Taylor Polynomials

Approximating a function with a Taylor Polynomial
More free lessons at: www.khanacademy...

[@lightzebra]

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.

[@chantastlc]

9:04

[@Quarker]

"I hope this video gave you some intuition on the Taylor Series. If it didn't, please ignore this video" HAHAHAHAHAHAH BEST ENDING EVER

[@nocturnalvisionmusic]

I was your 100th like! 🥳😁

[@KevinVandyTech]

Nice math lesson, but the most important lesson we all learned is to always drink water with your walnuts.

[@michiganfan725]

Math is really hard when you trippin on walnuts.

[@MR1stinga]

"Sorry I just had some walnuts" lol

[@paulblart4478]

"Sorry my brain is... I ate too many walnuts" -khan academy tutor

[@SP-qi8ur]

That's actually Sal Khan

[@vanessang4216]

go bless you idk why I'm paying so much money for uni when I just end up coming here

[@Glendragon]

where do you have to pay for uni? FeelsGoodMan

[@helmiazizm]

So you could have a bachelor degree certificate to apply to some shitty job with it, duh

[@ziyuchen3112]

Cuz u need a motive for coming here to exhaust ur brain

[@meeblings6]

US and Canada for sure

[@utsavshakya6891]

@@Glendragon almost everywher except for norway maybe

[@Denizen36]

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

[@HappyFaceXD]

Note to self: do not eat walnuts before exams.

[@liesalllies]

Reading the comments before watching the video was really confusing hahaha

[@Pikminiman]

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, KZbin could serve as a viable replacement for college. Cheers!

[@bigpharts]

o/

[@shankysays]

i pay you nothing yet you teach me a lot. i spend all my parents saving to my university and i get nothing.

[@sardarhedayati3842]

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!

[@andrewtcb1]

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!

[@khanacademy]

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).

[@pwownage]

I like your funny words, magic man.

[@rachel22cute]

"My brain had too many walnuts" lololol xD

[@MonsieurCashow]

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

[@StupidBadyXD]

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.

[@leerobbo92]

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.

[@entertainmentera6791]

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.

[@khanacademy]

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.

[@sandmitches]

2nd video ive seen where hes choking on walnuts

[@echoofsilence]

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

[@5hak3itup]

How the fuck do people just come up with this stuff? its amazing

[@alexhughes8211]

With walnuts

[@asp1346]

Jesus

[@AmitLavania]

I can understand the effect of walnuts

[@yungyb7527]

khan saved my linear algebra and now my calculus too. thanks alot haha

[@lpbug]

this. is. amazing. I completely understand how taylor polynomials work now. Taylor was a genius.

[@Presenter-A]

nice work you have changed my attitude to the tailors theorem

[@c00kiemonsters]

i love you. you are the reason a never ever have to go to math lectures/tutorials :D

[@lexinaut]

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!

[@SahaquieI3]

this 18 minutes teached me more than the 2 hours i wasted today in the school lyrbrary

[@SequinBrain]

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.

[@CallMeMantou]

oh my. Khan Academy's videos are something that i never regret watching! Best 18 minutes spent

[@valhalla4144]

This was the most beautiful thing I've ever seen in math

[@mokopa]

Once you GET the Taylor Polynomials...it really does blow the mind into orbit for a while.

[@1uk35j]

"my brain is a bit urgh, i ate to many walnuts" Educational AND amusing.. Win! ^.^

[@BoQuan22]

Just wanted to say, I bet the reason everybody is on this video is because their professors make it SO difficult to understand. This makes it look SO easy so thanks again Sal =)

[@pianodan763]

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.

[@KianwithaK]

My exam is in 2 and a half hours and i only just learned taylors method from this video. thanks man.

[@brco2003]

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!

[@KebleTar]

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!

[@watchingstupidshit]

wow thank you so much, my ap exam is in 3 days and i had NO idea how to do series before, just skipped all of the questions.

[@JayJaySaladBar]

yes it is the same level. maclaurin is just the special case where your center, a, is defined at 0, a = 0.

[@acmbhs1234]

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

[@tresusarinok4192]

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?

[@54huggybear]

Thank you so much man! I didn't understand this concept well during class and this really cleared up Taylor Polynomial's for me!

[@johnyapple8447]

This has been incredibly helpful-along with many of your other videos.

[@JCP598]

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!

[@Mondoshawn]

Khan should get a Nobel Peace Prize for giving people around the globe access to education for free

[@tresusarinok4192]

You have no IDEA how truthful that statement is.

[@nocturnalvisionmusic]

4:44 - 6:28 Best two minutes of this vid for me :D

[@kckdude913]

@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.

[@mcpearce]

Thanks, straightened the Taylor polynomial out for me in 20 mins... should have looked this one up sooner :)

[@VideoOfMike]

thanks! my text book didn't explain this very well and confused HELL out of me.

[@watchingstupidshit]

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?!

[@someonetoogoodforyou]

@kckdude2 Thanks for that kck. You're right, p(x) = f(x) at c if you take enough derivatives. But I don't understand how taking infinite derivatives will make 2 functions equal each other for all x. I can understand them being equal to either other at c or at c+epsilon or at c - epsilon. I think it's one thing to say they're the same at a point, but another to claim they're the same over the entire domain. I'm not saying you're wrong, I'm claiming I still don't quite get it :S

[@Ichimaru666Gin]

tanx for the lecture mr. khan, i like your teachin alot.... its helps me more than my boring ass lecturer

[@ny1fanta]

NOW thats the intuition behind the taylor! thx

[@MsOrangePen]

My school actually sent an email to everyone to watch your videos to prepare for our finals!

[@karamwahba1591]

You are blowing dust out of my 👂 , thank you

[@MrAlexhusa]

Thank you very much! I read the book but could not understand until I watched this video!

[@kellylouiseoneill1750]

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... :/

[@santiagoarce5672]

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.

[@myonlynick]

8 words: thanks very much for this video. sweet explanation.

[@MRAXELGRINDER]

Really Great lesson! So good! I recommend it to anyone trying to understand Taylor polynomials. Khan academy is amazing

[@c4chus]

love your videos!!! thank you very much!!! Knowledge is for humans!!! :D... greetings from mexico

[@S3thMusic]

you save my life consistently

[@adamsouljazz]

look for the video entitled Euler's Formula and Euler's Identity IT WILL BLOW YOUR MIND!!

[@prashantdubey9430]

i see a bit of remainder therorem here...where polynomials is evaluated at c and its easy to understand only if you know FIRST PRINCIPLE OF DIFFERENTIAL AND TRIGONOMETRY WELL. THE reason I say first principle is you need to know why derivative of sin cos tan cosec sec and all...

[@etiennesellar6065]

I never *saw* him type the taylor ploy into the calculator; I don't know if I "believe" you, mr.khan.

[@sereda008]

Aww thank you :) Exam tomorrow... This really helped XD

[@PetStuBa]

omg math is so beautiful ... and extremely well explained, thanks a lot !!

[@Ambarenya13]

"I ate too many walnuts..." - Classic! Thanks for the help sal!

[@eggmachine27]

"sorry i had too many walnuts" LOLLL i died

[@pravithandstuff1908]

What’s good bro

[@davidpappaianni1306]

When you are computing the derivatives of cosine, aren't you forgetting to derive the argument ? If I have Cos(x-1)^4 / 4! , you write -sin(x-1)^4...shouldn't it be (4)-sin(x-1)^3?

[@jerrypower3430]

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.

[@akrm2944]

cos(1)=0.999 hence, y=0.99 so why is the first green graphed function equal to 0.5, ( y=0.5)?

[@vaanisingh6796]

Sir , i can't express my gratitude to you in words You help me a lot

[@jorgemercent2995]

definately confused me alright..what on earth do you mean approximate around c?

[@imegatrone]

I Really Like The Video From Your Approximating a function with a Taylor Polynomial

[@RAF0769]

Amazing, had no clue what was going on until this video.

[@OswaldChisala]

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.

[@domagojmarjanovic8824]

It is used primarily in computing, to make calculations faster!

[@passwifjreiguru5325]

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

[@shiza223]

God bless you!!!

[@shibanichakraborty7471]

Thank you sir

[@paoloparker8991]

Tank you very much sir, very simple and intuitive explanation

[@arghyadeepchatterjee6100]

Taylor series is so beautiful . I literally have tears in my eyes

[@cigxhang486]

wow that's... a deep understanding of it

[@duckboy81]

thanks for the video. I like how you apply the equation directly into a graph, thanks

[@kurrizzle]

@Dynamics18 type it in you calculator dude your mind will be blown

[@TotliTotli]

@makmegs Don't be selfish. Your not the only one that needs his help.

[@saionjik]

final calc exam tomorrow, never learned this stuff... combination of you + my book = win :>

[@someonetoogoodforyou]

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?

[@fatffatable]

Thank you

[@Hampeps]

I love it!

[@oskarengl8645]

thanks for this fun explanation!, a student from the future

[@jzz8860]

i can be pretty confident to say that I graduated not from my university but from Khan Academy.

[@Peter_1986]

I consider becoming a math teacher in the future some time, and if I do then I'll make absolutely sure to be just as clear as this. Math is supposed to be fun and awesome, too bad it has such a bad reputation.

[@angel209056]

"but hopefully that gave you some intuition" "if it didn't, ignore this video." HAHAH really helpful, btw. reading from the textbook, it's hard to understand.

[@OZZl3]

What about the Lagrange Remainder! I understand all this but I can't make sense of lagrange remainder.. I know it's supposed to give you a better approximation but I don't understand the equation for it! How do you actually use it?

[@diwr]

nice intuition btw!

[@jbsg01]

You make so much more sense than my Bus Cal 2 prof

[@sanujatharinda6525]

Thank you very much. This was so much helpful.