Taylor series | Chapter 11, Essence of calculus

[@chastgibson]

WOW! I'm a calculus teacher. I have watched hundreds of hours of calculus videos always looking for ways to improve my own methods of explanation. This is by far the best math video I have ever seen. I am in awe. It literally gave me goose bumps.

[@MrAlRats]

Is it possible to gain an intuition for what it means for a function to be classified as 'analytic'? I understand what it means for a function to be infinitely differentiable. I think it means that all its higher derivatives are continuos. However, roughly speaking, what property does a function need to possess, to be Taylor expandable in addition to being smooth?

[@MrAlRats]

@@vwlz8637 But the polynomials have turning points and points of inflection.

[@thelonegerman2314]

Like how the Taylor series and logarithmic forms are related to the golden ratio ,harmonic series Quadratic formula, and Prime numbers

[@kurtb.kaiser8699]

@@MrAlRats Intuitively, I think it means that the function is continuous: It doesn't jump abruptly from one value to another, no matter how closely you look at it. All derivatives must be finite.

[@SirNobleIZH]

@@MrAlRats it just means that no matter where you look, the function will have a derivative

[@gobbedy]

Omg. This has to be one of the most brilliant math videos I've ever seen. Not just beautifully explained, but with amazing moving graphs, perfect syncing between explanations and animations, perfect rate of explanation, perfect tone. I'm just sitting here in awe. So thankful. SO thankful!!

[@gobbedy]

Seriously. I just can't get over how amazing the animations are. How is this even possible? It would take me a decade to make a video like that. Just wow. I can't get over it.

www.udemy.com/diferansiyel-denklemler-differential-equations/?couponCode=DIFFOG

[@egorsilovs156]

Welcome to 3B1B youtube channel

[@abdullahx8118]

bruh he is on some type of adderall or something cos he's making these animations in the video editing software known as the python programming language A PROGRAMMING LANGUAGE this guy is in the next tier of brain ascension

[@aadityabhetuwal5990]

@@abdullahx8118 we all know this was made in PowerPoint

[@michaeladdis3323]

My Calc professor called them "tailored polynomials" in the sense that they are tailored to fit a function at a desired point

[@noamtashma2859]

Genius

[@lonestarr1490]

Yeah, but actually it's named after Brook Taylor who (partly) came up with this in the freakin' first two decades of the 18th century (and thus only a couple of years after differentiation itself has been discovered by Newton and Leibniz independently).

[@zss123456789]

Your prof deserves a medal

[@sblort123]

lone starr you bufoon its named after taylor swift

[@gregsavitt7176]

@@sblort123 No it's named after Taylor Lautner you dummy.

[@ParadoxPython]

Taylor Series are one of the things I just could not grasp in my uni calculus class because of how dry and abstract everything was. I understand abstraction is important, but it helps so, so much to be led towards it from concrete examples rather than being thrown into its cold rapids right away. Thank you so much for closing this gap for me, you are a gift to humanity.

[@codygoodman7909]

Same!

[@MegaMONI45]

Completely agree and had the exact same experience!

[@naturegirl1999]

Yes, examples are importnt

[@ujjwalyadav8780]

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[@bruhnling33]

Math teacher used this in class today instead of teaching it herself cause this video is THAT good, the teacher put aside her pride in favor of the amazing visuals. This is by far my favorite math channel and I was internally freaking out when she started playing it and I realized it was you. Probably the highlight of that class tbh

[@happysoul8760]

Bruhnling I wish my teacher did this. I was lost the whole lecture on this chapter

[@lonestarr1490]

That's maybe not the best thing to do for there's quite a bit of hand-waving in these videos, which is hard to spot for students and thus quite easy to draw misconceptions from. Grant is right to emphasize every now and then that those videos are only meant to provide you the intuition (and do an amazing job in this regard). But they're not sufficient in on itself for a study of the respective subject. The math has to be made explicit in a rigorous manner at some point.

[@RD-lf3pt]

@@lonestarr1490 But only after you get the intuition. Formulas without a way to visualise and use them priorly only click for Math teachers, who are passionate about the beauty of formulas. They should get this

[@eobardthawne6903]

Here I'm, recalling my 2 year old Mathematics classes. How pathetic she taught me!

[@kg3217]

I praise the teacher 👍

[@BlackwaterPark666]

Can we have a video where we just watch 3b1b animations of approximating functions with Taylor polynomials? That's so satisfying.

[@Ariana-dn4mm]

Blackwater Park Or any of his animations!

[@fossilfighters101]

++++

[@vivaldirules]

Yes, yes, and yes! And I'd be thrilled to have a piece of software where I could do something like that on my own functions without juggling a zillion display parameters and other stuff.

[@DaysNightsTeam]

Well, you could go search it on github anytime.

[@BigDBrian]

It made me think about the familiar series in a different way. Even though it's obvious if you think for a second about what adding more and more terms means! It also gives really really good insight about why the series for cosine and sine skip the odd and even terms respectively. This video was amazing

[@iandavidson5158]

Finally understanding a new math concept is a spiritual experience.

[@pampamdnf3422]

Best Comment

[@girirajyadav3206]

What a beautiful comment!!!!

[@khayalethumakosi6678]

this is beautiful!!!!!!!!!!!!!

[@bob_bobsen]

So true

[@hamzaiqbal7178]

You can say that again

[@davidmichels3089]

I am a calc 1 teacher for engineers and you just keep giving me amazing input to improve my lessons. Thank you!

[@mayankjain04]

I am a calc 1 engineering student and i want to think my professor does what you do too (even though i know he doesn't)

[@joeyaintwaffling]

Just thinking how mathematicians used to think all these, we need these extraordinary animation to just pick up the superficial part of it, truly they were marvelous.

[@chanakyasinha8046]

They simulate it in their brain...i have seen.

[@s.tahsin.r2288]

I wonder the same

[@ANIKETSONAWANE]

An on top of it Euler was blind when he made many breakthroughs .

[@karihotakainen5210]

And are!

[@audreyandremington5265]

@@ANIKETSONAWANE really? That was his secret? * proceeds to poke eyes out with pencil * *NOW I AM AN UNSTOPPABLE GENIUS!!!!!!!!!*

[@eriksundell1400]

@3Blue1Brown - I'm currently teaching students aged 16-17 about derivatives and integrals... The educational impact you make is immense! Please keep creating series about math! You have great narratives conveying beautiful insights in a time efficient manner with visualizations of highest quality. --- You are my educational hero. One Chan to rule them all, One Chan to find them, One Chan to bring them all and in the interest bind them In the Land of Math where the insights lie.

[@ogunsadebenjaminadeiyin2729]

❤️🙏

[@AdhiNarayananYR]

Shout out to my math teachers at school and jee coaching centre who just wrote the formula for the Taylor series and proceeded to solve some example problems that may or may not appear in jee exam. And that was the end of it. All this time I was looking at this series as an ugly series until I watched this video. Under the guidance of the right teacher even the most mundane things do become beautiful. Thank you grant Sanderson for making these videos! Love from India 🇮🇳

[@huzaifaabedeen7119]

jee selection hua ??

[@AdhiNarayananYR]

@@huzaifaabedeen7119 nope.

[@ashutoshmahapatra537]

Agreed! He makes mathematics look like an art which in essence it is. This channel will always remain a goldmine :)

[@ark5458]

Not defending him, but tbh, time is very low in jee training, putting this much work into the visuals to teach every single concept is really hard, you don't really go to jee coaching to learn stuff, you go to it to learn algorithms to crack entrance exams, sad ik

[@kl6544]

Wouldnt be a yt comment section without that one unrelated india comment

[@chanduiit42]

My 12th grade maths teacher used to teach us maths this way(on chalkboard) and his way was the only reason I still learn maths even at the age of 29. Imagine what effect your videos can have on people..I really hope this inspires youngsters to maths. Best explanation ever seen..wish i saw this years back..would have definitely been full time into maths research.

[@mat-hu5ys]

I, as a young person (9th grade) inspired by his videos can confirm thi sis truly amazing

[@ebarbere]

I graduated with a math degree in '95 and started watching your linear algebra series a couple of weeks ago for a refresher. I was treated to a view of the topic that I hadn't considered and revealed so much more to me than I had ever thought possible. This is no different. I had always loved the Taylor series in describing transcendental functions, and was vaguely aware of the relationships involved, but fuzzy on the derivation. This is the best and clearest explanation I have seen, and one I will not forget. You have a real gift. Thank you for sharing it.

[@911gpd]

Our math teacher speaks highly of your work and encourages us to watch your videos to learn more about the chapters we're working on. He's definitely right, congrats sir. Cheers from France :)

[@clementboutaric3952]

I'm french myself, and another froggy cheer you.

[@remus-alexandrusimion3439]

I would suggest you cherish such professors :)

[@samuelmcdonagh1590]

Your English is exceptional for a Frenchman!

[@jons2cool1]

I wish I would have had these videos when I was in calc 1 and 2. I hated taylor series and didn't really see the point in them other than proving integrals. If you ever take calc 3, try to find some videos helping to describe 3d graphs and planes, that is what I struggled with most conceptually in that class.

[@jeanmarabou9774]

T'es d'où pour avoir des profs qui recommandent ça ?

[@3blue1brown]

Thanks for watching, and thanks for such a warm reception of the series! For those just landing on the series through this video, the full playlist is at kzbin.info/aero/PLZHQObOWTQDMsr9K-rj53DwVRMYO3t5Yr Needless to say, there are many topics not covered in this series so far. Just think of how much was left unsaid about integrals! I do intend to revisit this playlist and add videos on simple differential equations (separation of variables), how and why substitution works in figuring out tricky integrals, and integration by parts. In the immediate future, however, there are other projects I'd like to sink my teeth into. Please do keep exploring math, whether that's delving more into calculus, linear algebra, number theory, taking my sincere recommendations about 3b1b.co/aops or 3b1b.co/brilliant, or even just sitting down in a quiet room with nothing more than a pencil, paper, and a supply of curious thoughts. And if you want to see the kind of thoughts that might lead you to a formula for pi, through a path that wanders quite close to the Riemann zeta function, keep an eye out for the next video on this channel: 3b1b.co/subscribe

[@joshuagross6022]

I loved the series! (Yours and Taylor's) I would really appreciate a video/series explaining the Millennium Problems. The animations are very helpful.

[@OzOz255]

3Blue1Brown i dont even know what to say, but thank you so much. this and your linear algebra videos have helped me greatly

[@JRush374]

3Blue1Brown I'd love a small series on fractional calculus if you're up for it. I discovered it on the internet one day and asked my professor about it. He wasn't aware of it but looked into it. We ended up meeting for a few weeks to discuss what he found and it totally blew my mind. It generalizes calculus in a beautiful way so that you can take fractional integrals and derivatives. For example, you can take the 3/2 integral of some function.

[@JuanLuisLealContreras]

This videos are incredible! They are better than any documentary I've ever seen. Thank you very much. I would like to know what programs are you using for the graphics and animations, they make the video amazing.

[@dijek5511]

For the ln(x) Taylor series, for values of x greater than 1, does the *average* of the outputs of the polynomials at least approach ln(x)? It looks like it may, but I don't know. If it does, are there continuous (and continuous on every derivative) functions where this does not happen somewhere they are defined?

[@rewtru8273]

I studied Taylor polynomial expansion almost 10 years ago. I remember seeing the professor write the factorial at the denominator and wondering "What does the factorial come out of?" and also "Why isn't the reason why it does part of the class?" "Why isn't it explained explicitly on my book?" And finally I see this video. I looked it up it because I was sure you were going to reveal this to me. THANK YOU GRANT

[@Hivlik]

13:30 watching the taylor polynomials of higher orders fit more and more closely to the original function is unbelievably soothing

[@aerodynamico6427]

...and revealing!

[@ineslima9723]

I'm studying calculus at the univertity and whenever i don't fully understand a topic i come here and it lights me up. Thank you for the excelent and interesting explanations and for the extremely useful visual approches. Helps a lot!

[@RavindraKempaiah]

What you are doing to educate all the science learners around the world is truly incredible. This generosity of heart, this dedication to share knowledge is a truly positive karma for your soul and will carry you across life. So happy to see someone explain things so incredibly well. This is what brings depth to life. Hope you have a long and healthy life.

[@prathyusha5393]

Well said ..

[@cheicktoure9586]

I am a graduate students in maths, and i am literally having tears in the eyes after watching the video toward the ends. In so many years I just could'nt fully understand the meaning of all this, even though i had excellent grades during exams, everything was so abstract. All this time, It was all that simple !? Thank you so much

[@pacogutierrez2484]

Tears? What a ....

[@SuryakantSingh5]

How beautiful! This is not just Math anymore it is art too. I envy young students who are just starting to study these topics and have access to such beautiful explanation.

[@mountainc1027]

Math is Art. It never was anything else but art

[@convergency1068]

@@mountainc1027 Yes Thank you!!! That's what I was about to say...

[@maurocruz1824]

Me too. This explosive "age" of such a high quality videos on internet (few but valuable youtube channels) catch me out in my thirthies.

[@freemind.d2714]

Beautiful and useful!!! The nature of the tech we call artificial intelligence(AI) just like it!!!

[@freemind.d2714]

@@mountainc1027 Must careful on what you just said, those day even silly girl who take a selfie will be consider as an art so... Math is The Real Art!!!

[@alimacoremor]

Not only have a FINALLY understood Taylor Polynomials, I am completely ecstatic. They are so cool!!

[@carultch]

Good way to remember it: Taylor polynomials are "tailored" to match the shape of another function.

[@cauchy3293]

These videos are helpful but don't be in an illusion that you understand them completely. You don't understand something unless you have done rigourous practice on the topic. Even after watching this video you won't be able to solve problems based on it. Hence to understand something in mathematics,you first have to go through the rigour.

[@tiemen596]

"The first time this clicked for me was in a physics class, not a mathematics class." As an engineering graduate I can say that almost all math clicked for me in physics or engineering classes. Complex numbers clicked for me when studying control theory. Differential equations clicked for me when studying vibrations, etc. Math teachers could take that message home.

[@amatya.rakshasa]

I think pure math students and engineering/applied physics students are fundamentally different types of people. Most math professors cater to math students ,have been math students themselves, and what they consider to be concrete examples that make things click are perhaps a bit different from what makes things click for engineering students.

[@friedrichknauss6260]

This is why I teach both physics and calculus. I've convinced my school to let me teach it as a combined course.

[@xXxIMMORTALxXx]

Hey I was in a similar situation here. Maths clicked for me in Computer Science classes especially when proofs came along. I think this probably aligns with what @Anshuman Sinha said engineering students perhaps need physical applications, movements, natural phenomena to see how maths make sense whereas pure maths or CS students would find maths make sense in a more abstract way since everything we do is intangible. However, I think really the best way for anyone to appreciate maths is when it's put in a context like physics.

[@xXxIMMORTALxXx]

@@PersimmonHurmo Yea mathematicians from the past when there were no computers are those with a substantial amount of brainpower to do crazy abstract imagination/thinking. I'm in awe.

[@rithikradhan367]

Noice. In fact I was taught the basics of calculus in physics class.

[@CapsCtrl]

Thanks!

[@lightning_11]

What does the thing in front of your message mean?

[@CapsCtrl]

@@lightning_11 it's because I donated money to him

[@janaosea6020]

I'm still crying from the beauty in this video. I just fell in love with the Taylor Series.

[@ehza]

still crying

[@olemikole1997]

Math is amazing

[@yahia2601]

Truly beautiful

[@convergency1068]

I see a potential math major here

[@gaelbgalbar9307]

You should try MacLaurin series too, this is as fascinating.

[@jean-marcrocher1463]

Back in college, Taylor polynomials/series, and how they related to the rest of calculus, left me completely baffled. You've made clear in 20 minutes what a month of Math 1b lectures and problem sets didn't.

[@ricardoescobar109]

anyone who has been watching these videos from the beginning can easily appreciate the amazing visuals but I think an under-rated aspect of these video's is the verbal elegance used to explain these abstract concepts... the phrase "derivative information propagating out from the radius of convergence" was never mentioned when I first learned this stuff and it took my understanding and appreciation of the subject to a whole new level.. thank you!!

[@misted3508]

I love the way the little Pi characters have little eyes that follow whats going on above them. Great work with this video, your attention to detail is immaculate and the content is flued and intuitively understandable.

[@hanskywalker1246]

Thus true

[@Truth4thetrue]

On behalf of all students who've had dumb maths teachers that never reached us things right, thank you soooooo much

[@electronmechanicalcorporat2143]

Abody Aref I had dumb math teachers past. This channel forces you to love math by thoses PI 's with the eyes! Math is hard, breaking it down makes it easier.

[@lachlanmann4511]

Even the best you can expect to find in high school aren't this good for understanding. My teacher taught me everything by it formal definition and I managed to do well but seeing it all from this perspective makes it so much easier to remember and use appropriately.

[@justrinat2207]

If your excuse for not grasping mathematics is a bad teacher, then you aren't too bright yourself.

[@emf6621]

If you're mean to your maths teacher now, you'll regret it when you grow up. I promise

[@KFlorent13]

@@justrinat2207 Yes

[@gautamsethi3751]

Wow, wow, wow! I thought I knew Taylor polynomials well but the visuals are just gorgeous and helped me understand Taylor's polynomials deeper than ever before.

[@abhishekshankar1136]

MIND = BLOWN , i cant explain my happiness right now , 3 years of frustration with taylor and laurent series !!!!!!!!!!!!!!!!!! i always knew i lacked the intuition behind the purpose of these series , i knew how to derive and everything else , but the intuition part just makes it a 100 times better for me to appreciate these important concepts!!!

[@theseusswore]

I have always seen, and painfully memorized the general formulas for the value of e^x or anything of the sort related to e. I could never have imagined that Taylor Series could be used for something like this, I have always found calculus to be interesting but this...new..perception - it just takes my thought process to a hole new level and my excitement to study maths more rigorously in the future continues to grow. I have watched countless videos of yours, and NONE of them have bored me. All of them were MAGNIFICENTLY visualized and I felt kinda happy when I realized hard concepts were actually pretty easy! All you needed was a different way to view the problem. Thank you, 3b1b. Truly thanks, from the deepest part of my heart.

[@Anife69]

we are both autistic right?

[@theseusswore]

@@Anife69 god reading my 2 year old comment is painful but yeah im pretty autistic when it comes to maths

[@Anife69]

@@theseusswore XDD me too but dont worry your comment might be old but its really true and good for reading :3

[@theseusswore]

@@Anife69 hehe thank you

[@zuccx99]

"cos(x)=1 is a good approximation too"-some engineer

[@prathamyadav3105]

Touche...

[@deepthakur14916]

for x

[@boggless2771]

""Cos(x) =/= 1" - some math mathmatician or physicist" - some engineer.

[@aadityabhetuwal5990]

@@boggless2771 Starting with double double quotes and ending with only single double quote. Absolutely barbaric.

[@ishworshrestha3559]

Yt

[@pratyushmenon2455]

Having just finished high school calculus, this series was brilliant for me to review for exams and actually understand calculus instead of mindlessly applying it. So thanks a lot! I'm pretty sure I aced my exams thanks to you! :)

[@brewer0215]

Pratyush Menon wow, how early do you bring calculus up in US? I live in Sweden, and calculus is almost NEVER tought before our 10th, 11th or sometimes even 13th year in school...

[@pratyushmenon2455]

Brewer021 Well, I do the IB program (Higher Level Math) which is a lot more advanced than the regular curriculum in Canada, but we started calculus in Grade 11.

[@blownspeakersss]

In the US, many students don't take calculus until university. But some take "Advanced" mathematics in High School (year 11-12), which is calculus.

[@mpardalos]

IB math HL student here too! The timing of this series has been amazing. My calculus paper 3 is literally tomorrow :D.

[@pratyushmenon2455]

Michalis Pardalos Haha good luck! My school does discrete math for the option (which I'm doing tomorrow as well) but I've been self-studying the calculus option for fun and to better understand the problems on Paper 1 and 2.

[@deborahossai1435]

I’ve never commented on a post before but you did a bang on job. Absolutely clear. To the point. Easy to understand. Life saver

[@ayrapetoff]

Спасибо!

[@fallout3freak360]

Wow. I just finished calc 2. And this was explained in a COMPLETELY different way. This is much more intuitive, and actually explains the reasoning behind taking multiple derivatives of the same function.

[@Niyudi]

I've recently been doing some random derivatives as I learned how they worked in the begining of this series, but I wanted to do it algebraically as well, because it just feels nice to see the numbers crunching and canceling to a nice formula. I was stuck on proving that e^x was its own derivative when I stumbled upon the exponential function described as an infinite series. When I realized it came from this Taylor series, my jaw just dropped in amazement as my brain tried to process all this information. It's kind of hard for me to study calculus, because I'm in 9 grade (which is middle school here where I live) and teachers don't really have time to really help me in the short period I spend in school, so the internet, specially your series, has helped me A LOT. Thank you for the amazing content!

[@jmcsquared18]

Beautiful. I always tell me calculus students, don't try to imagine the second derivative of a curve algebraically. Just think: would a parabola approximating the curve at that point be opening upwards or downwards? It helps so much with understanding what the second derivative is and why it is important, namely in finding extrema and solving optimization problems. Understanding mathematics is always better than mindless computation.

[@abhishekcherath2323]

jmcsquared18 thanks very much for that tip, it finally got the idea of concavity to click in my head.

[@bigfatfannie]

Anybody from you guys care to explain why at 16:05, the Height= Slope times (x-a). P.S. Sorry for the unrelated comment. It's just that this has been bothering me, and if I write it as a separately, it probably won't see any attention.

[@bigfatfannie]

Hahaha. I can't believe I missed that. Thanks for your reply! Also, I dare say that most of the concepts in the series were clear to me before starting to watch (watching just consolidated my understanding), but I never quite understood why slope equals height over length?! I know, pretty ironic. I would appreciate it if you could explain once again!

[@bigfatfannie]

Thanks for your fast reply. However, I already understand that, as I stated I know what a derivative is. The thing that I don't understand is - why do you describe/express the slope as height/length ratio (dy/dx). For example - I think that the slope should be calculated via the pythagoras theorem (slope^2=length^2+height^2). Again, thanks for your time, it is much appreciated.

[@DDranks]

That would be the *length* of the hypothenuse of the triangle representing the slope. But the length has nothing to do with how *acute* the slope is. You can't calculate that with Pythagoras theorem. What the word "slope" means is the angle of the hypothenuse expressed as a ratio of the tangents.

[@kasperholck5928]

I first saw this video when you posted it four years ago and didn't really derive much from it. Now I'm a uni student, and I can tell you with absolute certainty, that this video should grant you an eternal afterlife and a golden casket.

[@sando_7]

I finally realized what "radius of convergenc" is. It's literally just beautiful. Thank you for your hard work😊

[@lex33122]

You have demonstrated that mathematics is an art form! This was wonderful entertainment. I thoroughly enjoyed this the same way I'd sit back and watch an anime series. I was genuinely excited and engrossed by this entire series. I first found your channel a few months ago while I was looking for some basic information on neural networks. I'm chemical and process engineering masters student and at the time, I was studying process control featuring a neural network controller. That video was great and insightful. Fast forward to today; this was beautiful! To unwind and relax with this series was like a neural massage. Keep up the great work! I love your appreciation for math as philosophy and art that forms part of our lives. Its an approach that is being lost among the masses and I fear that one day math will just be viewed as "that subject in school we need to pass and will never be used in life". I look forward to be further entertained by you.

[@RD-lf3pt]

If all teachers were like him (and some other), imagine what we could learn and accomplish in our lives. We have about 17,000 hours of school in our lives. This video is 22 minutes.

[@captainwalter]

this is a good example of how much there resistance there is to doing things better. especially in education, people are slow to change bc they know it would mean they would have less work. if people could learn math at 10x the rate, then that would mean 1/10 the jobs (all other things constant), or at least thats how people see it. i think its the reason we need universal basic income- people would be able to move out of the way of innovation bc they wouldnt be so reliant on the paycheck

[@aventurasenlauni7772]

Walter I agree. We spend a lot of time just trying to survive. Many times I’ve thought: Here’s your mansion and your food for the rest of your life. Now do something productive for humanity!Obviously it is an exaggeration but I agree things would be better with a minimum income or guarantee for everyone. Having just the basics to survive, even if it is 10m2 and a baguette or pizza a day, would mean we could focus so much time on productive things.

[@glitchystar1336]

edu__ceo And we have 17500 hours of school till High school only..

[@RD-lf3pt]

Still Unknown Young Gamer yep I messed that up🙈 At 6 hours / day, 5 days per week, 4 weeks per month and 9 months that is a rough estimate of 1080 hours per year. With 15-16 years until university that is 16,200-17,280 hours.

[@RD-lf3pt]

Still Unknown Young Gamer I think I meant 3,000 hours of university: 4 hours/day, 5 days/week, 4 weeks/month, 9 months/year and 4 years for roughly 3,000 (2,880) hours of university!

[@mckayanderson2046]

Was just learning about Taylor Series and needed to know why the hell we were doing what we were doing. This video summed it up perfectly and the dynamic visuals really propel this content to the best possible explanation of the topic. Great work

[@markkennedy9767]

It's incredible how Grant approaches and motivates these topics. I always learn something new watching them. And by learn I mean really internalise a particular concept. He's got an amazing ability to teach and is a genuine treasure.

[@WellventilatedPianis]

My classes have all moved online because of a certain infectious disease making the rounds and I am more grateful than ever for your videos. Thank you for refining and sharing your gift for communicating complex topics.

[@timkarl4099]

Hey 3Blue1Brown, I'm studying mathematics in my 2nd semester right now, and obviously we did Taylor series, but I was always kind of weirded out by it. I was just told it exists, and, well, we calculated around a bit with Taylor series. But, I swear, this video completely opened my eyes about how exactly, or rather, why exactly it looks like it does. Thanks a lot for that, that was extremely helpful!!

[@maibster]

THIS IS INCREDIBLE! that taylor polinomial for e^x just BLEW my mind! THANK YOU! So many things just clicked all at once in 2 minutes. The value ur videos have for humanity is immeasurable!

[@gogl0l386]

Enginers after skipping through the video: "alright got it cos (x) =1"

[@NovaWarrior77]

@Another Random Cuber maybe. However, this only really becomes helpful if made clear to the students why the approximation works, and when it doesn't, so if they run into the need for more accurate approximations in their work, they know HOW to get more information, i.e. add another term. I know nobody asked but this bothered me when it was taught to me.

[@NovaWarrior77]

@Another Random Cuber you're right about that.

[@skpcboy]

@Another Random Cuber got it, we'll target physicists too

[@angelmendez-rivera351]

M J Well, the point behind the meme isn't that the approximation is overused, the point behind the meme is that it is misused, because the approximation is only really good for small x, but it gets treated almost as if it holds for all x.

[@baburayhanshaik5121]

XD

[@pharynx007]

i just started on taylor series today in calculus class, and i sent this video to my professor. i had watched it before, but after his lecture, i feel i understand this video better, and because i understand this video better, i understood his lecture better.

[@BWAcolyte]

imagine if all math textbooks were this interactive and visual. We could be doing rocket science in 8th grade

[@official-obama]

kindergarten

[@fatitankeris6327]

You can, by the way. If you study impulses, basic Newton physics, then you'll know a lot about rocket science.

[@GodzillaGoesGaga]

This why Star Trek is right when they showed Spock on Vulcan learning in a interactive environment (and then the other kids teased him afterwards because of his human mother).

[@Diaming787]

100 years from now, it has to. At that time, the vastness of human knowledge means that we have to learn and understand the essentials more quickly.

[@sanstheblaster2626]

*...THIS VIDEO IS SPONSORED BY BRILLIANT!*

[@-An_Idiot-]

This was actually the only video of this series that stumped me. I needed some time to understand how it behaves and stuff. I was very confused at first but when I rewatched it a couple of times I actually understood it and how awesome it is! Your level of teaching is amazing and the way you explain everything is very comprehensive and visually pleasant. I deeply appreciated your channel.

[@tiagoverissimo1404]

This one of the most intellectual beautiful things that I have seen in my career as a student, math is awsome.

[@TechToppers]

Sir, you are are not of this world. You explain everything that it is addition. It was pleasing and a convincing explanation.

[@spelunkerd]

What a brilliant series, many issues fell into place for me. Like completing a puzzle, the last few steps can be very satisfying. I hope your next series touches on the binomial theorum, another area that can be conceptually sticky.

[@migueld2456]

Every time I watch a 3B1B video I think it's the best explanation I could ever hear on the topic.

[@sirfist198]

In AP Calculus BC, when Taylor Series were introduced, I was simply confused. It seemed as if my teacher was simply getting formulas out of thin air. I proceeded to memorize the formulas and do well in the class. But not until watching this amazing video did I really understand what was going on! The idea of approximating a function through taking many higher order derivatives at one point is simply mind blowing. After thinking about the video, I now realize the importance of the many tests for series convergence that we had to learn. Taylor polynomials are created to model functions that have real life applications in physics and engineering, and the best approximations we have are Taylor series. We need all the tests for series convergence in order to determine whether or not the Taylor series that we create will actually provide an approximation that will be accurate at a given point! If the Taylor series is divergent then it won't approximate at all, if it is conditionally convergent it will approximate only within the interval of convergence, and if it is convergent then it will approximate everywhere. Awesome stuff! And people say math isn't fun...

[@commie281]

Do you know if there has been a situation where people need to get the Taylor Series throughout a certain interval but couldn't because the function they were trying to approximate didn't work? Genuine question.

[@nolanfaught6974]

@@commie281 Try making a Taylor series of f(x) = 1/x centered at x = 0

[@swatir.567]

True mister! watching and understanding this 22 min video and Your comment too makes me appreciate my engineering course SO MUCH MORE. thanks for the elaborate and informative comment :D

[@DotcomL]

What a week. My first paper got accepted in a nice journal, France elections went well given the options, and this series has helped me not only understand better the concepts, but think differently about math, physics, and maybe the world in general. 3Blue1Brown, thank you. Excepect a patron soon.

[@3blue1brown]

Congrats on the paper, that's great!

[@shanesellman8487]

This series has been such a big help to me, I am going back to college and my first math class in a decade is calculus 1, I was terrified about failing but after watching these videos everything just clicks so well, thank you so much for the high quality and excellent explanations.

[@suparnaprasad8187]

Somebody please give this man a Nobel prize. He truly deserves it. I had not understood head or tail of this concept in class, because, well, the teacher never even described what a Taylor's series is, instead just started writing on the board. You are a saviour, looking forward to more of your amazing videos!!!!

[@chanakyasinha8046]

The way you fluently communicate math hits right into the hypothalamus.

[@Amb3rjack]

Yes, and the poor thing just wanted to wallow in a pool of soothing mud . . . . . .

[@tanmay996]

What a fantastic video! Who else agrees he saved the best for the last?

[@dlevi67]

I don't know. I think this is "the best" if it is because it's where things start to get interesting and relatively "unintuitive" (even if still relatively easy to follow). However, as with any great building, the architecture is interesting because of its foundations and plan, not by chance...

[@revooshnoj4078]

nice name

[@MenkoDany]

Me

[@AayamS]

You are a god. This AND linear algebra have been amazing. Although it takes only a fraction of the time i spend on the course, i get just as much insight from ur videos (if not more) than from class.

[@anonjan82]

Now I am left with the question why some functions can be approximated completely by derivatives at one point and others cannot. So I am going to find out by studying Taylor series. I love it.

[@Crazy_Diamond_75]

I've never seen a video present this topic so... beautifully. Honestly, I went to uni for engineering and I still remember how to construct Taylor series, but I never really understood where the mathematics came from. This is so clear and so concise... I just...gaah. Thanks for the vid haha.

[@aravindhsm1287]

I was asked by my teachers to just memorize the Taylor series expansion for some standard functions which has a higher probability to be asked in the examination. Sad truth:This is very common in India. Thank you Grant,this video felt like you were opening the cave in which i was living in.

[@puli36]

that's some very strange wording at the end there

[@Thelostdreamer21]

in 11th and 12th they dont teach talyor series but use its expansion, mainly in limits so solve questions, we are told that it will be taught in higher classes and taylor explansion is not in JEE syllabus. i came here for extra knowledge and loved the video

[@Thelostdreamer21]

@@puli36 🤣

[@Omar-bi9zn]

@@puli36 it is an allusion to Plato's cave allegory I believe

[@anch8064]

You have an amazing skill of making the most difficult topics of Mathematics easiest by right illustrations. Thanks.

[@pramodvora2067]

Words of thanks are just too little to express my gratitude for reveling the beauty of calculus. The graphic illustration is just out of the world to reveal the philosophical nature of mathematics. There is much more to learn with this inspiration. My humble thanks and great appreciation!

[@seersam]

Ok, I really like calculus. But this video was more than epic. I actually paused when he got the aproximation for cos(x) up to x^2 and tested what happens when I add more terms. I found the sequence 1 0 -1 0 and that factorial thingy and managed to write the actual sum of this for k=0 to infinity. When he introduced the Taylor series into the video I googled it for cos(x) only to be left shocked. It was THAT easy??! Now I may become more into calculus than ever!

[@luca0ben]

I can say I could watch these videos just for the pleasure of watching them, as long as they are so enjoyable. And I can learn or "just" understand something amazing (I'm no longer a student, but I thank you for these gems).

[@Piffsnow]

I'm a maths teacher and your videos make me appreciate ever more the beauty of maths. Sadly, I can't really share them with my friends since the few who speak english don't care about maths... I didn't learn as much with this series than with the previous one (which totally made linear algebra a sense to me : I (re)discovered everything on the topic !). I really enjoyed it nonetheless, and this particular video kind of blew my mind : I think I never really understood what was the point of Taylor series before... Thus I really want to express my gratitude : Your videos are clear, super interesting and astonishingly well done ! It's a real pleasure. I might give some lectures in english in a few years. Be sure that I'm gonna talk about your channel to my students ! I wish you all the best for the future and hope you will keep illustrate maths for long. :) I'm already looking forward to the probability series. I'll miss you ! Thank you !

[@gdfyredragan2270]

I was so confused as to how the hell did this seemingly arbitrary summation approximated any function, but after seeing this if makes so much more sense. Your ability to explain topics with such intuitive ease is awe-inspiring, and to believe all of this content is free blows my mind. Thank you so much.

[@MrRishik123]

Hey guys 3Blue1Brown here WITH A DOUBLE UPLOAD TODAY.

[@alexsere3061]

Mr Rishi The Cookie Hell has frozen over XD

[@shugaku2461]

Mr Rishi The Cookie When are the 4 horsemen of the apocalypse coming?

[@batubulgur]

Hey guys! It's Scarce here. Today we have a double upload!

[@jibran8410]

more like... bubble upload

[@danielsick424]

hey vsauce, michael here

[@SamiyuruSenarathna]

The only channel I have enabled notification on.

[@danolez1]

I swear

[@sabarishr381]

JEE 2024 aspirant here. Today I have learn that the Taylor series ain't just a bunch of formulae that we had to memorise but a result of a beautiful way the creative mathematicians had devised to calculate trignometric, exponential functional values of weird values that are close to 0. THANK YOU SOOO MUCH for this elegant explanation and captivating Animations !

[@AniketKumar-dl1ou]

Jee 2018 cracker here. I was always confused about the series during my jee preparation. Teachers weren't able to answer from where these equations came and it pissed me off so much. Internet was not so prevalent then. What I did back then was to relate the kinematic equation derivation ( from HC verma ) and these series and formulated the taylor series myself. if acc is constant s = s0 + ut + 1/2 a t^2 which is simply the taylor series for displacement s = s0 + s' * t + 1/2 * s'' * t^2 This way I was also able to solve the questions including jerk ( accn non constant ) by easily writing the equations directly. Feels so good to watch you all being able to form a intuition with such great videos and not being limited by the teacher teaching you. All the best for your exams 😄

[@chadroose5404]

Rank kya aayi Bhai?

[@basabromander2778]

It still baffles me how this guy summarizes things I've been cracking my head at trying to learn for hours into an in depth yet brief video that makes it crystal clear. Pedagogy is an art and these guys are virtuoso of the craft

[@artisticnoob5003]

Our teacher taught this series to us..not the derivation, he just told us to memorize it! And i kept mugging the series expansion of sins,cosx,tanx etc. I finally came across this video and you,sir, did a great job! Omw to write the expansion of e^× by myself :)

[@sebastiannagel180]

so 23 years ago, a somewhat desperate math teacher in highschool (with a specialisation leaning towards math and pyhsics over languages) tried to tell us about the usefulness of taylor polynomials ... he was very fascinated by them, we were very underwhelmed as 17-years-olds ... now watching this, i understand his fascination and i wish my kids will learn this one day too, just for the sake of it, just like for the sake of it to learn latin to understand and approximate modern languages better (i expect they will be very underwhelmed :-)

[@Al.Quraan30]

This is how maths should be taught

[@tomepsilon]

*Everyone’s* talking about you right now, man. Good job

[@xer_t3661]

💯💯

[@CharlieScarver]

I’ve never seen complex math explained so well. Mind blowing and wonderful to watch. This has to be among the best pieces of content on YT.

[@avengerofthedead142]

First video I've ever "fully" (I use that term loosely) understood. I'm so happy, I have a great calculus teacher and just enjoy math so much.

[@mrdyo94]

I'm currently studying electrical engineering and we were thought that this technique works, but they didn't teach us why would they! But it's very simple! If our polinom function changes like our original function then they will end up looking alike. I'm seriously thank you for that video!

[@yikes7918]

My math teacher sent me there after I just told him that in a certain way the functions sin(x) and cos(x) could be considered as polynomials with a degree tending to the infinite.

[@angelmendez-rivera351]

That is very often how mathematicians treat these functions. In fact, there is a rigorous formalism in which that makes some sense.

[@dasav6724]

Nice. He must be a good teacher.

[@yikes7918]

@@dasav6724 He sure is.

[@KT-bg8wj]

I'm a second year physics student currently taking Calc 2 for the third time. I always had trouble grasping the idea of series, but it's starting to make sense now. I was terrified of redoing Taylor Series because that always seemed like a different language that I was never taught. This is the best explanation by far on this. Thank you so much

[@Aycore2011]

Wow....I feel high right now, this is the purest drug I've ever had.

[@BraveGisgo]

3BL1BR IS DOING THE LORDS WORK

[@yzyzyz44]

You mean Crystal Math?

[@prabhatthapliyal9160]

bhut bhyanker nsa hai ye

[@PsyCodeqz]

oxygen is probably the purest drug ever

[@Phoeniiks.E]

@@yzyzyz44 Hell yeah bruther

[@TheJacobshapiro]

This is a much, MUCH better explanation than how it was taught to me in college. I learned it and knew it could be used for approximations but never really got a good explanation as far as how it actually worked.

[@nasirmijinyawa6360]

You are simply the best of the best philanthropists, keep giving us the intuitions in mathematics

[@pinetreegang5232]

Thank you so much for posting this. I am currently learning about Taylor and Maclauren series in my calculus course, and I have been doing fairly well in it, but I didn't understand how all the concepts were connected or important/relavent. I feel like my Calculus course was telling me how to use taylor series but this video gave me the sense of how they actually work. I don't think you will ever see this, but your work is appreciated. : )

[@algerois96]

Every video of yours I watch gives me that incredibly sweet feeling of satisfaction when stuff clicks in my head like: "oh wow, I got it! that makes sense"

[@tradetor]

Can’t believe I almost finished my master degree without this video, this is amazing, thank you!

[@ml.9106]

OMG. I cannot believe I finally understand Taylor series after graduating from college 8 years . Thanks!!!!!

[@TheCalcSeries]

Seeing how a formula translates into the simplest and most accurate picture is why math is so satisfying to learn. This video is an absolute masterpiece that demonstrates what an "intuitive" explanation is.

[@thatoneguy2169]

Plebian: T-series Me: *Taylor series*

[@Cjnw]

Music: #TaylorSwift

[@alfiealfie35]

I actually thought that

[@nikhilnegi9446]

Trailer scene

[@peggyfranzen6159]

Robots need upgrading too!

[@peggyfranzen6159]

" We" employee it. Stalin said " Who is Nikola Tesla?"-J.P. Morgan " My money." Ford said " Great!", what is that about it.

[@w04h]

"You can do even better approximation by adding c4" _FBI wants to know your location_

[@QuantCake247]

ded

[@pardeepgarg2640]

UnUnderstandable Wut

[@heavennoes]

it's 123 likes, it's to good to be broken.

[@pardeepgarg2640]

@@heavennoes it's broken lol

[@Patrik2569]

Omg. This is the moment it clicked for me👀 Unbelievable how well made visuals can help

[@rahulc0dy]

This is the best understanding of the Taylor's theorem. Starting my first year, I couldn't understand a thing about the Taylor's Theorem because I didn't understand what the theorem was doing to a function. Now I know what each of the terms mean. Thank you very much

[@matgg8207]

Have been spending tons of efforts to study the machine learning stuff and watching this guys' video to strengthen my understanding of math behind it. I purchase your music album to support you !

[@leftfield00]

Best video on KZbin.. I call on KZbin to award this video..

[@matheuscardoso1]

If I become rich one day, I'll make you rich as well.

[@christinedai1817]

haha

[@pedronunes3063]

Matheus Cardoso Tinha que ser br.

[@xgalarion8659]

This comment made me remember the plot of DIckens Book 'Great Expectations'

[@apolloniuspergus9295]

It had to be a BR.

[@AlchemistOfNirnroot]

@@xgalarion8659 Is that a spoiler?

[@harishankarkarthik3570]

My teacher made the Taylor Series sound so complicated but you just made it look so natural and intuitive... congos... your channel just earned a new subscriber!