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Quoting the prerequisites: “On the matter of prerequisites, this book assumes you are competent, if not a Jedi, at basic algebra and arithmetic. Specifically,

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.
    - Sniffnoy 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? No. We're talking about instantaneous velocity. You know, the thing the speedometer displays. How fast is the car moving at any given moment? Like, a car doesn't need to be moving at constant speed for a speedometer to give meaningful information, right? Sometimes it is moving faster and sometimes it is moving slower. Sometimes it is moving at a rate such that if it stayed at that rate it would go 60 miles in an hour, and sometimes it is moving at a rate such that if it stayed at that rate it would go 30 miles in an hour. This is the informal notion of instantaneous velocity you should already have. Now the question becomes, how do we formalize this? Which is what the page is trying to answer.
- tannrckb 7y ago
  If it's a curve, that means that the growth rate of the function is itself changing. And using the approach of finding slopes of lines incrementally closer to the tangent line through a point, you can naturally identify the value that the slopes approach. This is the most basic way to demonstrate taking a limit
  - kstenerud 7y ago
    But if the growth rate is changing, then how could a line be equivalent to the growth rate? Actually, what does growth rate even mean?
    - gregpetrics 7y ago
      Hiya. I wrote a more detailed response above. Thanks for sharing your struggle.
- chongli 7y ago
  The best way I can come up with to describe it is this: Think of your morning drive to work. Your distance from the office is always changing as you accelerate and decelerate along the way. It may even stop changing at various times (when you're parked, at stop signs or red lights). If you were to record your distance over time, it would be a curve that goes down (and up when you go in reverse), and sometimes remains level. The slope of that curve at a given point, that is the derivative, corresponds to the speed displayed by your speedometer at that particular moment in time. This is differential calculus in a nutshell: given a curve representing your distance over the entire trip, find the speed at any instant in time. We can then relate this to integral calculus in the following way: given a way to record your speed at every moment in time (speedometer), determine the total distance you travel. If it sounds like two sides of the same coin, well it is! This is the brilliant discovery of the fundamental theorem of calculus.

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.
  - Sniffnoy 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? No. We're talking about instantaneous velocity. You know, the thing the speedometer displays. How fast is the car moving at any given moment? Like, a car doesn't need to be moving at constant speed for a speedometer to give meaningful information, right? Sometimes it is moving faster and sometimes it is moving slower. Sometimes it is moving at a rate such that if it stayed at that rate it would go 60 miles in an hour, and sometimes it is moving at a rate such that if it stayed at that rate it would go 30 miles in an hour. This is the informal notion of instantaneous velocity you should already have. Now the question becomes, how do we formalize this? Which is what the page is trying to answer.
tannrckb 7y ago
If it's a curve, that means that the growth rate of the function is itself changing. And using the approach of finding slopes of lines incrementally closer to the tangent line through a point, you can naturally identify the value that the slopes approach. This is the most basic way to demonstrate taking a limit
- kstenerud 7y ago
  But if the growth rate is changing, then how could a line be equivalent to the growth rate? Actually, what does growth rate even mean?
  - gregpetrics 7y ago
    Hiya. I wrote a more detailed response above. Thanks for sharing your struggle.
chongli 7y ago
The best way I can come up with to describe it is this: Think of your morning drive to work. Your distance from the office is always changing as you accelerate and decelerate along the way. It may even stop changing at various times (when you're parked, at stop signs or red lights). If you were to record your distance over time, it would be a curve that goes down (and up when you go in reverse), and sometimes remains level. The slope of that curve at a given point, that is the derivative, corresponds to the speed displayed by your speedometer at that particular moment in time. This is differential calculus in a nutshell: given a curve representing your distance over the entire trip, find the speed at any instant in time. We can then relate this to integral calculus in the following way: given a way to record your speed at every moment in time (speedometer), determine the total distance you travel. If it sounds like two sides of the same coin, well it is! This is the brilliant discovery of the fundamental theorem of calculus.