honestly, this guy may not show difficult examples that I may be tested on, however his theory lessons are what makes hime so great! Thank you very much Sal :)
Hey Sal, I think I am in love with you. This is me shooting my shot. I have a winning personality, fun hobbies, and I will spend every moment with you like it is my last. Much love xoxo
"...IN A NEW COLOUR!" *Fails to click on a colour* "That is not 'a new colour'!" 😂😂😂 PS. This was the best explanation of MacLaurin's Series I've seen till date. When we did this in college, our prof seemed to use Aladin's magic lamp to bring this series into existence. Thank you for this.
Amazing! When my college professor explained this he didn't even bother to tell us what we were actually doing/finding. He just basically gave us the formula. Thanks so much!
I was really bad at Maclauren and Taylor series when I was in Calc 1, and now I'm learning about Laurent series in Complex Analysis. This video helped me tremendously to understand it because I'm too lazy/exhausted to read about it. Thanks again, Mr. Khan.
this is indeed around first year uni math in the usa, but they do start teaching about polynomials in algebra 2 which was taught in 10th or 11th grade i forget
@dickie4thepeople i think of it as an approximation. First we write an equation, then we add an equation for the gradient, then we add an equation for the rate of how the gradient changes (using differentiation) etc... so our approximation for the rest of the line is becoming better.... sorry i dont completely grasp this either. but i hope this kinda helps!
@Yakushii not really, because you wouldn't be able to find the second derivative etc, what it does allow you to do is turn a function into a polynomial ie one just involving x to certain powers, so you can make sin(x) into a function of lots of x's of different powers added together.
You can evaluate at any number, but a Maclaurin series is evaluated at zero. That's the only distinction. Zero can be replaced with 'a' or any variable holding the place for any number.
They are both awesome. Sal tends to focuses a little more on theory/understanding while Patrick is usually more direct and demonstrates how to apply the concepts.
Basically the binomial expansion method that we are told to cram is also a case of the Taylor series just like the Maclaurin series is but not on basis terms. One is centered around avalue and one just aims at expanding a function.
Thank you so much!! I have finals later this week, and I was sick when this concept was introduced, and now I understand it!! Thank you thank you thank you!!
Where does it come in 11th?? I am also an Indian and I studied this only while doing my B-Tech. Yes, you have some specific functions whose Taylor's expansions you have to learn in 11th and 12th, if you are preparing for the JEE exam, but you still don't learn the derivation and that it comes from Taylor's and Mclaurin series....You just memorize the expansions of specific functions
Khan academy as I know it in the past was made so that people can watch the videos and understand the concepts without getting bored and without getting confused and before they were doing well in staying consistent with the curriculum they are tackling... but as of now, when I watch Khan academy videos and I easily get bored, I easily get confused, and there is no consistency with any curriculum that I know of. I don't know if it has to do anything with that I am in college or if my understanding of the topic is low, but if I know something, it is that Khan academy has fallen in my opinion
When I first saw this, I was puzzled by the fact that the sequence of derivatives at ONE point only can approximate the WHOLE function when keep adding terms. I still don’t get the intuition here.
Woah man, who knew that it is THIS intuitive. Well, I guess most things are... but if it wasn't for you I would've gone my whole life thinking that this is some kind of elusive abstract thing that some guy thought of and it's just the way it is, and we should accept it and use it. Man, why does no one tell us this, insted of just giving us the formula. Stupid. I was first 'taught' Taylor series without drawing any graphs whatsoever :/
Awesome! The best interpretation of Maclaurin series! BUT i would really appreciate if you would show How we get Taaylor from Maclaurin. And why we use (x-a) in Taylor.
@qwobify If you differentiate f^4(x)*x^4/24. You get f^3(x) * 4x^3/24= f^3(x) * x^3/6. Differentiating that gives f^2(x) * 3x^2/6 = f^2(x) * x^2/2. And finally you get f(0) at the end, just as before.
at that particular point (doesn't have to be zero) we want our function and the actual function to have the same value, then the same derivative, then the same second derivative, then the same 3rd derivative etc.. and the more you do that, the more likely it will look like the actual function Basically we are saying you can approximate a function if you keep taking the derivative of a certain point
Ahhh this would've been helpful about 2 months ago! I eventually got it but this would've made the process much faster. I take Calc 3 in the fall... Hint hint... But thanks for all you do your vids are an amazing help!
I still don't understand i get lost at 3:06 when he makes p(x)=f(0)+f'(0)x. Why did he add f'(0)x?? how does adding f'(0)x make p(x) and f(x) have he same first derivative. Also how does adding all these extra derivative terms give us a better approximation?? Thank you for any clarification!
Take the derivative of p(x)=f(0)+f'(0)x and notice that it is precisely the first derivative of f(x). This means that we have improved our approximation slightly. Therefore it can be improved further by matching higher order derivatives of the approximation to that of the original function.
+purplefire5 "how does adding f'(0)x make p(x) and f(x) have he same first derivative?" -> just derive it man, you'll see they do have the same derivative (remember that f(0) is a constant)!
Thanks so much for this video Sal... is there a video for proof of Taylor Series Expansion for function with 2 variables? Thanks again for all the videos.
Well I don't think that the line he draws at minute 5:00 should be linear. Because p'(x) is a linear line and p(x) should be something like y=x^2. Am I wrong?
Google Taylor or Maclaurin proofs. The full definitions of these series expansions have them going to an infinite number of terms anyway. Sal just stopped after a few derivations when in reality, these keep going forever.
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Sometimes you need these tiny details, these tiny bits of intuition to get the picture. You really help with this. Thanks.
honestly, this guy may not show difficult examples that I may be tested on, however his theory lessons are what makes hime so great!
Thank you very much Sal :)
Hey Sal, I think I am in love with you. This is me shooting my shot. I have a winning personality, fun hobbies, and I will spend every moment with you like it is my last. Much love xoxo
I hope I'm not the only one who burst out laughing at "well Sal that's a hoarrible approximation".
I chuckled too
YOU try doing better with a straight line 😂😂
I love the "aha!" moment you get in the middle of watching the video. Nothing quite like it!
Khan Academy just has this magic that touches parts of everyone's brains and gets it working like no one else could and would ...
That sounds weirdly romantic.
Hahahah man you just gotta ruin it!
I dont know what I would do without you :')
"...IN A NEW COLOUR!"
*Fails to click on a colour*
"That is not 'a new colour'!"
😂😂😂
PS. This was the best explanation of MacLaurin's Series I've seen till date. When we did this in college, our prof seemed to use Aladin's magic lamp to bring this series into existence. Thank you for this.
Aladdin's magic lamp 😂😂😂
Amazing! When my college professor explained this he didn't even bother to tell us what we were actually doing/finding. He just basically gave us the formula. Thanks so much!
I was really bad at Maclauren and Taylor series when I was in Calc 1, and now I'm learning about Laurent series in Complex Analysis. This video helped me tremendously to understand it because I'm too lazy/exhausted to read about it. Thanks again, Mr. Khan.
Sal! You are just amazing. You know what? You are on the list of my most most most favourite teachers in my life.
You are that amazing.
“TRY TO DO ANY BETTER USING A HORIZONTAL LINE THEN” 😂 loved that
okay but did i ask
@@andrewmontoya8511 Yes you did ask
Khan breaks everything down in understandable chunks yet without losing the generalization rigor of Maths - wonderful
this is indeed around first year uni math in the usa, but they do start teaching about polynomials in algebra 2 which was taught in 10th or 11th grade i forget
@dickie4thepeople i think of it as an approximation. First we write an equation, then we add an equation for the gradient, then we add an equation for the rate of how the gradient changes (using differentiation) etc... so our approximation for the rest of the line is becoming better.... sorry i dont completely grasp this either. but i hope this kinda helps!
i cant's take it in once,,,,, but watching again again & finally get it
@Yakushii not really, because you wouldn't be able to find the second derivative etc, what it does allow you to do is turn a function into a polynomial ie one just involving x to certain powers, so you can make sin(x) into a function of lots of x's of different powers added together.
I don't know what I'd do without you sal😭😭😭😘😘
The best teachers aren't the smartest ones, it's the ones that doesn't require their students to be very smart to learn.
You can evaluate at any number, but a Maclaurin series is evaluated at zero. That's the only distinction. Zero can be replaced with 'a' or any variable holding the place for any number.
After reading the comments I envy all of you that you all have mastered the Taylor theorem but I am still struggling!!😂😂
They are both awesome. Sal tends to focuses a little more on theory/understanding while Patrick is usually more direct and demonstrates how to apply the concepts.
You're the world's best teacher, undoubtedly Mr. Sal Khan, Masha-Allah!
I really, really like this.
Derivation of the Maclaurin Series from "CORE MATHS for A-level" by L. Bostock and S. Chandler, published by Stanley Thornes (Publishers) Ltd:
A power function of f(x) = (a + x)^n
Where, when considered with the general binomial theorem, gives:
a^n + a^(n-1) * x + a^(n-2) * x^2 + ...
Where a^n, a^(n-1), a^(n-2)... are all constants, to be reconsidered as:
a0 + a1 * x + a2 * x^2 + ...
f(x) = (a + x)^n = a0 + a1 * x + a2 * x^2 +...
f'(x) = a1 + (2)a2 * x +...
f''(x) = (2)a2 + ... (3)(2)a3 * x +...
f''(x) = (3)(2)a3 + (4)(3)(2)a4 * x +...
When x = 0:
f(0) = a0
f'(0) = a1
f''(0) = (2)a2 -> a2 = f''(x) / 2 = f''(x) / 2!
f'''(0) = (3)(2)a3 -> a3 = f'''(x) / (3)(2) = f'''(x) / 3!
...
f^n(0) = f^n(x) / n(n-1)(n-2)... = f^n(x) / n!
Therefore:
f(x) = (a + x)^n = a0 + a1 * x + a2 * x^2 +... = f(0) + f'(0) * x + [f''(0) / 2!] * x^2 + [f'''(0) / 3!] * x^3 + ... [f^n(0) / n!] * x^n
Bro what are u talking about
Basically the binomial expansion method that we are told to cram is also a case of the Taylor series just like the Maclaurin series is but not on basis terms. One is centered around avalue and one just aims at expanding a function.
Colin Maclaurin is credited with the Maclaurin Series
Thank you so much!! I have finals later this week, and I was sick when this concept was introduced, and now I understand it!! Thank you thank you thank you!!
How are you doing bud
you are a star, and a hero, thank you for this
Funny how Indian students do this in class 11 while in the US college students struggle in AP calc 😅
I live in England and im also doing this in highschool lol. Im also Indian btw :D
Where does it come in 11th?? I am also an Indian and I studied this only while doing my B-Tech. Yes, you have some specific functions whose Taylor's expansions you have to learn in 11th and 12th, if you are preparing for the JEE exam, but you still don't learn the derivation and that it comes from Taylor's and Mclaurin series....You just memorize the expansions of specific functions
Nice Video Dude
Khan is king!
Yes, more stuff on Series and Sequence. It was lacking in your Calculus Playlist.
Khan academy as I know it in the past was made so that people can watch the videos and understand the concepts without getting bored and without getting confused and before they were doing well in staying consistent with the curriculum they are tackling... but as of now, when I watch Khan academy videos and I easily get bored, I easily get confused, and there is no consistency with any curriculum that I know of. I don't know if it has to do anything with that I am in college or if my understanding of the topic is low, but if I know something, it is that Khan academy has fallen in my opinion
Wow. So well and simply explained. Thankyou.
when he says "I don't have the computer power of my brain to draw it properly..." damn!! this guy has some imagination I really envy of him >.
When I first saw this, I was puzzled by the fact that the sequence of derivatives at ONE point only can approximate the WHOLE function when keep adding terms. I still don’t get the intuition here.
wow what concept! thanks do much for explaining it so simply i feel like a have a good understanding of it now
So helpful thanks!
I HAVE EXAM ON THIS ON MONDAYYYYY ahhhhhh THX SALLLL
Luv u lot. Best explanation for Maclaurin series ever.U doing great work ...
"That's not a new color" 😂
Luv u guys
Woah man, who knew that it is THIS intuitive. Well, I guess most things are... but if it wasn't for you I would've gone my whole life thinking that this is some kind of elusive abstract thing that some guy thought of and it's just the way it is, and we should accept it and use it. Man, why does no one tell us this, insted of just giving us the formula. Stupid. I was first 'taught' Taylor series without drawing any graphs whatsoever :/
I Really Like The Video From Your Approximating a function at 0 using a polynomial
Thanks. This was very helpful in trying to understand my not so helpful textbook.
Thanks for saving my grades!
Sal, you're so good at explaining. Thank you so much
Awesome teaching i love your teaching style
Thank you Sal
This is gold. Thank you again. :)
This is pretty good. I got the idea about how Mclaurin Series generated. Thumbs up.
Awesome! The best interpretation of Maclaurin series! BUT i would really appreciate if you would show How we get Taaylor from Maclaurin. And why we use (x-a) in Taylor.
I have my final in two days and I absolutely love you
Hey are you alive
Thanks Sal!
amazing explanation. thanks.
this guy, is the saviour!
.
Many,many thanks to khan academy!!
not very formal, but quite clear and intuitive. thank you!
Got a vid on the radius of convergence?
i love you, Sal
thank you khan
@qwobify If you differentiate f^4(x)*x^4/24. You get f^3(x) * 4x^3/24= f^3(x) * x^3/6. Differentiating that gives f^2(x) * 3x^2/6 = f^2(x) * x^2/2. And finally you get f(0) at the end, just as before.
at that particular point (doesn't have to be zero) we want our function and the actual function to have the same value, then the same derivative, then the same second derivative, then the same 3rd derivative etc..
and the more you do that, the more likely it will look like the actual function
Basically we are saying you can approximate a function if you keep taking the derivative of a certain point
Ahhh this would've been helpful about 2 months ago! I eventually got it but this would've made the process much faster. I take Calc 3 in the fall... Hint hint... But thanks for all you do your vids are an amazing help!
i never tought taylor series would recieve a 1.1 million views
Good example, great explanation
Great explanation sir ❤❤
Well, I finally understand it... 3 years after finishing my calculus course.
Without Sal I wouldn't have passed high school
This was a really helpful video.
This video was Awesome
i wish i saw this video back when i was 20 -__- nice and easy explanation
very helpful thanks alot
This guy know bio,chem,math and physics. damn
Nice one!
Thank you so much.
thanks for the proof!
"That's not a new color...."
Pretty much describes my understanding of Taylor series
saved my day
@MilitaryMan006 This is first year uni math.
some people get so excited at things
I think he differentiated the second video one time to many. ;O)
@Mugwump720 Agreed! Finals this week, and I've been sick recently, so this is really making it all make sense!!
Thats actually genius.
Well you use (x-a) in Taylor because you can choose any center point (a). So in the case of the Maclauren Series, a=0 so it' would just be (x).
very helpful........thanks a lot!
ultimate, fantastic, mind blowing Boss.
Thank you soooo much!! I love your explanations in all your videos! Thank you for making these videos available for us!
How f(0) and its derivatives can have all the information of the entire curve? The curve could go in any direction after f(0).
that's pretty interesting, but how are you suppose to know the values of f'(0), f''(0), f'''(0), etc... if you aren't given the function f(x)???
Sal FTW! Thank you so much buddy :D
Yeay more Calculus :D
Yeah, thanks. I can never seem to freaking understand anything in calc2, I aced the first calc, but it's like there's a brick wall in calc 2.
thanks
I still don't understand i get lost at 3:06 when he makes p(x)=f(0)+f'(0)x. Why did he add f'(0)x?? how does adding f'(0)x make p(x) and f(x) have he same first derivative. Also how does adding all these extra derivative terms give us a better approximation?? Thank you for any clarification!
Take the derivative of p(x)=f(0)+f'(0)x and notice that it is precisely the first derivative of f(x). This means that we have improved our approximation slightly. Therefore it can be improved further by matching higher order derivatives of the approximation to that of the original function.
+TheNuclearpolitics Is it necessary to know the infinite higher order derivatives of the line?
+TheNuclearpolitics
Great explanation.
+purplefire5 "how does adding f'(0)x make p(x) and f(x) have he same first derivative?" -> just derive it man, you'll see they do have the same derivative (remember that f(0) is a constant)!
Could you do a video explaining time travel?
Beautiful.
are there any Khan academy videos for convergence tests for series?
Now I know where the kinematic equations came from!
Thanks so much for this video Sal... is there a video for proof of Taylor Series Expansion for function with 2 variables?
Thanks again for all the videos.
Better Call Sal
Well I don't think that the line he draws at minute 5:00 should be linear. Because p'(x) is a linear line and p(x) should be something like y=x^2. Am I wrong?
He is drawing P(x) = f(0) + f'(0) x which is eqn of line
So is it possible to find the function of any given graph, just by reading the graph, using MacLaurin series?
Does anyone know where I can find the proof Sal is referring to at 12:20 ?
Google Taylor or Maclaurin proofs. The full definitions of these series expansions have them going to an infinite number of terms anyway. Sal just stopped after a few derivations when in reality, these keep going forever.