 So let's talk about graphs and rates of change. Algebraically, the average rate of change over some interval corresponds to some formula. We can compute this, and if we ever need to, we can apply the formula and find it. But algebra is a learned response, and sometimes it's nicer to be able to see something, to visualize something, and we can generally do that through some sort of geometric interpretation. So the question is, what does this rate of change look like if we view it geometrically? And in order to do that, what we can do is we can consider the graph of our function. So algebraically we have some function, geometrically we can talk about the graph of y equals f of x, and it may look something like that. Now here's an important idea to keep in mind. This equal symbol means that left and right hand side are completely and totally interchangeable. That if I talk about one, I can replace it with the other and have exactly the same sense. So when I say y equals f of x, well this is the graph of y equals f of x, what that means is my y values are my function values. Well my y values are heights above the x axis, so the height at any point is a function value. The height is a y value, is a function value. Key idea to keep in mind because it's going to be critical to all of our interpretations of anything we get from a graph. So if I want to talk about change, well the value of a function, f of a, there's a function value and that's going to give me a point, x is the argument of the function, so x is equal to a, y is the function value, so my y value, f of a, and so the point on the graph is going to be some point a, f of a, and since b is greater than a, then the other point f of b is going to give me a second point and that's going to be some place further along the graph. So what about that rate of change? The rate of change, the ratio of the change of the function divided by the change in the independent variable, that ratio of change, consists of two components. The numerator, f of b minus f of a, now let's figure that out, remember y is the function value, so this is a y value, height above the x axis, this is a y value, height above the x axis, and so this difference here is going to correspond to a difference in y values, it's going to correspond to the difference in height between this point here and this point there. So there's my difference in height, there's my difference of f of b minus f of a, likewise the denominator here, b minus a, that's going to correspond to the x values, the difference in the x values, and we can say that is a difference between the extension of the two points, how far over we have to go. So for the first point we have to go over this far, for the second point we have to go over this far, and that difference b minus a corresponds to that distance, that difference in extension, and so when I look at the ratio between those two, when I look at f of b over b minus a, well that's how far up we've gone, over how far over we've gone, that's rise over run, and that's just the slope of the line between those two points. So algebraically the average rate of change corresponds to the geometric slope between the two points on the graph of y equals f of x. So there's our interpretation for a rate of change, is also a slope of a line, and because this line runs between two points on a curve it's called a secant line. So the average rate of change is the slope of the secant line between two points. What about the instantaneous rate of change? Now the two key things to remember is the average rate of change approximates the instantaneous rate of change, and the closer my second point is to the first, the better the approximation is going to be. So let's take that apart. The algebraic rate of change corresponds to the geometric slope between the two points, so what about that instantaneous rate of change? Well, again the slope of the line, the rate of change, the average rate of change, is an approximation to the instantaneous rate of change, because these two points are pretty far apart, the approximation is going to be mediocre, but I can get a better approximation by picking a point that's closer to a f of a. So I'll pick a closer point, draw the secant line, figure out what the slope of the secant line is going to be, and this is going to be an approximation to the instantaneous rate of change, and it'll be a better approximation than what I had before. But maybe that's not good enough. So let's pick a point that's even closer. And again I pick a point, draw the secant line, find the slope, and I get something that's an even better approximation to the instantaneous rate of change. And if I wanted to get an even better approximation, I could pick a point that's even closer to a f of a. So let's see what happens. Well, imagine that we pick a point that's so close to a f of a that it would be very hard to see the difference. Well, let's talk a somewhat more sophisticated language. How about differentiate between, I don't like that word, how about to distinguish between the two points? So our graph might look something like this. So here I have my point, a f of a, and a very, very, very, very close by. I have another point, b f of b, and I have two points, so I can run the line between the two of them, and I get the secant line. And it looks something like that. And what's worth noting here is that that secant line that runs between two points on the graph looks an awful lot like the tangent line through just the point a f of a. And because the slope of the secant line approximates the instantaneous rate of change, whatever the slope of this line is, it's going to be close to the instantaneous rate of change. Because the tangent line looks very close to this line, the tangent line is going to have a very similar slope, and so that says the slope of the tangent line is also going to be a good approximation, possibly even the best approximation to the instantaneous rate of change. And so what that suggests is I can take the instantaneous rate of change as the slope of the line tangent to the graph at x equals a. So if I want to find the instantaneous rate of change, well I approximate it by finding the slope of the secant line between a point and something so close by that I can't distinguish between the two of them, or I might take the slope of the line tangent to the graph of y equals f of x at the point where x is equal to a. And this leads us to the first problem of calculus. The first central problem of calculus is how do I find the slope of the line tangent to the graph of y equals f of x at a point where x is equal to a. So you can think about this as the problem that's going to occupy the first half of the semester. How do I find the slope of a tangent line? And we'll take a look at that in the next few videos.