I'm a high school student from Brazil, and I can say that you've inspired a generation of people to pursue math and physics for their own sake. I started to learn calculus through these videos 4 years ago and now I'm studying for the AP Physics C exams (wish me luck lol). I know you'll probably never see this, but if you do, just know that we really appreciate your work.
Excellent! How I wish I'd had access when I was in high school and college to this information. We were 'taught' by Rote.. but it never made sense to me because the Purpose was hidden away.. Good luck with your studies! Blessings!
As a school going kid, I always wanted to visualize math like this animation. But my teacher could never provide with any material like this. Now, several years after finishing my bachelors I can finally feel these ideas for real. Thanks a lot for the hard work and this fantastic motion animation. I wish, in future you will cover each and every aspect of math, science and technology. The world needs these kind of content more and more for the people to better understand the underlying rules of our world. Cheerio from Germany. :)
Calculus was the definitely the strangest thing I had to go through during college and I feel silly after watching this series. Such elegance and I missed it completely.
ich weiß nicht, aber unser Lehrer hat uns das nicht so sehr vor Augen geführt, vor allem nicht mit Kreide und Tafel. Aber trotzdem hat er es irgendwie genauso heruntergebrochen mit Worten. Man musste dann natürlich das ganze selsbt machen, und da kannste ja dir alles vorstellen. Aber an alle Kernaussagen hier in dem Video erinnere ich mich. Komisch.
Very true. It's thankfully easier considering we can work with geometry where all objects have equal height which thankfully allows us to calculate a change 'dr'
Im a physics student and I had to learn calculus like this. But when calculus was used in QM and EMT I shit my pants cause I couldnt picture anything. Nothing made sense. Now im going through calculus again by myself. -__-
@@RanEncounter why would mit even do this they have a face to save, greedy school and way worse in college hire teachers at low cost and are generally of these sort
This video blew my mind. I took calculus in college 45 years ago, and it nearly killed me. Dropped the class a couple of times before finally squeaking by with a sufficient grade to satisfy my geology degree. I haven't had much need to apply calculus since, but my lack of understanding has always bothered me. Our professors just blew through it so quickly, I never had a clear understanding about why some factors were important and others could be diminished and discarded. Grant's simple explanation of the area under the curve equating to the piRsquared formula actually caused me a physical reaction! Why hadn't that been shown in any of my classes? Now, at age 67, math comes a little harder for me, but I still plan to work through this entire series!
In a crowded field, Grant has the best maths channel on TH-cam. I highly recommend going through all his videos. 'PBS Infinite Series' and 'Mathologer' are also great
Man I’m half a century behind you, I have to take this next semester! Would be nice to have a professor as good as 3b1b, but I’ll probably just have to settle for more videos
I'm almost 30. Never in my whole life I had calculus explained to me in such an intuitive (and therefore effective) way. I'm watching the whole series and taking notes. Just priceless. Thank you!
@@ananyasaha12345 Like it was an online course, drawing most of the graphs and writing down most of the equations and conceptual analyses. I use a tablet but pen and paper would be roughly the same. I think it took a bit too much time but I like the result.
@@juancharovsky252people often tell me that as they became adults, they lost the ability to learn new things without a lot of effort. Is that the same case for you? From when you were a child and now, is there a drastic difference?
Brilliant: not just "what's the formula" but the deeper understanding of "how they figured out the formula"! This is the way to learn not only math but how to do analytical thinking!
@@H3XED_OwO the point here is to learn about the model and not to recreate it. These videos help in sparking the fire of curiosity and its the viewers choice how far they wanna take things. If you are really interested then after learning the model you will march forward and learn everything there is to recreate the model even if you have to learn whats under the hood
"Math has a tendency to reward you when you respect its symmetries." "Mathematicians don't just care about finding the answer but developing general problem solving tools and techniques." "Transitioning from something approximate to something precise is pretty subtle and it cuts deep to what calculus is all about." As much as I love your clear cut explanations of the topic - these hidden gems of wisdom is what attracts me to the videos you create and helps me build an intuition around the subject which my university professors failed to. This is the sort of insight which is developed when you've spent enough time with the subject and examined it thoroughly from all perspective. I am very grateful of the effort you've put into making these series' and I will be coming back to it from time to time to jog my memory.
@@abhaysharmafitness I love 2 play with my little stink hole it's kind of itchy bc I bleed when my turds are too large in diameter but it just feels oh so wonderful boy
My issue with college courses, very few of them you can see applications of and therefore never stick with you after graduation. It was only when I started working as an engineer I began to see some math concepts coming back and me going back to refresh my memory.
Bizarre notion that a abstract thing can reward someone: sounds like a human need projected onto an abstract thing - sounds like childhood trauma of longing for respect and love from parents... Math doesn't even know you, let along reward you, it is what it is.
Coming to this video after my final class in my first calculus course is mind blowing. Spotting the connections earlier on would’ve saved me so much sleep. This is an amazing video thank you!
This is a fantastic lesson. It is interesting that he starts with areas and integrals. All calculus course start with differentials followed by integration. This approach, with its simple clear explanations building up from basic math gives the student a clear explanation of the processes involved. It really accomplishes its goal of presenting the material so that the student says, "I could have invented Calculus." I have never seen anyone start with the circle and turn the areas into small rectangles before. It is a great starting place, and accomplishes multiple goals. The students learn the thought processes used to develop mathematics in general and Calculus specifically. They gain deeper insights into how the integral is developed and it relation to real world scenarios. And they learn why the formula for the area of circle looks like it does as an added bonus. Understanding why the calculations are done the way are and how they relate to real world problems give the student the deep understanding of Calculus so often lacking in traditional classrooms where the teacher just drones on about equations and applications. I wish math teachers did better jobs of teaching calculus in this manner because it would have made my job of teaching Physics so much easier. As it was, I had to teach Calculus before and alongside Physics. I had a policy of teaching the students how the equations they used were derived and how they related to the real world situations they described. Just like this video, I started with simple concepts and built on them until we arrived at the result we wanted. It was more about the process and understanding of fundamental concepts than the final answer. I have not seen any other episodes, but if they are all like this, I would recommend using this series in place of the distance learning now going on during the Covid-19 crisis. Wayne Adams B.S. Chemistry (ACS Certified) M.S, Physics R&D Chemist 9 yrs. Physics Instructor 33 yrs.
Thank you Wayne for your extremely insightful comments here as I start to enter this world for the first time at 65 to help my son see the deep beauty of math. When experts like yourself write these comments it just inspires us to find more meaningful ways to learn and to understand.
Thank you for this - I am a distance learner through Open University and luckily my tutor uses the same approach. Seeing the videos really helps me - and I can run the visuals through my head. Wow, 33 years of teaching experience - that is quite an endorsement! I dropped math in Grade 11 and regret - in my 40s now and using the pandemic time to catch up.
I'm a Computer Science student at University, and I've watched your videos on Linear Algebra and now this series. I cannot overstate how much of an impact these videos have had on my learning. I have genuinley learned more from you than my lecturers.
I just broke down in tears. Why isn't math taught this intuitively at all levels, at all schools, by all professors? Math is so beautiful; It is upsetting it took me so long to realize it. P.S. Grant, thank you for your earnest passion for teaching and mathematics, you've introducing me to world I did not know existed!
Because it is difficult to teach math. What 3b1b is doing is incredibly impressive, and there are simply not that many people that are able to break it down as well as this.
Can you please help... 13:57 but doesn't it depend on the angle of the graph (line) to the x axis , I mean it seems to be more correct in this case as it's almost parallel to x axis but not so much for a line which is raising rapidly, even though dx is very small but it is different although slightly but it is, isn't it?? ( I know I am wrong but how someone please elaborate)
@@harsharya545 going back the the triabgle shaped graph at the beginning, as dx gets smaller the left and right side of the thin rectangle get closer in height to the point where slant of the graph is negligible
I’m currently taking calculus in high school and I can confidently say, they do not teach you how to understand the concepts like this video does. In ten minutes, I was able to fully understand the countless hours of plugging in numbers my teachers had me do. I am deeply grateful for this video.
Agree. Logic to derive, in my opinion, is the understanding at it finest. A computer can calculate...it takes a consciousness to UNDERSTAND...to comprehend MEANING. This is overlooked these days.
Yeah thats great and all, but after the first couple of weeks in maths (University) you will have to memorize a lot too to formally prove things... unless you always want to start from scratch and that won't work out very well
this just proves that maths isn't boring, unenthusiastic teachers are boring. Literally 25 minutes ago I was in a lecture being taught this and almost falling asleep, watching this, it's literally like a completely different subject.
sometimes it's because your subconscious "recorded" the lecture for you even if you weren't consciously paying attention so when you watch something related to that lecture, you'd tend to understand it better. that or it's because you actually wanted to watch this vid instead of the boring lecture. sometimes wanting to learn or do something makes us pay more attention to it because we have a genuine interest rather than being obligated to learn or do that thing.
In many Western countries we spend a lot of money on fairly mediocre teachers. When I see material like this I wonder why we waste our money on a lot of the weaker teachers we employ. Kids would learn so much more from just watching videos like this. It is about time we had a complete rethink about education and moved forward from the 19th century approach which we are clinging on to.
pretty much. I felt like I was going to die, even with my calc teacher loading a bunch of TI Nspires with programs to help us halfway skate through it. I feel like we have hope now
A few years ago, I got a bit bored, so tried my hand at deriving the formula for the area of a circle. I got there by slicing it into smaller and smaller pie sections where each pie section approached a right-angled triangle with the short side being some fraction of the circumference, and the long side being the radius (radius and hypotenuse would approach equality as the angle got smaller and smaller too). So the area of the triangles would be 1/2(r.((pi.2r)/x)).x with x being the number of slices. Simplified out it became pi.r^2 :D Just this week I got bored again... and started wondering how pi got calculated - had some fun working that one out!!! Got to pi=(360/x).sin(x/2) with it getting more accurate as x approaches 0 :D Sure, I could've just googled them, but it was so rewarding to have that AHA! moment when I figured it out! I'd encourage everyone to try to figure things out themselves every once in a while, because that is a very special feeling!
My good friend. You have been more influential in my life than my father my dog and all my teachers of the past combined. I won’t ever meet you but I embrace you with gratitude and respect. Thanks
Interesting, I've never seen integration being taught before differentiation. I wasn't feeling super crazy about this video at first, but when you got to dA over dx I was genuinely impressed.
As History of Mathematics says, Newton found the derivative first mostly because your intense concerns about motion and derivatives is really related to that (and as well he did claims a special operator for it, the upper dotted notation). However, Leibniz found first the integral and later the differentiation!
Integration was generally discovered before differentiation - some integration processes (more aptly called "method of exhaustion") appeared even in ancient Greece (thanks to Eudoxus and Archimedes.) If I recall correctly, both were developed before limits were a well defined and established method of understanding derivatives/integrals as well!
Thanks for clarifying! I really wished my university offered a history of mathematics class since I know a huge number of mathematicians without actually knowing anything about them (Leibniz notation, Euler's constant, Simpson's Rule, etc).
The quote "The art of doing mathematics is finding that special case that contains all the germs of generality." really touched the deep of my heart. Excellent deep meaning in that. Thanks Grant for the wonderful work that you put in
Mister 3Blue 1Brown you deseve a nobel, for real! I don't understand why universities don't teach math like you do here. You make it look so easy and instresting. We have somehow to promote those videos on every univestity in planet. I hope you'll never loose interest in making those amazing videos. You make the world better. (just a guy from Greece)
Moreover your forefathers from Greece are responsible for the mess that is caused today in maths...how unnecessary and complicated have your ancestors from Greece have been...they should have been very easy and simple...they would have been really honoured by millions of students across the globe ... instead of abusing cus words🤣😃😅😄😉😋😆🤔🤗😝😜😛😝😝😥😥😓😐😶🙄😏😣😣
Thank you for making this video. As an AP Calculus teacher I love it! Was looking for something interesting to suggest to my students as we work from home during Coronavirus. Thank you!
as an engineering student working my way through an accelerated calc sequence having taken 0 calc in HS, this video is filling in the gaps for me with these visual aids. your answering all the side questions I wondered and small mistakes in my ideas as you go along. (and of course the prof is too busy to answer while they speed through problem explanations.) amazing supplemental videos here! still holding up 5 years later. it's the high level concepts & gap-fillers tips/hints I knew I needed but couldn't find.
I'm 15 and your essence of calculus series has really helped me understand calculus before school even started teaching it. Teachers in the lessons i've attended as extra activities in the past just made me remember the power rule and i thought it was the definition of a notation called dy/dx, which I was quite wrong with. Thanks to your incredible visualization and awesome method of teaching I'm now fully prepared to start doing questions from school and exploring further!
Wow, I’m stopping half way through this video as I have to comment on how well done the animations are and how perfectly in sync they are with the explanations.. Really amazing work all around. Thanks so much for making this content freely available!
Thank you (and your team if any) SO MUCH for this super intuitive visual explanation! But I also have a question: when lookin at the yellow approximation rectangle (at 12:21), I though about something; if the smaller the rectangles are, the higher their more precise their approximation, would those be sufficiently precise for all applications of this math where absolute precision is needed as the rectangle "width" approaches 0? Or would it then (for absolute precision) require to also consider the adding to the rect. area the area of the triangle the forms (in this case) above the rectangle? Or how else would one go about determining the absolutely precise/correct values here?
Thank you! To your question, it's worth keeping in mind that these rectangles (known as Riemann rectangles) are not necessarily to directly approximate the given area. In this video, the serve primarily as a conceptual tool to explain the key idea of how integrals and derivatives are opposites. That is, how if you want to know the area under a curve (in full unapproximated precision), you can reason you're way to it using a knowledge of derivatives. This is covered more later on in the series. That said, sometime you _do_ approximate areas under curves with an approach like this, and you're question about quantifying that error becomes very pertinent. Later in this series, in the video on Taylor series, there's a discussion involving a role that triangle plays which you may enjoy.
You know, it's funny. I'm a math major and I'm already well beyond the calculus subset of math classes for the collegiate level - I felt I had a solid, *solid* grasp on this content. I mostly clicked on this video because 3Blue1Brown made it and his videos are just amazing - my linear algebra professor was terrible at giving us intuition, even if the calculations were rather easy, so his essence of linear algebra playlist actually helped me understand a *lot*. I honestly didn't expect to come here and actually get a key bit of intuition that I've been missing for a little while - well, not so much "missing" as much as I looked it over because my AP Calc class was a little rough on me initially. ^_^; Keep up the good work, 3Blue1Brown. I look forward to your future videos in this playlist and hope they can be even more enlightening for me. :)
Thanks so much Daniel, I'm glad you got something out of it. I always find it challenging to target a math video to a wide variety of backgrounds, so your words mean a lot to me.
Im in college right now, and exactly the same thing goes with me. I've finished my algebra and calculus courses, but came here to become more intuitive about these subjects.
same here, i am a physics student(european student, i don't know if its the same) and i had a realy blunt way of doing mathematics, untill i started watching you're videos. thank you so much, you opend new dimensions for me.
I've now taken five Mathematics units at university on my way to becoming a Maths and Chemistry teacher. It blows my mind that I'm paying thousands of dollars for my 'education' when there is far superior content on TH-cam that is essentially free. The worst part of it all is I've scored really well on all of my exams without truely understanding some basic calculus concepts. I've since forgotten most of the content from only two terms ago and I'm starting to realise it's because I've been remembering rules and not attaining a deep foundational understanding which cements it all together. I'm mad because it's not even my fault. I'm clearly striving for proficiency (I'm sitting here, in my own time, watching a Maths video while eating breakfast) but my university never explained calculus this well, or any Math concepts in fact. Grant, as soon as I have the means I will be supporting you and other cannels on Patreon so you can continue to produce quality content.
Can you help me ? 13:57 but doesn't it depend on the angle of the graph (line) to the x axis , I mean it seems to be more correct in this case as it's almost parallel to x axis but not so much for a line which is raising rapidly, even though dx is very small but it is different although slightly but it is, isn't it?? ( I know I am wrong but how someone please elaborate)
@@harsharya545 Even if the graph were to be rising rapidly, if you get an infinitesimally small dx you will get values that are very precise and can account for even the steepest of slopes. Yes, the example shown on 13:57 shows a big dx and maybe that would not work with that graph in the right extremities where the graph is increasing rapidly, but he only showed this big of a dx for visual purposes. He could have used a very small dx for that parallel section to be more accurate as well. Dont know if this clarifies much, but yeah
In many Western countries we spend a lot of money on fairly mediocre teachers. When I see material like this I wonder why we waste our money on a lot of the weaker teachers we employ. Kids would learn so much more from just watching videos like this. It is about time we had a complete rethink about education and moved forward from the 19th century approach which we are clinging on to.
BEST.MATH.TEACHING.EVER! I always had trouble with math because teachers, even the good ones, couldn't find the time or (in case of the bad ones) didn't have the talent, to explain math in a simple, intuitive way. Breaking down the basic assumptions and connecting those ideas to the big picture was something I always missed and was a main obstacle for me. Remembering math as a collection of rules and tricks gave me a weird distrust for the application of math in new problems, it just never felt quite natural (and was also just hard to remember). Your videos made me like math as I never really could before. Keep up the good work. If you ever need help making more vids just ask your fans, I'm sure you have a lot ^^ - and now you have one more
I literally changed browsers from incognito to my acc just to comment on this video and I usually NEVER COMMENT, EVER. I want to THANK YOU infinitely much for your hard work making this series. It is literally the most well made, visual, audio math explanation video I have EVER seen and i'm sure a lot of people agree. Your approach of explaining it; us being the sort of mathematician and like trying to come up with it, explaining it to us from the inside out is just brilliant, and you nailed it. Intuitive, simple, beautiful visuals, beast voice ... just 5 stars. GREAT WORK, PLEASE KEEP DOING THIS STUFF. Love u.
These 17 minutes gave me more understanding then 3 years of studying calculus at university. Can't be thankful enough, imagine how smart I'll be after all episoeds :>
These video series are just becoming Gems in my learning of ML. When my teacher in college started differentiation he just a book and said memorize. But no, now that I can understand from where they arrive, it;s just GREAT! A THANKS!!! for making these series
I just started my Calculus 1 course at my university, and my professor sent us a link to this series (and to the essence of linear algebra series). I had already watched the series, and now I'm going to watch it again, hopefully, this time I will learn things I missed the first time!
It's great that your professor recommended it........ if I had enough authority, I would have made it compulsory to include these videos as part of all calculus courses
great job! looking forward to witness whole subject of higher math visualized by you, no matter how long it take to be done) It's the moment where all traditional courses lacks whole point of view! That leads to real understanding through dynamic of geometry.
Impressive! My son will be starting college calculus later this month, and we're watching this together as a way to get a head start. The animations are very well done and add a lot for visual learners.
My first ever youtube comment: Thank you so much for these videos! Very rarely nowadays math is taught in such an intuitive way. If only this was how math was taught in school! Keep up your amazing work.
I watched your calculus videos a couple of years back and you genuinely inspired my interest in mathematics. You taught me to look past the formula and into the patterns and i'll be forever thankful for that!
"Whenever you come across a genuinely hard question in math, a good policy is to not try too hard to get at the answer directly since usually just end up banging your head against a wall. Instead play around with the idea." I wish someone had said this to me in college... I would have had a much easier time with calculus.
Can you help me ? 13:57 but doesn't it depend on the angle of the graph (line) to the x axis , I mean it seems to be more correct in this case as it's almost parallel to x axis but not so much for a line which is raising rapidly, even though dx is very small but it is different although slightly but it is, isn't it?? ( I know I am wrong but how someone please elaborate)
@@harsharya545 The dx being a tiny nudge is not a specific value. It's just a very very small amount. So small that no matter what the slope of that line is its still a very small change. Remember the answer gets better and better for smaller choices of dx. So we keep making dx smaller until it bares no significant difference on the answer at the precision we're looking at. While dx is not an infintessimal, it is also not an actual number. It is a concept invented to be the arbitrarily small number represented by a limit approaching zero.
@@TheMagicalTaco I do all my lessons outside of class and ignore the teacher and just do the homework. You learn it easier and instructors are terrible. Also when you try to take notes its difficult.
I study Civil Engineering. Today I was brainlessly doing homework. Determining the center of mass of a circle section. So I had to get the mass first. Without really thinking about it I used polar coordinate double integration on it and ended at R^2*alpha... Then noticed...Wait...that looks familiar... So I plugged in pi for alpha and realized that thats what it would look like if it reached all the way around and....wait...thats the formula for a circle Suddenly i had an understanding of what the fuck I was doing all the time and all clattered pieces that were just formulas in my mind came together in a beautiful moment where I suddenly had determined the size of a circle and actually understood why Thats the beauty of Maths. Its so much fun when you understand what youre doing
Can you help me ? 13:57 but doesn't it depend on the angle of the graph (line) to the x axis , I mean it seems to be more correct in this case as it's almost parallel to x axis but not so much for a line which is raising rapidly, even though dx is very small but it is different although slightly but it is, isn't it?? ( I know I am wrong but how someone please elaborate)
@@harsharya545 You are not wrong in the sense that for relatively "flat" graphs (equations with slow change like (x+a)) the error for this approximation is smaller than for equations with rapid change (like exponential functions, b^2x). But as dx gets smaller, this becomes more or less irrelevant since when dx aproaches zero, soo does the actual change and thus the error also approaches zero, which means a better aproximation. The fact that the error is larger for more steep graphs can thus be viewed as irrelevant for the purposes of what is discussed in this video. Not sure if this was the kind of answer you were looking for :/
@@harsharya545 As the dx gets smaller and smaller, the error gets smaller simultaneously, For very-very small value of dx, Error is soo small like it does not even existed.
I'm so proud to be the 314th like. :) Genuinely hope this comment gets 31415 likes as well. I was going to flex my 9265358979323846 knowledge but let's not go overboard with the likes, I said. Still it's good to imagine.
It honestly takes some great skill to be able to explain everything from such a sueprficially daunting topic while *also* never falling two steps behind on the questions a viewer might want to ask. This man is a legend.
When I learned calculus, it was as a mechanical process. I learned, and applied, the rules. I became quite good at it but never questioned the way it was constructed. Thank you for this. It has plugged a hole I never knew existed until I stumbled upon your channel.
Can you help me ? 13:57 but doesn't it depend on the angle of the graph (line) to the x axis , I mean it seems to be more correct in this case as it's almost parallel to x axis but not so much for a line which is raising rapidly, even though dx is very small but it is different although slightly but it is, isn't it?? ( I know I am wrong but how someone please elaborate)
@@harsharya545 I suppose so, dA as a whole approximated as a rectangle so graph's tiny little curve has been overlooked. Like 1)Trapezoid shape was assumed a rectangle 2)The Rectangles don't fill the graph but about to fill it for dr ->0. There are always assumptions but I'm still on my way to explore it. Cheers
Maybe this is the best video i ever watched on youtube. I'm not a english speaker and I could understand every line and every thought better than any calculus classes i ever had. I already subscribed the channel and I can't wait to see the other videos. And please, I stand for 'Essence of Probability' as well! haha
I left school at 15 to start an apprenticeship in the Navy. Much later as an adult, I started Computer Science undergraduate studies at uni. I had zero exposure to calculus before. Wish I had seen this. It makes sense.
I feel you. After 4 years in Marine Corps intelligence, I got a C in pre-calc and didn't want to even attempt calculus. I changed major because of that, but this makes it look... tolerable.
@@billandpech I did 5 yrs Navy, started college, did precalculus and then did 2 semesters Calculus. Do not give up. I got a C in all 3 courses. That's all you need.
the sync between verbal and visual explanations you have here makes it feel so intuitive and easy to understand - this is awesome!!! Thank you so much for this great explanation!
Funny thing is: I just learned about the fundamental theorem of Calculus last week at College. I already new about it from previous studies, from private teachers or not, but this shows just how high-leveled this intuitive explanation is. You're brilliant.
17 min of this video gave me better insight into Calculus than 17 hours on this subject in school... I'm literally crying at how beautiful this is. Thank you so much, @3Blue1Brown!
Wow! Wow! Wow! I'm an electronics engineer turned Software Development Manager over the course of 20+ years and I decided to go for a Masters in AI, mostly for intellectual interest in the area. So, I started revising my Maths when I stumbled upon this and was left literally speechless with the clarity, insights and clear explanations of the fundamentals of one of the toughest areas of mathematics. How so much fun and productive my uni years would have been had I had access to this type of free (!!!!) content. What difference it would have made in the training of engineers, many of which just developed strategies to squeeze through their exams, never having properly grasped the subject and its relation to their areas! Congratulations and thank you so much!
in my math class we have done nothing but algebra and geometry and when it came time for calculus, our teacher didnt spare any time for introduction. when i asked why the derivative of some function had that result, the teacher just responded: "there we go with those phylasophical questions". im not sure if the problem is in the teacher of the school system but either way im glad there are other sorces where i can learn and not memorise a chart
... He would need to do quite a bit more analysis, to dive into probability. Like what means measure and to get a decent understanding of that topic, he would need to build a little foundation on topology, set theory and rings. I would say, he should first go into topology and measure itself, before starting with something like probability.
PLEASSSE haha I would love to have a great understanding of probability. I was very disappointed by my university's highest level probability class and didn't get much out of it. They just didn't get to in depth.
You are not thinking. He wouldn't teach you like this. He would have many students to teach and focus on. Also he would be restricted to the circulum to teach. Also he has to make sure people remember which in high school and would have to change the way he teaches. Also you are assuming your teachers are bad without thinking how hard there jobs are or the handicaps they have.
00:00 Learn calculus core ideas through visual approach 02:22 Approximate the area of a circle using thin rings 04:22 Approximating the area of rings using rectangles 06:25 The formula for the area of a circle is pi times R squared. 08:42 Many problems in calculus involve finding the area under a graph. 10:46 Finding the area under a graph is a hard problem 12:50 The derivative is a measure of how sensitive a function is to small changes in its input. 14:52 Calculus ties together the two big ideas of integrals and derivatives. Crafted by Merlin AI.00:00 Learn calculus core ideas through visual approach 02:22 Approximate the area of a circle using thin rings 04:22 Approximating the area of rings using rectangles 06:25 The formula for the area of a circle is pi times R squared. 08:42 Many problems in calculus involve finding the area under a graph. 10:46 Finding the area under a graph is a hard problem 12:50 The derivative is a measure of how sensitive a function is to small changes in its input. 14:52 Calculus ties together the two big ideas of integrals and derivatives.
@Michael Gayle either works tbh. If he stuck with calling it a trapezoid then you'd have the height dx and the top and bottom lengths a and b. As dx becomes smaller a and b approach each-other. Putting this in the area for a trapezoid: 1/2(a+b)*h 1/2(2pi*r+2pi*(r-dx))*dx where dx -> 0 1/2(2pi*r+2pi*r)*dx 1/2(4pi*r)*dx 2pi*r*dx So you'd game the same answer.
I’ve loved math ever since my childhood and I’d been out of touch since I started with SW engineering. One of the things I miss is the pure joy I used to savour after running into a little discovery of my own. I just had a similar epiphany when you delineated a way to visualise a function as a rate of change of area under it. For years I’ve been trying to visually relate a function and it’s integral, and your lucid explanation made me truly enjoy this tiny moment. Thank you for the work!
this just showed me how much highschool and uni just doesn't cover the explanations expecting students to understand without question - this changed my view on calculus and how it all works at 1:30 in the morning, thank you so much
Me few weeks ago: has Math homework to do My brain: *Let's procrastinate by watching video compilation of famous authors quotes* Me now: has English essay to do My brain: *This is it! Time to learn the Essence of Calculus*
if only this video was available 10 years sooner, maybe i would survive in my engineering class. Now, i'm stuck doing spreadsheet 9 to 5. Aiming to get master of data science though, and I think this series is a good start. Wish me luck guys
I just finished watching this video, and wow. It's the most entertaining math video I have ever watched. I've never been a math person, and I never really enjoyed learning about the beauty and intricacies of math. This video, however, has changed me. I have never before felt this interested in learning math. This video is so well made, and I'm so glad I was able to see it. You have done a truly amazing job, and have managed to do what I thought was impossible, getting me interested in math. I sincerely thank you for making this video.
I've almost finished an 8-week calc course, it's frustrated me because I couldn't understand why things work and haven't had time to play around with the ideas. The first animations with the explanation of the development of the (pi)r^2 function and the derivative absolutely blew me away. Everything makes so much more sense and I can start to see where it connects. Thank you so much for this video series. I think you just made me love math even more. ❤
was there not this one researcher in some field not entirely within maths that reinvented calculus (or at least integration by summing areas of rectangles), without knowing calculus was a thing ? (it apparently passed peer review too)
it might seem like a lot to say this but this video is so spiritually uplifting. I do want to feel as if I could have come up with these concepts. I'm getting calculus next semester and I'm beyond excited! Thanks for your dedication!
I've just started properly learning calculus at school over the past week. I found these videos at the end of last year, and found them quite interesting. Now, I find them very useful. You explain things so much better than my maths teacher.
This is the best video that explains a complex subject like calculus in such a simple manner. You are a great teacher. I liked when you said "and my goal is for you to come away feeling like you could have invented calculus itself"- great words !!! it will remove a lot of fear for Mathematics. Great work
Im so glad Micheal Stevens from Vsauce recommended this channel a couple years ago when I was only in junior high school. So now at university my brain just remembered this channel from nowhere and I had to look it up! Amazing video! The pedagogy is astounding!
Leading students up to discovering/inventing concepts by themselves is such a progressive approach. Hats off to you, Grant. You make both science and education beautiful!
That’s a pretty uniformed and moronic comment. True teachers crave new ways to present materials. You just sound like a bitter calculus student who struggled to barely pass.
@Davey Jones If this bitter student had trouble understanding what the teacher was trying to tell him, but had no problem understanding this video ... wasn't the teacher inadequate by definition? Probably not for all students, but for some at least ... maybe even for most of them. What you said sounds like an insult to the student, but you are just reinforcing his point. And if you think being bitter is not the adequate reaction ... maybe you should try being less bitter too. Yes, I get it is a joke. But the best jokes always have a grain of truth.
Billions and billions are spent because: #1 teachers need to get paid. #2 buildings don't get built for free. #3 buildings don't get cleaned for free. #4 buildings don't get warmed and cooled for free. #5 Nor repaired for free. I'm sure there's a jillion tiny little other expenses I forgot...
yes, there is a massive cost behind school but it must stand to reason that better facilities such as physical school should be better than a video floating on the internet that you literally pay 10 cents for(if you watch the ads) why it isn't? school has waaaaaaaaaaaaaaayyy more resources than him, and you can argue that he won by the economy of scale (he can reach millions when normal teacher can only reach thousand) but the argument still holds, school has more facilities to use and teacher to ask, why isn't it better
Martin Watson yes they pay their school, that's why it suprise me that it is worse than europian school Paid stuff is usually better but apperently most thing USA is an exception to this concept, expensive stuff that is also not good
going into college and they never take the time to explain it in depth like this because it would take forever and they have to make deadlines for tests and what not. I'm in Calc 2, but it's always good to go back to the basics real quick
Okay... I don't know how I took so long to get here, but you just explained, so elegantly, so beautifully, visually and without random, unnecessary jargon, like the first 5 chapters of most Caluclus books, if not more? This is insane. Thank you so much. I think you're about to save me soooo much time! With this, I'll be able to tackle pretty much all my course's problems. Thank you so much!! I've never had a clearer understanding of the relationship between the area under the curve and the derivative!!
⏱Timestamps for this video! 0:00 - Introduction to the essence of calculus series 1:32 - Introduction to the area of a circle 2:40 - Finding approximations of the area of a ring 4:14 - Finding the area of the circle by adding the areas of many rings 9:47 - Introduction to thinking about integrals 11:53 - Finding a property of the mystery function 13:16 - Relationship between tiny changes to the mystery function and the values of x-squared 13:41 - All functions defined as the area under some graph have a special property 14:06 - Derivatives 14:30 - The importance of derivatives in solving problems 15:13 - The fundamental theorem of calculus 15:40 - A high-level view of calculus 16:02 - A thank you to supporters 🧙♂✨ Generated with Houdini Chrome extension.
At 5:24 , How do you know the concentric circumferences form a straight line when you stack them up side by side? Each subsequent circle is 2*pi*dr longer than the previous one. Since thickness dr is constant throughout this, so is the angle of elevation between two subsequent slices. This makes the line connecting the tip of the slices straight and gives us a nice triangle to get our formula :)
thank you for your addition. One really can see how this difference of lenght between slices is constant, and when you find constants on math the path clears, it gets way easier to find ways to take advantage of that constant and to discover your formula. Very nice.
I'm heading back to school in January to finish my engineering degree and I'm particularly worried about the maths. From the way math is taught to us in school (do this, do that, calculate the exact value, any deviation from the answer is wrong) it seems like an oxymoron that one can "play" with math, and yet this video just showed me clear-as-day that true understanding comes from the ability to hold the ideas with a degree of flexibility. I hope that I can stretch my mind enough to break the rigidity of my foundational maths education and be able to think about these problems the same way you're explaining. Looking forward to the rest of the series, thank you for making this!!
I'm 16, I really love to prove many theorems with logic or axioms, not only memorize it. With your channel, it helps me to know how math works. I really appreciate it.
I'm also 16 and I try to do this too, but damn, my brain can't even go 2 steps into a single idea I have for such proof, I can't do nothing and honestly I'm not atracted by the idea of using paper to write down those things either, I know it's the logical thing to do, but it just doesn't feel right for some reason.
@@eterty8335 if you want to learn and understand more about mathematical thinking I would recommend you to read the book "Discrete mathematics with applications" by Susanna Epp. It's a awesome book and it covers since the beginning of logical thinking, which is great for people like us, who don't understand much of this subject. Btw I found a pdf version of it for free on the internet, so if can't buy the book you only have to search for its pdf and study using it.
I haven't studied traditional math since highschool about 12 years ago, but I've done plenty of programming since. So returning to this format is rather challenging, but I figure this series is a good place to jump back into it. Your explanation of PiR^2 broken into the area of a triangle blew my mind. It was like i suddenly saw the whole thing in 3 dimensions, and finally that formula MAKES SENSE. I'm sure other parts will have their own "Ahah" moments, but I just want to say thanks for all your incredible work.
A stunningly good explanation of basic calculus in less than an hour. Hopefully math professors everywhere will watch these videos to better understand what it means to be a teacher.
Thank you so much for this series (and others). I never went to college, and my highest level of math in high school was technically pre-calculus, but honestly geometry. It is difficult for me to retain information that doesn't apply to my everyday life, so I didn't try to retain it then. Now, I am a data scientist. These concepts matter to me now, so I am motivated to learn them. This series has given me the confidence to approach these subjects on my own. Again, thank you so much!
When you showed the (A(3.001) - A(3)) / 0.001 I instantly understood lim(x -> 0) (f(x + h) - f(x)) / h Grant has such a fundamental understanding of math and we're lucky to have him guide us through it. Thank you
When I was in high school, my friend's dad, a biologist, told me that he integrated graphs by cutting them out and weighing them on an analytical balance. (It was quicker than walking across campus to the computer center or using a calculator.) That was more effective than the hours of effort my calc teacher put into explaining it. Sometimes it just takes another perspective, and we didn't really have anything like this video back in the day. Mechanical Universe, I guess.
@@raymondfrye5017 don't generalize. I personally love physics, and am currently studying engineering at university. I didn't do physics in high school because of how infamous our teacher was for making the subject boring. I forced myself to learn physics myself instead, because I still wanted to keep up with physics, but I didn't want to do it at school. Schooling =/= education I was interested. There are kids like me. Rare, perhaps, but still existent.
@@raymondfrye5017 Teachers suck at making it interesting. They do not combine our different interests with our long term wants acting as if we are all the same kinds of people that work in the same way, thus making the class fairly unengaging and boring.
my whole life my math teachers have taught only the formulas and spent all class doing fentanyl and I never understood the math until I cam across this video. Your video enlightened the neurons in my brain and gave me a new will to live along with curing my fentanyl addiction. If it weren't for you I would have never escaped my junkie math professor's calc class. Much love, - a concerned mother
Sounds like things were really hard. Jello can come in a lot of colors. Just like pills. Get them mixed and you’ll be drying your clothes with the moon light. -anna (Luisiana swamp gater)
Thanks! I'm a math tutor, and I must say that 3Blue1Brown is a remarkably interesting, insightful series. Thank you, Grant.
I'm a high school student from Brazil, and I can say that you've inspired a generation of people to pursue math and physics for their own sake. I started to learn calculus through these videos 4 years ago and now I'm studying for the AP Physics C exams (wish me luck lol). I know you'll probably never see this, but if you do, just know that we really appreciate your work.
Boa prova mano (se já não fez kkk)
Good luck buddy
Como assim tem AP no Brasil?
How did the exam go? I hope well :)
Excellent!
How I wish I'd had access when I was in high school and college to this information. We were 'taught' by Rote.. but it never made sense to me because the Purpose was hidden away..
Good luck with your studies!
Blessings!
As a school going kid, I always wanted to visualize math like this animation. But my teacher could never provide with any material like this. Now, several years after finishing my bachelors I can finally feel these ideas for real. Thanks a lot for the hard work and this fantastic motion animation. I wish, in future you will cover each and every aspect of math, science and technology. The world needs these kind of content more and more for the people to better understand the underlying rules of our world. Cheerio from Germany. :)
Calculus was the definitely the strangest thing I had to go through during college and I feel silly after watching this series. Such elegance and I missed it completely.
Same feelings for me and all people like us who need to visualize in order to fully understand the nature of things.
Jawad Tahmeed
I agree visualizing is the best tool
just did 900th like to your comment. :)
ich weiß nicht, aber unser Lehrer hat uns das nicht so sehr vor Augen geführt, vor allem nicht mit Kreide und Tafel. Aber trotzdem hat er es irgendwie genauso heruntergebrochen mit Worten. Man musste dann natürlich das ganze selsbt machen, und da kannste ja dir alles vorstellen. Aber an alle Kernaussagen hier in dem Video erinnere ich mich. Komisch.
“Math has tendency to reward you when you respect it’s symmetry” - has easily became my favorite math quote!
That quote is very relatable with my work.
Very true. It's thankfully easier considering we can work with geometry where all objects have equal height which thankfully allows us to calculate a change 'dr'
Yess.. Perfect lines said..
If it's your favorite quote then maybe you should use the correct "its".
Great to hear
Can we all appreciate how much effort Grant puts into the animations in his videos so we can understand?
of course
It's Python! He animates everything using python! SOOO COOLLL!
A python tool he created btw
Actually!!!
@@adarshas8454 really? Which one is that?
My calculus professor usually just says “lol here are some formulas and rules to remember”. Higher education at its finest.
Everywhere to be honest not only the US. Its damn sad.
Im a physics student and I had to learn calculus like this. But when calculus was used in QM and EMT I shit my pants cause I couldnt picture anything. Nothing made sense. Now im going through calculus again by myself. -__-
@@OmarChida Not where I am studying. Where do you get these garbage professors? Even online courses in MIT do not do this.
@@RanEncounter why would mit even do this they have a face to save, greedy school and way worse in college hire teachers at low cost and are generally of these sort
@@kaus05 Have you even looked at the lectures I mentioned before you commented?
This video blew my mind. I took calculus in college 45 years ago, and it nearly killed me. Dropped the class a couple of times before finally squeaking by with a sufficient grade to satisfy my geology degree. I haven't had much need to apply calculus since, but my lack of understanding has always bothered me. Our professors just blew through it so quickly, I never had a clear understanding about why some factors were important and others could be diminished and discarded. Grant's simple explanation of the area under the curve equating to the piRsquared formula actually caused me a physical reaction! Why hadn't that been shown in any of my classes? Now, at age 67, math comes a little harder for me, but I still plan to work through this entire series!
In a crowded field, Grant has the best maths channel on TH-cam. I highly recommend going through all his videos. 'PBS Infinite Series' and 'Mathologer' are also great
Man I’m half a century behind you, I have to take this next semester! Would be nice to have a professor as good as 3b1b, but I’ll probably just have to settle for more videos
never too old to learn :D
Good for you, man!
Psalms 34
I'm almost 30. Never in my whole life I had calculus explained to me in such an intuitive (and therefore effective) way. I'm watching the whole series and taking notes. Just priceless. Thank you!
How have you been taking notes? Please guide me.
@@ananyasaha12345 Like it was an online course, drawing most of the graphs and writing down most of the equations and conceptual analyses. I use a tablet but pen and paper would be roughly the same. I think it took a bit too much time but I like the result.
Same. None of the calculus books I've studied explain it so effectively. So grateful for this content.😊
I'm 36 now and doing the same.
@@juancharovsky252people often tell me that as they became adults, they lost the ability to learn new things without a lot of effort. Is that the same case for you? From when you were a child and now, is there a drastic difference?
Brilliant: not just "what's the formula" but the deeper understanding of "how they figured out the formula"! This is the way to learn not only math but how to do analytical thinking!
correct
You can learn how to create a specific computer model, but if you don't know the computers parts work, you cannot improve the design.
@@H3XED_OwO the point here is to learn about the model and not to recreate it. These videos help in sparking the fire of curiosity and its the viewers choice how far they wanna take things. If you are really interested then after learning the model you will march forward and learn everything there is to recreate the model even if you have to learn whats under the hood
@@bavidlynx3409, I think he was agreeing
A much clearer way to phrase it is: "how to think analytically". How are you not embarrassed to have such clumsy and sloppy thinking?
"Math has a tendency to reward you when you respect its symmetries."
"Mathematicians don't just care about finding the answer but developing general problem solving tools and techniques."
"Transitioning from something approximate to something precise is pretty subtle and it cuts deep to what calculus is all about."
As much as I love your clear cut explanations of the topic - these hidden gems of wisdom is what attracts me to the videos you create and helps me build an intuition around the subject which my university professors failed to. This is the sort of insight which is developed when you've spent enough time with the subject and examined it thoroughly from all perspective. I am very grateful of the effort you've put into making these series' and I will be coming back to it from time to time to jog my memory.
are u a JEE aspirant?
@@abhaysharmafitness I love 2 play with my little stink hole it's kind of itchy bc I bleed when my turds are too large in diameter but it just feels oh so wonderful boy
@@abhaysharmafitness i am a JEE aspirant
💪😭
My issue with college courses, very few of them you can see applications of and therefore never stick with you after graduation. It was only when I started working as an engineer I began to see some math concepts coming back and me going back to refresh my memory.
"Math has the tendency to reward you when you respect its symmetry" - Grant Sanderson
My new favorite Grant quote!
Bharath Bhat I was going to say the same thing... brilliant!!
That is an excellent quote.
"The character of a country depends upon the racial character of the men and women who dominate it."
-Madison Grant
My favorite Grant quote.
Bizarre notion that a abstract thing can reward someone: sounds like a human need projected onto an abstract thing - sounds like childhood trauma of longing for respect and love from parents... Math doesn't even know you, let along reward you, it is what it is.
It’s so weird seeing favourite without a u, but I guess that’s just the American spelling.
Coming to this video after my final class in my first calculus course is mind blowing. Spotting the connections earlier on would’ve saved me so much sleep. This is an amazing video thank you!
ho
This is a fantastic lesson. It is interesting that he starts with areas and integrals. All calculus course start with differentials followed by integration. This approach, with its simple clear explanations building up from basic math gives the student a clear explanation of the processes involved. It really accomplishes its goal of presenting the material so that the student says, "I could have invented Calculus."
I have never seen anyone start with the circle and turn the areas into small rectangles before. It is a great starting place, and accomplishes multiple goals. The students learn the thought processes used to develop mathematics in general and Calculus specifically. They gain deeper insights into how the integral is developed and it relation to real world scenarios. And they learn why the formula for the area of circle looks like it does as an added bonus.
Understanding why the calculations are done the way are and how they relate to real world problems give the student the deep understanding of Calculus so often lacking in traditional classrooms where the teacher just drones on about equations and applications.
I wish math teachers did better jobs of teaching calculus in this manner because it would have made my job of teaching Physics so much easier. As it was, I had to teach Calculus before and alongside Physics. I had a policy of teaching the students how the equations they used were derived and how they related to the real world situations they described. Just like this video, I started with simple concepts and built on them until we arrived at the result we wanted. It was more about the process and understanding of fundamental concepts than the final answer.
I have not seen any other episodes, but if they are all like this, I would recommend using this series in place of the distance learning now going on during the Covid-19 crisis.
Wayne Adams
B.S. Chemistry (ACS Certified)
M.S, Physics
R&D Chemist 9 yrs.
Physics Instructor 33 yrs.
Thanks Wayne
Thank you Wayne for your extremely insightful comments here as I start to enter this world for the first time at 65 to help my son see the deep beauty of math. When experts like yourself write these comments it just inspires us to find more meaningful ways to learn and to understand.
He also has a linear algebra series.
Thank you for this - I am a distance learner through Open University and luckily my tutor uses the same approach. Seeing the videos really helps me - and I can run the visuals through my head. Wow, 33 years of teaching experience - that is quite an endorsement! I dropped math in Grade 11 and regret - in my 40s now and using the pandemic time to catch up.
yessirrrr
I'm a Computer Science student at University, and I've watched your videos on Linear Algebra and now this series. I cannot overstate how much of an impact these videos have had on my learning. I have genuinley learned more from you than my lecturers.
I just broke down in tears. Why isn't math taught this intuitively at all levels, at all schools, by all professors? Math is so beautiful; It is upsetting it took me so long to realize it.
P.S. Grant, thank you for your earnest passion for teaching and mathematics, you've introducing me to world I did not know existed!
Because it is difficult to teach math. What 3b1b is doing is incredibly impressive, and there are simply not that many people that are able to break it down as well as this.
Can you please help...
13:57 but doesn't it depend on the angle of the graph (line) to the x axis , I mean it seems to be more correct in this case as it's almost parallel to x axis but not so much for a line which is raising rapidly, even though dx is very small but it is different although slightly but it is, isn't it?? ( I know I am wrong but how someone please elaborate)
bro calm
@@harsharya545 going back the the triabgle shaped graph at the beginning, as dx gets smaller the left and right side of the thin rectangle get closer in height to the point where slant of the graph is negligible
very true
I’m currently taking calculus in high school and I can confidently say, they do not teach you how to understand the concepts like this video does. In ten minutes, I was able to fully understand the countless hours of plugging in numbers my teachers had me do. I am deeply grateful for this video.
When I took AP Calculus I was able to pass the exam but understood nothing I was doing haha- I just was taught to get the answer.
What other Math topics are covered in your syllabus?
@@manahil558 He pretty much covers them all in this series
The reason I like math is because I’m obsessed with using logic to find answers rather than memorizing. That’s why I loved these type of videos.
Winter Hippo same,
I also like physics
Agree. Logic to derive, in my opinion, is the understanding at it finest. A computer can calculate...it takes a consciousness to UNDERSTAND...to comprehend MEANING. This is overlooked these days.
Exactly! Logically understanding how math and physics work the way they do >>>>> cramming formulas
How do you feel about quantum mechanics then ? If you are a true beaver it will drive you nuts cause you can't find the missing stick....
Yeah thats great and all, but after the first couple of weeks in maths (University) you will have to memorize a lot too to formally prove things... unless you always want to start from scratch and that won't work out very well
this just proves that maths isn't boring, unenthusiastic teachers are boring. Literally 25 minutes ago I was in a lecture being taught this and almost falling asleep, watching this, it's literally like a completely different subject.
sometimes it's because your subconscious "recorded" the lecture for you even if you weren't consciously paying attention so when you watch something related to that lecture, you'd tend to understand it better.
that or it's because you actually wanted to watch this vid instead of the boring lecture. sometimes wanting to learn or do something makes us pay more attention to it because we have a genuine interest rather than being obligated to learn or do that thing.
@@unrested7294 Yes. Repetition is the mother of all mastering
@@LM-he7eb Repetition is based on rote memorization, not actual understanding.
@@KameraShy Not entirely.
Sometimes through repetition you fill blanks that were left on first attempt.
I hear you though
In many Western countries we spend a lot of money on fairly mediocre teachers. When I see material like this I wonder why we waste our money on a lot of the weaker teachers we employ. Kids would learn so much more from just watching videos like this. It is about time we had a complete rethink about education and moved forward from the 19th century approach which we are clinging on to.
Dude. AP Calculus BC Exam is 11 days away. You're saving lives out here.
Art Schell - Underrated comment
pretty much. I felt like I was going to die, even with my calc teacher loading a bunch of TI Nspires with programs to help us halfway skate through it. I feel like we have hope now
Tristan Saldanha unfortunately, the series won't be finished in time for my physics exam
Tristan Saldanha i dont think hes going to hit on bc topics much. looks like mostly ab, he did mention taylor series though so idk
proffessor leonard will save yo ass. just make sure to do the practice problems along with the video.
5 years have passed but this video still brings inspiration to many to love calculus even now❤️
for sure.....
does anyone know the name of the song that played at the intro
thats gay
@@QmVuamFtaW4 Definitely gay, but not the Homosexual sodomy type,
@@martinladley lmao
A few years ago, I got a bit bored, so tried my hand at deriving the formula for the area of a circle. I got there by slicing it into smaller and smaller pie sections where each pie section approached a right-angled triangle with the short side being some fraction of the circumference, and the long side being the radius (radius and hypotenuse would approach equality as the angle got smaller and smaller too). So the area of the triangles would be 1/2(r.((pi.2r)/x)).x with x being the number of slices. Simplified out it became pi.r^2 :D
Just this week I got bored again... and started wondering how pi got calculated - had some fun working that one out!!! Got to pi=(360/x).sin(x/2) with it getting more accurate as x approaches 0 :D
Sure, I could've just googled them, but it was so rewarding to have that AHA! moment when I figured it out! I'd encourage everyone to try to figure things out themselves every once in a while, because that is a very special feeling!
I aspire to work like this.
Similarly, but less interesting, is the insight that the golden ratio is (1 + sqrt(5))/2...
Technically using sin(x) to calculate pi is cheating
How long did it take?
@@idiotidiot2805 Not terribly long, maybe an hour or so
My good friend. You have been more influential in my life than my father my dog and all my teachers of the past combined. I won’t ever meet you but I embrace you with gratitude and respect. Thanks
Wow that's... More influence than dad and teachers, ok I guess. More influence than doge tho? I mean that's... ruff.
+Guillaume Perrault Archambault LMAO
+aboctok I hop you spilt too wrong on porpoise.
@@vwlz3603 lol
THIS MAN IS A GODDD
Interesting, I've never seen integration being taught before differentiation. I wasn't feeling super crazy about this video at first, but when you got to dA over dx I was genuinely impressed.
As History of Mathematics says, Newton found the derivative first mostly because your intense concerns about motion and derivatives is really related to that (and as well he did claims a special operator for it, the upper dotted notation). However, Leibniz found first the integral and later the differentiation!
Integration was generally discovered before differentiation - some integration processes (more aptly called "method of exhaustion") appeared even in ancient Greece (thanks to Eudoxus and Archimedes.) If I recall correctly, both were developed before limits were a well defined and established method of understanding derivatives/integrals as well!
Thanks for clarifying! I really wished my university offered a history of mathematics class since I know a huge number of mathematicians without actually knowing anything about them (Leibniz notation, Euler's constant, Simpson's Rule, etc).
Integration is presented first in the book by Apostol.
The quote "The art of doing mathematics is finding that special case that contains all the germs of generality." really touched the deep of my heart. Excellent deep meaning in that. Thanks Grant for the wonderful work that you put in
Mister 3Blue 1Brown you deseve a nobel, for real! I don't understand why universities don't teach math like you do here. You make it look so easy and instresting. We have somehow to promote those videos on every univestity in planet. I hope you'll never loose interest in making those amazing videos. You make the world better. (just a guy from Greece)
Dimitris Tzimikas too bad there are no nobel prizes for mathematics!
There's Abel prize for maths which is equivalent to Noble prize in maths....
Moreover your forefathers from Greece are responsible for the mess that is caused today in maths...how unnecessary and complicated have your ancestors from Greece have been...they should have been very easy and simple...they would have been really honoured by millions of students across the globe ... instead of abusing cus words🤣😃😅😄😉😋😆🤔🤗😝😜😛😝😝😥😥😓😐😶🙄😏😣😣
This is probably my favourite moment of this week
Ilia Boitsov yup
Me,too.😊
Thank you for making this video. As an AP Calculus teacher I love it! Was looking for something interesting to suggest to my students as we work from home during Coronavirus. Thank you!
as an engineering student working my way through an accelerated calc sequence having taken 0 calc in HS, this video is filling in the gaps for me with these visual aids. your answering all the side questions I wondered and small mistakes in my ideas as you go along. (and of course the prof is too busy to answer while they speed through problem explanations.)
amazing supplemental videos here! still holding up 5 years later. it's the high level concepts & gap-fillers tips/hints I knew I needed but couldn't find.
I studied up to advanced calculus. But it was never this well explained nor intuitive. My grandchildren will have it easier. Thank you.
I'm 15 and your essence of calculus series has really helped me understand calculus before school even started teaching it. Teachers in the lessons i've attended as extra activities in the past just made me remember the power rule and i thought it was the definition of a notation called dy/dx, which I was quite wrong with. Thanks to your incredible visualization and awesome method of teaching I'm now fully prepared to start doing questions from school and exploring further!
Yeah, same
Yeah, same
Yeah, same
14 year old here- same
Yeah, same
Wow, I’m stopping half way through this video as I have to comment on how well done the animations are and how perfectly in sync they are with the explanations.. Really amazing work all around. Thanks so much for making this content freely available!
Thank you (and your team if any) SO MUCH for this super intuitive visual explanation!
But I also have a question:
when lookin at the yellow approximation rectangle (at 12:21), I though about something; if the smaller the rectangles are, the higher their more precise their approximation, would those be sufficiently precise for all applications of this math where absolute precision is needed as the rectangle "width" approaches 0? Or would it then (for absolute precision) require to also consider the adding to the rect. area the area of the triangle the forms (in this case) above the rectangle? Or how else would one go about determining the absolutely precise/correct values here?
Thank you! To your question, it's worth keeping in mind that these rectangles (known as Riemann rectangles) are not necessarily to directly approximate the given area. In this video, the serve primarily as a conceptual tool to explain the key idea of how integrals and derivatives are opposites. That is, how if you want to know the area under a curve (in full unapproximated precision), you can reason you're way to it using a knowledge of derivatives. This is covered more later on in the series.
That said, sometime you _do_ approximate areas under curves with an approach like this, and you're question about quantifying that error becomes very pertinent. Later in this series, in the video on Taylor series, there's a discussion involving a role that triangle plays which you may enjoy.
You know, it's funny. I'm a math major and I'm already well beyond the calculus subset of math classes for the collegiate level - I felt I had a solid, *solid* grasp on this content. I mostly clicked on this video because 3Blue1Brown made it and his videos are just amazing - my linear algebra professor was terrible at giving us intuition, even if the calculations were rather easy, so his essence of linear algebra playlist actually helped me understand a *lot*. I honestly didn't expect to come here and actually get a key bit of intuition that I've been missing for a little while - well, not so much "missing" as much as I looked it over because my AP Calc class was a little rough on me initially. ^_^;
Keep up the good work, 3Blue1Brown. I look forward to your future videos in this playlist and hope they can be even more enlightening for me. :)
Thanks so much Daniel, I'm glad you got something out of it. I always find it challenging to target a math video to a wide variety of backgrounds, so your words mean a lot to me.
Im in college right now, and exactly the same thing goes with me. I've finished my algebra and calculus courses, but came here to become more intuitive about these subjects.
same here, i am a physics student(european student, i don't know if its the same) and i had a realy blunt way of doing mathematics, untill i started watching you're videos. thank you so much, you opend new dimensions for me.
I've now taken five Mathematics units at university on my way to becoming a Maths and Chemistry teacher. It blows my mind that I'm paying thousands of dollars for my 'education' when there is far superior content on TH-cam that is essentially free. The worst part of it all is I've scored really well on all of my exams without truely understanding some basic calculus concepts. I've since forgotten most of the content from only two terms ago and I'm starting to realise it's because I've been remembering rules and not attaining a deep foundational understanding which cements it all together. I'm mad because it's not even my fault. I'm clearly striving for proficiency (I'm sitting here, in my own time, watching a Maths video while eating breakfast) but my university never explained calculus this well, or any Math concepts in fact. Grant, as soon as I have the means I will be supporting you and other cannels on Patreon so you can continue to produce quality content.
Can you help me ?
13:57 but doesn't it depend on the angle of the graph (line) to the x axis , I mean it seems to be more correct in this case as it's almost parallel to x axis but not so much for a line which is raising rapidly, even though dx is very small but it is different although slightly but it is, isn't it?? ( I know I am wrong but how someone please elaborate)
@@harsharya545 Even if the graph were to be rising rapidly, if you get an infinitesimally small dx you will get values that are very precise and can account for even the steepest of slopes. Yes, the example shown on 13:57 shows a big dx and maybe that would not work with that graph in the right extremities where the graph is increasing rapidly, but he only showed this big of a dx for visual purposes. He could have used a very small dx for that parallel section to be more accurate as well. Dont know if this clarifies much, but yeah
most of us have done the same thing ,just memorise it ,never understood it , the fault lies with the teacher too .
How convenient, I’m watching this video while eating breakfast too
In many Western countries we spend a lot of money on fairly mediocre teachers. When I see material like this I wonder why we waste our money on a lot of the weaker teachers we employ. Kids would learn so much more from just watching videos like this. It is about time we had a complete rethink about education and moved forward from the 19th century approach which we are clinging on to.
BEST.MATH.TEACHING.EVER!
I always had trouble with math because teachers, even the good ones, couldn't find the time or (in case of the bad ones) didn't have the talent, to explain math in a simple, intuitive way. Breaking down the basic assumptions and connecting those ideas to the big picture was something I always missed and was a main obstacle for me. Remembering math as a collection of rules and tricks gave me a weird distrust for the application of math in new problems, it just never felt quite natural (and was also just hard to remember).
Your videos made me like math as I never really could before.
Keep up the good work. If you ever need help making more vids just ask your fans, I'm sure you have a lot ^^ - and now you have one more
NORMIE.
Thank you for this amazing series.
No amount of thank you will do justice to this series.
Yes 👍
I literally changed browsers from incognito to my acc just to comment on this video and I usually NEVER COMMENT, EVER. I want to THANK YOU infinitely much for your hard work making this series. It is literally the most well made, visual, audio math explanation video I have EVER seen and i'm sure a lot of people agree. Your approach of explaining it; us being the sort of mathematician and like trying to come up with it, explaining it to us from the inside out is just brilliant, and you nailed it. Intuitive, simple, beautiful visuals, beast voice ... just 5 stars. GREAT WORK, PLEASE KEEP DOING THIS STUFF. Love u.
Does that mean that I just read one of the very few comments you've ever posted? Wow, I feel special now
Incognito mode doesn't stop google from tracking what you view on their website...
Porn and math best combo ever
@@banemen27 maybe he felt this was easy to explain to his mom. He just had to switch the tabs.
@@Rojoyerf How can google track in incognito. You are not even signed in. There are no saved cookies or history.
These 17 minutes gave me more understanding then 3 years of studying calculus at university. Can't be thankful enough, imagine how smart I'll be after all episoeds :>
LOL I'm 12
mario mario cool?
Yeah this video gives us an other view of calculus like we invented it. Like if we invented it we should understand it!
Which university lol
Using derivatives you will be able to collapse the economy again.
These video series are just becoming Gems in my learning of ML. When my teacher in college started differentiation he just a book and said memorize. But no, now that I can understand from where they arrive, it;s just GREAT! A THANKS!!! for making these series
This channel is like listening to Mozart.
EagleSlightlyBetter and watching him too
"Screw Mozart" - Salieri
Or Bach
You should watch sciencephile AI
Yup
I just started my Calculus 1 course at my university, and my professor sent us a link to this series (and to the essence of linear algebra series). I had already watched the series, and now I'm going to watch it again, hopefully, this time I will learn things I missed the first time!
It's great that your professor recommended it........ if I had enough authority, I would have made it compulsory to include these videos as part of all calculus courses
Next video will be on "The paradox of the derivative". You can follow the full playlist here: th-cam.com/play/PLZHQObOWTQDMsr9K-rj53DwVRMYO3t5Yr.html
3Blue1Brown Euler or gauss?
great job!
looking forward to witness whole subject of higher math visualized by you, no matter how long it take to be done)
It's the moment where all traditional courses lacks whole point of view!
That leads to real understanding through dynamic of geometry.
I love your videos, I've never had calculus in school.
This is super interesting :)
3Blue1Brown Amazing work!!!!!!!!
and next month we will dive into differential geometry and measure theory! :D
Would have never imagined myself crying because of the eloquence of one's teaching.
Amazing video!
Impressive! My son will be starting college calculus later this month, and we're watching this together as a way to get a head start. The animations are very well done and add a lot for visual learners.
Just dont force ur son and be a helicopter parent lmao
other than that u good fr!!
My first ever youtube comment: Thank you so much for these videos! Very rarely nowadays math is taught in such an intuitive way. If only this was how math was taught in school! Keep up your amazing work.
I said it before and I'll say it again: best educational channel ever.
* there's always a prime between _n_ and _2n_
Jam where n is an integer
What if n is 1?
not an integer. n must be natural and it's inclusive until n=2.
Is 'There's always a prime between n and 2n' inclusive? Because it'd still work for 1 if it were inclusive.
I watched your calculus videos a couple of years back and you genuinely inspired my interest in mathematics. You taught me to look past the formula and into the patterns and i'll be forever thankful for that!
"Whenever you come across a genuinely hard question in math, a good policy is to not try too hard to get at the answer directly since usually just end up banging your head against a wall. Instead play around with the idea."
I wish someone had said this to me in college... I would have had a much easier time with calculus.
True
.
Can you help me ?
13:57 but doesn't it depend on the angle of the graph (line) to the x axis , I mean it seems to be more correct in this case as it's almost parallel to x axis but not so much for a line which is raising rapidly, even though dx is very small but it is different although slightly but it is, isn't it?? ( I know I am wrong but how someone please elaborate)
@@harsharya545 The dx being a tiny nudge is not a specific value. It's just a very very small amount. So small that no matter what the slope of that line is its still a very small change. Remember the answer gets better and better for smaller choices of dx. So we keep making dx smaller until it bares no significant difference on the answer at the precision we're looking at. While dx is not an infintessimal, it is also not an actual number. It is a concept invented to be the arbitrarily small number represented by a limit approaching zero.
Me: "What's a derivative?"
My calculus teacher: "Derivative is Derivative!"
Can't relate more
Its like im right back in the classroom again. Man.
I'm glad I saw this video before I got to calculus in school. Now I won't be drowning in confusion
@@TheMagicalTaco I do all my lessons outside of class and ignore the teacher and just do the homework. You learn it easier and instructors are terrible. Also when you try to take notes its difficult.
@@gregorybattis9588 thanks for the info, I'll remember that when I'm a senior
I study Civil Engineering. Today I was brainlessly doing homework. Determining the center of mass of a circle section.
So I had to get the mass first. Without really thinking about it I used polar coordinate double integration on it and ended at R^2*alpha...
Then noticed...Wait...that looks familiar...
So I plugged in pi for alpha and realized that thats what it would look like if it reached all the way around and....wait...thats the formula for a circle
Suddenly i had an understanding of what the fuck I was doing all the time and all clattered pieces that were just formulas in my mind came together in a beautiful moment where I suddenly had determined the size of a circle and actually understood why
Thats the beauty of Maths. Its so much fun when you understand what youre doing
Can you help me ?
13:57 but doesn't it depend on the angle of the graph (line) to the x axis , I mean it seems to be more correct in this case as it's almost parallel to x axis but not so much for a line which is raising rapidly, even though dx is very small but it is different although slightly but it is, isn't it?? ( I know I am wrong but how someone please elaborate)
@@harsharya545 You are not wrong in the sense that for relatively "flat" graphs (equations with slow change like (x+a)) the error for this approximation is smaller than for equations with rapid change (like exponential functions, b^2x). But as dx gets smaller, this becomes more or less irrelevant since when dx aproaches zero, soo does the actual change and thus the error also approaches zero, which means a better aproximation. The fact that the error is larger for more steep graphs can thus be viewed as irrelevant for the purposes of what is discussed in this video.
Not sure if this was the kind of answer you were looking for :/
@@johnnygustafsson525 Great answer!
@@harsharya545 As the dx gets smaller and smaller, the error gets smaller simultaneously, For very-very small value of dx, Error is soo small like it does not even existed.
I'm so proud to be the 314th like. :)
Genuinely hope this comment gets 31415 likes as well. I was going to flex my 9265358979323846 knowledge but let's not go overboard with the likes, I said. Still it's good to imagine.
It honestly takes some great skill to be able to explain everything from such a sueprficially daunting topic while *also* never falling two steps behind on the questions a viewer might want to ask. This man is a legend.
When I learned calculus, it was as a mechanical process. I learned, and applied, the rules. I became quite good at it but never questioned the way it was constructed.
Thank you for this. It has plugged a hole I never knew existed until I stumbled upon your channel.
Can you help me ?
13:57 but doesn't it depend on the angle of the graph (line) to the x axis , I mean it seems to be more correct in this case as it's almost parallel to x axis but not so much for a line which is raising rapidly, even though dx is very small but it is different although slightly but it is, isn't it?? ( I know I am wrong but how someone please elaborate)
@@harsharya545 I suppose so, dA as a whole approximated as a rectangle so graph's tiny little curve has been overlooked. Like 1)Trapezoid shape was assumed a rectangle 2)The Rectangles don't fill the graph but about to fill it for dr ->0. There are always assumptions but I'm still on my way to explore it. Cheers
Maybe this is the best video i ever watched on youtube. I'm not a english speaker and I could understand every line and every thought better than any calculus classes i ever had. I already subscribed the channel and I can't wait to see the other videos. And please, I stand for 'Essence of Probability' as well! haha
I left school at 15 to start an apprenticeship in the Navy. Much later as an adult, I started Computer Science undergraduate studies at uni. I had zero exposure to calculus before. Wish I had seen this. It makes sense.
Please don't build the Deathstar after learning calculus Darth!
I feel you. After 4 years in Marine Corps intelligence, I got a C in pre-calc and didn't want to even attempt calculus. I changed major because of that, but this makes it look... tolerable.
@@billandpech I did 5 yrs Navy, started college, did precalculus and then did 2 semesters Calculus. Do not give up. I got a C in all 3 courses. That's all you need.
@@brendandrury2177 Thanks, good advice for a younger person.
the sync between verbal and visual explanations you have here makes it feel so intuitive and easy to understand - this is awesome!!! Thank you so much for this great explanation!
Funny thing is: I just learned about the fundamental theorem of Calculus last week at College.
I already new about it from previous studies, from private teachers or not, but this shows just how high-leveled this intuitive explanation is.
You're brilliant.
17 min of this video gave me better insight into Calculus than 17 hours on this subject in school... I'm literally crying at how beautiful this is.
Thank you so much, @3Blue1Brown!
Wow! Wow! Wow! I'm an electronics engineer turned Software Development Manager over the course of 20+ years and I decided to go for a Masters in AI, mostly for intellectual interest in the area. So, I started revising my Maths when I stumbled upon this and was left literally speechless with the clarity, insights and clear explanations of the fundamentals of one of the toughest areas of mathematics. How so much fun and productive my uni years would have been had I had access to this type of free (!!!!) content. What difference it would have made in the training of engineers, many of which just developed strategies to squeeze through their exams, never having properly grasped the subject and its relation to their areas! Congratulations and thank you so much!
in my math class we have done nothing but algebra and geometry and when it came time for calculus, our teacher didnt spare any time for introduction. when i asked why the derivative of some function had that result, the teacher just responded: "there we go with those phylasophical questions". im not sure if the problem is in the teacher of the school system but either way im glad there are other sorces where i can learn and not memorise a chart
Do you learn derivatives at school?
@@ElioSch1423 everybody does. In my country, India, for example we are taught it in the final year of high school.
@@thesocialpi9451 I asked here, and o found out high school teach derivatives here too, but public schools here are pieces of shitt by the way.
the problem is not the school system (which also) but your teacher. a man who loves maths would NEVER give that answer to the question
little does he know that philosophy is the mother of all science and intelligence, without it there's no motivation to learn anything.
I love when people put effort into their videos.
Make the "Essence of Probability" after!!! I love your animations!
I'm totally with you there. Probability is imo a very interesting, but also pretty hard field of mathematics. I would love to see a series on it.
... He would need to do quite a bit more analysis, to dive into probability. Like what means measure and to get a decent understanding of that topic, he would need to build a little foundation on topology, set theory and rings.
I would say, he should first go into topology and measure itself, before starting with something like probability.
PLEASSSE haha I would love to have a great understanding of probability. I was very disappointed by my university's highest level probability class and didn't get much out of it. They just didn't get to in depth.
And then essence of complex numbers
I think that the "Essence of Statistics" would be better, as it includes probability
One word for you is "Awesome!", If there were teachers like you when i was in school... I would have been really educated.
Sirus Das
Awesome factorial? Hell yes please
TheLenny27 Lmao
Lol you take your own responsability on your education. Studying is always active not passive
You are not thinking. He wouldn't teach you like this. He would have many students to teach and focus on. Also he would be restricted to the circulum to teach. Also he has to make sure people remember which in high school and would have to change the way he teaches. Also you are assuming your teachers are bad without thinking how hard there jobs are or the handicaps they have.
I mean, if you had put more effort in, you could have. If you changed this attitude, you could have. Hell you still can!
00:00 Learn calculus core ideas through visual approach
02:22 Approximate the area of a circle using thin rings
04:22 Approximating the area of rings using rectangles
06:25 The formula for the area of a circle is pi times R squared.
08:42 Many problems in calculus involve finding the area under a graph.
10:46 Finding the area under a graph is a hard problem
12:50 The derivative is a measure of how sensitive a function is to small changes in its input.
14:52 Calculus ties together the two big ideas of integrals and derivatives.
Crafted by Merlin AI.00:00 Learn calculus core ideas through visual approach
02:22 Approximate the area of a circle using thin rings
04:22 Approximating the area of rings using rectangles
06:25 The formula for the area of a circle is pi times R squared.
08:42 Many problems in calculus involve finding the area under a graph.
10:46 Finding the area under a graph is a hard problem
12:50 The derivative is a measure of how sensitive a function is to small changes in its input.
14:52 Calculus ties together the two big ideas of integrals and derivatives.
I'm so glad that my Calculus teacher is using your videos to help us understand what we're learning. You're doing amazing work!!
2:26 "Math has a tendency to reward you when you respect its symmetries"
*30 seconds later*
2:56 *RECTANGLE-ISH*
hahaha
@Michael Gayle either works tbh. If he stuck with calling it a trapezoid then you'd have the height dx and the top and bottom lengths a and b. As dx becomes smaller a and b approach each-other. Putting this in the area for a trapezoid:
1/2(a+b)*h
1/2(2pi*r+2pi*(r-dx))*dx where dx -> 0
1/2(2pi*r+2pi*r)*dx
1/2(4pi*r)*dx
2pi*r*dx
So you'd game the same answer.
Yeah LOL
@Michael Gayle trapezoids would work the same exact way but be waaay less visual and ultimately no further help as dx approaches zero!
I’ve loved math ever since my childhood and I’d been out of touch since I started with SW engineering.
One of the things I miss is the pure joy I used to savour after running into a little discovery of my own.
I just had a similar epiphany when you delineated a way to visualise a function as a rate of change of area under it.
For years I’ve been trying to visually relate a function and it’s integral, and your lucid explanation made me truly enjoy this tiny moment.
Thank you for the work!
this just showed me how much highschool and uni just doesn't cover the explanations expecting students to understand without question - this changed my view on calculus and how it all works at 1:30 in the morning, thank you so much
Me few weeks ago: has Math homework to do
My brain: *Let's procrastinate by watching video compilation of famous authors quotes*
Me now: has English essay to do
My brain: *This is it! Time to learn the Essence of Calculus*
Lol
So true
I have an english presentation to do but nope, calculus
and I only have calculus and vectors next semester lol
Oh, boy. I'm feeling this on another level.
is your name istvan?
if only this video was available 10 years sooner, maybe i would survive in my engineering class. Now, i'm stuck doing spreadsheet 9 to 5. Aiming to get master of data science though, and I think this series is a good start. Wish me luck guys
I just finished watching this video, and wow. It's the most entertaining math video I have ever watched. I've never been a math person, and I never really enjoyed learning about the beauty and intricacies of math. This video, however, has changed me. I have never before felt this interested in learning math. This video is so well made, and I'm so glad I was able to see it. You have done a truly amazing job, and have managed to do what I thought was impossible, getting me interested in math. I sincerely thank you for making this video.
I've almost finished an 8-week calc course, it's frustrated me because I couldn't understand why things work and haven't had time to play around with the ideas.
The first animations with the explanation of the development of the (pi)r^2 function and the derivative absolutely blew me away. Everything makes so much more sense and I can start to see where it connects. Thank you so much for this video series. I think you just made me love math even more. ❤
15:20 Please someone tell me I can't be the only one who wanted to cry at this point, the music, the explanation, everything is just too beautiful.
Same here bro. Infact everytime this music comes, it really hits me.
0:58
STUDENTS HATE HIM!!!
See how this guy invented calculus with one simple trick:
Lmao good gone
Good one*
Meryl nst as gold or g/bx or underx commenx or not, anyx can b perfx
this comment deserves 5,000 likes lmao
was there not this one researcher in some field not entirely within maths that reinvented calculus (or at least integration by summing areas of rectangles), without knowing calculus was a thing ? (it apparently passed peer review too)
it might seem like a lot to say this but this video is so spiritually uplifting. I do want to feel as if I could have come up with these concepts. I'm getting calculus next semester and I'm beyond excited! Thanks for your dedication!
I've just started properly learning calculus at school over the past week. I found these videos at the end of last year, and found them quite interesting. Now, I find them very useful. You explain things so much better than my maths teacher.
This is the best video that explains a complex subject like calculus in such a simple manner. You are a great teacher. I liked when you said "and my goal is for you to come away feeling like you could have invented calculus itself"- great words !!! it will remove a lot of fear for Mathematics. Great work
Im so glad Micheal Stevens from Vsauce recommended this channel a couple years ago when I was only in junior high school. So now at university my brain just remembered this channel from nowhere and I had to look it up! Amazing video! The pedagogy is astounding!
Leading students up to discovering/inventing concepts by themselves is such a progressive approach. Hats off to you, Grant. You make both science and education beautiful!
Your channel is A GEM. I passed Engineering without truly grasping calculus and you make it seem so simple...This is beautiful work, THANK YOU!!
The dislikes must have been from calculus instructors, who feel inadequate after watching this.
I am a calculus instructor and I love these videos.
That’s a pretty uniformed and moronic comment. True teachers crave new ways to present materials. You just sound like a bitter calculus student who struggled to barely pass.
@@DaveyJonesLocka it's a joke.
@@DaveyJonesLocka You have no idea how some teachers can be
@Davey Jones If this bitter student had trouble understanding what the teacher was trying to tell him, but had no problem understanding this video ... wasn't the teacher inadequate by definition? Probably not for all students, but for some at least ... maybe even for most of them.
What you said sounds like an insult to the student, but you are just reinforcing his point.
And if you think being bitter is not the adequate reaction ... maybe you should try being less bitter too.
Yes, I get it is a joke. But the best jokes always have a grain of truth.
billions and billions (-; are spent on "education" systems in countries around the world.
This was practically free. I feel conned.
Billions and billions are spent because:
#1 teachers need to get paid.
#2 buildings don't get built for free.
#3 buildings don't get cleaned for free.
#4 buildings don't get warmed and cooled for free.
#5 Nor repaired for free.
I'm sure there's a jillion tiny little other expenses I forgot...
yes, there is a massive cost behind school
but it must stand to reason that better facilities such as physical school should be better than a video floating on the internet that you literally pay 10 cents for(if you watch the ads)
why it isn't?
school has waaaaaaaaaaaaaaayyy more resources than him, and you can argue that he won by the economy of scale (he can reach millions when normal teacher can only reach thousand) but the argument still holds, school has more facilities to use and teacher to ask, why isn't it better
Read Bryan Caplan's "The Case Against Education". He will confirm your concern satisfyingly.
Unfortunately the us educational system is paid for by the students
Martin Watson yes they pay their school, that's why it suprise me that it is worse than europian school
Paid stuff is usually better but apperently most thing USA is an exception to this concept, expensive stuff that is also not good
going into college and they never take the time to explain it in depth like this because it would take forever and they have to make deadlines for tests and what not. I'm in Calc 2, but it's always good to go back to the basics real quick
Okay... I don't know how I took so long to get here, but you just explained, so elegantly, so beautifully, visually and without random, unnecessary jargon, like the first 5 chapters of most Caluclus books, if not more? This is insane. Thank you so much. I think you're about to save me soooo much time! With this, I'll be able to tackle pretty much all my course's problems. Thank you so much!! I've never had a clearer understanding of the relationship between the area under the curve and the derivative!!
"Math has a tendency to reward you when you respect its symmetry" Beautifully put
It's poetic
How does this guy make a 20 minute long math video that feels like it went by all too fast?
Because it's 17 minutes?
Haha!
+RonJohn63 - I believe with higher "dx" values, that error is acceptable.
he's a Professional...- A Master....and LOVES his Subject....
If you take 17:03 and multiply it by the height of the video, you won't learn anything.
These are the best math videos anywhere.
⏱Timestamps for this video!
0:00 - Introduction to the essence of calculus series
1:32 - Introduction to the area of a circle
2:40 - Finding approximations of the area of a ring
4:14 - Finding the area of the circle by adding the areas of many rings
9:47 - Introduction to thinking about integrals
11:53 - Finding a property of the mystery function
13:16 - Relationship between tiny changes to the mystery function and the values of x-squared
13:41 - All functions defined as the area under some graph have a special property
14:06 - Derivatives
14:30 - The importance of derivatives in solving problems
15:13 - The fundamental theorem of calculus
15:40 - A high-level view of calculus
16:02 - A thank you to supporters
🧙♂✨ Generated with Houdini Chrome extension.
At 5:24 , How do you know the concentric circumferences form a straight line when you stack them up side by side? Each subsequent circle is 2*pi*dr longer than the previous one. Since thickness dr is constant throughout this, so is the angle of elevation between two subsequent slices. This makes the line connecting the tip of the slices straight and gives us a nice triangle to get our formula :)
thank you for your addition. One really can see how this difference of lenght between slices is constant, and when you find constants on math the path clears, it gets way easier to find ways to take advantage of that constant and to discover your formula. Very nice.
I like your mathematical rigor 👍
Really nice brother
Well it could be more straight forward by thinking of it as 2πr (a linear equation)
@@x_ployt4086 You're right, it's easier to remember it that way. Thank you
I'm heading back to school in January to finish my engineering degree and I'm particularly worried about the maths. From the way math is taught to us in school (do this, do that, calculate the exact value, any deviation from the answer is wrong) it seems like an oxymoron that one can "play" with math, and yet this video just showed me clear-as-day that true understanding comes from the ability to hold the ideas with a degree of flexibility. I hope that I can stretch my mind enough to break the rigidity of my foundational maths education and be able to think about these problems the same way you're explaining.
Looking forward to the rest of the series, thank you for making this!!
I'm 16, I really love to prove many theorems with logic or axioms, not only memorize it. With your channel, it helps me to know how math works. I really appreciate it.
I'm also 16 and I try to do this too, but damn, my brain can't even go 2 steps into a single idea I have for such proof, I can't do nothing and honestly I'm not atracted by the idea of using paper to write down those things either, I know it's the logical thing to do, but it just doesn't feel right for some reason.
Same 15 here.
@@eterty8335 if you want to learn and understand more about mathematical thinking I would recommend you to read the book "Discrete mathematics with applications" by Susanna Epp. It's a awesome book and it covers since the beginning of logical thinking, which is great for people like us, who don't understand much of this subject. Btw I found a pdf version of it for free on the internet, so if can't buy the book you only have to search for its pdf and study using it.
@@enricoboldrini5350 Noice, thx
I'm almost 18, and I recently started learning math deeply, it's a lil bit late but I think it doesn't really matter lol
I haven't studied traditional math since highschool about 12 years ago, but I've done plenty of programming since. So returning to this format is rather challenging, but I figure this series is a good place to jump back into it.
Your explanation of PiR^2 broken into the area of a triangle blew my mind. It was like i suddenly saw the whole thing in 3 dimensions, and finally that formula MAKES SENSE. I'm sure other parts will have their own "Ahah" moments, but I just want to say thanks for all your incredible work.
A stunningly good explanation of basic calculus in less than an hour. Hopefully math professors everywhere will watch these videos to better understand what it means to be a teacher.
Thank you so much for this series (and others). I never went to college, and my highest level of math in high school was technically pre-calculus, but honestly geometry. It is difficult for me to retain information that doesn't apply to my everyday life, so I didn't try to retain it then. Now, I am a data scientist. These concepts matter to me now, so I am motivated to learn them. This series has given me the confidence to approach these subjects on my own. Again, thank you so much!
This guy alone has the power of increasing the effectiveness of education by 100%
More like its over 9,000 !!!!!!
Vegeta
No because a 100% increase in a value of 0 is still 0
@@awaken6094 it's still a 100% increase
@@lamichhane so it's a 0% increase?
@@awaken6094 so 100% = 0%
1 = 0 ?
When you showed the (A(3.001) - A(3)) / 0.001 I instantly understood lim(x -> 0) (f(x + h) - f(x)) / h
Grant has such a fundamental understanding of math and we're lucky to have him guide us through it. Thank you
This should give 2x right instead of x^2
When I was in high school, my friend's dad, a biologist, told me that he integrated graphs by cutting them out and weighing them on an analytical balance. (It was quicker than walking across campus to the computer center or using a calculator.) That was more effective than the hours of effort my calc teacher put into explaining it. Sometimes it just takes another perspective, and we didn't really have anything like this video back in the day. Mechanical Universe, I guess.
This 17 minutes video was better than 17 years of studying calculus at university. Thanks😀
_-it's actually just 3..?-_
Are u a multiple PhD holder or a multiple drop out 😀
17 years or 17 months😂
teachers at school always explain the "How" but never the "Why"
Kids aren't interested.
@@raymondfrye5017 but if they explained, kids would better understand and it wouldn't be such a struggle for them to get good grades
@@raymondfrye5017 don't generalize. I personally love physics, and am currently studying engineering at university.
I didn't do physics in high school because of how infamous our teacher was for making the subject boring. I forced myself to learn physics myself instead, because I still wanted to keep up with physics, but I didn't want to do it at school.
Schooling =/= education
I was interested. There are kids like me. Rare, perhaps, but still existent.
@@PsychonautTV You are right. Some kids love it but inept teachers can't convey the problem-solving skills needed.
Regards
@@raymondfrye5017 Teachers suck at making it interesting. They do not combine our different interests with our long term wants acting as if we are all the same kinds of people that work in the same way, thus making the class fairly unengaging and boring.
my whole life my math teachers have taught only the formulas and spent all class doing fentanyl and I never understood the math until I cam across this video. Your video enlightened the neurons in my brain and gave me a new will to live along with curing my fentanyl addiction. If it weren't for you I would have never escaped my junkie math professor's calc class. Much love,
- a concerned mother
Sounds like things were really hard. Jello can come in a lot of colors. Just like pills. Get them mixed and you’ll be drying your clothes with the moon light.
-anna (Luisiana swamp gater)