Alright. So now we're going to calculate the slope of a curve that's not a straight line, using the arc method. What I mean by the arc method is basically we're going to find the slope between two points, right? When we did the point method, it was just one point. Right? So now we can find the slope over a region like this. Okay. I've got instructions here on the right where you want to draw a line connecting the ends of the arc. Right? So the region that you want, to calculate the slope over. Right, I've got 2 points on the graph there. I could easily calculate the slope over, you know, this whole region here or this region like that or this region here, right. You're going to get different answers in all those cases because the slope is constantly changing and you can pick any points right. I just picked points where we've got intersections, it's just going to make the math easier.

So here this is the slope, this slope is the average. Average slope over the region, over that arc. So between those two points that you select and calculate, you're going to be calculating the average slope over that region, not just, what is the slope. Remember that slope is constantly changing, so we're doing our best, and we're going to find an average. So what I'm going to do is I'm going to pick these two points right here, and I am going to draw a line connecting those points. So from here to here we'll draw a straight line. So you can see that that almost approximates what the graph is actually doing. Right? So this is why we're kind of finding an average. We're doing our best to estimate what that slope is. So it's almost like we've got a line here going like this, right? We calculated those points and we're going to have this line going like that.

So let's go ahead and calculate the slope of that line. So same thing. We're going to do our rise and our run. Let's see what our rise was between these two points. Looks like it goes up. We started at, excuse me, at 4 and we went up to 5. So it looks like our rise was 1. And let's do the same thing for our run. We started at 3. It looks like we went over to 6. So it looks like our run was 3. Right? So using our same formula for slope,m=riserun, so the slope of that line connecting that arc is going to equal our rise over our run. Our rise was 1, our run is 3, our slope is one third. So that is the average slope over that arc. Right? And if we had picked 2 other points, we would have got a different answer, but the method stays the same. You draw a line, you calculate the slope of that line, and that will be the average slope over that section. This is the method that we'll use more often in this class. I'm not expecting you to have to be drawing tangent lines and stuff like that, so just be pretty comfortable with this, being able to pick 2 points, draw the line, and calculate the slope. Cool. Let's move