4 ms·

"You might notice that when h is very close to 0, the slope of the line very closely matches the graph of f(x)" Huh? "and therefore, the slope of the line ver

"You might notice that when h is very close to 0, the slope of the line very closely matches the graph of f(x)" Huh? "and therefore, the slope of the line very closely matches the growth rate of the function as well." Growth rate? What's that? "Notice that as h gets close to zero, the secant line almost perfectly matches the growth of f at point A. " Not sure what this means... "For instance, in this situation we can study the limit of the slope of g when h tends to 0. As we can see, the limit of the slope of g as h tends to 0 is 4." Wait... where is this 4 coming from? "From this, we can conclude that the growth rate of the function f at x=2 is 4." What the hell is growth??? "Sometimes limits are obvious like this one" And now I give up.

tannrckb 7y ago
To find the slope of a line tangent to a point (x, f(x)) on a line, you can "draw" a secant line through two points (x, f(x)) and (x+h, f(x+h)). Then, identify the slope of the line passing through these two points. This gives an approximation of the slope of the tangent line passing through (x, f(x)). To get a more and more accurate approximation, you can look at what value the slope tends to as h approaches 0. So, (x+h, f(x+h)) gets closer and closer to (x, f(x)), the slope of the line passing through those two points tends closer to the tangent line passing through (x, f(x)). In other words, we are identifying the limit of the slope as h approaches 0. Based on the points of confusion you mentioned, I recommend a refresher on algebra. I think that will clear up your confusion
- garmaine 7y ago
  I think he is pointing out that this “for the people” tutorial uses math jargon without introducing it.
slumenta 7y ago
Quoting the prerequisites: “On the matter of prerequisites, this book assumes you are competent, if not a Jedi, at basic algebra and arithmetic. Specifically, an understanding of lines, their equations, slope, y-intercepts, x-intercepts, and so on is more or less assumed. I think this is reasonable.”
- kstenerud 7y ago
  And I do understand those. What I don't understand is growth rate and the slope of a function that is a curve, not a line, therefore not a slope.
  - Sniffnoy 7y ago
    It's using a naive, informal notion of those. If you were to define it formally, well, you'd have the derivative. Which is what he does quite soon after. This is how definitions frequently work in mathematics -- they're meant to take some naive informal notion and formalize it, by coming up with a formal definition that matches how it should work. So, it's assuming you already have some informal notion of growth rate in your head, like being able to talk about the velocity of an object even when that velocity is not constant. (Imagine the x coordinate is time, and the y coordinate is position (we'll work in one spatial dimension here); then the "growth rate" is velocity.) Then it discusses how to define this formally.
    - kstenerud 7y ago
      So you're just drawing a line from start to end and calling that the velocity? That just averages the whole thing out, doesn't it? Unfortunately, I don't have an informal notion of growth rate in my head :/
      - mac01021 7y ago
        Yes, that's right. The growth over any finite window, if you partition it into smaller windows, is the sum of the growth within each partition. Draw enough pictures and you'll develop the intuition that, if you keep partitioning smaller and smaller, you'll reach a point where the average growth rate across a partition is never going to change very much by subpartitioning further. If instantaneous growth rate is going to be defined at all, it has to be very close to the average rate over that tiny interval, no?
      - nabla9 7y ago
        If you would learn this rigourously like mathematicians do, you learn it trough infinite series and limits. The derivate is defined as limit for the slope when the endpoints of the slope get closer and closer together () Lots of work, proofs are needed to show that this actually works. Common person or engineer only needs to accept/trust that point in a curve has well defined 'slope' and it's not an approximation.
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!

tannrckb 7y ago

To find the slope of a line tangent to a point (x, f(x)) on a line, you can "draw" a secant line through two points (x, f(x)) and (x+h, f(x+h)). Then, identify the slope of the line passing through these two points. This gives an approximation of the slope of the tangent line passing through (x, f(x)). To get a more and more accurate approximation, you can look at what value the slope tends to as h approaches 0. So, (x+h, f(x+h)) gets closer and closer to (x, f(x)), the slope of the line passing through those two points tends closer to the tangent line passing through (x, f(x)). In other words, we are identifying the limit of the slope as h approaches 0. Based on the points of confusion you mentioned, I recommend a refresher on algebra. I think that will clear up your confusion

garmaine 7y ago
I think he is pointing out that this “for the people” tutorial uses math jargon without introducing it.

slumenta 7y ago

Quoting the prerequisites: “On the matter of prerequisites, this book assumes you are competent, if not a Jedi, at basic algebra and arithmetic. Specifically, an understanding of lines, their equations, slope, y-intercepts, x-intercepts, and so on is more or less assumed. I think this is reasonable.”

kstenerud 7y ago
And I do understand those. What I don't understand is growth rate and the slope of a function that is a curve, not a line, therefore not a slope.
- Sniffnoy 7y ago
  It's using a naive, informal notion of those. If you were to define it formally, well, you'd have the derivative. Which is what he does quite soon after. This is how definitions frequently work in mathematics -- they're meant to take some naive informal notion and formalize it, by coming up with a formal definition that matches how it should work. So, it's assuming you already have some informal notion of growth rate in your head, like being able to talk about the velocity of an object even when that velocity is not constant. (Imagine the x coordinate is time, and the y coordinate is position (we'll work in one spatial dimension here); then the "growth rate" is velocity.) Then it discusses how to define this formally.
  - kstenerud 7y ago
    So you're just drawing a line from start to end and calling that the velocity? That just averages the whole thing out, doesn't it? Unfortunately, I don't have an informal notion of growth rate in my head :/
    - mac01021 7y ago
      Yes, that's right. The growth over any finite window, if you partition it into smaller windows, is the sum of the growth within each partition. Draw enough pictures and you'll develop the intuition that, if you keep partitioning smaller and smaller, you'll reach a point where the average growth rate across a partition is never going to change very much by subpartitioning further. If instantaneous growth rate is going to be defined at all, it has to be very close to the average rate over that tiny interval, no?
    - nabla9 7y ago
      If you would learn this rigourously like mathematicians do, you learn it trough infinite series and limits. The derivate is defined as limit for the slope when the endpoints of the slope get closer and closer together () Lots of work, proofs are needed to show that this actually works. Common person or engineer only needs to accept/trust that point in a curve has well defined 'slope' and it's not an approximation.

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!