Introduction to Calculus (1 of 2: Seeing the big picture)

What I need to say before I start delving into this is that I know many of you have already heard about this topic. Okay, I just want you to forget everything you've ever heard about this blank slate. Well, the biggest danger, especially with a class like this, is to approach this topic as if it were a set of rules, basically a set of rules, memorize the rules, learn to apply them, know hundreds of thousands, maybe situations, and that It's all, you know, calculation, nothing could be further from the truth. Well, the design and use of

calculus

is such a creative process that if we boil it down to well, you will learn rules, you will learn rules, but if learning rules is what

calculus

is for you, then you don't know what calculus is.

Okay, now grab your pen and let's think about what this is: It's actually an abbreviation, the full name of calculus is infinitesimal calculus. Now the reason why this is what we're actually going to do is an infinite test of small calculus, but as you can see it doesn't exactly roll off the tongue, due to such a long and awkward phrase, calculus is just what it came to. know each other, but it's a little strange, calculus is a very general word, it's the same word from which we get the word. calculator and calculate actually um before the 1600 calculus if you said calculus that would basically mean the same thing as math, it's a calculus thing, right? um the reason calculus means mathematics is because calculus is actually a Latin word, it means um it's Latin um what does that have to do with anything? and um, small pebbles, small pebbles are what they used to calculate with in the early days in ancient Rome, right, they get a bunch of, a bunch of small pebbles, right, and some, um, some smart guy.

In some country, I don't know its name, put all these pebbles like roses and move them and call them an abacus. Okay, small pebbles are the ones used for counting and calculations. By the way, it's also where we get the The same word chalk calculus, you get it, um and in fact, if any of you ever become calcium, that's exactly right, if any of you ever become a dentist or You go to the dentist, they might say, "You have calculus on your teeth, which, by the way, you don't." one good thing because you're not meant to have little pebbles in your teeth like a, you know all the gross stuff that shouldn't be there and that's damaging your teeth, okay, so calculus, is really this word for calculus. and the fact that it took over like what we're about to do, this little subtheme took over like, can you imagine?

Well, math is a really big word, right? What if only a small topic in mathematics was called mathematics, right? What does that say about all the other things? He says this is really very important. We have many branches of mathematics. Correct calculation is just one of them. Each one of them is connected to a single big idea. Alright? Think about this. with me geometry, for example, with familiar geometry when you think about geometry, you think about shapes, right, you think about shapes, things that take up space, about shapes, plane and space, that's what geometry is about, like dimensions and all that kind of things, algebra, this is a little more complicated, right? but all those It's a little bit easier, it's the mathematics of uncertainty probability, just when you think I'm not sure if something is going to happen, but I can know that the basics of this are more likely. less likely to happen, am I going to win the lottery or not, will we get a tower of 99.95, etc., each branch of mathematics has its key idea, calculus, calculus is the mathematics of all things that change, all things that change and you know.

There are many things in the world and in the universe that change, so you can imagine why calculus actually took over number one. The hsc preliminary courses you are doing in mathematics and mathematics extension one are called calculus courses. They are a good 70 80 percent more calculation and they are like subtopics. Well, in fact, the equivalent of what you are learning in the United States is called AP calculus. Ap means advanced placement. don't mess with you, you are serious about mathematics, okay, the course is called calculus somehow, like ours, it just doesn't happen, yes, that's right, there is precalculus and then calculus, right, that's how it is . the things that prepare you for this, okay, now, calculus, you need to know, you need to know, it was developed not jointly but independently by two guys in two countries at exactly the same time and there is a big argument between these two about who invented it. by the way, before it was called calculus it was called calculus the b is like a capital you know cum the two guys are one who i have heard of a guy called isaac newton guy in england right you may have heard of him , the other, the other, you probably haven't heard of gottfried livenitz, the german, newton, in england, leibniz, in germany, and both of them, together, sorry, usually did not simultaneously develop this great idea, it is Well, and I'll tell you later, when we get there.

Can you remind me why it's unusual that no one has heard of lightning and everyone has heard of Newton. Well, here's the problem and this is the famous part I was trying to solve. So Newton, well actually yes, Newton was sitting underneath. a tree, more or less, the story goes well and he saw the apple fall from the apple tree, it fell to the ground, right, that's what he really wrote in his papers, no, that's right, uh, the apple fell to the ground, where it really is , you already know. The hit is up for debate or whether it was a real apple or not, but the point is that he saw the apple fall and he watched it from the ground and said, "Okay, gravity, gravity is one thing that makes the apple fall, but then as he watched the apple falls, he thought that the moon was very high up and is also held in place by the same thing that makes the apple fall, gravity, but it doesn't, it doesn't hit the earth, why? what doesn't it?

Why does gravity? Do this to this object, but that object just sits up there and he says, How can I solve this properly? So what he did was try to, like we've been doing, solve it. a problem that you don't know how to express it and turn it into a problem that you can express because then you can solve it you ran into a problem gravity does something unusual does something like this gravity like most forces in the universe is inversely proportional to the force you exert on something is inversely proportional based on um I was going to say distance but it's actually the square of the distance you are away from something okay the further you get something from something so if I have here the distance is the same as your distance increases the force of gravity and in fact everything, really, if you think about the electrostatic force, it's Coulomb's law, I think it gets smaller and smaller and falls like this, okay , it falls pretty quickly, okay, now here's your problem, right? trying to figure out well, how do these things relate to each other?

How does gravity change with distance? That's the problem he was trying to solve and um, this is complicated because this thing is changing at different speeds everywhere. you look good, for example, if you just compare it to a straight line, you can draw me one of these. Okay, for a straight line, if you want to know how this straight line is changing. Okay, that's not hard to figure out, that's not hard to work with. Just think, for example, of a particular point in distance or time or whatever value, you compare it to another point, okay, and if you have two points there, you can just say look, I'll compare how much it changes by one. amount versus how much it changes in the other, you just get them as a proportion, right, we know that nature's proportion from the name starts with the gradient g right on a um in a Cartesian plane, you would say look, this is vertical, so you call that rise and this is horizontal, we call that run, okay and the gradient is still who knows why, okay, the gradient is just the relationship between those two things, okay, but there's a problem here, There is a problem here, no matter which two points you choose.

We will always get a different value. Look what happens with this. The reason this works is because I can calculate that or I can calculate between these two points or I can calculate between these two points and you will always get the same one. value, you always get the same gradient, it will be that constant, right, we know how to read it, but we have some problems here, so this is what Newton did and it's ingenious, okay, think in a circle, okay, yeah. I've done better now, what I really wanted was how much am I changing at any given time right now by making green, you guys already know in terms of similar geometry, what it looks like, is the slope of the graph. right, if you can tell how steep something is at a certain point, that's its grain, so I wanted to know like I know, obviously, it's steep over here and it's not so steep over here, but how much is okay, so what?

It was really after the amount it was. next was the gradient of the tangent, okay this is a really critical idea, the gradient of the tangent, the problem with calculating the gradient of a tangent is that the gradient increases throughout the run, you have to have some run , you must have two correct points to compare between y2 minus y1 over x2 minus x1 you need y2 and x2 but a tangent by definition does not have two points, it only has one correct, in fact, that is why it is the word tangent, does anyone know that we just like all this?

We have another word in our English languages ​​that comes from the same root as tangent. They don't know what it is. They don't use it very often. It is true that tangerine is tangible. What does tangible mean? It means you can touch it, you can touch it. okay, so a tangent is something that just touches a point, okay, this presents problems for us and Newton trying to work and Leibniz trying to calculate what the gradient is at that point because there are no two points to compare by definition, so that this is what they both do