Εισαγωγή στους ρυθμούς μεταβολής

- [Voiceover] What I want to explore in this video is the notion of a "rate." So, let's look at some examples of rates that you've probably encountered in your everyday life. So, if you're driving in your car down the road, and you're looking at the speedometer, you might see that it says that you are going 35 M-P-H, where the M-P-H stands for 35 miles per, per hour. Well, what's that saying? That's saying, well, every hour, how many miles are you going if you were to stay at that current rate. So, it could be a measure of speed. How much distance are you covering per unit time? And, most typically, when people talk about rates, that's what they're talking about. They're talking about how much of something that is happening per unit time. And, it doesn't have to be even distance per unit time, you might have a, you might have your hourly rate for someone who is doing some type of a job. They might say that they're making, they're making $10, so they're making $10. And, actually, let me write the dollars out so the units become a little bit more obvious, 10 dollars, dollars per hour, dollars per, dollars per hour. And so, once again, this is how much money. It's not talking about distance anymore. How much money is being earned per unit time? And, so, even though rates are often associated with how much something is happening per some unit time, and it could be miles per hour, or it could be meters per second, or, in this case, it could be a wage, it could be dollars per hour. Rates don't have to be just in those terms. In fact, you might say, "All right, "I have a dessert that I really enjoy, "but I'm very conscientious "about, about the number of calories that I consume." And, you might, you might see something like, there are 200 calories, calories per serving, per serving. And, so, this is telling us the number of calories per a serving. And they'll tell us what a serving is. A serving might be a cup or eight ounces or whatever else. And, so, I could say, "Okay, look, if I have two servings, "then I'm gonna have 400 calories. "Same way, if I work two hours, I'm gonna have 20 dollars. "If I, or if I go two hours, "I'm gonna go 70 miles." So, rates give you a sense. It's like, how fast is something happening? Or how much of one thing is happening for every time something else happens? Now, I can write rates so they look an awful lot like a ratio. And, these words are, actually, very related, 'cause you see that even how they're written. R-A-T, R-A-T. Their roots are coming from the exact same idea. In fact, this rate over here, 35 miles per hour, it could come from, "Hey, I just, I just went 35 miles in one hour, "what's the ratio?" So, the ratio of miles to hours. And, then, you could say, "Well, I went 35, "the ratio miles to hours "was 35 to one." Or it could have been, maybe it was 70 to two or something like that. But, that could have been reduced to 35 to one. So, as a ratio, you would typically see it written like this... Or maybe see it written like, see it written like this... And, sometimes, you might even see it written like this, 35 miles to one hour. But, now it's starting to resemble more of the special case of a ratio, which we call a "rate." Because, this is the same thing as 35. Instead of writing it out "miles per hour," you'll often see it written like this, miles per, miles per hour. So, these are very, very related ideas. If you find the ratio between calories and servings, well, then, you're going to be able to write, you're going to be able to express it as a rate and vice versa. Now, why do we care about rates? Well, especially if we're thinking about things like speed, without rates, it would be hard to quantify how fast things are happening. Otherwise, we'd be in a world where we're saying, "Hey, I'm faster than you," or "She's faster than me." But we wouldn't be able to quantify exactly how fast they are. But with rates, we can say, "Hey, that person ran "a hundred meters in 10 seconds, "they run 10 meters per second." We can quantify exactly how fast that thing is happening, the rate at which it is happening. Here, instead of saying, "Hey, a cup of that "is gonna give you, is gonna give you more energy, or, maybe, contribute more to your weight than a cup of that, and making these relative comparisons, here, you can actually, you can actually quantify things. And when we study rate, we're gonna study rate a lot in mathematics. It's gonna be essential in algebra when we look at the rate of change of a line, how far it moves in the vertical direction relative to the horizontal direction. We're gonna call that "slope". And you can even imagine the slope of a hill as how fast is it climbing for as much as you move forward. But we're also gonna study rates in detail when we go to calculus. In fact, the whole basis of differential calculus, that you might see later in high school and early college, is all about measuring instantaneous rate. How fast is something going right now? So, rates are really, really interesting, really, really important. And, I would guess that, if you just look around your life, even over the next few hours, you're going to encounter many, many, many rates.