Calculus 1 Lecture 0.1: Lines, Angle of Inclination, and the Distance Formula

Well, welcome to the official start of calculus. Congratulations, you made it in the class.

Today we're going to talk about a review, a review of a lot of the Math 2 concepts and some basic algebra concepts that you really need to have down in order for you to be successful in calculus.

The first place we're going to start is Section 0.1, Chapter 0. We're going to talk about lines, just basic lines. We're going to go through some families of curves.

We'll get into some trig functions, your favorite, right? And then we'll continue calculus.

Lots of weird lines. What's special about a line? What do you know about lines?

Well, it has infinite points, but specifically you need at least two, right? What else do you know about lines? Do they curve at all?

Do they curve? Lines. Do the end. What do you need to know about a line in order to graph it?

You need two points. You need two points, or you need one point and specifically one other thing. Slope. You've got to have slope. Lines have slope.

They're straight. They don't end. They have slope. In fact, the slope is what we're going to talk about for the first part of today.

The slope is pretty much how a line rises or falls. Now, you might have been walking through this a long time ago.

We're going to go really quickly through how to find the slope of a line. And we're going to invent the formula ourselves.

So let's take a generic line, and we'll pick two random points on it.

We're not going to be specific on it, because in order to create a formula, you can't really say something specific like any particular point.

We need to work for all points. Can you tell me, if I've got two points, how can I make a distinction between these?

What does every point have? Coordinates. Coordinates. What are the coordinates? Someone over here, what are the coordinates generally for a point?

X, Y. X, Y. OK. So we know that any point that we draw is going to have the coordinates x, y.

This one will, and this one will. But the problem is we need some way to tell a difference between those two points. How are we going to do it?

Yeah, with some numbers. What are those? What numbers? Are they going to go on top of our x or below our x?

In place. Where now? If I put them up here, we're talking about x to the first power.

So we're going to put an x1 and an x2, a y1 and a y2, saying that this is our first point and our second point, whatever those points may be.

We can't use real numbers because then it wouldn't work in general for any two points.

Can you tell me along the x-axis, how far is this?

One? One. If this is the point x1, y1, how far is that?

Yeah, it's x1 for sure. How far is this point?

Yeah, are you guys with me on this? No, you're all right so far. If you're not, go like this. Are we okay or no?

All right. If this was like the point 3, 5, to get to 3, 5, you'd go over 3 and up 5, right?

So then this would be 3. This is not 3.5. It's X1, Y1. So we're going over X1.

How far are we going up? Good. And here?

Now, when we talk about slope, typically a long time ago when you were first introduced to slope, the teacher probably said, yeah, it's how your line rises or falls.

But then they also said slope is defined as what over what? Let's go ahead.

Let's try to identify what our rise is and what our run is.

What would you say would be our rise? This way? This way.

So if we find the difference between those two numbers over there, we're going to find the rise for our line.

What is the difference between those two numbers over there? How do you find the difference, this distance here?

Sure, yeah. If this was 10 and this was 3, the distance between them would be what?

Yeah, you'd subtract them, right? You'd do 10 minus 3. So here we're going to go, well, it's not 10 and 3. It's y2 and y1.

So our rise, we're going to call y2 minus y1.

Can we do the same thing with the run?

How far is our run? What's that distance represent?

I thought you were going to answer. Instead, you sneezed. I was like, she's off.

And then, darn it. So, for a good one. Yeah. We got x2 minus x1 for sure.