Mean value theorem

Let's see if we can give ourselves an intuitive understanding of the mean value theorem. And as we'll see, once you parse some of the mathematical lingo and notation, it's actually a quite intuitive theorem. And so let's just think about some function, f. So let's say I have some function f. And we know a few things about this function. We know that it is continuous over the closed interval between x equals a and x is equal to b. And so when we put these brackets here, that just means closed interval. So when I put a bracket here, that means we're including the point a. And if I put the bracket on the right hand side instead of a parentheses, that means that we are including the point b. And continuous just means we don't have any gaps or jumps in the function over this closed interval. Now, let's also assume that it's differentiable over the open interval between a and b. So now we're saying, well, it's OK if it's not differentiable right at a, or if it's not differentiable right at b. And differentiable just means that there's a defined derivative, that you can actually take the derivative at those points. So it's differentiable over the open interval between a and b. So those are the constraints we're going to put on ourselves for the mean value theorem. And so let's just try to visualize this thing. So this is my function, that's the y-axis. And then this right over here is the x-axis. And I'm going to-- let's see, x-axis, and let me draw my interval. So that's a, and then this is b right over here. And so let's say our function looks something like this. Draw an arbitrary function right over here, let's say my function looks something like that. So at this point right over here, the x value is a, and the y value is f(a). At this point right over here, the x value is b, and the y value, of course, is f(b). So all the mean value theorem tells us is if we take the average rate of change over the interval, that at some point the instantaneous rate of change, at least at some point in this open interval, the instantaneous change is going to be the same as the average change. Now what does that mean, visually? So let's calculate the average change. The average change between point a and point b, well, that's going to be the slope of the secant line. So that's-- so this is the secant line. So think about its slope. All the mean value theorem tells us is that at some point in this interval, the instant slope of the tangent line is going to be the same as the slope of the secant line. And we can see, just visually, it looks like right over here, the slope of the tangent line is it looks like the same as the slope of the secant line. It also looks like the case right over here. The slope of the tangent line is equal to the slope of the secant line. And it makes intuitive sense. At some point, your instantaneous slope is going to be the same as the average slope. Now how would we write that mathematically? Well, let's calculate the average slope over this interval. Well, the average slope over this interval, or the average change, the slope of the secant line, is going to be our change in y-- our change in y right over here-- over our change in x. Well, what is our change in y? Our change in y is f(b) minus f(a), and that's going to be over our change in x. Over b minus b minus a. I'll do that in that red color. So let's just remind ourselves what's going on here. So this right over here, this is the graph of y is equal to f(x). We're saying that the slope of the secant line, or our average rate of change over the interval from a to b, is our change in y-- that the Greek letter delta is just shorthand for change in y-- over our change in x. Which, of course, is equal to this. And the mean value theorem tells us that there exists-- so if we know these two things about the function, then there exists some x value in between a and b. So in the open interval between a and b, there exists some c. There exists some c, and we could say it's a member of the open interval between a and b. Or we could say some c such that a is less than c, which is less than b. So some c in this interval. So some c in between it where the instantaneous rate of change at that x value is the same as the average rate of change. So there exists some c in this open interval where the average rate of change is equal to the instantaneous rate of change at that point. That's all it's saying. And as we saw this diagram right over here, this could be our c. Or this could be our c as well. So nothing really-- it looks, you would say f is continuous over a, b, differentiable over-- f is continuous over the closed interval, differentiable over the open interval, and you see all this notation. You're like, what is that telling us? All it's saying is at some point in the interval, the instantaneous rate of change is going to be the same as the average rate of change over the whole interval. In the next video, we'll try to give you a kind of a real life example about when that make sense.