I am so... confused. How did this get recommended to me? Why did I watch it? Why are AI Elon and Swift explaining mathematics to me? Who is this man in the hoodie? How does this exist? .....This is so f**king chaotic and I love it.
One thing the video got wrong: if you infinitely continue the Taylor series it won’t become e^x. The left side limit of e^x is obviously zero,and the left side limit of the Taylor series is indeterminate. Also after x amount of terms the Taylor series can no longer follow its function.
@@tiziocaio101 no, it doesn’t. There is a limit to how precise the Taylor series can become. There are several ways to prove this, but one obvious one is stated in my comment above. Top answer here covers a good bit about limits and Taylor series and polynomials.
@@joshualee1595 As far as i know it does converge to the actual funcion, thats why e^x is called an analytic function. The difference between de Taylor polynomial and the actual function tends to zero as the degree of the series tends to infinity. Tell me if im wrong, but im quite sure i got it proved using the Lagrange remainder.
@@tuandingkang3263The tone of the voice is bland. It would be possible to differentiate if it's real or not even if the topic was different. However, what's concerning is that this was probably made by some 12 yo kid (no offense), the point being there are probably people who can make something far more convincing...
@@dariobarisic3502i do agree that this probably wasn’t made by an expert but do you really think a twelve year old made a deep fake about taylor series in calculus
@@hirandompeopled4968 Well, a 12 yo can for sure make a deep fake of this quality. But no, I don't literally think that this was done by a 12 yo considering it's about calculus. It was more like a hyperboly.
I want to add something : if adding an infinite number of terms allows to fully recreate the exponential everywhere (for all real numbers and even complex numbers) it is NOT the case for every function. If the function is "cool" enough (ie differentiable a bunch of times), you will be able to approximate it very well at a certain point or even on a interval but not necessarily everywhere it is defined. Take for example ln(1+x). If you consider its "infinite" taylor expansion around 0 it will actually be valid on ]-1,1[ but not for all x > -1.
The ratio test can be used for determining the radius/interval of convergence of the general power series representation of the Taylor series; if the radius of convergence is infinity, then the original function is exactly equal to the infinite series and not just the best local approximation. :)
i NEVER understood why each term in the taylor series/any series in general “makes up” the function we’re modeling until the last 10 seconds of this video. like genuinely.
It’s setting the initial value at a point a to be the same as the target function. Then the value of the first derivative is equated to the target function. Then the second, the third, and so on. All the derivatives of a function completely determine what the future values will be. So once all the values of the derivatives of the Taylor series match up with the target function at a point, you will know that the two functions are equal. Well, except if the target function is discontinuous with any derivative.
The “why” of calculus and this is explained (through a rigorous proof) in a more advanced class called “Real analysis” usually taught in 2nd year of a math degree in college. But u can see it kind of holds true through the visualisation. again the actual full mathematical proof though requires a course in real analysis.
@@onlocklearning I'm so glad that you answered me, right away. Otherwise, assuming your video format, I would have to wait until a Pop Star by the name of Maclaurin came along.
Was just about to ask this. I haven’t learnt about the Taylor series yet ,but did see a resemblance to a maclaurin series so I was wondering if it was a different name or something.
@@B3N_J1 Calculus textbooks usually have two separate sub-chapters (one for Maclaurin series; another for Taylor series.) If you are interested, it was actually a student of Gauss, specifically Riemann, who discovered the Taylor series. In my opinion, if we insist upon naming 50 or so significant math-methods after only one guy, then it makes it really hard for subsequent students to remember each distinct math-method. So, by all means, continue to name significant math discoveries after other people (who were not the actual discoverer.)
Wow, I'm shocked and grateful. My Calc 2 final's in a couple days, so this served as the perfect refresher; it's mind-numbing that a 88 second video voiced over with actual brainrot could go more in-depth than my Professor ever did. You genuinely have my thanks, and I hope you keep doing what you're doing; you're underrated man!
Bro I am so happy it actually helped! Thanks for the support I will keep going my goal is to cut all the inbetween yap and just have it as short as possible but still make sense
Really cute! I see you've done several of these, and I think it's a great idea. I especially think it will grab people's attention to get them to work through it their own heads, which is what's required for comprehension.
If anyone is still confused about how it actually works, you’re equating the all the derivatives of the Taylor series with those of the target function when x=a. Ie the values of each function are set to match at x=a. then the first derivatives are the same for each function at x=a. Same for second derivative and the third, and so on, until the last term in the Taylor series is reached. This is why functions with vertical asymptotes or discontinuities tend to not able to be approximated by a Taylor series for every value of x. The derivatives stop being able to completely determine the value of the function beyond the discontinuity.
Thank you so much for your educational video which made me realise that I do not fully comprehend maths! You indeed have a concrete understanding of them.
as a math major litteraly everything is crystal clear and its fucking brainrot, I am in admiration Only thing id say is that you shouldve explained that 99% of time we stop at X order but its so awesome I love it
Basically you have a function f(x) which can be expressed in form of x the pattern for each term will that the term(x) with nth power will be multiplied to nth derivative of f(x) where you need to put rhe value ,and the term will be divided by n factorial
I just wanted to say I took the Taylor (and MacLaurin) series in a 50-minute class, yet you simplified it much easier. I definitely thought of Taylor Swift too, when reading the name 😂.
It works, it makes you pay attention to see if the AI actually got everything right, the lip movement and the sound and meaning of what they are saying. Not kidding, this is incredible
Can you also explain what it used for in real life. like what does it do ? I really wanted to understand math but I don't know what a certain diagram can do or represent in real life.
Well, for example you can't tell what sin(x) is equal to without using a calculator. Using a Taylor series, you can approximate what sin(x) is equal to with some good precision. In fact, Taylor series is what calculators use behind the curtain to tell you what sin(x) is equal to.
Taylor series can be used to calculate functions like e^x, sinx or any other nice function. You can also use it to prove stuff like e^ix = cosx + isinx. Sometimes its easier in physics or engineering to work with taylor series than the original function. And you can use it to show that for small angles sinx ≈ x, which is a really useful and important approximation in physics and engineering
hahaah, but this series is really a life saver. especially in my field of embedded systems where i sometimes do really need to get an approximate plynomials for easy implementation of scary functions
"Eventually we will end up fully recreating e^x" Ehm actually 🤓 taylor serie APPROXIMATE the original function, meaning if you keep going on forever, you will keep getting closer and closer to the original function, but you will never get end up recreating it, as by definition you can't reach infinity One simply does not reach infinity, because the moment you reach it, it means you can add one and infinity isn't the biggest value anymore. Also i need to correct myself: infinity is not a number! It's just a concept!
The precise statement is captured by limits. Let S(N) be the partially summed Taylor series, i.e the sum of the first N terms. In the limit that N goes to infinity S(N) will tend to the function. Effectively what this means is that you can always compute |f(x) - S(N)|, and that you can always make this difference smaller by making N larger.
I am so happy knowing that Elon is adding all those terms, forever, on some stage in front of an audience that are sure they paid to hear something else.
![Limits using Taylor series[Calculus]](http://i.ytimg.com/vi/gXlXSaXMH1U/mqdefault.jpg)
![Limits using Taylor series[Calculus]](/img/tr.png)
![BLACK BAG - Official Trailer [HD] - Only in Theaters March 14](http://i.ytimg.com/vi/Du0Xp8WX_7I/mqdefault.jpg)
Taylor serie (taylor's version)
Underrated comment
@@Neonb88obvious comment
@@pabloasrin2211 Obvious yet necessary
You sir have won the internet
I am so sharing this with my Calc II students.
Coolest teacher ever
Please do
Why? It is not a good explanation.
不建议你这样做
@@thinkandmove479Yes it is LOL
I am so... confused. How did this get recommended to me? Why did I watch it? Why are AI Elon and Swift explaining mathematics to me? Who is this man in the hoodie? How does this exist?
.....This is so f**king chaotic and I love it.
Frrrr dude
I thought it was smin else
ahahaha
As a massive swiftie and physics nerd, this is exactly what I would expect to be in my recommended. Dunno what Elon is doing here though.
@@doxasticcelon mos and trailer swipt are married
@@doxasticc W I am a swiftie too
As someone who is learning Taylor series in a week, this is actually helpful
One thing the video got wrong: if you infinitely continue the Taylor series it won’t become e^x. The left side limit of e^x is obviously zero,and the left side limit of the Taylor series is indeterminate. Also after x amount of terms the Taylor series can no longer follow its function.
@@joshualee1595 the radius of convergence of e^x is infinite, shouldn’t that mean that the series converges for every real value?
@@tiziocaio101 no, it doesn’t. There is a limit to how precise the Taylor series can become. There are several ways to prove this, but one obvious one is stated in my comment above.
Top answer here covers a good bit about limits and Taylor series and polynomials.
@@joshualee1595 ok I see
@@joshualee1595 As far as i know it does converge to the actual funcion, thats why e^x is called an analytic function. The difference between de Taylor polynomial and the actual function tends to zero as the degree of the series tends to infinity. Tell me if im wrong, but im quite sure i got it proved using the Lagrange remainder.
This is the final form of math RUclips.
This is definitely not
more like final brainrot.
as someone who is working on a sine function for a vintage processor this is genuinely useful
it's in its prime but it's ain't even final form
deep fakes are becoming more and more scary
I thought this was real NOT KIDDING
If they were talking about something more believable, very tricky to tell
Are you retarded? Neither taylor swift nor elon musk know shit about math
@@tuandingkang3263The tone of the voice is bland. It would be possible to differentiate if it's real or not even if the topic was different. However, what's concerning is that this was probably made by some 12 yo kid (no offense), the point being there are probably people who can make something far more convincing...
@@dariobarisic3502i do agree that this probably wasn’t made by an expert but do you really think a twelve year old made a deep fake about taylor series in calculus
@@hirandompeopled4968 Well, a 12 yo can for sure make a deep fake of this quality. But no, I don't literally think that this was done by a 12 yo considering it's about calculus. It was more like a hyperboly.
quite literally helped me grasp the basics of taylor series despite multiple efforts before idk how u do it man but keep it going
I want to add something : if adding an infinite number of terms allows to fully recreate the exponential everywhere (for all real numbers and even complex numbers) it is NOT the case for every function.
If the function is "cool" enough (ie differentiable a bunch of times), you will be able to approximate it very well at a certain point or even on a interval but not necessarily everywhere it is defined.
Take for example ln(1+x). If you consider its "infinite" taylor expansion around 0 it will actually be valid on ]-1,1[ but not for all x > -1.
thank you! that is very true :))
Another example:
1/(1+x^2)
(Secretly has singularities at i and -i)
idk about maths but can we pull it out in the imaginary realms?
The ratio test can be used for determining the radius/interval of convergence of the general power series representation of the Taylor series; if the radius of convergence is infinity, then the original function is exactly equal to the infinite series and not just the best local approximation. :)
So true
That was actually well explained, please do a longer version of this. I'm actually learning shit I'm supposed to know
i NEVER understood why each term in the taylor series/any series in general “makes up” the function we’re modeling until the last 10 seconds of this video. like genuinely.
Im sorry but you still doesnt fully.
It’s setting the initial value at a point a to be the same as the target function. Then the value of the first derivative is equated to the target function. Then the second, the third, and so on.
All the derivatives of a function completely determine what the future values will be. So once all the values of the derivatives of the Taylor series match up with the target function at a point, you will know that the two functions are equal. Well, except if the target function is discontinuous with any derivative.
The “why” of calculus and this is explained (through a rigorous proof) in a more advanced class called “Real analysis” usually taught in 2nd year of a math degree in college.
But u can see it kind of holds true through the visualisation. again the actual full mathematical proof though requires a course in real analysis.
My recurring confusion happens when I try to remember what the difference is between a Maclaurin series and a Taylor series.
Dont worry bro just remember mclaurin is taylor series when a is set to 0 - so this one technically was mclaurin
*maclaurin
@@onlocklearning I'm so glad that you answered me, right away. Otherwise, assuming your video format, I would have to wait until a Pop Star by the name of Maclaurin came along.
Was just about to ask this. I haven’t learnt about the Taylor series yet ,but did see a resemblance to a maclaurin series so I was wondering if it was a different name or something.
@@B3N_J1 Calculus textbooks usually have two separate sub-chapters (one for Maclaurin series; another for Taylor series.) If you are interested, it was actually a student of Gauss, specifically Riemann, who discovered the Taylor series. In my opinion, if we insist upon naming 50 or so significant math-methods after only one guy, then it makes it really hard for subsequent students to remember each distinct math-method. So, by all means, continue to name significant math discoveries after other people (who were not the actual discoverer.)
How did you guys turn math into brainrot
getting ready for generation α 🔥
One channel can turn math to brainrot?
You brainrot
Brainrot? This stuff is braingrow
@@onlocklearning the oldest of gen α are still in grade 8, you have plenty of time to prepare
Taylor Swift swiftly explains the Tailor series.
Wow, I'm shocked and grateful. My Calc 2 final's in a couple days, so this served as the perfect refresher; it's mind-numbing that a 88 second video voiced over with actual brainrot could go more in-depth than my Professor ever did. You genuinely have my thanks, and I hope you keep doing what you're doing; you're underrated man!
Bro I am so happy it actually helped! Thanks for the support I will keep going my goal is to cut all the inbetween yap and just have it as short as possible but still make sense
i genuinely cant believe that this helped me understand the taylor series better than any lecture video
Oh man am obsessed with your content❤️
2 years in engineering college, and this is the first time I'm actually understanding Taylor series...
Dude this seriously helped a lot! Thankyou!!!
noway we got TS explaining TS
type shit
This is actually a really good explanation of taylor series
C’mon, was Elon’s part actually ai generated?
🤣
Do you think Elon knows this much maths
@@azertyuiop61473 This is not complex math, I would not be surprised if he knows this
Elon is an AI
@@azertyuiop61473 This is like Calc-2 level math so for Elon this is late high school to early college
Really cute! I see you've done several of these, and I think it's a great idea. I especially think it will grab people's attention to get them to work through it their own heads, which is what's required for comprehension.
I like how the orchestral version of "Wildest Dreams" from the Bridgerton soundtrack came in toward the end. By the way, why is Elon here?
ahaha it's a banger, and just to spice things up tbh loool
If anyone is still confused about how it actually works, you’re equating the all the derivatives of the Taylor series with those of the target function when x=a. Ie the values of each function are set to match at x=a. then the first derivatives are the same for each function at x=a. Same for second derivative and the third, and so on, until the last term in the Taylor series is reached.
This is why functions with vertical asymptotes or discontinuities tend to not able to be approximated by a Taylor series for every value of x. The derivatives stop being able to completely determine the value of the function beyond the discontinuity.
As someone who is preparing to study calculus, thank you so much for making these!
Love your math vids , pls keep em comin 😭❤
That's the most I've heard Elon speak without taking like ten breaks between each sentence
Unironically i use your videos to help me retain calculus stuff better
Its also very interesting to note that the graph looks like a thread sewing a cloth which is what tailors do so the name tylor series is perfect
crazy how i acrually learnt what it is through this video of all the videos on it ever
Honestly, if Taylor Swift were ACTUALLY to explain the Taylor-series, that would be pretty wild
Im so confused on how swift is explaining this. This chaos is lovely
On how swift she's explaining it
honestly pretty good explanation
Taylor Swift swiftly explains the Taylor series
Might have to change the title to that
This feels just like scrolling through an electronic copy of a school textbook instead of tiktok, pupils dilated.
I couldn't understand Taylor series back then. Thank you.
I'm honestly not going to lie, Taylor is helping refresh my understanding!
Gen alpha: Who is taylor swift, Elon Musk
Gen Z: our math teachers😊
Can you please do the arf invariant next?
Looks like a graph of Taylor's net worth ;-)
Clever and entertaining vid Onlock.
Pls make more of these
It has a fundamental flaw actually like at one point it just starts breaking up not be able to follow e^x completely at infinity ends ...
the first thing that came up on my mind when i learned about taylor series was taylor swift lmao
This is actually pretty well made, and if it can make gen zers/alphaers learn something, I'm all for it!
This actually helped me understand the topic and my exam is in 6 days. Brain rot is turning into brain development
Thank you so much for reminding me of my calculus class i need it for my masters courses
i literally just learnt about taylor series w this
It's interesting to see such people explain this concept in 1 min while our teacheee couldn't in 2hours😢
Thank you so much for your educational video which made me realise that I do not fully comprehend maths! You indeed have a concrete understanding of them.
if only I got this recommended to me a week ago (before my final exam)
Wish I find this video earlier, will help me a lot in calculus
i'm still in precalc but this explanation was so brilliant that i kind of got it
Thank you bro!!
Where I am from we are only tested on the Maclaurin series which is a special form of Taylor series.
as a math major litteraly everything is crystal clear and its fucking brainrot, I am in admiration
Only thing id say is that you shouldve explained that 99% of time we stop at X order but its so awesome I love it
Bro please one videos on binomial theorem with questions include permutations and combination also
As a Calc 2 student who’s a swiftie, I actually like this 😂
Please do something like this in relation to relativity, quantum physics and timespace of a blackhole etc.
i can't believe that a video of ai taylor swift and elon musk explaining the taylor series would ever exist,
Basically you have a function f(x) which can be expressed in form of x the pattern for each term will that the term(x) with nth power will be multiplied to nth derivative of f(x) where you need to put rhe value ,and the term will be divided by n factorial
I just wanted to say I took the Taylor (and MacLaurin) series in a 50-minute class, yet you simplified it much easier. I definitely thought of Taylor Swift too, when reading the name 😂.
POV: it's the year 2035 and you're learning for your math exam
It was made in 2023 Sooooo
I truly can not believe this exists, but I am glad it exists.
A polynomial is a finite combination of power terms; in general, a power series is not a polynomial.
*e ^ x* is the GOAT 🗣🔥
Next have them explain residue calculus and analytic functions
It works, it makes you pay attention to see if the AI actually got everything right, the lip movement and the sound and meaning of what they are saying.
Not kidding, this is incredible
everybody gangsta until non-analytic smooth function
Am I watching taylor swift and elon musk teach me Taylor Series? Yes.
Am I going to get an A in calc? No, but this makes it worth it.
Please, make explanation of Fourier Series.
Remove those background sounds man. They are really distracting. Your content is gold.
you literally can make a whole series for lets say a levels math like this and sell it, this will be a nice thing to scroll before going to class
i unironically want to see the real taylor swift explaining the taylor series
I understood this better than the profesor explaining it in 10 hours
Isn't there for some functions edge effects with oscillation with a too high polynomial approximation?
This seems like its derived from linear approximation or some kind of calc aproximation methods?
Memes aside, this is actually so helpful.
okay... getting her to explain the Taylor series is hilarious
Can you also explain what it used for in real life. like what does it do ? I really wanted to understand math but I don't know what a certain diagram can do or represent in real life.
Well, for example you can't tell what sin(x) is equal to without using a calculator. Using a Taylor series, you can approximate what sin(x) is equal to with some good precision. In fact, Taylor series is what calculators use behind the curtain to tell you what sin(x) is equal to.
Taylor series can be used to calculate functions like e^x, sinx or any other nice function. You can also use it to prove stuff like e^ix = cosx + isinx. Sometimes its easier in physics or engineering to work with taylor series than the original function. And you can use it to show that for small angles sinx ≈ x, which is a really useful and important approximation in physics and engineering
can you explain surjective injective bijective? i still don't understand also the explanation that i found it's not understandable
Man this is better than my maths classes, and this is the kind of shit I'm not learning in school for like 2 or 3 something years
Hey this is actually working on my mind 💀💀
Just took the AP BC Calc test today, hopefully I’ll get a 5.
I could have used this video right before the test
i cant believe im really using this for study XD
hahaah, but this series is really a life saver. especially in my field of embedded systems where i sometimes do really need to get an approximate plynomials for easy implementation of scary functions
Teach us how you do these kind of videos :)
thank you taylor
How get a initial a for every function
How do you make this? I would like to do with some popular figure in my country
Bro forshadowed the Elon Musk and Taylor Swift Drama
I'd like to be a part of Taylor's series!
"Eventually we will end up fully recreating e^x"
Ehm actually 🤓 taylor serie APPROXIMATE the original function, meaning if you keep going on forever, you will keep getting closer and closer to the original function, but you will never get end up recreating it, as by definition you can't reach infinity
One simply does not reach infinity, because the moment you reach it, it means you can add one and infinity isn't the biggest value anymore.
Also i need to correct myself: infinity is not a number! It's just a concept!
The precise statement is captured by limits. Let S(N) be the partially summed Taylor series, i.e the sum of the first N terms. In the limit that N goes to infinity S(N) will tend to the function.
Effectively what this means is that you can always compute |f(x) - S(N)|, and that you can always make this difference smaller by making N larger.
i did need this for analysis 2 exam thanks taylor 👍
Correction: Taylor series are not polynomials. Polynomials are finite.
this combo of elon and taylor is insane now 💀
Didnt really explain but still cool video.
When I was in college I was wondering why they call it Taylor series . Now I figured out .😊👌
I am so happy knowing that Elon is adding all those terms, forever, on some stage in front of an audience that are sure they paid to hear something else.
Really Impressed. Its not really brain rot
Next: Adam Lambert on lambert w
Please do complex expansion of trig functions sin(z)