we obtain
If the two points have coordinates and in the -plane, then this distance formula turns into
This is because or , depending on which is positive, and similarly or .
But how exactly does all of this relate to a circle? Well, the definition of “circle” involves the notion of distance. A shape is a circle if it is built out of all points that are the same distance from a central point. To visualize why this works, imagine taking a piece of string. Attach one end of the string to a fixed point on a piece of paper. Put a pencil in the other string, and move the pencil around the paper, keeping the string tightly pulled the whole way. If you do this correctly, you will draw a circle. The length of that piece of string is the “distance” involved in the definition, which we call the radius of the circle. The center of the circle is point that was pinned down to the paper.
Let’s use this to make an equation for circles. We want to know whether a point is on a circle centered as with radius . That is, we need to know whether and are distance apart. Using the distance formula we previously came up with, this becomes
If we square both sides (and swap the right and left-hand sides), we come to the easiest-to-look-at version of the circle equation:
This completes the discussion of “pre-calculus” topics. You don’t have to know how to do detailed work with these ideas in order to understand what calculus is about. But I’ve found through teaching that these concepts are quite helpful and are worth thinking about.