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Slope of a line secant to a curve
So let's review the idea of slope, which you might remember from your algebra classes. The slope is just the rate of change of a line. Or the rate of change of y, with respect to x, as we go along a line. And you could also view it as a measure of the inclination of a line. So the more incline the line is, the more positive of a slope it would have. So this right over here, this has a positive slope. It's increasing as x increases. And if this had an even higher inclination like this, if it increased even more as x increased, then it would even have a higher slope. So this right over here is some line. So that's some line. And just as a reminder, we can figure out the slope between two points. Two points define a line. And between those two points, we can find the rate of change of y, with respect to x. So let's put two points on here. So let's say that this point right over here, this x value, is x sub-- well, this is pronounced x naught, or x sub 0 is just x naught-- and when x is x0 for this line, y is y0. So this is a point x0, comma y0. And let's say we have another point all the way over here. And let's say that this x value, this x value right over here is x sub 1. And the y value over here is y sub 1. So this is the point x sub 1, y sub 1. So just as a review, the slope of this line, and a line by definition, has a constant slope between any two points that you pick. The slope of this line, which is often denoted by the letter m, is your rate of change of y with respect to x. Or another way of thinking about it, for a given change in x, how much are you changing y? Or a change in y divided by a change in x. Just as a reminder, this triangle, that's the Greek letter delta. It's shorthand for change in whatever. So change in y over change in x. So let's think about what this is going to be for this example right over here. Well, let's think about change in x, first. So we are moving from x0 to x1. So our change in x. So this is our change in x, right over here. We're starting at x0 and going to x1. That is our change in x. I'll put that in that pink color. That is our change in x. And what is it equal to? Well if we're ending here and we started here, let's just do ending point minus starting point. So it is x1 minus x0. And that way, doing it this way, I would have made sure that I have a positive value. I'm just assuming that x1 is larger than x0. And what is my change in y? Well, once again, ending point, ending y value minus starting y value. y1 minus y0. Now, you might be saying hey, could I have done y0 minus y1 over x0 zero minus x1? Absolutely. You could have done that. Then you would have just gotten the negative of each of these values in the numerator and denominator, but they would have canceled out. The important thing is that you're consistent. If you're subtracting you're starting value from your ending value in the numerator, you have to subtract your starting value from your ending value in the denominator as well. So this right here you probably remember from algebra class. The definition of slope is the rate of change of y with respect to x. Or it's the rate of change of our vertical axis, I should say, with respect to our horizontal axis. Or change in y, or change in our vertical axis over change in a horizontal axis. Now I'm going to introduce a little bit of a conundrum. So let me draw another axis right over here. Scroll over a little bit just so we have some space to work with. So that was for a line. And a line, by definition, has a constant slope. If you calculate this between any two points on the line, it's going to be constant for that line. But what happens when we start dealing with curves? ' When we start dealing with non-line or non-linear curves. So let's imagine a curve that looks something like this. So what is the rate of change of y with respect to x of this curve? Well, let's look at it at different points. And we could at least try to approximate what it might be in any moment. So let's say that this is one point on a curve. Let's call that x1, and then this is y1. And let's say that this is another point on a curve right over here, x2. And let's call this y2. So this is a point x1, y1, this is a point x2, y2. So we don't have the tools yet. And this is what's exciting about calculus, we will soon have the tools to figure out, what is the rate of change of y with respect to x at exactly this point? But we don't have that tool yet. But using just the tools from algebra, we could at least start to think about, what is the average rate of change over the interval from x1 to x2? Well, what's the average rate of change? Well, that's just how much did my y change-- so that's my change in y-- for this change in x. And so we would calculate it the same way. y2 minus y1 over x2 minus x1. So our change in y over this interval is equal to y2 minus y1, and our change in x is going to be equal to x2 minus x1. So just like that we were able to figure out the rate of change between these two points. Or another way of thinking about it is, this is the average rate of change for the curve between x equals x1, and x is equal to x2. This is the average rate of change of y with respect to x over this interval. But what if we have we also figured out here? Well, we figured out the slope of the line that connects these two points. And what we call a line that intersects a curve in exactly two places? Well, we figured out, we call that a secant line. So this right over here is a secant line. So the big idea here is we're extending the idea of slope. We said, OK, we already knew how to find the slope of a line. A curve, we don't have the tools yet, but calculus is about to give it to us. But let's just use our algebraic tools. We can at least figure out the average rate of change of a curve, or a function, over an interval. That is the same exact thing as the slope of a secant line. Now just as a little bit of foreshadowing, where is this all going? How will we eventually get the tools, so that we can figure out the instantaneous rate of change, not just the average? Well just imagine what happens if this point right over here got closer and closer to this point. Then the secant line is going to better and better and better approximate the instantaneous rate of change right over here. Or you could even think of it as the slope of the tangent line.