What does area have to do with slope? | Chapter 9, Essence of calculus

One view on why integrals and derivatives are inverses.
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Thanks to these viewers for their contributions to translations
Hebrew: Omer Tuchfeld
Vietnamese: ngvutuan2811

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3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with YouTube, if you want to stay posted about new videos, subscribe, and click the bell to receive notifications (if you're into that).

Various social media stuffs:

[dabeer]

i have fallen in love with maths again. I teach physics and as an educator, i have genuine respect for your creativity and lucidity. BRAVO!

[louisng]

I think about the average of a continuous variable as shaking up a bucket of sand until it levels. Since the amount of sand is constant, the area under the curve stays the same. New shape is a rectangle with the area being the base (interval length) times the height (average value). Therefore, the average is the integral divided by the interval length. This can be extended to multiple variables.

[franzluggin]

Those pi-creatures are a subtle, but at least for me very effective way to make the parts of the video that are non-visual have more of an impact. They do so many things, emoting how you think the audience feels at the moment (and asking questions when you want to tackle them), tracking what's going on on the screen with their eyes, pointing at stuff, ...

This alone must have been so much work! You also made their various poses very expressive, and not to forget cute.

I'm not really going anywhere with this other than a big "Thank you!". It's so rare so see something of such consistently high quality, much less for free!

[MrBebopbob]

Absolutely beautiful. I showed this to my teenager that just finished college calculus and he was blown away at the intuition it gave him. We need to clone you, and replace all the uninspiring math teachers in this country so kids will understand the amazing beauty of mathematics. Thanks for the hard work. Bob

[FacultyofKhan]

Who needs Saturday-night nightclub escapades when you have 3blue1brown's videos to keep you entertained?

[lcarsos]

At some point I understood math's symbology, and got good at moving around all the variables and bits of notation. This meant that I could answer homework questions, and do most test problems, but I had no intuition for what was happening. It took me many years (and failing classes a few times when I ran into a professor that cared more about understanding than good test takers) to build up an understanding for what was happening when I was shuffling all those bits around on the paper.

Thanks for your really excellent videos.

[jonathan.gasser]

No kidding, you are an absolute inspiration for me (I'm planning on becoming a high school teacher).
This is math taught the way it should.

[cluelessinky]

Great stuff, thank you. I took calculus courses nearly 60 years ago and while I passed the courses I just didn’t quite get what the instructors were trying to prove. Seeing your courses has made those objectives plain. I’m well into my 70’s and am feeling a sense of loss at having missed the beauty of math but I’m taking as many of your videos and getting to appreciate what math has to offer. Again, thank you

[SamuelAndradeGTutos]

You are actually the best professor i ever had. *-*

Tks from Brazil.

[Twisol]

For maximum enjoyment, take the equation at the top-left of 10:50 and ask what happens as you take (b - a) closer to 0. Well, on the left, you get the integral from a to almost-a of f(x), over a very tiny amount approaching zero; this ends up being just f(x). And on the right, you get the limit of (F(b) - F(a)) / (b - a) as (b - a) goes to 0, which is just the definition of dF(x)/dx. So f(x) = dF(x)/dx, exactly as expected!

[zairaner]

Its nice that you also show this "interpretation" of integration, because it generalizes far better to curve integrals than the normal "area" intepretation

[matthewfuerst]

8 minutes in and I finally had my aha moment. This is the first thing that I’ve come across in my actual math class that I couldn’t quite reason out why it worked. It was that little animation where you had the average slope of the anti derivative graph that did it. Thank you so much

[Jj-kflq]

I'm in s1 (12 years old) and don't understand any of this but I still think all your videos are amazing

[rishavdhariwal]

"when you reframe the question of finding an average of a continuous value as instead finding the average slope of a bunch of tangent lines it lets you see the answer just by comparing endpoints." This was the highlight for me in this video

[alfonshomac]

Of the 4 people I support on Patreon, you're the one I've never considered dropping. You are one hi-fivable person.

[ultimate]

This series is so impressive, in normal calculus courses it’s just taught how to solve problems without making the students visualise what is actually happening . But that is really important and moreover everything starts to make sense .
Thanks for making this really awesome series :)

[dannyundos]

Make videos about:
- Set Theory
- Non-classical Logic
plz

[Hivlik]

1:30
It seems to me that this is because, when approaching an average of a finite set of values, we have been drilled to treat it as the sum of the set over the number of terms - finding the mean of the set. This, in the cases that we've always seen in algebra, works perfectly fine, but is not the most general concept of an average. An average value is, most generally the "middle-est" value of a set - which, in the simple finite case, is very well described as a simple mean, but in finding the average of a continuous variable of course can't be approached the same way.

My point is this - the definition of an "average value" does not break down when asked about an infinite set, but rather asks you, the mathematician, to broaden your idea of what an average is. In any case, your video is exceptionally made as always. Thank you for the content.

[renzo]

This series really is how I perceive math. First, you are following everything and then wonder how you got where you are. You think about it for a little and suddenly it clicks and everything makes sense and you appreciate the genius behind it.

[ultravidz]

Hey man if you ever stop making videos ima track you down