Use Calculus, NOT Algebra

[@keheck]

This is like playing a card game where you could have defeated your enemy long ago but want to show off how nicely your cards synergise together and pulling off a huge combo

[@ProgThoughts]

Lmao!!

[@thomascioban3997]

You play YuGiOh? Lol

[@clemverpomtato]

my legend of runeterra ass is laughing rn

[@incription]

MISSED LETHAL

[@mikplaysthings3228]

Magic: The Gathering players be like

[@txikitofandango]

Now go in the opposite direction: explain integrals not in terms of Riemann sums, but only with binomial expansion

[@herbie_the_hillbillie_goat]

That sounds like fun. 😁

[@juanpedro19840914]

And then, explain tetration in terms of arithmetic sums.

[@bprpcalculusbasics]

😮

[@da4762]

Lol, that's basically how Newton did it XD.

[@luisapaza317]

Eternal suffering

[@mtrichie111]

I honestly think this is hilarious, because its ironic how thats the long way to do something simple, however, its a smart way of learning more tools of calculus using such fundamental equations that you know if you're doing it wrong or right. I love this !

[@hannevanderven5230]

Honestly I got kinda annoyed watching this video but I like your perspective so maybe I should calm down a little xD

[@kinshuksinghania4289]

Now that's what I call 'Calculus'

[@corymitchell3228]

What's calcoolus?

[@bprpcalculusbasics]

@@corymitchell3228 is that a stand and deliver reference?

[@corymitchell3228]

@@bprpcalculusbasicsabsolutely 😂

[@charleswolfe619]

No one is talking about how smoothly the color of the markers is swapped out

[@Insightfill]

It's been "his thing" for a long time now (bprp=black pen red pen) and still remains a beautiful thing to see. Agreed!

[@herbie_the_hillbillie_goat]

This is some serious outside-the-box problem solving.

[@spudhead169]

I've seen a few ways to show this, never with integrals though. Fascinating.

[@jackiesharp018]

This actually taught me alot more about the content of my first quarter of AB AP CALCULAS than the class and the 36 hours of homework every two days. Seriously. That uses something I understand and just expands upon it with new ideas that the new ideas fit in perfectly!

[@localverse]

Merely curious (watched video without understanding it), does the video show a better way than the FOIL method? (first, out, in, last) Is the video showing a better alternative? (a calculus method)

[@bulldozer8950]

@@localverse no. Potentially in some specific situations it could be simpler, but in the vast majority of cases, no. It would take longer because foil is just doing multiplication 4 times, which is not that hard by comparison to what he showed. However, this would be a pretty good way to introduce some concepts of integrals, particularly because it makes use of so many concepts

[@localverse]

@@bulldozer8950 oh, because of video's title, I thought he was saying to use this method instead of FOIL

[@baptiste5216]

You inspired me, I'll try do do the same for (x+y)³ Edit : I did (x+y)³ but i have no clue about how to prove (x+y)(x-y) = x² - y² using integrals

[@skylardeslypere9909]

Challenge: try to do it for (x+y)^n, where n is any natural number

[@degeneratedeuterium5164]

@@skylardeslypere9909 extend the set of n variable to the reals, and then prove that 😈

[@samuelgunter]

@@degeneratedeuterium5164 let n be any complex number ...

[@notdumbrella6399]

@@samuelgunter ....

[@francescocostanzo8225]

@@samuelgunter I don't even know how(or what it means) to take a natural number to a complex number let alone a binomial of the set of complex numbers

[@kuzhii8416]

Pascal’s triangle: “am I a joke to you?”

[@Jaekyan]

please do one with (x+y)^n i think that would be so cool!

[@sh0ck3r48]

dudes marker swapping game is on point!

[@arimermelstein9167]

You can tell students until they get used to the “dummy variable” idea that you can just do a trivial substitution of t=u. Obviously, the bounds don’t change and the dt=du.

[@OrigamiCL]

I also like the explanation of not putting too much stock in what each variable is called since it was a 'definition integral' so to speak

[@stewartzayat7526]

I personally think of the variable of integration as a parameter, or a real valued index - it has a similar interpretation as a sumation index. It's basically a formal variable with no real meaning or value

[@txikitofandango]

How do you get 2u + 2x from 2(u + x)?

[@SupraaHero]

Lol, multiply the 2 two each of the components inside the parentheses. For example: 3(2 + 2) = 3*2 + 3*2 = 12. Although intuitively this is not how it's normally done because you would normally first compute 2+2 and then multiply but this is just factoring out the expression which is what was done in the original 2(u + x). You can always check this work by multiplying and then factoring out. 2(u + x), multiply and expand, 2u + 2x. Notice how each component has a factor of 2, you can divide and "pull out" a 2 from each component. So we now have the original 2(u + x).

[@BroArmyCommander]

@@SupraaHero This may have been a diss on FOIL

[@txikitofandango]

@@BroArmyCommanderGlad someone got my point. Although, as someone below pointed out, you don't have to distribute to find it. You can just add a copy of u + x to itself, reorder the addends, and "combine like terms" which is really a form of factoring.

[@BroArmyCommander]

@@txikitofandango Yeah I guess it would have to be two binomials to call it FOIL so I understand somebody might not get it haha

[@jancermak1988]

@@txikitofandango Is not necessary to use FOIL. 2(u+x)=(u+x)+(u+x)=u+u+x+x=2u+2x

[@That_One_Guy...]

Next do : how to prove e^(x+y) = e^x * e^y using calculus and only left hand

[@rileyborgard8020]

Now how to show that the derivative of x^2 is 2x without applying the FOIL method on the (x + h)^2 term?

[@nobivy3524]

lol.

[@bprpcalculusbasics]

😂

[@bprpcalculusbasics]

I am thinking to use logarithm differentiation. But I think there’s a better way.

[@Anonymous-df8it]

@@bprpcalculusbasics *WITHOUT* applying FOIL!

[@nikolaimerritt9228]

Product rule on x² = x * x

[@maowiisenseii1470]

Oh man! I never thought of this. Change of space. This might be nice with some tedious algebraic work.

[@_wetmath_]

im gonna assume that "cannot use foil" means cannot use distributive law. then how do you get from 2(u+x) to 2u + 2x?

[@nicholashalliday8423]

2(u+x) = (u+x) + (u+x) = u+u+x+x = 2u + 2x

[@txikitofandango]

It's strange that someone would know how to factor out 2 but not know how to distribute 2

[@anshumanagrawal346]

Well, your assumption is wrong

[@JJCUBER]

No, cannot use foil means cannot distribute for something of the form (a+b)(c+d); foil stands for first outside inside last. Regardless, the whole point of BPRP’s exercise is to better understand how to manipulate integrals, so he can define the exercise however he wants.

[@francescocostanzo8225]

That's what I'm thinking!

[@lukeeatschips6324]

Calculus : 5+ minutes to just solve simple multiplication Algebra : Does it in half a minute at most

[@dużapoduszka]

Why would I want to use this long and convoluted method if standard FOIL method (or other methods like FOIL) are much quicker?

[@NoovGuyMC]

Because memes are memes and the meme flies over your head, in fact fly higher by lim X->0 (170/X)

[@dużapoduszka]

I don't think the guy did it just for memes.

[@NoovGuyMC]

@@dużapoduszka eh I assume jjust because of his name

[@yea9597]

It’s just a demonstration that calculus can be used in a bunch of problems even when not seemingly related. The purpose isn’t to make a new way of expanding brackets but rather a practice problem to improve calculus skills.

[@jakegmnlo6411]

This is circular logic because the definition of dx^2/dx requires you to use FOIL when expanding (x+h)^2

[@tuan_ha]

Maybe we shall replace x and y by a and b, respectively, so that there is no confusion on x and y being constants.

[@jackkalver4644]

This works for higher powers too! Can you use this to prove the binomial theorem? How about for fractional values?

[@panostriantaphillou3702]

Confucious, you are amazing as always!

[@jamisonhodges4169]

Cool question! Just a nitpicky thing, but shouldn't we clarify that y>0?

[@141Zero]

What about +C?

[@thesnakednake]

Since it’s definite integration, when going from a to b, it’s F(b) - F(a). That makes it so you have C - C

[@sremagamers]

Interesting. Is there any way to define integration without talking about R being a ring? If not then this is interesting, but unfortunately circular.

[@rusokana3642]

Why is x considered a constant?

[@virajagr]

When defining x, we are using x in the upper bound of the integral, isn't that wrong?

[@baptiste5216]

The equality in itself is true. However indeed, I don't think this is a valid definition of x. But the point is not to rigourosly prove this equation, but rather proving it in a fun way.

[@tyler_schecter5805]

No

[@matthewfuerst6456]

x and y were just constants throughout this video so i think defining x in terms of itself is fine? it was only used to show a useful substitution anyway

[@dVPulse]

When writing t = u + x, wouldnt dt/du = 1 + dx/du? Edit: x and y are not variables here, they will be some fixed numbers, so dx/du = 0 for any constant x

[@benjaminkern2586]

I understand maybe two words a video and yet I’m subscribed

[@cbayoo9405]

ok ok, i got one. Which came first? The foil way? Or the Elite 4 member's way? Serious question tho lol. Is this a way to mega prove it orrrrr just another way to do things. Like dexters lab opening scene?

[@Wayyan-v]

When the teacher asks to show your work.

[@bug5654]

This feels wrong because it doesn't appear to break with non-abelian groups? (still gives x^2+2xy+y^2 instead of x^2+xy+yx+y^2). Or am I missing something like integration itself changes enough to fix it in non-abelian groups?

[@rainbowunicode8352]

Commutativity is assumed when rewriting the integral int(2(x+y)) as int(2x + 2y), as we are avoiding distribution and have to fall back on the definition of integer multiplication as repeated addition. I believe that's where things fall apart for non abelian groups; there's not an obvious way to simplify x+y+x+y.

[@mattwyes]

I need an example

[@bprpcalculusbasics]

See description I have an intro there

[@Jonathondelemos]

Nicely done

[@NeverSaid-]

What is the problem with x*x+y*y+x*y+y*x. Why do you even need the integral and pseudovariables.

[@irwandasaputra9315]

x^2+2xy+y^2

[@Sheikxlove123]

Try higher powers and complicated variables

[@DetectiveAndrey]

Why is dt equal to du exactly? Also, could one do a general case this way? (x+y)^n?

[@ZipplyZane]

You start with t=u+x and differentiate both sides (using implicit differentiation). That gives you dt = d(u+x). x is a constant (as it does not change when you calculate the integral) so the derivative of u+x is just du. Or, to out it another way, you can say you get dt = du + dx. But x doesn't change since it's not inside the integral. Thus the change in x or dx = 0.

[@DetectiveAndrey]

@@ZipplyZane, ah thx, understood.

[@JM-us3fr]

Unfortunately this is circular, since the power rule for derivatives and integrals requires the binomial theorem.

[@bprpcalculusbasics]

I did not take the derivative or the integral tho. I only used the properties of the integral.

[@JM-us3fr]

@@bprpcalculusbasics But evaluating the integral of 2t to get x^2 uses the power rule for integrals, which uses the binomial theorem. I suppose you could do it using Riemann Sums and a limit, but that's a bit ad hoc.

[@chasewilber9325]

@@JM-us3fr he was given the definition in the problem, so he did not use the power rule for integrals, even if it would have led to the same answer. he was only applying the definitions

[@johnbutler4631]

Even if he did use the power rule, it can be derived without the binomial theroem. It can be derived using exponentials and logarithms.

[@JM-us3fr]

@@chasewilber9325 But you can’t define the integral that way while also using other properties of the integral, such as linearity and u-substitution. These suggest a different definition of the integral; one that would require the power rule to give the x^2 identity.

[@Nyx_Room]

Everyone : Solving math problem Me: Staring at his Poke Ball

[@richardtrager7125]

Binomial Theorem: Am I a joke to you?

[@donrumata5299]

There's a small problem: how would you get derivative for x^2 formula?

[@bprpcalculusbasics]

😆

[@carultch]

You can prove the power rule for all real bases and all real exponents, through the lograrithmic differentiation.

[@Anonymous-df8it]

FOIL. Oh, wait. Circular logic!

[@bathysphere1070]

Leave it to a mathematician to make things far more complicated then they need to be. In the time it to took to do this for one problem anyone could have used FOIL on a dozen. Just use FOIL for order 2 polynomials, for anything higher just use the binomial expansion. It is far more efficient. Let mathematicians continue to engage themselves in theorems that have little bearing on practical application.

[@DaFrancc]

Engineer getting paid by the hour be like

[@joelyaw5389]

I just opened another world for me to explore

[@stapleman007]

Instead of walking to my mailbox to get my mail, I call an ambulance to go to the hospital, take a taxi to the airport, fly to a different city, take an Amtrak train back to my city, get an Uber ride to my house, get dropped of next to my mailbox, collect the mail, and ride my Segway back to my house.

[@bprpcalculusbasics]

😆

[@alexbair2542]

ood?

[@glitterishhh]

but u substitution uses algebraic techniques...

[@hasanplaster1510]

Just use (a+b) ^2=a^2+2ab+b^2

[@NoovGuyMC]

That's FOIL and also against what he wanted

[@armpitpuncher]

@@NoovGuyMC How can you tell he used FOIL? All he showed was the solution.

[@NoovGuyMC]

@@armpitpuncher implying lol

[@PhilippeCarphin]

That was very neat.

[@nablahnjr.6728]

but why though

[@muhamadsadulla]

Thank for you

[@costelnica3988]

Super, but how do I solve n=? so that a=sqrt(n×radical of order3(n)) to be natural number?

[@xinpingdonohoe3978]

Radical of order 3? Do you mean cube root? Just use cbrt. And do you want a or n to be natural? Anyhow, n*cbrt(n)=n^(4/3). √(n^(4/3)) = n^(⅔). If a is in N, then n^⅔ is in N. Therefore, n must be the cube of an integer. So, if n=x³, a=x². Choose x is a natural number to have it. I think that's right.

[@costelnica3988]

@@xinpingdonohoe3978 Thank you very much! I understood the reasoning . I'm blocking it a=n^(2/3). I'm happy now!

[@xinpingdonohoe3978]

@@costelnica3988 no problem

[@eliteteamkiller319]

This is amazing.

[@JiffyJokes]

Nice gains bro

[@toftimsecondaire]

Liked it !

[@bprpcalculusbasics]

Thanks!!

[@Andrumen01]

I hope that you understand that you can do the same with Taylor series and all you are doing is making a sophisticated Taylor expansion...

[@ayrthhhn]

WHAT???? I thought FOIL!!! was ALL there is

[@Harison.]

I always watch these videos and have no idea what he is doing

[@rainbowunicode8352]

Good on you for watching anyway.

[@kamikeserpentail3778]

It's neat to see the calculus of it and all, but I'm pretty sure students learn foil before calculus. Besides, if you just draw a square of length and width x+y, you can clearly see the x^2 part, the y^2 part, and the two xy parts, so it should be pretty hard to get that wrong after a few times looking at what it is.

[@bprpcalculusbasics]

Oh yea. That’s in my first part kzbin.info/www/bejne/r4OumZ-dltdgprs

[@reidflemingworldstoughestm1394]

This is what they should teach in Alg I

[@official-obama]

let's assume that x+y = z so it's z*(x+y) using the distributive property, we can expand to (z*x)+(z*y) replace z with x+y so ((x+y)*x)+((x+y)*y) expand with distributive property again ((x*x)+(x*y))+((x*y)+(y*y)) clean up and rearrange x^2+2xy+y^2

[@Anonymous-df8it]

You're only allowed to use calculus

[@official-obama]

@@Anonymous-df8it calculate the integral of (x^2+1)*3

[@official-obama]

@@Anonymous-df8it nowhere did he say we were only allowed to use calculus

[@Anonymous-df8it]

@@official-obama x^3. So?

[@Anonymous-df8it]

@@official-obama The title literally says 'Use Calculus, NOT Algebra FOIL'!

[@solomonkim7285]

Lowkey triggered cuz this made me remember that I had this question on a Calc exam back in high school (I’m graduating college soon) and I only know understand how it works…

[@GDPlainA]

how it looks when i first learned (x+y)^2:

[@irok1]

This is just a "can we beat the game without pressing a" but without foiling

[@rainbowunicode8352]

Perfect metaphor for why a mathematician might enjoy this and so many others just think "what in tarnation"

[@pandabearguy1]

Just let (x+y)=1

[@einsteingonzalez4336]

Now the Freshman’s dream is finally dead.

[@crep50]

Bruh (x+y)(x+y) is literally just x^2+2xy+y^2??

[@oak3785]

oh man do i wanna go back to school?

[@BigKnack]

This is awesome

[@UnordEntertainment]

doesnt "t = u+x" imply "dt = du + dx"? havent done substitutions in a while and this part is confusing me, could someone explain? my working starts from "t = u + x" method 1: differentiate with respect to u then multiply by du t = u + x dt/du = 1 + dx/du dt = du + dx method 2: differentiate with respect to x then multiply by dx t = u + x dt/dx = du/dx + 1 dt = du + dx so both methods (if theyre right) lead to dt = du + dx not dt = du what am i doing wrong?

[@sayarsine6479]

Powerful

[@skylight6904]

Why on earth is this on recommendend? Google should know by now that I'm mathematically challenged.

[@johnchristian5027]

This is super overkill XD

[@gabey.youngblood2613]

Anybody else notice that foil had to be used after he used the u substitution?

[@jaredjones6570]

Add (u+x) to (u+x). Apply the associative property. So 2(u+x) = (u+x)+(u+x) = (u+u) + (x+x) = 2u + 2x by definition. The distributive property would have only been invoked if two polynomial factors were multiplied.

[@joaobringel4712]

Nice

[@ronwilliams4184]

I honestly loled...

[@aBradApple]

Imagine teaching your kids calculus while they learn algebra…

[@stevetabor2605]

Where is the foil in your diagram? You show black and grey, but aluminum foil is shiny, like polished silver. You'll have to be more diligent about your representations. For many of us coming in from decades past, we were not trained to use aluminum foil in math, we had to use pencil and paper!

[@monke3559]

ah yes, my "t = x integration technique" i havent used this since the heian era.

[@ll-lw8vn]

is math all about making a elementary school level question to college school question？

[@Anokosciant]

man, just use taylor formula with integral rest to do it

[@planomathandscience]

How to waste time on a 2 second solution by making it overly complex.

[@samuellima5968]

Do the hard way instead easy way

[@redacted3097]

I mean.. that works too I guess 😂

[@alkharage7863]

What with the pokemon ?

[@carultch]

It's his microphone.

[@ChavoMysterio]

x^2+2xy+y^2...now let's see the proof using Calculus.

[@justingreen8006]

I'll stick with FOIL.

[@danieljohnson411]

Distribution please no FOIL I fought this all the way out the door while teaching. I used to teach calculus as well. Just NO FOIL

[@rainbowunicode8352]

Lmao, right? Students understand foil, then see product of a binomial and trinomial and the engine stalls. Distributivity is my favorite arithmetic property; don't call it the ugly name FOIL.

[@danieljohnson411]

I was trained to teach for understanding algorithmic learning although fast to get the answer it causes more problems down the line. Things like foil are there were introduced to help someone get their ph.d with no thought of what it does to the student in the long term

[@landy4497]

woah

[@alexandrenascimento5601]

How to complicate easy things. Seems shallow this type of computation.

[@ryanbeaucage1068]

40 seconds in and i'm lost.

[@FroggieIsHere]

what

[@bmar1784]

This is a repeat isn't it. I feel like I've seen you do this before

[@luisaleman9512]

Yes, he did it before on his other channel

[@bprpcalculusbasics]

Yes. A fast version 😆

[@bmar1784]

@@bprpcalculusbasics I watched again anyways

[@charlesaston6546]

FOIL is quicker

[@gregd335shot9]

Now show me how to do calculus from step one 😹

[@Firesawdust]

Oh no, Calculus is too powerful! 😂

[@GuyFromJupiter]

This is most definitely not easier than foil