 One way to describe calculus is that it is the mathematics of how things change. So let's take a deeper look at this notion of change. When a quantity changes, we define the change in the quantity to be the difference between the final amount and the initial amount. For example, at the start of a vacation you might have $300 and at the end you might only have $10 and so the amount that you had changed by 10 minus $300 or minus $290, or maybe you're climbing a mountain. At the start of a climb your elevation is 5,400 feet and at the end it is 5,500 feet and so your elevation has changed by 5,500 where you end up minus 5,400 where you started 100 feet. Before we continue, it'll be worth talking about the grammar of change. How you speak influences how you think and in mathematics not speaking precisely leads to not computing precisely. And one of the problems with change is we have to be very careful because there are two slightly different conventions. So we have to choose one. We could either explicitly indicate the sign of the change plus or minus and simply call it a change or we can drop the sign and choose an appropriate verb. Since the change has been completed the verb should always be in the past tense. So in our vacation example, we might say that the amount of money changed by minus $290 or we could say the amount of money decreased by $290 and it is vitally important not to mix the two things. If you're going to use an appropriate verb increase or decrease never, never, never, never, never, never, never, never, never combine that with a sign. Only use the signs if you use the word changed by. Now in addition to the amount of change, how rapidly a change occurs makes a difference. For example suppose your vacation lasted 20 days, so the amount of money changed by minus $290 over 20 days or minus $14.5 per day. If that happened, you probably had a good vacation. On the other hand, suppose you lost your wallet 10 minutes into your vacation and ended it earlier, then the amount of money still changed by the same amount minus $290, but that change occurred over a period of 10 minutes and you lost $29 per minute. If that happened, you probably had a bad vacation. Because these amounts represent a ratio of two changes, the amount of money and the amount of time, we call it a rate of change. Well, let's take a look at an example with an actual function. Suppose the height of an object, t seconds after it is thrown, is given by some function. Let's find the rate of change of the object during the first 10 seconds of motion. And so if we want to find a change, it's going to be the difference between the final amount and the initial amount. And here's an important idea, how you speak influences how you think. When we're calculating the rate of change, we are talking about amounts of something, but amounts of what? In this case, we're given a formula for the height of the object. So the only thing it would be reasonable to ask about is how the height has changed during those 10 seconds. And so we need to know the final amount of height and the initial amount of height. Well, at t equals zero, the height is going to be given by our formula. And we can calculate the three 500 meters. Meanwhile, at t equals 10, the height will be 200 meters. And so this tells us the change in height is final amount, 200 meters, minus initial amount, 500 meters, negative 300 meters. And finally, we want to find the rate of change that's the ratio of two changes. And in this case, one of those changes is this change in height minus 300 meters. And the other thing that changed was the amount of time. And since we looked at the interval during the first 10 seconds of motion, the amount that time changed by was 10 seconds. And so our rate of change, our ratio of the changes is minus 300 meters over 10 seconds. And that works out to be minus 30 meters over seconds. Now, just to recap what we did in this problem, we found the value of our height at the start, t equals zero, and at the end, t equals 10. And this allowed us to find the change of our height. And that was going to be the difference between where we were at the end minus where we were at the beginning. We also found the change in t, and that was also end minus beginning. That was 10 minus zero. And what this suggests is the following definition, the average rate of change. So remember, definitions are the whole of mathematics, all else is commentary. This is an important thing to remember. So, let F be a function. The average rate of change of F over the interval between A and B is defined as the ratio of the change. Between F of B minus F of A over B minus A. And importantly, be sure to keep those appropriate units. Now, if the universe was consistent, or I was a kind and gentle math teacher, then whenever we wanted an average rate of change, we'd ask for an average rate of change. But the universe isn't consistent, and I am not kind and gentle. And so sometimes, when we're looking for an average rate of change, we might not ask for an average rate of change. We sometimes omit the word average. So, while we might want the average rate of change of F over the interval from A to B, this can also be called the rate of change of our function over the interval, or the rate of change of our function from T equals A to T equals B. And in the special case, if A is equal to zero, we might actually speak of the rate of change during the first B units of T. In fact, we just did. But wait, it gets worse. We sometimes omit the word rate of change. And so we might ask questions like, how rapidly did our function change during the first B units? Or maybe, how quickly was F changing between T equals A and T equals B? And it's important to understand that sometimes we are looking for an average rate of change, but we might not use the word average or rate of change. For example, but A of T be the altitude of an airplane in meters, T hours after departure. And some of the values of A of T are given in the following table. And we might ask the question, how rapidly was the plane descending during the last two hours? So even though the question doesn't use the word average or rate of change, the phrasing how rapidly during suggests that we're actually looking for an average rate of change. There is unfortunately no algorithm, no infallible way of determining whether a question is asking for an average rate of change, because what we're dealing with is a question that is asked in a natural language. And every natural language question requires interpretation. And that is beyond mathematics. It requires an understanding of human language and of the real world. The good news is, once you've determined what the mathematical question is, the mathematical process is actually very easy. So once we've identified that we're looking for an average rate of change, we just need to figure out what's changing and what the average rate is. So during the last two hours suggests we're looking at this interval between three and five hours after the plane departed the airport. And so we can find the change in height, so remember the change is going to be the final amount minus the initial amount where we ended up minus where we started. And that'll be minus 12, minus 83, 86, or minus 83, 98 meters. And again, because we're using the word change and not increase, not decrease, we'll keep that negative sign. Next, the rate of change is going to be the ratio of the change amounts. So our change in height was negative 83, 98 meters. Our time changed by five minus three, two hours. And so our rate of change, our ratio of change, is the quotient of the two, which works out to negative 4,199 meters over hours. And we can summarize our results. We should actually go one step further. The question used a change verb, descending. And that's because the altitude of the plane was actually decreasing during those last two hours. Our answer gave a rate of change, and because it didn't use a change verb, we included that sign negative 4,199 meters per hour. But in recognition of the fact that the question specifically uses the word descending, we might want to rephrase our answer to use the word descending. And because this is a change verb, we need to drop that sign. And so we can say that the plane was descending by 4,199 meters every hour.