 So one of the key questions of calculus sent is around something called an instantaneous rate of change. And so let's take an example. Suppose I'm driving along a road 200 miles and the road has a posted speed limit of 65 miles an hour. And let's say that f of t is the distance that I've traveled along the road after some amount of time. And suppose I have the following information. I suppose that at t equals 0, I've traveled 0 miles. So I'm just starting my trip. And so that tells me that f of t, so f of t is the distance that I've traveled after t hours. f of 0 is going to be 0. And let's say that at t equals 4, I've traveled 200 miles. So again, f of 4 is 200. So this is just function notation from pre-calculus or from algebra. And I might note that, again, the calculus topic, they change in distance. It's just how far I am at 4 minus how far I'm at 0 200 minus 0, and I'll do that computation there. And the average rate of change of distance is going to be the ratio between that change in distance over the change in my independent variables, 200 minus 0 over 4 minus 0, which after all the dust settles, works out to be 50. f is measured in miles. These values 4 and 0, those are t values. Those are measured in hours. So my rate of change is 50 miles per hour. And so what that says is I've driven along this road at an average speed of 50 miles an hour. And that's great, except I got a speeding ticket. How do I manage to do that? I was only traveling 50 miles an hour, and the speed limit was 65 miles an hour. How could I possibly have gotten a speeding ticket? Well, to answer that question, I need a little bit more detail in our function. So let's say I have some additional function values. So I know that at 0, my f of t values would be 0 miles. At 4, I'm at 200 miles. And then let's fill in some of these intermediate points there. At 1 hour, I've gone 25. At 2 hours, I've gone 80. At 3, I've gone 150 and so on. Now in some of those intervals, for example, the interval between t equals 0 and t equals 2, then the average rate of change is going to be my function values, 80 minus 0, divided by my time values, 2 minus 0. It's 80 minus 0 over 2 minus 0. And so my average rate of change, my average speed, is going to be 40 miles an hour, well under the speed limit. But on the other hand, if I take a look at some other intervals, for example, this interval from t equals 2 to t equals 3, the average rate of change in that interval is going to be 70 miles per hour. And that does exceed the speed limit. So maybe in this interval, I was going faster than the speed level. I had to have been going faster than the speed limit. And so maybe that's when I got my speeding ticket. Well, at some point, the highway patrolman is going to have to provide some proof that I was exceeding the speed limit. So maybe that can happen in this way. Suppose they followed me for this one full hour, 60 minutes. They followed me from t equals 2 to t equals 3. And they determined that in that one hour period of time, I went 70 miles. And since I went 70 miles in one hour, that means I had an average speed of 70 miles per hour. And they have the grounds for giving me a speeding ticket. Well, two things. It's very unlikely that a highway patrolman will follow you for an hour to determine how fast you're going. And second, if you do have a highway patrolman following, you probably aren't going to regularly exceed the speed limit. More commonly, they're stationary. They don't move. And so the question is, well, how could they determine how fast you're going? Well, again, let's go back to this idea that we might have more detailed information on f of t. So let's have information like this. And I'll fill in some of the values. So I may know that here are some of the other values of time and distance. I'll pick some point. Oh, how about here? Between t equals 1.9 and 1.91, I've gone 1 mile in 0.01 And that gives me an average speed in this interval of 100 miles an hour, which far exceeds the speed limit. And the patrolman only needs to observe us for a whole 0.01 hours that works out to about 36 seconds. And again, I can find even more information. Suppose I know my speed and my distance at 1.801 hours. And I know the distance at 1.8 hours. So here I have two times very close together. And so I can find even more information. And it turns out that my average speed over that short interval of time is about 80 miles an hour. And here, the interval of time corresponds to 0.001 hours, about 3.6 seconds. And I can imagine measuring the average speed over even shorter intervals, 1 second, 1,000th of a second, a 10 billionth of a second, and so on. I can imagine using shorter and shorter intervals to find the average speed. Now, this takes a little bit of a leap of intuition. Imagine I could measure how far you travel in an instant of time. That is informally defined as an interval of time so short that we can't even measure it. And we can't actually do this, but we can imagine doing that. And if we found the rate of change over this instant of time, we would have what we would call the instantaneous rate of change. And this leads us to two of the important concepts in differential calculus. First off, what we've been doing, the average rate of change of a function over some interval, is going to approximate the instantaneous rate of change at the point. And the closer x equals A is to x equals B, the better that approximation is going to be. The better the average rate of change approximates the instantaneous rate of change. Again, in practice, we can only find the average rate of change over some interval. But what we are interested in is the instantaneous rate of change. So the key concept here is that our average rate of change approximates that instantaneous rate of change. And the closer my two points are to each other, the shorter that interval, the better the approximation. Now, grammar is important if we are talking about the language of change. So important note, the average rate of change is always defined over some interval. So we can describe that in a couple of different ways. We might say the average rate of change between x equals A and x equals B. Or another possibility, we could talk about the average rate of change from x equals A to x equals B. Or we might talk about the average rate of change over, and in this case, we'll give the interval A less than or equal to x, less than or equal to B. The instantaneous rate of change, on the other hand, is always measured at a point. So we talk about the instantaneous rate of change at x equals A, or we might say when x is equal to A. And part of the significance of this is that we don't always specify average rate of change or instantaneous rate of change. Remember, the world is not so accommodating as to ask us in mathematical language. Frequently, the world will ask us questions in English. And so we might be asked to find a rate of change, or even more generally, how rapidly is something changing? And we know that's a rate of change of some sort. And one of the ways we have of distinguishing between whether we are looking for an average rate of change or an instantaneous rate of change is where we're looking at that change. So if we have an average rate of change, it's a question that could ask us between, from, or over. So how rapidly are you moving over the next hour? That's an average rate of change. On the other hand, if we're looking for an instantaneous rate of change, the grammar that's used with that add a point when something happens. How fast were you moving when you got that speeding ticket? And that refers to an instantaneous rate of change. Grammar is essential, even in mathematics.