 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 three hundred dollars and at the end you might only have ten dollars. And so the amount that you had changed by ten minus three hundred or minus two hundred ninety dollars. Or maybe you're climbing a mountain. At the start of a climb your elevation is fifty four hundred feet and at the end it is fifty five hundred feet and so your elevation has changed by fifty five hundred where you end up minus fifty four hundred where you started one hundred 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 two hundred ninety dollars. Or we could say the amount of money decreased by two hundred ninety dollars. 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 two hundred ninety dollars over 20 days. Or minus fourteen point five dollars per day. If that happened, you probably had a good vacation. On the other hand, suppose you lost your wallet ten minutes into your vacation and ended it earlier. Then the amount of money still changed by the same amount minus two hundred ninety dollars. But that change occurred over a period of ten minutes and you lost twenty nine dollars 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 ten 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 ten 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 that to be five hundred meters. Meanwhile, at t equals ten, the height will be two hundred meters. And so this tells us the change in height is final amount, two hundred meters, minus initial amount, five hundred meters, negative three hundred 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 three hundred meters. And the other thing that changed was the amount of time. And since we looked at the interval during the first ten seconds of motion, the amount that time changed by was ten seconds. And so our rate of change, our ratio of the changes is minus three hundred meters over ten seconds, and that works out to be minus thirty 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 ten. 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 ten 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.