Next up is integrals. Follow the full playlist at 3b1b.co/calculus By the way, there is a piece of math, commonly called "non-standard analysis", which makes infinitesimals a rigorous notion, thereby avoiding the need to use limits. That is, in the real number system something like 0.000....(infinitely many 0's)...1 doesn't make sense, it's not an actual number. But the "hyperreal numbers" of non-standard analysis are constructed so as to include a number like this. I have no problem with that system. I think it's great to invent new math and new number systems meant to rigorously capture a useful intuitive notion, although the construction of the hyperreal numbers requires some questionable usage of the axiom of choice. But I do think it's important to first learn about limits, and how mathematicians made sense out of calculus using the standard real number line without resorting to infinitesimals. It's not a matter of clinging to old systems, it's because limits help to gain a deeper appreciation for structure and character of the real numbers themselves, which in turn will help to understand any extension of those numbers.
I always have an issue with infinitesimals, I do not understand what it means. To me, the concept of limits seems more reasonable, more understandable, and more intuitive. Can someone care to explain to me, why some people think of derivatives as infinitesimals, and what does it mean to have infinitesimals instead of limits?
i find the notion of numbers "approaching" far more nebulous and unconfortable than infinites and infinitesimals.. i recently had an discussion with a mathy friend of mine, and after a few hours of staring at (for my eyes) convoluted equations (involving absolutes and "for all" statements) i eventually found it "kinda agreeable", but i could never reconstruct it.. with infinitesimals there is no hopping around between approximations and exact values, and no pretending that i could somehow shove all convergent series into it. less crying and headaches. and i guess the "lim (x -> y, f (x))" can just be expressed as "st (f (y + ε))" and "st (f (y - ε))" with ε being infinitesimal and st the standard function. i don't see what else there is to gain from limits.. and while i have no idea of what the axiom of choice entails, i don't think any use of it is any more questionable than any other. :P
If it was infinite, we’d have infinite amount of videos to enjoy. But we would also never get to watch all of them, this way, we can see everything they make
I have read The Feynman Lectures on Physics. Waiting for The Sanderson Lectures on Mathematics. Grant Sanderson teaches really way, and what I LOVE about him is that he teaches visually, for the sake of learning and understanding, not just for the sake of covering a topic. Thanks a lot. Ur long-time fan, Grant Sanderson.
I absolutely love this channel. it is very selfless of you to create such great content for learners across the globe. The animation, examples and script all reflect the amount of effort you put in to truly inject your passion and expertise in the video. Keep up with the great work!
Just look at his subscriber count. Even though his work is definitely amazing, it's quite far from selfless, since with that subscriber count, he can probably live off of his work. Not saying this is a bad thing, because he does deserve that, but if you can comfortably live off of your work, it's not really selfless, but more of a job.
@@mozesmarcus6786 You definitely aren't wrong, but arguably this particular job creates more total value in the world than many of the other equally or better paying jobs a smart dude like him could do. In that sense it's selfless of him to pick a career that creates such a positive externality.
@@mozesmarcus6786 it's selfless anyway With that skill he could be doing the double the money he makes rn not posting his videos for free, not spending valuable time making accurate and intuitive presentations etc. ...
This is the video that has the most utility and is the most accessible. My 12 year old son watched it, understood it, and was profoundly more interested in math after seeing it. And he was already a mathy kid. Well done sir.
Finally! My problem with your previous videos were, that they convey the right intuition for the general case, but leave out the dangerous edge cases! Only in university, through infinite series and limits I began to apply these rules with confidence. It's great to be sure you don't break anything, and I think it is one of the most crucial things to ponder about. What can be safely ignored (and what cannot) for arbitrary choices of some value within a given (changing) range.
Excellent video, as always. Limits are *the* essence of Calculus. The single most important concept to learn in a Calculus course. The only thing I'm "complaining" is: why didn't you actually write down the eps-del definition of limit? You had the ground work all laid down, you just needed to finish it off by giving the actual definition!!! :)
I've trying to fully understand all this concepts for several years (+5), you know the feeling of joy that is close to tears? That's how I felt at 17:23. Thanks for the videos, 引き続き頑張れ！
I don't know why, but I have learned about derivatives and limits a bit different in my country. We started with limits to be able to define derivatives and we were solving them differently, which really showed the beauty of it. We mentioned l'Hopital's rule briefly, but I had to see for myself how useful it is in harder limits. Also we didn't write the fractions df/dx at the end of everything, the calculations were much more clear thanks to that. However, we started using it at the end of integral, to know what is the variable. I guess we would have used in normally if we were planning to get to multivariable function derivatives. Also, implicit differentiation was a bit different too. This series is great, but it really shakes with my view on calculus, that I have built for the last two years.
Judging from your way of writing (in particular the commas before "but" and "that"), I would've guessed that you were German like me which is apparently not the case. However, it is the "usual" way of introducing limits like you said since it is a quick way to transport rigorous definitions. Unfortunately, in doing so, much of the intuition is lost or hard to associate with the definitions. (And one loses the reason why there is a dx in the end of the integral). As soon as one gets to multidimensional calculus/analysis, it is quite important to give directions via df/dx...
They taught is the wrong way round in the UK, power rule and chain rule with some vague mention that limits is how it all hangs together but no detail.
You are totally rocking it with these newer videos. It's not just nostalgia either. I love your earlier videos. They are amazing. Seeing actual geometry explained by calculus using animations is a true stroke of genius. I imagine these new calculus videos were inspired by your deep learning research. I would love to buy your merch, but I haven't had a job for almost a year and a half. I have 150K (and growing) in student loans and just completely drained my 401k of 60k just to make payments and pay rent. I have no idea if posting this on RU-vid poses any risk. However, if you are looking for inspiration in your future videos. I would appreciate any financial mathematics animations you would dream up. I am not the same person who took out so much money. I however still a person. Thus, I need all financial advice I can consume. I don't think there are haves and have nots, I think each person's finances are either accelerating up or down. The rich get richer because they are accelerating up, the poor get poorer because they are not accelerating up consistently. Right now, I am still accelerating down. Compounding interest is a lot like gravity. One last thing. If you were to self publish a book using some of these animation frames. You might usurp all other math books because you would have a link to the videos the pictures are from. Thus, giving the reader the option to test their knowledge on the text and if that is not enough intuition then they would have the convenience of watching these artful videos.
Thank you very much for your French subtitles . I am learning French and these subtitles are a Big help . Also your wonderful graphics increase my understanding of complicated Math .
Thank you for making the subtitles available in Portuguese, here, people who study automation and engineering generally turn to channels like this, because in Brazil we are "poor" in digital content. some here have even resorted to Indian videos hahaha
4:45 perhaps an analogy? Imagine you have a friend whom you have no idea where she lives onand you want to meet her, though you're not allowed. Met at college. So you ask a friend, and she points out it's in the next city over, so you head there. Then arriving at the city, you ask another friend, she says it's in a particular neighborhood, so you go there Finally you ask again and you're told on what street she's in. Now you can't go there yourself and go into her house, since her father so grumpy, but at least you can see from a distance where the house is. Limit is seeing the house in a distance and saying I know enough And infinitely small is when you're welcomed to the house. Both scenario leads to knowledge of the location but You don't need to enter the house to know where it is, so you wouldn't need GPS to tell you next time. *I know it sounds creepy but it's a weird analogy I come up with that might help? Like limit is the direction of which everyone is pointing at The GPS is manually checking actual values nearing up to the location
I just can't help but comment. No need to mention the mathematical lecture is superb. On top of that, I just get mesmerised by the facial expressions of the pi creatures, every time. The three blue puzzled blinks, the curious blue stares, the brown gentleness walking thru the concepts, the typical nerdy smile of the brown... lol
If we were stuck using the limit process for derivatives during Calculus, I don't even want to know what it would be like having to do that! Great videos as always!
Nice video but I don't like the bashing of infinitesimals. Infinitesimals are rigorously constructed and proved by Robinson's nonstandard analysis There are no logical fallacies with infinitesimals. And given the history of calculus, perhaps more intuitive. Also within constructive mathematics (specifically smooth infinitesimal analysis), nilpotent infinitesimals are valid. Computationally, these infinitesimals have been reified as dual numbers or nilpotent matrices. Limits do make sense in numerical computational methods
Limits were invented late in the history of calculus. Just as calculus was being used before real analysis (Cantor's set theory and Frege's logic), infinitesimals were used before nonstandard analysis. Infinitesimals are to negative numbers what limits are to subtraction, an operation reified as a number
1:16 L'Hopital's rule wow my entire life was a lie, I have been taught it's pronounced L Hospital Rule. And I always found it funny but never questioned it and now I'm finding out what's it's actually pronounced.
There are two mistakes. First one is: using l'Hospital for sin(x)/x, which is circular reasoning. cos'(x)=limit you want to find. Second one is: df/dx is the slope of the tangential. df is finite and dx is too. The limit slope is calculated through the limit process yes, but the final dx need not to be small. Vladimir I. Arnold told us in his textbook 'Ordinary Differential Equations', second edition, page 17 proposition 1, that dx is finite arguing with differential forms. And one can see it easily one the graph that one can take a finite step to the right along the tangential.
These classes are wonderful. Unhappily at the university I had lousy classes about these matters. Now I am starting to understand these matterd. Congratulations, dear Professor.
I can't believe that so many people disliked this video! Even if you're a calculus master, you have to respect 3Blue1Brown's amazingly comprehensible explanations of indispensable topics.
I m much lucky that utube has u type of explainer...I wasn't able to understand calculus in mine's institute...but urs vids r the best to know about calculus THANK U SO MUCH
At about 16:54 You address the question of whether one can apply L'Hospital's Theorem to the limit definition of the derivative, saying "that would be cheating" & writing on screen "That requires knowing d(sin)/dx". Well, yes & no. We CAN apply L'Hospital to the definition of the derivative and we learn a tiny bit: First re-write the RightHandSide (RHS) of the definitional equation as lim_{h-->0} N(h)/D(h), where N(h) = f(h+x) - f(x) and D(h) = h. At h = 0, the quotient becomes 0/0, so let's apply L'Hospital to get: lim_{h-->0} (dN/dh)/(dD/dh). Note that dN/dh = f '(h+x) and dD/dh = 1, so the definitional equation transforms to be (df/dx)(x) = f'(0 + x)/1 = f '(x), which is true but doesn't teach us anything new except that L'Hospital's Theorem is consistent with the definition; that's not much but it's nice to see that these puzzle pieces do indeed fit together nicely. So it seems to me that the 16:54 quote bubble on-screen should really read "That requires knowing d(sin)/dh".
Please cover enough topics before the AP Exams on Tuesday for Calculus lol. These are great in understanding what all those formulas and tools one has been memorizing and learning he entire year.
Thank you so much. I‘m currently in 10th grade (Germany) so I have barely come in touch with calculus yet. We already learned the concept of derrivatives but without dx and df. So I was really confused when you mentioned them. But after watching the video around 6 times It finally made sense. Thank you very much for your helpful videos. I appreciate it!
0:28 I often will joke that limits are arguably one of the most intuitive concepts in calculus, or possibly even in all of math: so much so that newton and liebniz were able to manage to explain calculus without bothering to formally define limits (which, like proving that 1+1=2, is a whole lot of effort just to convince people of something they probably already knew intuitively)
I don't know if it's something you'd be able to add to your store easily, but notebooks would be awesome. And as a side note (pun not intended) my favorite type is the one with a dot grid instead of a square grid. Less clutter, but the dots are there as a guide if you need them.
lol this entire series could have been titled "derivatives seen as finitely small nudges". As always love your videos, thanks for doing this full time.
I really like dual numbers; take some function f(x) and some number t such that t != 0 but t^2 = 0 (and ignore that using standard arithmetic that can't happen), and plug in f(x+t) and the output can be expressed as f(x)+f'(x) (at least when t doesn't end up stuck in the argument to some function in f, but Taylor series might be able to help there)
That is good because I don't understand the overarching problem without each nuance explained. It is hard for myself to grapple the equations that I do not understand the quest of why, and how. I guess being some what odd has it's drawbacks. But wrapping my mind around something I can't make sense out of often repels the outcome of curved learning. Thank you for the explanation as I hope for a better outcome.
Just wanted to point out something: To find the derivative of the function f(x)=sin(x) by the formal definition, you are forced to use the fact that lim x-->0 sinx/x=1 (try it). Therefore, you can't use L' Hopital's rule to find the limit of sinx/x as x approaches 0, since you use the derivative of sin x (which is cos x). Basically you prove something using that same something in an indirect way. Besides that, great vid!
You have successfully expanded my _LIMITED_ knowledge on *Limits* which was earlier _limited_ by my professors. Enough of this i guess i will limit myself now.
I think there is a error at 6', I got the idea, but to match the equation with the graphic, you need to place the empty dot at x=2, not at zero. The h approachs zero, but the x remains at 2. Anyway this is a outstanding work, just love your contents.
Did you make these to line up with the AP Calculus examine or is that just a coincidence? Either they are very helpful for studying for the exam so thanks!
Another way of looking at the issue where the denominator being equal to zero is the infinitesimally small value you can reach before the derivative "flips" to other side of the function in question.
Not gonna lie. I've always thought of infinitesimal nudges whenever I see dx. Never really stopped to think that it's actually nonsense. Well played 3b1b !
I was kinda pissed when I saw your video get hated on in r/math because you treated "dx" like a number. I think the way you explained derivatives was clear, accurate, and intuitive. I'm glad you included a little rebuttal to their argument in this video. Also, awesome content as usual. I look forward to the integral videos -- Reimann sum are awesome!
hello, I'm a high school student in korea. I am making a powerpoint for my presentation at school, but it's hard for me to create a moving graph. Would it be okay if I use your video in my powerpoint? thank you for uploading this great video!
Good work, Does limit solve undefined or indeterminate form types of problems or both ? For instance if a function is undefined 0/0 or indeterminate 1/0 Then can limit be appllied on both bases? Another point is that Do we use limits to determine the y-output where the ACTUAL FUNCITON was unable to determine the y-output? Could we say that both function formula and limit formula ARE NOT SAME? Why we say a function is undefined at a specific x value?? A function is a function when all of its domain members have unique images BUT at a specific x value the function does not have image so its not a function anymore because it is not following the definition of a function.
I'd wish you would do a series on anything EXCEPT probabilities. Not only is probability the most boring subject, it is also the only one which doesn't help you understand any other subject. On the other hand there are so many interesting things you could do a series on, all the series which would be direct follow uos to this one (real analysis, differential equations, multivariable calculus, complex analysis, topology, differential geometry) or all the algebraic ones (group theory, ring/field theory and more linear algebra)
I am a bit confused with the chain rule. Sometimes it is mentioned with dx at the end (@12:48) Cost(pi x) pi dx and other times is it omited for example in chapter 4 definition of chain rule the derivative of sin(x^2) is cos(x^2) 2x.
Let f(x) = sin(πx). df represents the tiny change in the function with respect to the tiny change in x. df = cos(πx)π*dx, so that is the tiny change in the function. Dividing both sides by dx, we get df/dx = cos(πx)π. df/dx is the ratio between the tiny change in the function and the tiny change in x, which is the derivative of f with respect to x. In summary, cos(πx)π dx represents a tiny change in the function; cos(πx)π represents the ratio between the tiny change in the function and the tiny change in x. Or, more accurately, it represents whatever that ratio approaches as dx approaches 0.
I badly needed that how L'hôpital's rule makes intuitive sense ..I know Thanks is not enough for what you did.. i wish i could do more for you than giving a poor thanks 😞
SIR PLEASE PROVIDE THIS PRICELESS CONTENT IN HINDI LANGUAAGE BECAUSE YOUR TEACHING WITH ANIMATION SKILL IS JUST INCREDIBLE SO PLEASE SIR DUB ALL VIDEOS IN HINDI TOO.........
I would define something like: the limit of a function is said to be L if for every sequence, and every epsilon there exist at most a finite number of indices k such that |f(x_k)-L|>epsilon and 0
For the sake of curiosity, the thing about using epsilon and delta for the definition of limits seems to be because some french mathematician (I don't remember exactly who... Cauchy, maybe?) called those intervals as "error" and "difference". So that's the possible origin of epsilon and delta in this topic.
