3 ms·
Good point. Thanks. I do think that limit is obvious and equal to 4. The point tends to 4. That said, I also agree the "growth rate" thing is coming in a littl
by gregpetrics 7y ago
Good point. Thanks. I do think that limit is obvious and equal to 4. The point tends to 4.
That said, I also agree the "growth rate" thing is coming in a little too quickly there. It's meant to foreshadow derivatives in the next chapter, but it seems like maybe it's introducing confusion to the reader.
I went ahead and made some revisions to try to ease that connection of the "slope of a secant line" as an estimate of "growth rate" of a function.
That said, no matter what I do, this is one of those "object equivalencies" in calculus that there's no way to really make for someone. At the end of the day "slope of secant line" and "growth rate" are two different objects that in the context of a mathematical model are equivalent, but in a mathematical vacuum, are not. I write about this a little bit at the end of the book here: https://www.geogebra.org/m/x39ys4d7#material/fxpkwpt7 https://www.geogebra.org/m/x39ys4d7#material/fxpkwpt7
Sadly, the resolution for you isn't really very concrete. To get another person to "learn" an object equivalence is a challenging thing. There's really only two options: 1. tell them. 2. put evidence in front of them and hope they make it themselves. I went for option 1 after sprinkling in a bit of option 2. I've tried to slow it down a bit more, but of course, every learner will be different on when they're ready to make this important connection.
So at some point or another, this speed-bump needs to get hit.
If you have more thoughts let me know!