Real Analysis 47 | Proof of Taylor's Theorem

[@VisuallyExplained]

I found your channel by chance, and I was simply amazed by the quality and the consistency. Keep up the good work!

[@brightsideofmaths]

Thanks! Welcome aboard!

[@ieppun1342]

Thank you so much for this video! I couldn't get where the derivitive of the auxiluary function comes from, but you've shown it so plainly and explained it so easily!

[@brightsideofmaths]

Thanks a lot :)

[@Hold_it]

I'm happy to finally be able to understand the proof for Taylor :) Thank you very much for this❤️

[@douglasstrother6584]

Countless Physics and Engineering Majors saved from intractable problems!

[@lucaug10]

Always such a joy to see a new real analysis video! Thank you for the lessons!

[@sinanakhostin6604]

The way you prove these theorems is similar to creating an art work. I do not have any clue how you come up with these neat proofs! I hope at some point I can also be able to come up with proofs by myself. At this point I only follow your beautifully explained ones.

[@nm-de3bw]

lol he didnt come up with all the proofs in analysis

[@Independent_Man3]

At 5:08, the t and x0 switched places in the definition of capital F subscript n, h

[@brightsideofmaths]

Yes, that is a mistake there. Sorry.

[@speedbird7587]

Excellent , very nice, short, and instructive proof

[@brightsideofmaths]

Many thanks!

[@cptiglo5632]

I passed analysis 1 year ago but I Still love ur videos

[@pan19682]

Congratulations you are a real very good teacher we are looking forward expanding your video playlists thanks alot

[@brightsideofmaths]

Thank you! 😃

[@JojiThomas7431]

Nicely done

[@johnstroughair2816]

Really nice proof!

[@weiweng183]

At 2:31, the professor said “when k is equal to zero, this factor here is defined to be 1.” Is the professor also saying that (h + x_0 - (x_0 + h))^0 is equal to 1 by definition? Do we define 0^0 to be 1 in this series? I thought 0^0 is usually not defined

[@brightsideofmaths]

We don't define 0^0 but the symbol x^0. In other words, we write x^0 but mean the constant function 1.

[@awesomecraftstudio]

That's crazy. Easy to understand when explained like this, but no idea how people figure this out by themselves. Anyway great video as always tysm.

[@brightsideofmaths]

Glad you liked it!

[@ahmedamr5265]

Great video as usual! Quick question: is it sufficient that the remainder of a Taylor polynomial up to 2nd order includes h^3 to prove that it is the best quadratic approximation? If not, how do I prove that it is the best quadratic approximation to a function?

[@brightsideofmaths]

The 2nd order polynomial is the best quadratic approximation in the sense described in the video. So it's already proven. What exactly do you want to show now?

[@ahmedamr5265]

@@brightsideofmaths is that 2nd order polynomial the best because the remainder is of a higher order?

[@brightsideofmaths]

Yes, the remainder goes faster to zero as stated when x approaches x_0. This is actually what we mean by best quadratic approximation in this context.

[@sslelgamal5206]

Thank you! So Taylor expansion does not guarantee that the approximate polynomial $T_n(h)$ is the best fit, i.e. some difference like $|f(x+h)-T_n(h)|^2$ is minimized! Then why is it the best fit?!

[@brightsideofmaths]

It's the best fit in the sense of the remainder term around the expansion point x_0.

[@TheSandkastenverbot]

Polynomials that minimize a certain norm are in general different from a Taylor expansion. So if you want a "best fit" polynomial over a finite interval, Taylor expansions are usually not a good bet. But they are an invaluable tool for all kinds of convergence analyses or analyses around a small neighborhood of a point.

[@light_rays]

Thanks!

[@svenlovell]

Tut mir leid, wenn das der falsche Platz für so eine Frage ist, aber du hast einen Kommentar geschrieben, dass du das Boox Note Air 2 für Mathe nutzt. Ich bin Mathestudent und spiele schon länger mit dem Gedanken mit ebenfalls ein E-Ink Tablet zu kaufen. Das Remarkable 2 gab es für 260 Euro und so musste ich zuschlagen. Jetzt meine eigentliche Frage, geht es dir generell um das Schreiben auf so einem E-Ink Tablet oder hat das Note Air 2 besondere nützliche Eigenschaften die es besonders gut für mathematisches macht, dass du so begeistert bist? Danke.

[@brightsideofmaths]

Danke für die Frage! Das Remarkable 2 ist mit Sicherheit genauso gut für deinen Anwendungsfall. Letztendlich hat mich das Note Air 2 überzeugt, da ich alle PDFs problemlos darstellen kann und direkt Kommentare dazuschreiben kann :)

[@svenlovell]

@@brightsideofmaths Danke dir :)

[@minglee5164]

First time I understood

[@pishleback6151]

Are you going to be covering integration proofs in the future? Very nice videos btw

[@brightsideofmaths]

The next videos will be about integration :)

[@pishleback6151]

@@brightsideofmaths brilliant stuff :)

[@MOVIES57264p]

I want proper statement of this proof

[@brightsideofmaths]

Proper in which way?

[@minglee5164]

The author must be a mathematician or a PhD in mathematics at least

[@MOVIES57264p]

Can you help me

[@lueelee6063]

ngl that capital F looks alot like a T...

[@brightsideofmaths]

I see the problem but fortunately I chose another colour there :D