What Sal does at 9:04 is one of the things that make it so easy to learn from him. He clarifies why he is using sin(1) even though it would be obvious to a lot of people. Teachers generally assume that the students know the reasoning behind every single step in their calculations, which isn't always the case. Thanks a lot, Sal.
I seriously can't thank you enough. My math professor's ineptitude is rivaled only by your competence in explaining the same material. I went into that lecture less confused than I was leaving, whereas this video provides a crystal-clear explanation. Who knows? If there were someone like you for every major subject, UA-cam could serve as a viable replacement for college. Cheers!
Wow. This really helped me better understand the concept. I've watched another popular teacher on youtube, but visualizing it with the help of a software and the way you explained it, really helped me understand it better. Thanks
I had the assigned notes on this for my calculus course, but I couldn't understand. I read it over, and over, and gave a week space in between so I can have a fresh look at the concept of Taylors Polynomial. I heard your site, checked out on how you were teaching the concept... and wow... best 18 minutes of my life spent ACADEMICALLY! Thank you.
Don't be sorry Sal, you deserve all the walnuts you could ever want for explaining this so well! I wish more people would take the time to explain the reasons behind maths functions, it makes it so much easier to see why and what's happening.
I did the former. Each added term contributes to the approximation and doesn't replace it. So the approximation with 100 terms would be much better than the approximation with 2 terms.
This is just fantastic! At my university I was only taught how to use the Taylor Series like it's some magical formula you just have to remember, but don't need to understand. In less than twenty minutes you managed to explain what exactly it is and how it is used. Thanks a lot!
To answer a question that's popped up a couple times below: Doing a Taylor expansion for certain functions makes evaluating them around a certain point easier than evaluating the actual function. For example, doing this for sin(x) or tan(x) for SMALL values of x, the later terms of the expansion are so small that you can approximate sin(x) [or tan(x)] to equal x. Cool, right? Here's what I mean: en.wikipedia.org/wiki/Small-angle_approximation
this is amazing since, after seeing so many classes, and knowing that males are visual, only this guy decides to make taylor visual. Thx for putting "visual males" and "math" together.
I reviewed these videos again, and understand it now. Indeed, even the approximation at n=0, that is, where it's just a straight line or constant, is equal to the function at point c. Making the approximations better by adding more terms gives you better approximations at points "near" c and, with even better approximations with more terms, further away from point c. THANKS SAL!
Had Numerical Analysis class for the first time at the start of semester today. He made me feel like a complete moron because he sped through Taylor Polynomials in class as if it was the simplest thing in the world to just pick up. This taught me more in 18 minutes than my teacher could in an hour and fifteen. Seriously, why can't more professors be this good?
Nicely stitched video. Taylor taylored a polynomial fabric that overwhelms the imagination. What is the meaning of an "nth derivative," for example? The graphs help you begin to see what's going on! Taylor Polynomials aren't just "sew sew." They are awe-inspiring! May the nth derivative be with you!
Hey Sal! Amazing video. Why did you make the assumption that if the 0th, 1st, 2nd, 3rd, 4th, 5th derivative is = to the function, then it perfectly = the function? I get the intuition behind it, and I can see it work very well on the graph. But surely there must be a proof, right?
What I really think is mind-blowing is how you can write so clearly with a mouse. People can't even tell what I'm drawing on Draw My Thing, and I bet if you played that, you'd be drawing the Mona Lisa left and right.
thank you so much...i just watched every series video and it makes perfect sense now. my ap calc test is this Wednesday and i was flipping out cause everytime i took a practice free response i just completely skipped the series question and was getting no points for it, not to mention the multiple choice. thses videos are great and e^(ipi) + 1 = o ...wtf?!
when u add each term, are you adding it to all the other terms or are u just graphing it by itself? that is, was the second approximation p(x) = cos1 - sin1(x-a) or was it p(x)=-sin1(x-a)??? i think ur doing the latter... in which case, wouldn't the, say, 100th derivative be a really small constant times (x-1)^100?
lol ahah me too. random question though cuz its insane we have a math final on the same day with the same stuff... do you happen to go to the university of saskatchewan? that would be kind of insane.
Could you possible explain the error term when approximating a function only to the nth derivative. I've got something about it in my notes but it doesn't really make sense to me.
Love the video, thank you soooo much but there's one thing I don´t understand.. What's the purpose of the approximation when we already have the function?? dont get it... :/
Holy shit O.O I've not gone to a single lesson we have so far because the teacher is so bad, and i think i'm about to pass the whole course just by your videos.
@someonetoogoodforyou The nth derivative is not equal to the function. It's equal to the function at c. As the nth derivative approaches infinity, not only does p(x) equal f(x) at c, but some of the terms near c of p(x) are close to the terms of f(x) near c. The more derivatives, the closer the values near c of p(x) are to the values near c of f(x) and the farther from c you can approximate. Theoretically, taking infinite derivatives will make the two functions equal for all x.
Your presentation was fine. However, I would like to know why we go through the hastle of defining a function around a particular point for a stated function when we have THE actual function. I guess an application is in order so if you could get that on a video sometime in the not-too-distant future, that‘d be great. Thank you.
Oswald Chisala can you do cos(1) of the top of you head? No you cant. Thats why we use taylor polynomials. You can actually sit down and find cos1 with a reasonable degree of accuracy without a calculator when you use taylor polynomials. Also your calculator is actually using taylor polynomials to calculate trig functions and other weird functions like e^x when you put them in the calculator
Nice math lesson, but the most important lesson we all learned is to always drink water with your walnuts.
Math is really hard when you trippin on walnuts.
"Sorry my brain is... I ate too many walnuts" -khan academy tutor
That's actually Sal Khan
"Sorry I just had some walnuts" lol
What Sal does at 9:04 is one of the things that make it so easy to learn from him. He clarifies why he is using sin(1) even though it would be obvious to a lot of people. Teachers generally assume that the students know the reasoning behind every single step in their calculations, which isn't always the case.
Thanks a lot, Sal.
9:04
"I hope this video gave you some intuition on the Taylor Series. If it didn't, please ignore this video" HAHAHAHAHAHAH BEST ENDING EVER
I was your 100th like! 🥳😁
Note to self: do not eat walnuts before exams.
go bless you idk why I'm paying so much money for uni when I just end up coming here
where do you have to pay for uni? FeelsGoodMan
So you could have a bachelor degree certificate to apply to some shitty job with it, duh
Cuz u need a motive for coming here to exhaust ur brain
US and Canada for sure
@@Glendragon almost everywher except for norway maybe
Reading the comments before watching the video was really confusing hahaha
i pay you nothing yet you teach me a lot. i spend all my parents saving to my university and i get nothing.
I want all my uni fees to go straight to you because you're teaching me more than any of my lecturer's ever could dream of
"My brain had too many walnuts" lololol xD
I seriously can't thank you enough. My math professor's ineptitude is rivaled only by your competence in explaining the same material. I went into that lecture less confused than I was leaving, whereas this video provides a crystal-clear explanation. Who knows? If there were someone like you for every major subject, UA-cam could serve as a viable replacement for college. Cheers!
o/
The next time I'm not studying for a calculus exam, I'm going to try and computer formulas eating walnuts. Sal, you're the best!
this is truly amazing, with limited class time there is no way anyone can understand this shit. Thank you Khan Academy, for all the review.
Wow. This really helped me better understand the concept. I've watched another popular teacher on youtube, but visualizing it with the help of a software and the way you explained it, really helped me understand it better. Thanks
in 18 minutes you taught me a whole chapter of my maths book that my lecturer couldn't teach me in a 2 hour lecture.
thanks sal!
I had the assigned notes on this for my calculus course, but I couldn't understand. I read it over, and over, and gave a week space in between so I can have a fresh look at the concept of Taylors Polynomial. I heard your site, checked out on how you were teaching the concept... and wow... best 18 minutes of my life spent ACADEMICALLY! Thank you.
Adding more terms makes the approximation of the function better at all points (not just at C). Even with just the first term, P(c)=f(c).
Don't be sorry Sal, you deserve all the walnuts you could ever want for explaining this so well! I wish more people would take the time to explain the reasons behind maths functions, it makes it so much easier to see why and what's happening.
I did the former. Each added term contributes to the approximation and doesn't replace it. So the approximation with 100 terms would be much better than the approximation with 2 terms.
This is just fantastic! At my university I was only taught how to use the Taylor Series like it's some magical formula you just have to remember, but don't need to understand. In less than twenty minutes you managed to explain what exactly it is and how it is used. Thanks a lot!
I can understand the effect of walnuts
How the fuck do people just come up with this stuff? its amazing
With walnuts
Jesus
To answer a question that's popped up a couple times below: Doing a Taylor expansion for certain functions makes evaluating them around a certain point easier than evaluating the actual function. For example, doing this for sin(x) or tan(x) for SMALL values of x, the later terms of the expansion are so small that you can approximate sin(x) [or tan(x)] to equal x. Cool, right?
Here's what I mean: en.wikipedia.org/wiki/Small-angle_approximation
2nd video ive seen where hes choking on walnuts
This has been incredibly helpful-along with many of your other videos.
nice work you have changed my attitude to the tailors theorem
this is amazing since, after seeing so many classes, and knowing that males are visual, only this guy decides to make taylor visual. Thx for putting "visual males" and "math" together.
Thank you so much man! I didn't understand this concept well during class and this really cleared up Taylor Polynomial's for me!
oh my. Khan Academy's videos are something that i never regret watching!
Best 18 minutes spent
khan saved my linear algebra and now my calculus too. thanks alot haha
Really Great lesson! So good! I recommend it to anyone trying to understand Taylor polynomials. Khan academy is amazing
I reviewed these videos again, and understand it now. Indeed, even the approximation at n=0, that is, where it's just a straight line or constant, is equal to the function at point c. Making the approximations better by adding more terms gives you better approximations at points "near" c and, with even better approximations with more terms, further away from point c. THANKS SAL!
Amazing love your voice, you tought something which none else taking about .
amazing. i sat through an entire hour of this and learned literally nothing. then i watch this video and i understand perfectly. thank you kind sir
Sir , i can't express my gratitude to you in words
You help me a lot
this. is. amazing. I completely understand how taylor polynomials work now. Taylor was a genius.
Holy cow! This makes so much sense now. Thank you.
This was the most beautiful thing I've ever seen in math
Once you GET the Taylor Polynomials...it really does blow the mind into orbit for a while.
you're amazing!!!!! loved the calculation part at the end
Now you just taught me in 18 mins, what my Maths professor wasn´t able to teach me in like 3 lectures of 90 mins each! Thanks!
You have no IDEA how truthful that statement is.
You are blowing dust out of my 👂 , thank you
I Really Like The Video From Your Approximating a function with a Taylor Polynomial
thanks for the video. I like how you apply the equation directly into a graph, thanks
Tank you very much sir, very simple and intuitive explanation
Had Numerical Analysis class for the first time at the start of semester today. He made me feel like a complete moron because he sped through Taylor Polynomials in class as if it was the simplest thing in the world to just pick up. This taught me more in 18 minutes than my teacher could in an hour and fifteen.
Seriously, why can't more professors be this good?
Thank you SO MUCH
This Got cleared !!
omg math is so beautiful ... and extremely well explained, thanks a lot !!
Thanks, straightened the Taylor polynomial out for me in 20 mins... should have looked this one up sooner :)
My exam is in 2 and a half hours and i only just learned taylors method from this video. thanks man.
Brilliant video - makes it so easy to understand :)
Nicely stitched video. Taylor taylored a polynomial fabric that overwhelms the imagination. What is the meaning of an "nth derivative," for example? The graphs help you begin to see what's going on! Taylor Polynomials aren't just "sew sew." They are awe-inspiring! May the nth derivative be with you!
i love you.
you are the reason a never ever have to go to math lectures/tutorials :D
Aww thank you :) Exam tomorrow... This really helped XD
Thank you very much. This was so much helpful.
*Sal does a mistake*
"Sorry I uh *stutters* My brain is really... I ate too many walnuts"
What a classic line
May I ask you one question sir which software do you use to plot the graph ?
omw thank you so much. u just gave me hope
Hey Sal! Amazing video. Why did you make the assumption that if the 0th, 1st, 2nd, 3rd, 4th, 5th derivative is = to the function, then it perfectly = the function? I get the intuition behind it, and I can see it work very well on the graph. But surely there must be a proof, right?
I like your funny words, magic man.
you save my life consistently
do you have any videos explaining the taylor remainder formula from Sal?
What I really think is mind-blowing is how you can write so clearly with a mouse. People can't even tell what I'm drawing on Draw My Thing, and I bet if you played that, you'd be drawing the Mona Lisa left and right.
This is invaluable, thank you.
thank you so much...i just watched every series video and it makes perfect sense now. my ap calc test is this Wednesday and i was flipping out cause everytime i took a practice free response i just completely skipped the series question and was getting no points for it, not to mention the multiple choice. thses videos are great and e^(ipi) + 1 = o ...wtf?!
Thank you very much! I read the book but could not understand until I watched this video!
how do power series differ from the taylor or maclaurin series?
when u add each term, are you adding it to all the other terms or are u just graphing it by itself? that is, was the second approximation p(x) = cos1 - sin1(x-a) or was it p(x)=-sin1(x-a)??? i think ur doing the latter... in which case, wouldn't the, say, 100th derivative be a really small constant times (x-1)^100?
Remarkable Job!
yes it is the same level. maclaurin is just the special case where your center, a, is defined at 0, a = 0.
This is so cool!!! Thank you so much
Khan should get a Nobel Peace Prize for giving people around the globe access to education for free
"I ate too many walnuts..." - Classic!
Thanks for the help sal!
You make so much more sense than my Bus Cal 2 prof
Amazing, had no clue what was going on until this video.
final calc exam tomorrow, never learned this stuff... combination of you + my book = win :>
thanks a lot, you did a great job of explaining! A+ Video
tanx for the lecture mr. khan, i like your teachin alot....
its helps me more than my boring ass lecturer
My school actually sent an email to everyone to watch your videos to prepare for our finals!
lol ahah me too. random question though cuz its insane we have a math final on the same day with the same stuff... do you happen to go to the university of saskatchewan? that would be kind of insane.
Could you possible explain the error term when approximating a function only to the nth derivative. I've got something about it in my notes but it doesn't really make sense to me.
NOW thats the intuition behind the taylor! thx
well done sir
Love the video, thank you soooo much but there's one thing I don´t understand.. What's the purpose of the approximation when we already have the function?? dont get it... :/
In some cases you can use it because it simplifies the problem. Look up how it is used to calculate the period of a pendulum.
"my brain is a bit urgh, i ate to many walnuts"
Educational AND amusing..
Win! ^.^
thanks man i needed this
thanks for this fun explanation!, a student from the future
Holy shit O.O
I've not gone to a single lesson we have so far because the teacher is so bad, and i think i'm about to pass the whole course just by your videos.
thank you very much, big help on explaining, especially for a freshman first year =D
Thank you sir
Thanks for this.
8 words:
thanks very much for this video.
sweet explanation.
whoa, this is pretty awesome... thank you for this video! :)
@someonetoogoodforyou
The nth derivative is not equal to the function. It's equal to the function at c. As the nth derivative approaches infinity, not only does p(x) equal f(x) at c, but some of the terms near c of p(x) are close to the terms of f(x) near c. The more derivatives, the closer the values near c of p(x) are to the values near c of f(x) and the farther from c you can approximate. Theoretically, taking infinite derivatives will make the two functions equal for all x.
4:44 - 6:28 Best two minutes of this vid for me :D
Your presentation was fine. However, I would like to know why we go through the hastle of defining a function around a particular point for a stated function when we have THE actual function. I guess an application is in order so if you could get that on a video sometime in the not-too-distant future, that‘d be great.
Thank you.
It is used primarily in computing, to make calculations faster!
Oswald Chisala can you do cos(1) of the top of you head? No you cant. Thats why we use taylor polynomials. You can actually sit down and find cos1 with a reasonable degree of accuracy without a calculator when you use taylor polynomials. Also your calculator is actually using taylor polynomials to calculate trig functions and other weird functions like e^x when you put them in the calculator
I don't understand how you come up with the values you use to divide the function (2, 6, 24). Could someone please elaborate?
Fantastic video