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AP Calc: CHA‑2 (EU), CHA‑2.A (LO), CHA‑2.A.1 (EK)

- [Instructor] So right over here we have the graph of y
is equal to x squared, or at least part of the graph
of y is equal to x squared. And the first thing I'd like to tackle is think about the average rate of change of y with respect to x over the interval from x
equaling 1 to x equaling 3. So let me write that down. We want to know the average rate of change of y with respect to x over the interval from
x going from 1 to 3. And that's a closed interval, where x could be 1, and
x could be equal to 3. Well we could do this even
without looking at the graph. If I were to just make a table here, where, if this is x, and this
is y is equal to x squared, when x is equal to 1, y
is equal to 1 squared, which is just 1. You see that right over there. And when x is equal to 3, y is equal to 3 squared,
which is equal to 9. And so you can see when x is equal to 3, y is equal to 9. And to figure out the
average rate of change of y with respect to x, you say, "Okay, well
what's my change in x?" Well, we could see very clearly that our change in x over this interval is equal to positive 2. Well, what's our change in
y over the same interval? Our change in y is equal to ... When x increased by 2 from 1 to 3, y increases by 8, so it's
going to be a positive 8. So what is our average rate of change? Well, it's going to be our change in y, or our change in x, which is equal to 8 over
2, which is equal to 4. So that would be our
average rate of change. Over that interval, on average, every time x increases by
1, y is increasing by 4. And how did we calculate that? We looked at our change in x, let me draw that here ... We looked at our change in x, and we looked at our change in y, which would be this right over here, and we calculated change
in y over change of x for average rate of change. Now this might be looking
fairly familiar to you, because you're used to thinking about change in y over change in x as the slope of a line
connecting two points. And that's indeed what we did calculate. If you were to draw a secant
line between these two points, we essentially just calculated the slope of that secant line. And so the average rate of
change between two points, that is the same thing as
the slope of the secant line. And by looking at the secant line, in comparison to the
curve over that interval, it hopefully gives you a visual intuition for what even average
rate of change means. Because in the beginning
part of the interval, you see that the secant line is actually increasing at a faster rate, but then as we get closer to 3, it looks like our yellow curve is increasing at a faster
rate than the secant line, and then they eventually catch up. And so that's why the
slope of the secant line is the average rate of change. Is it the exact rate of
change at every point? Absolutely not. The curve's rate of change
is constantly changing. It's at a slower rate of change in the beginning part of this interval, and then it's actually
increasing at a higher rate as we get closer and closer to three. So over the interval, the change in y over the change
in x is exactly the same. Now one question you might be wondering is why are you learning this
is in a calculus class? Couldn't you have learned
this in an algebra class? The answer is yes. But what's going to be interesting, and is really one of the
foundational ideas of calculus is well what happens as these points get closer and closer together? We found the average rate
of change between 1 and 3, or the slope of the secant
line from (1, 1) to (3, 9). But what instead if you found
the slope of the secant line between (2, 4) and (3, 9)? So what if you found this slope? But what if you wanted to get even closer? Let's say you wanted to find
the slope of the secant line between the point (2.5, 6.25) and (3, 9)? And what if you just kept getting closer and closer and closer? Well then, the slopes
of these secant lines are going to get closer and closer to the slope of the
tangent line at x equals 3. And if we can figure out the
slope of the tangent line, well then we're in business. Because then we're not talking
about average rate of change, we're going to be talking about instantaneous rate of change, which is one of the central ideas, that is the derivative, and we're going to get there soon. But it's really important to appreciate that average rate of
change between two points is the same thing as the
slope of the secant line. And as those points get
closer and closer together, and as the secant line is connecting two closer and closer points together, that distance between the points, between the x values of
the points approach 0, very interesting things
are going to happen.

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