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kstenerud

"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.

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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, an understanding of lines, their equations, slope, y-intercepts, x-intercepts, and so on is more or less assumed. I think this is reasonable.”

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.

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.

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 :/

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?

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.

> 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.

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

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?

Hiya. I wrote a more detailed response above. Thanks for sharing your struggle.

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.

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

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!

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

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

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