 So Calculus is primarily concerned with how things change and importantly how quickly they change. So let's try and develop some notation, some framework for this discussion. So here's a quick review of some pre-calculus topics. If you don't remember these things, you may want to go back and look these topics up in your pre-calculus book or other sources. So we might have some function, we'll call it f of x, and we'll take any two values in the domain of f of x. And for convenience, we'll assume that a is always going to be less than b. So given this information, we can start out with a couple of ideas. The value of our function at x equals a, and we're going to write that as f of a. Again, this is a review from pre-calculus. We can also talk about the value of our function at x equals b, which no surprise, we'll write as f of b. The calculus topic that we'll start to introduce, we can talk about the change in value between x equals a and x equals b. And we'll have a lot more to say about the grammar and syntax here. But for right now, note that we're talking about the change in the value of our function between some point x equals a and some point x equals b. Well, that's just going to be the difference between the two function values, f of b minus f of a. And then finally, the average or the mean rate of change between x equals a and x equals b, because it does make a difference how quickly we change, that's going to be f of b minus f of a over b minus a. Here's a useful thing to remember, rate and ratio and rational number all come from the same root. And they all refer to this idea of a quotient of two things. So here we have the quotient of the change in value, f of b minus f of a, and the interval over which that change occurred from x equals a to x equals b. So we can take a look at a couple of examples. So let's say the population of a country and it might be modeled by some function 150 e raised to power 0.03 million persons, and t is the number of years since 2000. And we might find a couple of values, find the population in 2020 or in 2030, the change in population and the average mean rate of change between the two years. So first important thing to notice here is that t is not the year, but rather the number of years since 2000. So if I'm looking at a population in the year 2020 or 2030, I need to find the corresponding values of t. So the year 2020 is 20 years after 2000, so that means that t is equal to 20. And since I want the population then, I'm going to evaluate p of 20. So there's my formula for what p of t is. I'll substitute in my t value of 20, and after all the dust settles, I get an estimated population of 273 million persons. Likewise, if I look at the year 2030, that's 30 years after 2000, so t equals 30 corresponds to the year 2030, and the population will be p at 30. And again, I'll substitute that in get a population of 369 million persons. So again, these are computations that are typical of pre-calculus. With calculus, we'll actually look at the change in quantity. So here the change in population is just going to be the difference between where we were in 2030 and where we were in 2020. End minus beginning. And so that difference, I know those two values, 369 minus 273. I can do the computation, 96. And important note, don't forget to include those units. Population is measured in millions of persons, so the change in population is going to be measured in millions of persons. Finally, the rate of change is just going to be the ratio of the change as p of 30 minus p of 20 over 30 minus 20. So again, I can fill in my numbers there. I know p of 30 is 369, 273, 30 minus 20, and that's just a computation, 96 over 10, 9.6 is the numerical value, and the units are going to be millions of persons, which is what our numerators measured in, over years, which is what our t values are measured in. So this is going to be measured in millions of persons per year. And that's all the calculus to finding average rates of change. Now, if the world were a kind and gentle place and were perfectly accommodating, then any time we needed to find an average rate of change, the world would ask us, find the average rate of change. However, the world is not always so accommodating, and in many circumstances we need an average rate of change, but we're not asked to find the average rate of change, but we're asked something that's equivalent to it. So for example, consider this problem. We have an object thrown into the air, and its height is given by some function t seconds after it's thrown into the air, and how rapidly is the height changing during the first five seconds after it was thrown? And there... I don't know, how do we find this? There's nothing in here that says find the average rate of change, but what we have to do is we have to infer from the phrasing, and in particular, many, many, many, many, many synonyms refer to an average rate of change. This is a problem in English language. It is not a mathematical question. It is a question of identifying what English phrasing corresponds to the average rate of change. And so here we have something that suggests we're looking at, first of all, some sort of change. Well, first of all, there's this phrase change here, but that could refer to either a change or a rate of change. What tells us that we're looking at a rate is this, how rapidly? And that's a question of how fast does the quantity change? A further note here is that we are looking at that rate of change during the first five seconds after it was thrown. And again, what this suggests is that we're looking at the average rate of change between t equals 0 and t equals 5, because that is the first five seconds. Well, the world might not ask us to find the average rate of change, but we can say that that's what we're going to find. The average rate of change between t equals 0 and t equals 5 is, and then I know how to express that as an algebraic expression. That's h of 5 minus h of 0 over 5 minus 0. And we need h of 5 and h of 0, so I'll evaluate those two. And substituting in 0, so that gives me my expression. 300 over 5 is 60. And again, always give the units. So this 450 and 150, these are h of t values. These are measured in feet, this 5 and the 0. Those are t values, t is measured in seconds, and so our units here are 60 feet per second. So this is how we might want to answer the question. Now, some quick notes on format. If you're 100% certain that your work is 100% correct, you only have to write the final answer here, 60 feet per second. Do include the units, because otherwise your answer is meaningless. But if you're absolutely certain that your work is completely correct, the final answer is all that matters. And if you're doing this as part of your job, if you're doing this for a living, the only thing that really matters is your answer should be absolutely 100% correct. In the meantime, it's probably worth going through and writing out all of your steps this way. And there's two important parts to this. One is the actual computations here. So if you make a mistake up here someplace, it's clear where that mistake occurred. But the other thing that's worth writing down is what you're actually finding. So here, you're finding the average rate of change, 60 feet per second. And the reason that that's important is that if the problem does not actually look for a rate of change, if we had misunderstood the question, then this answer is an incorrect answer to this question. But if you indicate at least that what you're finding is this amount, then what you have is an answer that is correct just for a different question. The goal is to answer the question that's asked. An intermediate goal is to give a correct answer to some question. And by writing everything down, you can make it more likely that you'll at least give a correct answer to some question, which will hopefully be the question that's asked.