Hi, everyone and welcome. We're going to introduce the notion of limits to the calculus topics that will follow by first starting off with the problem that actually motivated the study of what's the fallen chapter 2. Here's the question for you, I want to find the slope of a line. You say silver line, how is that a problem? I know how to do that. Good. Well, let's see which particular line I'm talking about. As always, let's start off with a graph and let's just pick some generic function. Here you go, f of x. You don't need the formula for what we're about to do. Let's pick a specific point, x_1 on the graph. Here you go. Let's pick another point, picking things at random here there's no good or bad choices. Let's put the y-values on the chart as well. We have two points on the graph. Everything normal so far. When I have two points, it's a pretty natural thing to do to draw a line between them, so let's draw a line between them. This is a nice line, looks a little wavy in my picture, but imagine then it's a line. This line, when I have a line that goes through two points is called the secant line. Secant line through x_1 and x_2. Now I know secant is also used for trigonometry where it's one over cosine, but that is not what I mean here by secant. This is a line through two given points of a function, they call the secant line. When you have a line, it is pretty normal to ask what is the slope of that line? Slope is denoted by m. Well, because I'm talking about a given two points in the secant line between him, I'll say msecant. This is the same form of the slope that you know and love. Change of y over the change of x, y_2 minus y_1 over x_2 minus x_1. For our particular case, just to give you some notation, I don't use y-2 and y-1, although I certainly could. This is fx_2. You have y_2 over here, and y_1 over here. You normally see this because we're talking about the given function f. We'd like to use the givens. This will be f of x_2 minus f of x_1 divided by x_2 minus x_1. That's normally the expression you see for these things. Given a function, given points, you can plug in, you can get numbers, not a problem. I said before, well, there's a problem with this. The problem occurs with any fraction. If I happen to pick the same two points, what if I talk about the points when I imagine this sliding over to the left a little bit of x_1 equals x_2? Now in the picture, there is not a problem. If we imagine this point slowly walking over and the line connecting it, there would be some line, give or take, that goes through this point just like that. This point goes through the graph right at my x_1. We call this line the tangent line. The tangent line of the function at x_1. Hopefully from the picture you can imagine it is a perfectly good tangent line. It is a normal line that you can draw. If we had graph paper and everything like that, you can find the slope. The slope of the tangent line is a number. It exists, measure it, go do your thing. Not on my picture. My picture is terrible, but you get the idea. What's the formula? Well, it's change of y over change of x. Everything's great. Well, not really and here's why. If you do the same thing, if you try to do the same thing with x_1 equals x_2, you run to this problem, f of x_2 minus f of x 1, but they're exactly the same. What happens to my expression? Well, since they're equal, the numerator becomes zero and the denominator becomes zero. Now I don't know if you've seen this expression before, but zero over zero, it is this working move to be like, isn't that just one? No, no, no, you're dividing by zero. That's not good. You can't cancel these things. If you look at the fine print back in the day when we learned about fractions we were like, the denominator cannot be zero, you can't cancel. This is not equal to one. Also, the other mistake that a little less people, fewer people make is that they say that well the zero in the numerator is and this shows zero, No,no no. In fact, this is so important that we put it in red. If you have zero over zero, this is undefined. It's a made-up word, like blah, blah, blah. What does that mean? Nothing, it doesn't mean anything. It's undefined, is not in the dictionary. You can't tell me that zero over zero is equal to one or you can't tell me it's equal to two or whatever, it's undefined. Think of it as like a number, an expression not in the math dictionary, so it can't come back. This is undefined, super-important. Means it is nothing. But now I have a problem because algebra is failing me here. The algebra that I know and love to find slopes is unable to get me back, which would seem to be a relatively easy problem. How do I find the slope of a line. Algebra is failing, and this is a huge wall. I have a division by zero error. What do I do in this case? This is the problem that Newton was faced with and Liminisome, people who didn't have the calculus and we're about to study developed in front of them. They had to overcome this problem. Calculus will be the tool that lets you overcome this problem to find this line. Now you may say, well, what's the point? Who cares? It's just a line, I can get out the ruler. Well, you don't always want to do that. It's, sometimes it's trickier. You'll want to have algebra not stop. You want to have algebra keep going. This tangent line, and it isn't just a geometric little exercise. This tangent line means things in the physical world. For example, if it represented the distance over time, then the velocity of the function is represented as the slope of the tangent line, so the instantaneous velocity at a specific point. When your card has a speedometer and I'm going to have fastened ongoing right now, that is this velocity, a single instant in time. That's how fastest is going. Between two points, no matter how far apart they are, that doesn't matter. You can always do that gentle change of y over change of x. As long as the two points are different, whether it's 10 seconds or 10 milliseconds or whatever, you can find a number, find the calculator and work it out. The problem is if I make the two points the same and I want to in an instant, what is the slope of this line? What is the tangent line at instant, whether it represents something, velocity, acceleration, whatever algebra fails. Before you move on to the next video, we'll talk about how to solve this problem. If you want to pretend you don't know your Newton or live minutes, think about how you can overcome this problem. See if you could be as smart as Newton was and work on this. But this is the setting up the problem that we're going to solve in the next video. See you there.
