![]() It's at a slower rate of change in the beginning part of this interval, and then it's actually Is it the exact rate ofĬhange at every point? Absolutely not. Slope of the secant line is the average rate of change. Rate than the secant line, and then they eventually catch up. 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 And by looking at the secant line, in comparison to theĬurve over that interval, it hopefully gives you a visual intuition for what even average And so the average rate ofĬhange between two points, that is the same thing as Line between these two points, we essentially just calculated the slope of that secant line. ![]() Now this might be lookingįairly familiar to you, because you're used to thinking about change in y over change in x as the slope of a lineĬonnecting two points. In y over change of x for average rate of change. 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 And how did we calculate that? We looked at our change in x, let me draw that here. Over that interval, on average, every time x increases byġ, y is increasing by 4. So that would be ourĪverage rate of change. 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 overĢ, which is equal to 4. When x increased by 2 from 1 to 3, y increases by 8, so it's Y over the same interval? Our change in y is equal to. 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. And to figure out theĪverage rate of change of y with respect to x, you say, "Okay, well And so you can see when x is equal to 3, y is equal to 9. And when x is equal to 3, y is equal to 3 squared, Is y is equal to x squared, when x is equal to 1, y If I were to just make a table here, where, if this is x, and this And that's a closed interval, where x could be 1, and We want to know the average rate of change of y with respect to x over the interval from 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 xĮqualing 1 to x equaling 3. Is equal to x squared, or at least part of the graph So right over here we have the graph of y There are plenty of things in mathematics that have no real world application, though they are studied nonetheless. Lastly, "not having a purpose" (which is not the case with secant lines and average rates of change) is a poor argument for neglecting to study anything – especially in mathematics. Furthermore, if you are looking at discrete data (as is the case in every real world observation), there is no way to get an instantaneous rate of change from that data because it is not continuous. In particular, in physics, there are a lot of phenomena that occur that have to deal with average rates of change instead of instantaneous rates of change. ![]() As for "not bothering with the secant", there is no way to explain the process of finding instantaneous change without explaining how to find average change for the reason you just identified: the principles involved in finding average rate of change are part of finding instantaneous rate of change.Īlso, average rates of change have advantages in their own right. As you say, the process of finding the slope of the tangent line descends from the process of finding the slope of a secant line (the only difference is that a certain limit is taken of the difference quotient which is the expression for the slope of a general secant). Calculating average change using secant lines is actually an important intermediate step to finding instantaneous change via tangent lines (and thus also derivatives). ![]()
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