Welcome to Math 16A

Before we dive into mathematics, let's step back and think about what we're going to covering this semester.

There are roughly speaking just 3 topics in Math 16A:

  • Functions and their derivatives. (Chapters 0, 1, 2 and 3 of Goldstein.)
  • Exponential and logarithmic functions. (Chapters 4 and 5 of Goldstein.)
  • The antiderivative and definite integral. (Chapter 6 of Goldstein.)

Of course, there is a lot to elaborate on those three topics, but it's important you understand ahead of time what the broad outline is. 


Functions and their Derivatives.

Numbers, Sets and Functions

Numbers

Before concerning ourselves with functions, we should first start with things more fundamental to mathematics, numbers. It's complicated to define exactly that we mean by a number, so we're going to give a working definition that isn't exactly precise, but is good enough for our purposes. 

Working Definition: A real number is a number on the number line. The set of real numbers is denoted by \(\mathbb{R}\). (We'll talk about sets in a moment.)

  Figure 1: The number line.

Figure 1: The number line.

As you can see, the number line is a way of representing numbers. There are some special numbers that we might need to refer to later, so I might as well tell you about them now.

  • The natural numbers are the numbers \(1, 2, 3, 4, 5,\ldots\) and so forth. Some people also include 0 as a natural number, so it's best to be a little careful if it's important that there's no confusion. The set of natural numbers is denoted by \(\mathbb{N}\).
  • The integers are the all the numbers \(\ldots, -3, -2, -1, 0, 1, 2, 3,\ldots\) . They go on for ever in both directions. The set of integers is denoted by \(\mathbb{Z}\). (Why Z? Well, Z stands for zählen, which is German for counting.)

Those two types of numbers are easy enough. You can see them (well, some of them) in the picture of the number line in Figure 1. But what about all the numbers in between? Here things get a little complicated. Some of the numbers that lie between the integers are called rational numbers. Here's a precise definition for a rational number:

Definition: A rational number is a real number that may be expressed in the \(\frac{a}{b}\) where \(a\) and \(b\) are integers. The set of rational number is denoted by \(\mathbb{Q}\).

And now we come to the first really deep question about mathematics: Are there any other numbers? Well, if you look between two consecutive integers, say 1 and 2, you find that there are infinitely many rational numbers in between. For example there is one and a half, which I can write as \(\frac{3}{2}\), and there is one and two thirds, which I can write as \(\frac{5}{3}\). In fact, any finite decimal is a rational number because I can just write it as some integer over a suitably high power of ten. For example: \[17.57842 = \frac{1757842}{100000}.\]

It's a little harder to show that every repeating decimal is also a rational number. Maybe I'll come back to that. For example, think about 0.1111111... where the ones go on forever. What happens if you multiply that by 9? Well then you get 0.999999... where the nines go on forever. And now I will shock you when I tell you that that is exactly equal to one, but I won't be able to justify it very well because I haven't told you about the definition of the sum of an infinite series, so you'll just have to take my word for it. (Or, hold on to your hats: Let \(x = 0.\overline{9}\). That is a shorthand for 0.999999 with infinite nines. Then \(10x = 9.\overline{9}\). That means that \(10x = 9 + x\). Therefore \(9x = 9\). And so \(x=1\). So \(0.\overline{1}\) is exactly \(\frac{10}{9}\).)

I think that's enough about rational numbers. It's not a very big topic in Math 16A. But I just wanted to give you a proper definition because the definition in Goldstein on page 50 is wrong. I'm sorry to put it so bluntly, but there will be times in your life when people you should be able to trust are going to bullshit you, and it's best to come to terms with that fact soon rather than later. 

Now, there are also real numbers that are not rational. Those are all the numbers that sort of fill in all the gaps between the rational numbers. We call them irrational numbers. Famous examples are the ratio between a circle's circumference and diameter, \(\pi\), the base of natural logarithms, \(e\) (we'll come on to that later), and all roots that are not exact roots, like \(\sqrt{2}\) and \(\sqrt[3]{16}\). There's some interesting history about irrational numbers, which you can read about here: https://en.wikipedia.org/wiki/Irrational_number.

I think that's nearly everything I have to say about numbers. However there is one more important thing I want to mention because it sometimes causes confusion: We'll often be talking about concepts called infinity and minus infinity. We donate these as \(\infty\) and \(-\infty\) respectively. They are very useful concepts, and we'll spend some time figuring out how to make sense of them, but I should emphasize that they are not real numbers. That means that all the usual things you can do to real numbers, like add them together, subtract them from each other, multiply and divide, that doesn't make any sense for infinity and minus infinity. If you see something like \(\frac{2x+1}{x}\) you can't plug in \(x=\infty\) because it doesn't make any sense to talk about \(\frac{2\infty+1}{\infty}\). 

Okay, that's enough about numbers to be getting on with. I really want to get onto talking about functions, but first I promised that I would tell you what a set is.


Sets

We've seen above that \(\mathbb{R}\), \(\mathbb{N}\), \(\mathbb{Z}\) and \(\mathbb{Q}\) are sets. But what do we mean by a set? Well, I'm afraid that I can't give you a precise definition so I'm going to give you a working definition that will be okay for our purposes. 

Working definition: A set is a collection of elements?

But what's an element? Well it's one of the things that goes inside the set. But's what's a set? It's a collection of elements. We could go on all day. So I think I should just give you an example: 

Consider the set \(\{1,4,5\}\). That means the set which contains three elements, namely 1, 4 and 5. I generally wrap my sets in curly braces like that, so you should look out for that and not be thrown off. I can truthfully say that 1 is an element of that set, and I would write that as "\(1 \in \{1,4,5\}\)". The symbol \(\in\) means "is an element of", and is written a little like a letter "c" with a line coming out the middle. It's quite simple really. 

So when I say "\(x \in \mathbb{R}\)" you know what I mean. I'm saying that \(x\) is a real number, one of those numbers on the number line. 


FunctionS

I'd like you to look at the following sentence, which might not make sense:

Let \(f:\mathbb{R}\rightarrow\mathbb{R}\) be a function.

What does this mean? Well, let's start with the first word, let

The word "let" in mathematics means we are assuming something to be true. I might have used a different word instead, such as assume or suppose. They all mean the same thing which is that I'm about to say something with no justification whatsoever, and we as the reader are meant to just accept it without question. If I say "Assume \(x=2\)." or "Let \(x=2\)." then I shouldn't say "Wait, why does \(x\) equal 2?". I just say, "Okay! I'm assuming that x is the number two." After that, every time I see \(x\), know that it just means \(2\).

So in the sentence above, I'm assuming that \(f:\mathbb{R}\rightarrow\mathbb{R}\) is a function. What does that mean? Well, first of all, I think I should tell you how to read that in words. You read it like this:

"Let eff from ar to ar be a function."

You know what the two \(\mathbb{R}\)'s mean. They each refer to the set of real numbers. So \(f\) is taking the real numbers to the real numbers. But what does it mean to talk about a function?

Definition: A function is a rule that takes an element of the first set, called the domain, and sends it to an element of the second set, called the co-domain. (Not the range, that's something different which I'll come onto later.)

So when I say "Let \(f:\mathbb{R}\rightarrow\mathbb{R}\) be a function." what I mean is that \(f\) is a rule that takes an element of the first \(\mathbb{R}\) and sends it to an element of the second \(\mathbb{R}\). (Remember element just means one of the things in the set.)

That's rather abstract. I think I should give you an example so that you feel reassured that this is not that complicated. 

"Let \(f:\mathbb{R}\rightarrow\mathbb{R}\) be defined by \(f(x) = x^2\)."

Now we should be feeling more comfortable. Now I'm saying that \(f\) is a rule that takes as input a real number (from the first \(\mathbb{R}\)) and sends it to an output real number (in the second \(\mathbb{R}\)) and the way I figure out what the output should be is to simply square the input. When I say \(f(x) = x^2\) I read that as "eff of ex equals ex squared." The notation \(f(x)\) means the result of putting in input \(x\) into the function \(f\) and so when I say that that is equal to \(x^2\) I'm just saying that if the input is \(x\) then the output is \(x^2\).

I think I should give you a question to try. (Did I tell you that I'd be asking you questions? You can think of these questions as homework. I suggest you actually try writing out solutions to them. They are meant to be easy enough that you just have to slightly adapt something we've already seen.)

Question 1: Write a sentence that assumes that \(g\) is a function from the real numbers to the real numbers whose output is the result of multiplying the input by 4 and then adding 7. 

View solution.

We're nearly done talking about functions and ready to start talking about Calculus. What is Calculus? Well, there are many ways to describe it, but broadest definition I can think of is that it is the quantitive study of change. For example, if we go back to our function \(f\) that we defined above, the one given by \(f(x) = x^2\), we can ask the question: How quickly does \(f(x)\) change if we vary \(x\)? How quickly does the output vary as we vary the input. That is a very important question. If you are running a business, you might figure that you would get more sales if you reduced your prices, but how many more sales? If you are a doctor you might want to reduce a patient's blood pressure by prescribing some medicine, but how sensitive is the patients blood pressure compared to the does of the medicine? You can see this sort of thing if you adjust the sensitivity of the mouse on your computer. If the sensitivity is increased, then the pointer will jump around on the screen as you move your mouse (your input device) around. On the other hand, turn the sensitivity down, and you have to move the mouse much further to make the pointer move. This is what we are studying when we study Calculus. We are trying to understand not just how one quantity affects another quantity, but how sensitive the output is compared to the input. We see this all day, every day, every time one quantity affects another quantity. 


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