When giving a a list of data, their average can always be determined. Therefore even when there is no constant rate of change there always an average rate of change. A function is said to be increasing if y increases proportionally to x. The function is decreasing if y increases disproportionally to x. Because y changes with respect to x, as the x values increase the change in y is the change between the y coordinates those two x values. Periodic functions go through increases and decreases and there for concave up and down, respectively.
The instantaneous rate of change is the the velocity at a specific point. Its value must be calculated, because this value is based on average velocity, which gets more precise for a given point as the intervals decrease. Instantaneous velocity is also the smallest interval of the average velocity of the surrounding points, because two points are required to find an average.
As a tangent line becomes more horizontal, the interval between the two points it hits on the function line decreases, therefore indicating the derivative.
challenges
If values increase proportionally to each other, does that mean that they both increase or does is mean that they both increase by the same factor?
I know they are related because the points for corners of a triangle, but I don't really understand why the change in the function between two x values is delta y.
Why can't the actual velocity of the grapefruit be calculates, as opposed to estimated?
How exactly is the tangent the derivative. How can another line represent the velocity?
reflections
I think that the concept of average rate of change is very important because in real world situations, I assume that data actually literally increases/decreases with a constant rate of change. However a pattern an be gauged and assumptions can still be drawn about the function by using the average rate of change.
The visualizing rate of change section looks very similar to a diagram my old math teacher showed us when explaining derivatives, so I'm guessing they are closely related.
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