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Calculus For The People

geogebra.org

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Re: Calculus For The People

#11

Post by kstenerud »

tannrckb wrote: post #10
kstenerud wrote: post #8

Earlier quoted context omitted.

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.

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?

Re: Calculus For The People

#12

Post by kstenerud »

Sniffnoy wrote: post #9
kstenerud wrote: post #8

Earlier quoted context omitted.

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…

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 :/

Re: Calculus For The People

#13

Post by maxmunzel »

kstenerud wrote: post #7

A common failing of these "for the people" guides is that they fail the most basic UX test: User observation. When you build a UI, at some point you have to test it on actual people, observing them as they try to use it. Without fail, you'll discover a whole bunch of assumptions you'd made without even realizing it. It's only natural, since you've been working on this project for months and have intimate understandin…

Actually I think that GeoGebra is by far the most intuitive and for-the-people that a math toolbox can be. When I picked it up in Highschool, I didn’t need to google a thing about it, contrary to the CAS Systems I use nowadays... (Maybe not a fair comparison)

Re: Calculus For The People

#14

Post by kstenerud »

maxmunzel wrote: post #13
kstenerud wrote: post #7

A common failing of these "for the people" guides is that they fail the most basic UX test: User observation. When you build a UI, at some point you have to test it on actual people, observing them as they try to use it. Without fail, you'll discover a whole bunch of assumptions you'd made without even realizing it. It's only natural, since you've been working on this project for months and have intimate understandin…

Actually I think that GeoGebra is by far the most intuitive and for-the-people that a math toolbox can be. When I picked it up in Highschool, I didn’t need to google a thing about it, contrary to the CAS Systems I use nowadays... (Maybe not a fair comparison)

Except that there's no "I don't understand" button where you can tell the author what you don't understand. So it's basically a one-way textbook with no feedback mechanism for the author to discover his assumptions and clarify them.

Re: Calculus For The People

#15

Post by mac01021 »

kstenerud wrote: post #12
Sniffnoy wrote: post #9

Earlier quoted context omitted.

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…

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?

Re: Calculus For The People

#16

Post by pugio »

For a truly "from scratch" and deeply empowering introduction to the basic notions in calculus (and all mathematics), I've found nothing better than Burn Math Class[1] by Jason Wilkes. It assumes nothing but basic arithmetic, and proceeds to guide you through how to invent maths for yourself.

[1] https://www.amazon.com/Burn-Math-Class-Reinvent-Mathematics/...

Re: Calculus For The People

#17

Post by nabla9 »

kstenerud wrote: post #12
Sniffnoy wrote: post #9

Earlier quoted context omitted.

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…

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 :/

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.

Re: Calculus For The People

#18

Post by garmain »

pugio wrote: post #16

For a truly "from scratch" and deeply empowering introduction to the basic notions in calculus (and all mathematics), I've found nothing better than Burn Math Class[1] by Jason Wilkes. It assumes nothing but basic arithmetic, and proceeds to guide you through how to invent maths for yourself. [1] https://www.amazon.com/Burn-Math-Class-Reinvent-Mathematics/...

I agree
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