Welcome to Math 1A

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

There are roughly speaking just 3 topics in Math 1A:

  • Functions and Limits. (Chapter 1 and Sections 2.1 through 2.6 of Stewart.)
  • Differentiation and its applications. (Sections 2.7 through 2.8 and Chapters 3 and 4 of Stewart.)
  • Integration and its applications. (Chapters 5 and 6 of Stewart.)

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 form \(\frac{a}{b}\) where \(a\) and \(b\) are integers. The set of rational numbers is denoted by \(\mathbb{Q}\).

And now we come to the first really deep question about mathematics: Are there any real numbers that are not rational? 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 another time. To give you an intuition for what's going on, 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 \(\frac{1}{9}\). You might like to see if you can figure out how to generalize this approach to other repeating decimals, but only if you think that's interesting.)

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 for the time being. 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 for 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 \(\frac{2\infty+1}{\infty}\). I shouldn't even really write \(x=\infty\) because it makes no sense.

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. Look at this sentence:

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 \(x^2\). Please never make the mistake of thinking that \(f(x)\) means \(f\) multiplied by \(x\). I know it's confusing.

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 pretty much 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 dose 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. 


A peek ahead.

We're about to get on to a topic that is pretty difficult for most students. So difficult that a lot of professors just skip it altogether. But those guys are just losers who don't have enough confidence in you. 

Here's the motivation. We want to be able to understand how quickly something is changing at a particular time, or to be more precise  when the input takes a certain value. For example, I might ask, how quickly is \(x^2\) changing relative to \(x\) when \(x=2\)? Well, the trouble is that if I fix \(x\) to be 2, then \(x^2\) is stuck at \(4\), which isn't changing at all. On the other hand, if I allow \(x\) to go up a little bit, say from \(2\) to \(3\), then \(x^2\) goes up from \(4\) to \(9\), but that is a change that takes place over the whole range of values of \(x\) from \(2\) to \(3\). I could look at a smaller range of values of \(x\) near to \(2\), like from \(2\) to \(2.1\), but that only gets me closer to what I really want, which is how fast \(x^2\) is changing relative to \(x\) when \(x\) is precisely \(2\).

The way we do this is with something called a limit.

What's a limit? Well, it's a little like the example we saw earlier with \(0.9999999\) where the nines go on for ever. You might have been surprised to hear that that number is exactly 1. But if I tell you that the sequence 0.9, 0.99, 0.999, 0.9999, and so forth, where I put an extra 9 at the end of the decimal expansion every time, gets closer and closer to 1, as close as you like to 1, you will probably think that is pretty obvious. And that is all I am saying when I say that \(0.\overline{9}\) is equal to \(1\). I'm saying that \(0.\overline{9}\) is defined to be that number which the sequence with an extra 9 each time gets arbitrarily close to, and that number is exactly 1. That's what we mean when we talk about an infinite decimal expansion. The idea of the limit is a way of talking in a precise way about a number that we get as close as we like to. We'll be talking more about sequences of numbers like this, and related things called series, in Math 1B. 

So anyhow, let's get back to how limits relate to functions. Suppose I have a function \(f:\mathbb{R}\rightarrow\mathbb{R}\). You know what that means right? If not, go back and read the part about functions again to remind yourself.

Where was I? Oh yes, suppose I have a function \(f:\mathbb{R}\rightarrow\mathbb{R}\). Suppose that \(a\in\mathbb{R}\). We're going to be interested in what happens to \(f(x)\) when \(x\) gets as close as I like to \(a\). If there's a real number \(L\) that I can make \(f(x)\) as close as I like to just by making \(x\) as close as I like to \(a\), then I call that number the limit \(\displaystyle \lim_{x\rightarrow a}f(x)\).

That sounds very complicated, but it's not that complicated. I can make \(x^2\) as close as I like to \(9\) by making \(x\) sufficiently close to \(3\), for example. 

Don't worry if this doesn't make entire sense. I'm trying to explain something rather complicated without using words you don't know. Here is the formal definition of a limit, so you can see what we're going to be spending time trying to understand. 

Definition: Let \(f:\mathbb{R}\rightarrow\mathbb{R}\) and \(a, L \in \mathbb{R}\). (Don't be thrown off - that just means that both \(a\) and \(L\) are real numbers.) Suppose that for all \(\epsilon > 0\) there exists \(\delta > 0\) such that if \(0 < |x-a| < \delta\) then \(|f(x)-L| < \epsilon\). Then we say that \(L\) is the limit of \(f(x)\) as \(x\) tends to \(a\), and we write that as \(L = \displaystyle \lim_{x\rightarrow a}f(x)\).

Gosh, what a lot to understand. I expect it doesn't make much sense. Don't worry. It's just meant to get us ready for our next topic. But I think we've done too much reading for now, so here's a video so you can grab some popcorn and follow along. You can skip the first few minutes and catch up to where we're at:


Logic.

Did you watch the video? Perhaps I might quickly recap the key points:

  • In mathematics we study statement that have a precise truth value. That means, they are either true or false. They are never vague. Of course, in real life, most assertions that we make are a little vague. If I say that it is sunny today, I allow that possibility that there might be a few clouds in the sky, but how may clouds? It's not exactly clear. But if I say something in mathematics, it has to be either true or false. For example, as a mathematician, if I say "\(x=2\)", that could be true, or it could be false, depending on what \(x\) is, but it can never be "kinda true" or "sorta true". In this way, speaking mathematically is different to the way we speak in real life.  
  • We can make new mathematical statements out of existing mathematical statements. Let \(P\) and \(Q\) be statements. (Remember that means they have a precise truth value.) The the following are new truth statements that have truth values depending on the truth value of \(P\) and \(Q\):
    • "Not \(P\)." This is true precisely when \(P\) is false. It's false when \(P\) is true.
    • "\(P\) and \(Q\)." This is true precisely when \(P\) and \(Q\) are both true, and false otherwise.
    • "\(P\) or \(Q\)." This is true if at least one of\(P\) and \(Q\) is true. It could be that \(P\) is true and \(Q\) is false, or \(P\) is false and \(Q\) is true, or it could be that they are both true. All three possibilities result in "\(P\) or \(Q\)" being true. The only way "\(P\) or \(Q\)" can be false is if both \(P\) and \(Q\) are false.
    • "If \(P\) then \(Q\)." (Sometimes this is written as "\(P \Rightarrow Q\)", pronounced "\(P\) implies \(Q\)".) This one is more tricky. In daily language the words "If\(\ldots\) then\(\ldots\)" have a connotation that is a little different to what we mean in mathematics. In mathematics, the statement "If \(P\) then \(Q\)." is true if unless \(P\) is true and \(Q\) is false. It's a little counter-intuitive, so I'll give you some examples. You may find it surprising, but mathematically speaking the first three of the following are true, but the last one is false:
      • "If \(1+1=2\) then \(\pi\) is irrational."
      • "If Donald Trump is a woman then Barack Obama is a man."
      • "If the capital of Germany is Paris then the river Nile is in Canada."
      • "If Moscow is in Russia then the set \(\{1,2,3\}\) has \(25\) elements."
    • "\(P\) if and only if \(Q\)." or "\(P\) and \(Q\) are equivalent." This is true when \(P\) and \(Q\) have the same truth value. Otherwise it's false.

If any of this is confusing, I suggest watching the video again. If it's still confusing, you can always phone me. 

It turns out there's a lot you can study just by thinking about how to put statements together. The part of mathematics that deals with that is called "propositional calculus", which is a branch of a bigger field called "logic". You can read more about that here. For our purposes we just need a very basic understanding of putting statements together like we've described above. 

Now, if we go back to that long definition I showed you before, we still have a lot of work to do to understand the whole thing, but we understand two important words. I'll put them in bold so you can see:

Definition: Let \(f:\mathbb{R}\rightarrow\mathbb{R}\) and \(a, L \in \mathbb{R}\). Suppose that for all \(\epsilon > 0\) there exists \(\delta > 0\) such that if \(0 < |x-a| < \delta\) then \(|f(x)-L| < \epsilon\). Then we say that \(L\) is the limit of \(f(x)\) as \(x\) tends to \(a\), and we write that as \(L = \displaystyle \lim_{x\rightarrow a}f(x)\).

So you can see, we didn't have a chance of understanding this until now because we didn't know what "if\(\ldots\) then\(\ldots\)" meant. Now we do. 

But how can I be sure that an if then statement is true. Well, what I do is I suppose the "if" part, and I try to deduce with a rigorous chain of argument the "then" part. Here's an example:

Question: Prove that if \(3x+1 = 10\) then \(x=3\).

Answer: Suppose that \(3x+1 = 10\).

Then \(3x = 9\).

Therefore \(x = 3\).

Therefore if \(3x+1 = 10\) then \(x=3\). \(\square\)

Of course, this is the kind of algebra we learned some time ago in school. However the point I want you to focus on is not the algebra, but the logical structure of the argument. Notice how I assuming the "if" part at the start, and after three lines of reasoning I'd deduced the "then" part. The fact that I was able to do that meant that I'd proved the over all "if\(\ldots\) then\(\ldots\)" statement. Because I'd finished a proof, I put a little square at the end to say that I was done, which is a nice touch that mathematicians often do. 

There is another point about the structure of the argument I'd like you to attend to. Notice how I wrote everything on a separate line. This helps me break down my thoughts so that I know I'm just writing one statement. I introduce each statement by saying whether it is an assumption or a deduction. We talked about assumptions before. Do you remember other words that mean the same thing as "suppose"? Look back to the section on functions if you've forgotten. In the answer to the simple question we just did, only the first line was an assumption. I was assuming the "if" part of the "if\(\ldots\) then\(\ldots\)" statement. Remember, I don't need to justify an assumption, I get to say whatever I want after "assume" or "suppose". 

The next three lines are not assumptions. They are deductions. What is a deduction? Well, you may have heard that word before if you've ever read any Sherlock Holmes stories. A deduction is something that you've figured out based on the information you already have. When I say "Then\(\ldots\)" or "Therefore\(\ldots\)" or "It follows that\(\ldots\)" or "Hence\(\ldots\)" you know that I am about to say something that I know with absolutely certainty based on what I already know. Here's an example taken from the Sherlock Holmes story The Adventure of the Priory School:

"It is impossible as I state it, and therefore I must in some respect have stated it wrong."

It's exactly the same in the argument above. The word therefore, and other words that mean the same thing, is saying something very strong. It's saying that without introducing any more information, we can say a new thing that we no to be true. That's a very powerful idea. That is the power of logic. 

I'm going to give you some harder examples. Try to answer these questions very precisely:

Question: Prove that if \(x^2 = 4\) and \(x\) is positive, then \(x=2\).

Question: Prove that if \(a^2 - b^2 = 0\) then \(|a|-|b|=0\).

Let me know if you have any difficulties with those! I really believe in your ability to do a good job. 

I think we've spent enough time on the statements. The next thing to understand is the words "for every" and "there exists". That brings us on to our next topic, qualifiers.


Quantifiers


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