 Welcome back to our lecture series Math 1050, College Algebra for Students at Southern Utah University. As usual, I'm your professor today, Dr. Andrew Missildine. This is gonna be the first video in lecture six in our series, which we're gonna talk about rates of change. So before we define what the average rate of change is, let's first make sure we understand what a rate is. So in mathematics, when we talk about rate, rate basically means the same thing as a ratio, which essentially is a fraction of numbers. This is why the rational numbers are called the rational numbers because they're the numbers of ratios. They're fractions of whole numbers, right? And so a rate is a type of ratio that compares two quantities with probably different units, right? And so this might involve, you know, things like when we measure speed, speed is a rate of distance per time. So we measure this in like miles per hour, kilometers per hour. If you're going really fast, you might measure in a feet per second. Density is a type of rate. Density measures, you measure maybe like in pounds per gallon. So like mass per volume. We might measure something like with efficiencies, right? How many tasks can I perform per day or cost, right? What's the dollars per item if you were to purchase this, right? And so when one talks about a rate, you often use the word in English per, right? Per. So you have one unit per another unit. Unit per unit. And so that's usually a pretty good indicator that we have a rate of some kind. So for example, Karen bought seven used books, maybe from like her local thrift store like Deseret Industries or something like that. So Karen bought seven used books for a total of $2.45. Clearly she was not buying her math textbook here. She bought seven books for $2.45. Well, each book might have cost a different price, but we could calculate the ratio of dollars per book to estimate the average cost per book. So if we take the total amount of dollars she spent, which is the $2.45 or $2.45, and we divide that by the number of books she bought, which is seven books, that ratio if we just do a little bit of division would come out to be $0.35 per book. And therefore we could say that the books were priced at a rate of 35 cents per book. That's not saying the books cost 35 cents each, but on average, that's about how much they cost. And so this naturally leads into the idea of the average rate of change. This is a very natural usage of ratios when comparing quantities. So the rate of change between two quantities is a special kind of ratio that compares the change in those two quantities, right? So more specifically, we have two points. So these will be points we could graph potentially. X1, Y1 is one point, X2, Y2. And so the idea is there's some collection of quantities, X1, X2, X3. We have some data for X, and then associated connected to the X values is some Y value. So we have Y1, Y2, Y3, whatever. And we could plot this data on an XY plane. And so with these points, X1, Y1, X2, Y2, we have these two points on the graph. Then we say that the rate of change from X1 to X2 will denote this as delta Y divided by delta X. This little triangle symbol is actually the Greek letter delta. Delta in this case is like the Greek equivalent of letter D in English. And so delta here represents difference, the difference, it's a mnemonic device. So delta Y divided by delta X, this is equal to Y2 minus Y1 divided by X2 minus X1. So notice that delta Y here we're saying is Y2 minus Y1. It's a difference of the Y values and delta X is equal to X2 minus X1. It's the difference of the X coordinates right there. So this average rate of change would be the quotient, the ratio of delta Y divided by delta X. So we take the difference of the Y coordinates divided by the difference of the X coordinates. And this gives us this rate of change. This difference we see right here is often referred to as the difference quotient, which is perhaps the least clever name anyone could ever come up with in this situation. It'd be like calling a dog. Hey, look at that furry quadruped over there. It's not a very clever name, but yet it sticks around here in mathematics. So let's give you an example of how one would compute a difference quotient. Say Gregor is driving a Crossman Montana here. At 1 p.m., he checks his odometer, like his trip meter, and he sees that he's driven 180 miles from his starting location. And then three hours later at 4 p.m., he checks the odometer again. So at 4 p.m., he checks his odometer again and he's now driven 360 miles per hour. Could we calculate what Gregor's average speed is by using the rate of change? And so we think of there's a measurement of time. So we have like this X1. Sorry, that's the distance there. We have some X1, 1 p.m., X2. So we have some measurements of time, but then we also have measurements of distance, right? At the first time stamp, we were 189 miles away from home. And then at 4 p.m., at a different time stamp, Gregor was 360 miles away from home, so Y2. So if we think of it as these two points, we have one point of data, one comma 189. And so that would be, so in terms of time, this is the unit there, I mean, it's time, right, 1 p.m. And then the Y coordinate is a measurement of distance. Using that same relationship, the X coordinate gives us time and the Y coordinate gives us distance. We have a relationship between these two quantity, these two quantities here, and then we can calculate the average rate of change here, delta Y divided by delta X. This would be the difference of Y coordinates, 360 minus 189, that's 171. Then take the difference of time, right? Four minus one gives us three hours. What this is telling us is that in three hours, Gregor drove 171 miles. Well, if he were to drive at a constant speed throughout that entire journey, we could divide 171 by three, and this would give us his average speed of 57 miles per hour. So on average, that's how fast Gregor was driving. We're not saying he drove constantly 57 miles, but that's what average is all about. If everything was constant, which we know there's variability, right? He might have to slow down for construction. He might have to speed up to get away from the police, you know, things like that. We don't know what his speed was at any instant, but his average speed is gonna be this 57 miles per hour. This is actually why rate of change here is often we throw this adjective average in front of it, because in physics and in calculus, one gets into an idea of an instantaneous rate of change. Like, what is his speed at us instant in time, as opposed to an average over interval? We aren't gonna delve too much into the instantaneous rates of change in this course, but do be aware that we often see average rate of change so that when we say rate of change, it's not confused with the related, but much more difficult notion of instantaneous rate of change.