
Calculus Explained with GIFs and Pics (2014) - archibaldJ
https://0a.io/chapter1/calculus-explained.html
======
Tainnor
Unfortunately, parts of the section about integration are wrong and contribute
to some of the widespread confusion about the FTC, which often seems reduced
to a mere tautology (which it's not; it's a deep result).

First, the "indefinite integral" is really just the antiderivative (and as
such, "the" indefinite integral is only unique up to a constant). This doesn't
follow from the 1. FTC, but purely by definition.

OTOH, the definite integral is not "defined" as being the difference of
antiderivatives; it's defined in terms of Riemann sums (at least in elementary
calculus), as explained further below.

What the two parts of the FTC do is proving that those two notions, which have
no a priori reason to be related, are in fact related in a particular way.

The first part of the theorem says that, if f is "nice" (in particular:
continuous), the antiderivative exists and can be expressed through the
definite integral with a variable upper bound.

The second part says that, if f has an antiderivative, the definite integral
can be computed using that antiderivative.

It's important to keep this distinction because there are e.g. functions that
are integrable but don't have an antiderivative (e.g. a function with a
"jump"). In such a case, the FTC tells you nothing and you have to go back to
Riemann sums to compute the definite integral.

~~~
ajross
You never got to the parts of the section about integration that are wrong.

Broadly: your criticism is a mathematician's nitpicking. I don't know that
that really has much value to someone learning "Calculus" in a practical sense
about understanding and working with functions. And frankly... I mean, the
fundamental theorem of calculus is a historically important result and a
relatively profound truth, but it's not _that_ big a deal. If you really want
to play the "you really need to know this important truth more deeply" game
with amateurs, start with the incompleteness theorem or something.

~~~
Tainnor
I think I did, but I'll be more precise. The first problem is that the FTC is
reduced to a tautology. But the more serious problem is that the definite
integral is _defined_ via the antiderivative. This is wrong, as the existence
of integrable functions without antiderivative shows.

I'm all for simplifying material, and I'm not saying this needs to be a
mathematically rigorous treatise, but it should get the basic maths right,
because there is already too much confusion out there.

FWIW, I think the FTC is deeply important and used all the time, but the first
part of it - the one about writing antiderivatives as integral - could
probably be omitted for non-mathematicians.

It would probably enough to establish: \- Here's what differentiation is.
Here's some rules. \- The inverse of that is antidifferentiation. You compute
it by going in reverse, but note that this is harder than differentiation
(which is mechanical) \- Then we have the integral via Riemann sums. Maybe you
can show some simple example. \- Now comes the deep result: the integral can
be expressed via the antiderivative, if it exists.

You don't have to go into full mathematical detail, but this presentation
wouldn't be getting it wrong anymore.

~~~
ajross
So... if it were made clear that the discussion was specific to continuous
functions, your criticism would disappear? Is that really worth the level of
your dismissal?

I mean, that's how it's actually taught to high school students, after all.
Complexities like discontinuities are absolutely not part of the initial
curriculum.

Alternatively, if you think a simple introduction to calculus _requires_ a
discussion of discontinuous functions and the fundamental theorem of calculus
before it describes how to compute a definite integral, I retreat to my
earlier point: that's ridiculous nitpicking.

Basically: you want a textbook. They make those. This isn't that, it's an
intuitive guide for people who found the treatments in textbooks opaque. Those
people exist, they want to learn this stuff too, and they aren't well served
by people like you just yelling at them to read the textbooks they already
have.

~~~
Tainnor
I don't feel you've actually read and understood my criticism.

The fact that the theory falls apart for discontinuous functions is just a
visible symptom of the deeper problem that the author confuses a definition
with a theorem. I don't think this is just theoretically wrong, it also makes
no pedagogical sense. To say "we define the integral as being the difference
of the antiderivative at the two endpoints, but we can also define it as the
Riemann sum" doesn't tell the student anything except "mathematics is magic,
don't even bother to understand it". Whereas to be genuine and say "we can
prove that the integral can be expressed through the antiderivative (though we
choose not to do this here)" doesn't treat the reader like an idiot.

I don't think it's impossible to provide an intuitive and non-proof-based
account without sacrificing getting things right conceptually. In fact, I
provided a template for doing so above.

~~~
ajross
> To say "we define the integral as being the difference of the antiderivative
> at the two endpoints, but we can also define it as the Riemann sum" is
> [like, really bad!]

How about if there was a footnote on that point explaining that these two
definitions are actually equivalent (for the continuous functions we're
talking about) and that the proof is really interesting and can be found in
your textbook? Would _that_ meet your requirements?

Again, this is just nitpicking. In fact those two definitions _are_
equivalent, as you keep pointing out. You don't have to prove everything in an
introductory treatment, and in fact even high school textbooks on this subject
don't even try.

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blululu
This is very nice work. If your goal is disemminating this work for
pedagogical purposes I would strongly encourage you to add some of these
animations to the relevant Wikipedia pages. A lot of students turn to
Wikipedia and these kind of animations can be very helpful.

~~~
caro_douglos
Are there any self-taught type of classes that utilize Wikipedia for say
history?

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hota_mazi
Nice illustrations but that article makes the same mistake that a lot of posts
of that type often falls to, namely, assuming the reader has a similar
background as the author:

> We all know that the slope of a linear equation/function has a constant
> value

Er... no. We don't "all" know that.

Actually, I'd say that 99.99% of the population of the planet has no idea what
that means.

~~~
reubens
Is there a word for this effect in education? Where the teacher forgets the
context of the learner?

~~~
rohan_shah
"The curse of Knowledge"

~~~
dredmorbius
[https://en.wikipedia.org/wiki/Curse_of_knowledge](https://en.wikipedia.org/wiki/Curse_of_knowledge)

Also Shoshin, "beginner's mind":

[https://en.wikipedia.org/wiki/Shoshin](https://en.wikipedia.org/wiki/Shoshin)

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archibaldJ
author here; i wrote this quite some time back (2014). I'm now studying
Chinese and would like to write a Chinese version for Sinophone audience (with
more Chinese humor, etc). Anyone knows a good place to share articles like
this if it's in Chinese? Thanks!

~~~
NieDzejkob
Hi. Your math notation gets cut off on mobile. Consider allowing horizontal
scrolling.

~~~
jacobolus
Let me recommend scaling equations down to fit the viewport width instead of
scrolling.

If you shrink equations, then if someone wants a closer look they can zoom
in/out. But if you have them scroll off the side of the view, it is impossible
to see the whole equation at once.

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Mugwort
A bird in the hand is worth three in the bush. If I could give someone from a
non-mathematician background, e.g. programmer, who sincerely wants to get a
peek behind the curtain and know what calculus really _IS_ without diluting
anything in an unacceptable way I'd show them I. Gelfand et al "Sequences,
Combinations, Limits". Once you have the idea of a limit of a sequence down it
isn't hard to learn the limit of a function (epsilon, delta) and move on from
there. A good second step is Apostol's calculus vol. 1 just the first chapter
where he calculates an integral from scratch with no "fluff" (details you
don't need just yet). (Yes, learning integration first actually makes more
sense than learning derivates.) Once you do that you can continue with Apostol
(highly recommended.) or use Spivak's calculus (tougher but even more highly
recommended).

To wrap up Gelfand is one bird in the hand, Apostol chapter 1? with
integration from stratch is another bird in the hand. That would be a good
first two steps to get a sense of what calculus is really about.

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teichman
If you like this site, check out (a) 3Blue1Brown's youtube channel for
intuitive visualizations of (otherwise) hard math topics, and (b) the book
Infinite Powers, for more calculus intuition + history.

Both are outstanding.

~~~
rsa4046
+1 3Blue1Brown's channel is indeed a treasure.

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wilkystyle
As a visual learner, I have always struggled to intuitively grasp mathematical
laws and concepts from words or equations in a textbook. This type of
illustration makes so much sense to me.

~~~
GuB-42
You should try to do 3D graphics and maybe some demoscene/shadertoy stuff.
Helped me tremendously.

3D graphics is mostly vector maths and linear algebra but you can do some
calculus too. Derivatives are tied to continuity and continuity is tied to
smoothness, which make things easy to visualize.

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40four
Cool. Seems like a good, quick and dirty run down of all the core concepts.

Funny, I don't think I realized it at the time, but is calculus the first time
we are formally introduced to the concept of a function in math? If so, that
gives me a new found respect for calc class :)

~~~
billfruit
I think the concept of function is normally introduced during teaching set
theory rather than calculus.

~~~
p1esk
I think function is introduced in elementary algebra when dealing with
expressions like y = 2x

~~~
gizmo686
In elementary algebra, functions aren't really first class objects you can
manipulate. That is, if you have f(x) = 2x and g(x)=x^2 , you might ask
questions like "what do they evaluate to at A?", or "what is their minimum?",
but you wouldn't ask questions like: "what is f + g". Calculus is the first
time in the standard sequence when you start manipulating functions as first
class objects.

~~~
p1esk
But we are talking about introducing a concept of a function. Which is a
mapping between values of x to values of y. That's it. No need to manipulate
anything.

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larnmar
I’ve often thought it very unfortunate that we leave calculus so late in K-12
education, and for some students never get there at all (at least in my
country). I remember spending most of my childhood mystified about what this
mysterious “calculus” thing was, and then finally getting around to learning
it and saying “oh, it’s just that?”

And then once students are initiated into the (disappointing) mysteries of
calculus, what next? They spend the next few years mechanically carrying out
analytic differentiations and integrations of a bunch of ever-more-artificial
one-dimensional functions. Do you know what the integral of tanh (x + x^x) is?
You shouldn’t, because it doesn’t fricking matter! Most of the actually useful
integrals that anyone would ever calculate in real life turn out to be only
solvable numerically.

In conclusion, put an introduction to the intuitive ideas behind calculus
earlier (Year 7), emphasise numerical over analytic solutions, and use the
time saved to move on to things like multivariable calculus (and path
integrals, which I still don’t understand properly).

~~~
sukilot
Which path integral?

[https://en.wikipedia.org/wiki/Path_integral](https://en.wikipedia.org/wiki/Path_integral)

Path integral may refer to:

Line integral, the integral of a function along a curve

Functional integration, the integral of a functional over a space of curves

Path integral formulation by Richard Feynman of quantum mechanics

~~~
Armisael16
We’re talking about basic calculus, so it’s the first one.

~~~
sukilot
I'd be inclined to agree, but parent poster was concerned about solving
problems numerically, which is trivial for line integrals, as they are just
sums of triangle hypotenuses.

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unethical_ban
>Gottfried Wilhelm Leibniz, a great German mathematician, came up with this
notation in the 17th century when he was still alive.

I would be impressed if he came up with it before or after that time.

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ducaale
The essence of calculus series by 3Blue1Brown
[https://www.youtube.com/playlist?list=PLZHQObOWTQDMsr9K-rj53...](https://www.youtube.com/playlist?list=PLZHQObOWTQDMsr9K-rj53DwVRMYO3t5Yr)

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mathgenius
This presentation of calculus is bog standard. I can't quite fathom how a few
little gifs can make people so excited. Is it the Will Ferrel meme? Apart from
that, this is dull as dishwater.

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MrQuincle
It is also nice for yourself to do. One time I created this visualization
using vis.js of the Legendre transform:
[https://www.annevanrossum.com/blog/2015/08/08/legendre-
trans...](https://www.annevanrossum.com/blog/2015/08/08/legendre-transform/)

It made it click for myself and it's fun to do!

------
haolez
I loved the parts where the author shows who created the notations used and
what was their context and way of thinking.

I tend to get distracted when the notation in a field is awkward to me.
Statistics confuse me with notations that resemble function applications, but
are not. I like how physics customize the notation to their needs, like the
use of brackets in quantum mechanics.

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happy-go-lucky
The linked Calculus One by Jim Fowler is no longer available on Coursera, but
at [https://mooculus.osu.edu/](https://mooculus.osu.edu/) you still have
access to the lectures and the textbook.

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blazespin
I tried explaining this to someone but struggled a bit with Motivating. I
think in math it’d be nice to start out by explaining why limits and integrals
are useful.

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xerox13ster
I'm familiar with these concepts but the notation used in the article
ironically looks like Greek to me. I'm used to classic mathematical notation.

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mraza007
To be honest I love the resource man You have put great effort into this Love
it keep it up!!!

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bobharris
I need this for the umbrella of statistical signal processing.

~~~
behnamoh
umbrella?

~~~
bobharris
I used umbrella, because statistical signal processing seems to include DSP
concepts + probability concepts + random process + Control Theory (Markov
stuff)

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throwawayhhakdl
Calculus is super intuitive, but the algorithms are both difficult to memorize
and involve a lot of guesswork if your pattern matching isn’t well honed
(integration, specifically). I would advocate teaching calculus with charts
and stuff to demonstrate the logic of if, but not require the solving of
problems. Most of it is a waste of time, but the concepts are useful.

