What does area have to do with slope? | Chapter 9, Essence of calculus
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- Опубліковано 5 тра 2017
- One view on why integrals and derivatives are inverses.
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Thanks to these viewers for their contributions to translations
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The next and final (for now) chapter of the series will be on Taylor series, with a small footnote on higher order derivatives coming along with it. Follow the full playlist at 3b1b.co/calculus
Can you make a video on the arc length formula?
3Blue1Brown bg
+3Blue1Brown Thank you. Relearning things that I learned over a decade ago (athough not fully - had to memorize a lot of stuff) from a different perspective really helps. I remember when I cracked (before I learned about limits, only heard of them) by myself the derivative of ax^2+bx+c and was so happy and proud, but also unnerved, since c disappeared and I thought - We're losing information here people! Cheers!
Darn. The single most confusing thing in all of calculus is integration by parts. I was sorta hoping you could shed some light on that.
Serious withdrawals coming ;)
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!
Awesome!
Maths
@p s hey! Pretty sure you did not intend to tag ME as one of the replyers but I will give my opinion anyway: from what I have observed pure mathematics in higher math tend toward topics like group theory and analysis. Applied mathematics at a higher level tend toward topics in other sciences and lab data analysis. I am far from the end all be all for what each field does, but there ya go. You may choose depending on your interests. I am always happy to see those interested! Lots of encouragement from me!
Dabeer, whimp
@@billymays495 "Maths" is the British abbreviation. "Math" is American. There ya go.
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!
+Franz Luggin Thanks for noticing! I do indeed find them useful in numerous ways.
couldn't agree more
@@3blue1brown I read this as "thanks for nothing!" lol
@@3blue1brown noumerous ways... we see what you did there
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.
louisng114 that’s a very nice way to think about it!
louisng114 That kind of sounds like cutting all the rectangles in the integrals so that they have the same height and then lining them up to form one large rectangle and the height of that large rectangle is your average for f(x) on the interval [a,b].That height has to be just the right value for you to be able to do that otherwise you would end up with a few rectangles that wouldn’t have the same height and they would screw up the large rectangle.
louisng114 I had a similar thought process only I imagined the area under the graph was some kind of ice shape that would eventually melt into a rectangle of equal area where the base is the same but the height is now uniformly the average value of the function
Why integrals means area
This idea is how I found average height
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
Yep you are right. I think it's the primary fault of math teachers who disgrace maths by teaching it without any feel and underlying beauty.
@@yutaitadori7318 The problem it's that mostly math teachers don't understand math, they just repite. I always try to prove algeabric expression and then I never forgot about the formulas because I know how to get there.
@@rajinfootonchuriquen yeah had once a math teacher who admitted to just memorize the formula during the studies without knowing how it works
@@DarthZackTheFirstI very sad man. Just repeating the "bad' learning it's what those teachers do.
Who needs Saturday-night nightclub escapades when you have 3blue1brown's videos to keep you entertained?
I really enjoyed your Complex Variables and Functions playlist. I hope you keep adding more videos to your repertoire. Number Theory, Real and Complex Analysis, Topology and Differential Geometry would be great! Thanks.
Thank you! I will definitely keep adding more videos (starting May 15 when my thesis is submitted), and I will take your video series suggestions into mind as well. Though next time, I would recommend commenting on my channel's discussion page. Idk if we should intrude on 3b1b's video haha
Those who needs more than a 12 minute video to be entertained.
3b1b is great for kids, but for the Adults wanting a little more to chew on, I highly recommend Faculty of Khan's channel.
I came here looking for better deeper insights and found little I didn't already know - The faculty of khan channel is much better in my opinion.
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.
IF you teach like he does then you could be a fantastic teacher
You should study mathematics
YES!!!
Please become a math teacher!!!
That's awesome
You are actually the best professor i ever had. *-*
Tks from Brazil.
Sou do brasil também.
X1000 de México
Caramba, achei que não ia achar brasileiro aqui não kkk.
Cara fez eu finalmente entender integrais
Cálculo tá fazendo mais sentido do que nunca com ele
Hey man if you ever stop making videos ima track you down
This comment pretty much describes how i feel about this channel
how about encouragement instead of discouragement?
Hey man if you don't stop making videos ima not track you down
lol, I am stupid
+Symbiotic Coherence
What do you mean? You weren't wrong.
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.
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!
That's another good way to look at it but I still can't directly see how an integral of some function between two points equals the difference of its anti-derivative btw those points. Average, I got it but this idea, still not clicking..
how is the left side of your mentioned eqn = f(x)?
@@SunilSkanda try to make sense of this
f(x) = slope of antiderivative of f(x)
f(x) = slope of F(x)
Take average on both sides, we get:
Avg [ f(x) ] = Avg [ slope of F(x) ]
That's how he got the equation he wrote on 10:50 at top left. I guess he forgot to write the statement in words.
Once you get this equation LHS is area under curve (i.e. LHS is integral of f(x) ) , RHS is slope of the 'antiderivative' F(x) )
Since LHS and RHS are equal, area under any f(x) and slope of ITS antiderivative F(x) are equal.
"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
I'm in s1 (12 years old) and don't understand any of this but I still think all your videos are amazing
I started learning this last year (12) and you would end up better, this is an amazing way to start learning the topic.
Don't worry boi, all you need is first try to understand how some graph calculator functions work such as the greek E looking letter (sigma), and then you'll understand these videos when you'll be 15 like me.
I love maths and knowing I will be able to understand complecated things like this makes me very excited!
When you know this, it gets easy af. The integral is basically the average of an infinite amount of infinitely long vertical lines, all starting from zero on the horizontal axis, and that were all cut on the vertical one to where they touched the curve.
ayy same.. i try to understand this stuff but im pretty sure i need to upgrade my processor
Make videos about:
- Set Theory
- Non-classical Logic
plz
NDos Dannyu Amazing guess ;)
0
according to his patreon the next series is about probability
non euclidean geometry
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
Its nice that you also show this "interpretation" of integration, because it generalizes far better to curve integrals than the normal "area" intepretation
I love the π creatures!
I see 4 creatures instead of ~3.14 creatures.
@@JorgetePanete *Proceeds to include 14% of a pi figure in the next few videos*
Mojadar jinish...maths er moddhe erokom ektu humour na holey chole kemon kore😅
lol lol lol lol lol lol
whats with the other 0.0015..% .. I’ll show myself out
Essence of non euclidean geometry (hyperbolic geometry perhaps) or topology.
Medhat Mostafa I wouldn't say that's a very easy topic to do an essence series on. Non euclidean geometry and topology use a lot of advanced maths that require explanations of their own. Sure, he could probably do a video or two about it, but I doubt it'd be an essence series
Of the 4 people I support on Patreon, you're the one I've never considered dropping. You are one hi-fivable person.
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
2 upload in one day?
what did I do to deserve this wonderful channel?
Dương Hoàng
but... He only uploaded once...
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 :)
Before even watching this video, I will say ny thoughts, I have since I was in highschool thought of the fact that a derivative is based on a slope which is dividing and an integral is based on area which is multiplying, and those are inverses of each other, so it made sense, I mentioned that do one of my highshool math teachers.
What you just said here is more helpful to me than anything I've seen anywhere. You stripped away all the ultra inscrutable terminology and put it in words I can understand. My question is, why does it never seem to occur to math instructors that getting to the nut of things, using good old plain English, is the only way some people (like me) will ever understand? What you said here should be the first thing out of every calculus teacher's mouth. THANK YOU, IslandCave!
God, Grant is just brilliant. At 2:00, he stops to identify feelings that come up in maths (getting the average of continuous values indeed feels unfamiliar) letting the student know he feels this too and it's ok. You rarely see that in any maths. exposition.
I can't believe this is free...I genuinely thank you for this. Your whole series is really amazing, the animation is also great, thank u for this
This is beautiful. Please consider doing a series on the essence of Probability! At least, a video about the Normal Distribution, CLT, LLN! Keep up the good work.
This is one of the things my prof quickly brushed past. Since I’m studying math, my profs tend to go into detail about why things are the way they are, but this wasn’t one of them. When you look at the formula, it kinda just makes sense, but this video really helped me understand exactly why that is. I had to pause halfway through and go back a minute to fully understand a couple times during this video. But taking the time to pause and ponder was worth it. Thank you for these videos. They’re entertaining and I always discover something new when watching them.
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.
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.
Please make videos about essence of probability and statistics
Metin Ersin Arıcan Support this!
Yeah!
By the way, I hate statistics a lot and I don't understand a thing there.
My AP test for BC calc is on Monday. This series has really helped to ground many of the concepts I've learned this year in a more intuitive foundation. Thank you for all you do!
These videos are amazing, the most well made ones on math I've ever found. Please continue to make them!!
"The integral from zero to pi of half of sin(x) with respect to x"st
Congratulations on your videos being awesome, I recommend them to all my friends
I'm sad that this video series is ending so soon. I love watching the animations and hearing your soothing voice. I can't wait for what you're going to be doing next (topology would be cool)
I was almost brought to tears. Humanity does not deserve this channel. I sincerely thank you for all your efforts.
Really excellent. Math is so much, with the right teachers.
No body taught me better
Ur the first one to teach me calculus the best possible way
I have come back to this video 8 times or so over the past 4-5 years, just to get the feeling of satisfaction from watching 8:01 over again. It is the only thing that I reliably forget and then get the same joy from reunderstanding. I have attempted to explain this idea to other people, but have failed miserably. Thank you Grant for explaining it so well.
This is simply brilliant. If every teacher in the world was like you, every child would love to be a mathematician.
This was great I never appreciated this despite having an MPhys. The integral encodes the height of a function into the slope of another by taking the anti-derivative. It is much easier to find the average slope than the average height. We can therefore find the average height (or total area) by taking the average slope of the antiderivative. Amazing, kicking myself.
Everyone watching this should donate at least 1 dollar to his patreon. Well deserved
Elliot Dekota some people here are 14 year olds with no money sorry!
You being a math God between math youtubers is a fact.
I was struggling to understand the relation from discrete time convolution to continuous time and this just made my day :)
Thank you very much for existing!
SHIKHAR AMAR I don't think he can take credit for that one
Finally, integrals like these don't feel like something I'll only understand when I'm in 11th/12th grade, but like actual approachable and even solvable things
And 7 months after that (not now, now is 2 more months after that) I was doing calc 3
I feel like commenting the same thing under all your videos : THIS IS THE BEST CHANNEL
Wow, I have always been looking for intuitive explanations of the Fundamental Theorem of Calculus and this video has definitely fulfilled that desire. Best calculus video ever to be found. Thank you very much! Imo this is much easier to understand than any of Khan Academy's calculus videos :)
Okay when Im at school, I hate math, But when Im just bored on my computer, I love it. WTF!
There’s no intuition to math given to you in school. When I learned the quadratic formula, I was never taught why it was that way, my teacher just insisted that it was.
@@connorcriss I mean, there's not really a lot of intuition to give to the quadratic formula. It's just a general solution found through luck and algebra. The correct way to teach people why the quadratic formula is that way, is "it's simpler and some expressions can be hard to factor"
Actually it's because when you're watching math videos on your computer, you're not actually doing math
Its weird that when you translate the dutch name for the quadratic formula litterly to english, you end up with a squarerootformula...🥴
I think it's because 3blue1brown doesn't assign homework...
I loathed math through all of middle and high school. Now I'm watching and understanding your videos on integration and Taylor series after only four semesters of catching up at college. I have my lowest grade in calc. 1 at a C+ and funny enough I don't mind, because it's my favorite course.
I'm graduated in information engineering and in all my academic life i didn't see a better, clearer, more intuitive explaination of this problem (in fact, i've only been teached the theorems and proofs, without the intuitive idea, which i think it's the most important thing).
It have always been a curiosity in my mind of WHY should the anti-derivative be an area.
Thank you very much for the video and keep up your great work!
Perfect timing for this series! I have calculus finals day after tomorrow and although this series doesn't directly talk about higher order derivatives or multiple integrels and the like, the content of your videos really does give some useful insight into the whole thing, and at the very least it rekindles my enthusiasm to do problems.
I actually passed all my calc classes as an engineer just by watching and learning from your vids
Thanks for making math awesome.
Thank you professor 😄..
This Series is, by far the best in-depth explaination of calculus! Keep up the great work :) see you in the next video! Im curiouse what the next video will be about ! :)
amazing!!! i just keep thinking about the correlation to classical mechanics all the time and it really helps me to understand even more
"We often hear that Mathmatics consists mainly of proving theorems. Is a writer's job mainly that of writting sentences?" Tom Boy
You should make a video series on high grade trigonometry
Prabha Dsouza But it's just triangles...
Prabha Dsouza maybe he can explain the hyperbola trig functions. trig isnt just triangles it connects to allot of mathematics
Natanel Sharabi You are right. There is more to trigonometry then just triangles like hyperbolic functions and inverse functions.
THIS IS THE BEST SERIES ON UA-cam RIGHT NOW
Absolutely brilliant video succeeds at linking the use of integrals to areas under a curve. Your work is appreciated more than you may ever know!
12:16
"Markus Persson"
wat
twitter.com/notch/status/794886685851086848
That just means Notch has good taste
the mc billionaire.
he supports tons of UA-cam channels, I've seen his name at the end of Noah Gervais videos.
ya boi
Reason why I became a dentist was because I hated maths, reason why I am getting into quantitative research / stats / predictive modelling is because of you! 3Blue1Brown... this's how religions begin... !!
I just want to thank you for your videos where are AMAZING. I'm a 9th grader currently studying calculus and the lecture videos provided by my school are longer, duller, and filled with examples I don't need. Your videos help me visualize the theorems and equations that my school just expects me to understand without help, and it's so much easier to understand what I am doing when I really comprehend how and why I am doing it. Thank you so much!!!!
This is truly a Eureka moment for me. I, now actually understand why the summation of area under a graph equate to the slope of another graph instead of just blindly manipulating it and solving it. I thank you from the bottom of my heart
5:57
"look at this graaaph"
Its so sad that it ends tomorrow :(
???
Shubham Shinde It's*
Jorge C. M.
No.
It's (present progressive)
ends (present simple)
@@JorgetePanete No it is it's.
@@user-nm4ni5us8j Not that...
Without a single doubt, the best series on UA-cam.
I can't explain in words how good these videos are and how thankful I am to u.
We have to find those 25 people who have disliked this video...
No we dont. They have the right to give their opinion
He makes obvious math more complicated sometimes.
I'm in that bunch, except that I didn't give him a thumbs down. I didn't because I know what he's saying makes perfect sense to people who can follow all the arcane terminology, and its helpful to them. To me, the following makes ZERO sense: "By definition, sine of X is the derivative of the anti-derivative graph. It gives us the slope of negative cosine at any point." It might as well say, "the octo-quad Pi inverse anti multiplier quasi place splitting quadrant divider's anti-pi is, of course, as can easily be seen, the drawn downdrop of plus valued hyper-zip splitters." That's how insanely inscrutable all this language sounds to me, and even more depressingly for me, it's meant to turn the light bulb on in people's minds, its actually AN EXPLANATION! The animation of the slope line tilting one way and then the other is conveying something, but it isn't at all apparent to me what that might be. I realize the animation and what he's saying at the time the animation is moving go together...but nope...the light bulb did not go on for me. It's absolute jibberish to me.
@@johnnysparkleface3096
You can't get integrals without knowing derivatives. Heck I know derivatives and I'm still struggling. Don't feel bad if you don't get it, there's not a magic formula for understanding those things, no matter how talented is the one explaining them (and this guy is talented, I guarantee). You have to do a ton of work. It's true, in some people's minds it just clicks together as soon as they go along with him, maybe some are 15 year olds but who cares about them. Care about yourself.
Don't rush to higher maths because you see that some people get it with ease, take your time. My advice is (if you want it) work your way up from basic geometry, it takes patience but you'll find out that going over the old stuff you get so many more things, and by the time you get here again it will be a breeze.
@@pietrotettamanti7239 Thanks for the encouragement, Pietro. Just yesterday I read a note I wrote a while back, and though I didn't understand it at the time I wrote it, I think some of it made a little sense - maybe that's a bit of the clicking you mentioned. I sure hope so. By the way, I've watched Nancy Pi''s UA-cam videos, she's pretty good at explaining things. My biggest problem is me, I can think of 10 ways a person might have meant what they said. Thanks again, stay safe.
I started "studying" calculus(just to understand it, not even thinking to study all the formulas)on my own a few months ago (i'm 13 :P)...now i think maths books should be written by someone who really tries to teach it, not just to make money by putting a few formukas on a piece of paper
Elizabeth Fortanely I'm not saying that i skip all the formulaswithout even reading em, i really enjoy trying to understand them, just once i understood them I'm not interested to remember em (at the moment :D )...eg. most of the formulas for integrals(don't know their names in english) have tons things to remember (resolving an integral with the "simple fracts")nice to understand..awful to remember :D
Ciccio&Bombo You son, do not seem to understand calculus at all. When you call them formulas you prove me. Are you sure you were not just trying to show off to a bunch of friend if that you mean by studying calculus which I am sure you do not get.
Balen Amir he is 13 he isn't familiar with the notation
All "formulas" from highschool calculus are pretty easy to prove and most of them are mandatory. Actually, you need to know everything in math,every formula, everything you can get your hands on if you want to truly understand high level math.
Once you go into high school and learn it (again) the teachers will explain the formulas like you want and yes it is stupid for math books to do that but that would extra length to the book and essay upon essay just search it up if you don’t fully get it and get the proof on why if you are truly curious it may not be convienient but it’s the best way.Also use khan academy it’s fantastic!
Love this series!! And this channel!!!
Thankyou so much. You gave me enough in the start of the video to feel like I was "making it" myself, I felt asthough I actually understood, instead of remembering, this is an amazing feeling and truly something that would inspire younger people to pursue maths if taught in schools, I really appreciate you doing this.
Thank you so much for making these videos. The AP Calculus exam is on Tuesday and this series is an exceptional review.
Wow, this video opened a whole new meaning of calculus to me. Thank you so much! 😊
Best channel! This video was a little bit more confusing so I'm going to have to go over it again
Calculus is wonderful, it changed my life and I'm happy for it.
This was a beautiful explanation. Please keep making such videos!
6:04 if there were Oscar’s for UA-cam the animator and 3 blue 1 brown channel would deserve it. This is beautiful and super helpful !!!!
That actually blew my mind, thank you.
This really helped me wrap my brain around the FTC in my Real Analysis course! Another masterclass 🙌
I am so happy I stuck with learning math, I can understand all this and it's so amazing to see these relationships
Thank you 3Blue1Brown! It's is always hard for me to truly understand some concepts without having a clear picture in my mind.After watching this video a few times,I finally can make connections between Mean Value Theorem and Average Value of function.
That's an OMG. The answer I've been looking for years. It felt always so vague to subtract from boundaries when integrating. Math is not just some "technique" to master but the idea behind, geometrical interpretation... Thanks so much Grant!
Man for almost a week I was having so much trouble grasping the concept of an integral and an antiderivative, but you video gave me what I needed to solidify the concept. The visualization and the explanation helped me so much thank you🙌🙏
Yaaaaay! I want more of this after 10, it's so great!
I LIKE A YEAR AFTER FIRST SEEING THIS VIDEO just now needed to figure out a problem. First it struck me that I had that sense that I needed to use an integral somehow. THEN I realized I EXACTLY needed to know the average of a continuous variable! I came right here and here It was! If i had not watched this i would have no idea that integrals could even do that! Thank you so much, bless you Grant!
One of the best video to understand relationship between area and integration. I watched like hundreds of videos but this is the only one I finally understood how area is related to integration!
You're doing an incredible job , I find your videos entertaining and your way of explanation is absolutely fantastic imo. I'm 15 years old and next year I'm going to attend 11th grade , where calculus begins , I like maths and I'd heard from my seniors that calculus is easily one of the more tough topics that I'm gonna study in the future. But after watching Ur playlist on it im wayyyy more confident than before. Thanks a lot and keep up the good work.
Good video. I learned a lot about this on my own while reading and trying to make sense of monte carlo methods for ray/path tracing.
You're changing my life man
In fact, this is the best video explaining the relationship between area and slope.
Dude's mind blowing understanding of math and the editing skill...
Who are you
this video es a must see , to complete a deep understanding on the subject , thank you !
that s the best explantion if ever heared much clearer then adding up many peaces i never understod that
I am so grateful for being able to access this content
Wow was looking for a video that explains this, thank you
I am starting to genuinely love maths, consistently. thank you Grant. the insight is immeasurable. might join patreon when i've got cash
Thank you very much for this video! It is interesting to see how to understand the area is related to the derivative!
Excellent . There is a beauty to the way integrals and anti derivatives work . Great explanation
7:50 - 8:08. Wow! I had goosebumps. What an insight!