my all-in-one calculus question

Want to learn more about calculus limits, derivatives, integrals, and infinite series? If so, head to Brilliant brilliant.org/... (20% off with this link)
This all-in-one calculus question contains the limit definition of derivative for f(x)=x^3 (aka the first principle for derivative), the power series of -ln(1-x) by integrating the best friend (aka the geometric series), the improper integral of ln(t) from 0 to x by using integration by parts (DI method) and using L'Hopital's rule to check the limit of t*ln(t) as t goes to 0+, and all the differentiation techniques (namely quotient rule, product rule, chain rule, power rule) to differentiate this monster! Enjoy!!! #calculus #blackpenredpen
0:00 my all-in-one calculus question
0:23 limit definition of derivative of the function f(x)=x^3
1:29 power series of -ln(1-x)
3:12 integral of ln(x) with integration by parts
6:28 differentiate this monster!
13:04 check out Brilliant
14:00 (bonus part) how I came up with this problem
Another all-in-one calculus question: 👉 • my all-in-one calculus...
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[@blackpenredpen]

Another all-in-one calculus question (uncut): 👉 kzbin.info/www/bejne/aaSUiIyLjL54gbc

[@Jschlick100]

I love that he’s just holding a PokeBall while crushing my brain

[@shuem_]

He gotta crush'em all

[@elijahdelacruz3004]

Facts

[@adrienleblancpiette6239]

Think that it is his mike or something like that.

[@blue5887]

@@adrienleblancpiette6239 it is his microphone

[@sirmaxie5640]

@@adrienleblancpiette6239 his “mike”

[@MonsterIsABlock]

"If you're a calculus teacher, feel free to use this on your calculus test" Me: NOOOOOOOOOOOOOOOOOOOOOOOOOOOO

[@blackpenredpen]

😆

[@nerduto1]

At least you have the answer now xD

[@anshumanagrawal346]

Well, you know the solution now

[@aviralkumarbarnwal83]

@@blackpenredpen Bhaiya you are the one of the best maths teacher in the world.plz give me some tips to strong calculus and coordinate geometry. ❤️❤️❤️❤️ u unended.

[@rickshn6801]

@@aviralkumarbarnwal83 look up questions online and do them a lot that's how you engrave it into ur memory

[@lenskihe]

My suggestion: Let each student have the choice. Either they take a normal calculus test or they only have to do this problem

[@blackpenredpen]

😆

[@Vivek-io3gj]

Or take a test of problems like these (bonus points)

[@cjxchess17]

and if you get it incorrect you get a 0

[@null3007]

@@cjxchess17 Funny seeing you here haha

[@GhostHawk272]

@@cjxchess17did not expect you here

[@cookyum501]

Calc teachers DO NOT feel free to use this on a test

[@blackpenredpen]

Lol why

[@cocco_cola2989]

@@blackpenredpen it is a little bit difficult ahahahha, the limiti and the power series are not impossible but the rest is quite hard

[@yaelmartinez4374]

@@blackpenredpen its hard

[@Geo25rey]

@@blackpenredpen it's too long of a question with too many interdependencies. If you get 1 of the first 3 parts wrong, then you get the entire question wrong. Also, look how long it takes to complete. It took you over 10 minutes to complete so for students it might take 20-30 minutes. That's at least half the test. It would be so brutal if a silly mistake cost you 1/4 of your final exam grade, assuming there's reasonable partial credit.

[@TheZenytram]

@@Geo25rey good

[@amitl2308]

calculus teachers would be : "you didn't close the parentheses at 6:44 , B-". jokes aside, i love your channel!

[@complexcreations5309]

Noooo, I can't unsee it😭😭!

[@MrConverse]

I noticed that too and it was bugggggggging me! BPRP is still awesome!!

[@Fasteroid]

*anger sounds*

[@Gniaum]

Or that one student that did not pay attention to anything else and at the end, when you ask if there is any question and if the method is understood, raises a hand to finaly say "You forgot to close the parentheses!"

[@mizba7859]

I also had the same doubt

[@blackpenredpen]

If you also like filling the whole board with math 👇

[@azcollins9138]

I bought my own whiteboard because of you : D

[@kushaldey3003]

Ok, so now you use green pen as well

[@lowlife_nolife6047]

Can you create a Video about Non Dimensionalisation of Differential Equations... Have been encountering that in many Heat Transfer Problems??? Just want to Understand the Concept Behind doing so..

[@26-dimesional_Cube]

Fun fact: you can type this to recreate bprp's problem in PE keyboard This video: d/dx((lim(h->0,((x+h)^3-x^3)/h)*sum(n=0->inf, x^(n+1)/(n+1)))/integral(0,x,ln(t),dt)

[@smrutiranjanbehera9237]

@@kushaldey3003 bule pen too 😀

[@josegalvezmateos8153]

Be careful, not always the integral of the serie is the serie of the integrals. You should say that in this case it happens because is uniformly convergent when x is in (-1,1)

[@gasabbath]

I was about to comment that.

[@diego_sabbagh]

i was about to post this very same comment too ^^

[@scar6073]

Screw that I'll just say "assuming all the conditions are met we have" and proceed.

[@spelunkerd]

Not doing calculus every day, this was a refreshing way to quickly review many topics. Thanks!

[@reidpattis3127]

Definitely for exam season. 😁😅

[@TheRealUSArmy]

Totally not complicated 😳😳

[@hoid8069]

Now convince me that this is indeed the derivative using an epsilon-delta proof.

[@hazwi]

NO ILL LITERALLY DROP OUT

[@wavingbuddy3535]

Whole test is this one question: a) find the derivative [5] b) find the derivative by first principles [10] c) verify using epsilon delta definition [20] d) find the second derivative [35] Time: 1 hour ugh

[@JemCel03]

I miss calculus so much. You've grown btw! I first watched you in 2015-2016 era and now you're here making the most intimidating calculus question that is worth 5000 points in the exams lol.

[@yaskynemma9220]

Well, I am not only impressed about you coming up with the question, but also with the fact that I could understand you everything

[@fizixx]

I've always likes your small clarifications as you go along. You always have unusual math content which is refreshing and interesting.

[@lorenzovillani3102]

You can group the second and third term by the fact that the second is multiplied by 1-ln(x), but the other is mulplied by one. So the result is a bit smaller by multiplying the second by 2-ln(x)

[@ViguLiviu]

Exactly what I thought, but my conundrum is still not resolved since it's not useful.

[@lorenzovillani3102]

@@ViguLiviu?

[@PrudentialViews]

As a 13 year old, I find this very interesting and now I wanna learn calculus.

[@General12th]

In case you haven't yet, you should watch 3Blue1Brown's "introduction to calculus" series here on KZbin. It'll give you a wonderful and qualitative understanding of the subject.

[@PrudentialViews]

@@General12th ok

[@PrudentialViews]

@@armgord thats cool!

[@sciencegeeks1370]

@@armgord Yh me too.

[@greeeeeeeg]

@@armgord Yh me too. (no)

[@abdulmalek1118]

Hello ! I hope you see my comment I saw this nice question so that I recommend it The question is : solve the system of equations a = exp (a) . cos (b) b = exp (a) . sin (b) It can be nicely solved by using Lambert W function after letting z = a + ib Hope you the best ... your loyal fan from Syria

[@Benjax_95]

This is actually crazy and great at the same time, great idea, video, and performing! :D

[@blackpenredpen]

Thanks!

[@Ken-no5ip]

I open youtube after a 4 hour calc 2 test and this is the first suggestion 😭

[@blackpenredpen]

😆

[@johncarlodimaangay3812]

I remember having to answer these type of questions in my Integral Calculus days, separating the whole equation and integrating it 1by1 and having an answer 3x longer than the answer of this whole equation

[@Harkmagic]

When I saw "all in one calculus problem," I was envisioned something that required chain rule, product rule, quotient rule, trig substitution, hyperbolic trig substitution, etc. Kind of an absolute nightmare, but really satisfying to complete.

[@fariesz6786]

C=O d/dt so you even managed to smuggle some organic chemistry in respect!

[@fivestar5855]

d/dt in organic chemistry? I've never heard about it (I study a chemical Engineering)

[@fariesz6786]

@@fivestar5855 Dichlorodiphenyltrichloroethane (DDT) is an insecticide yeah, i know, lame joke.. but he literally said it as "dee dee tee"

[@manuelgandaras7871]

I thought organic chemistry too when i saw the C=O

[@vipuleshtiwari3679]

@@manuelgandaras7871 carboxyl bond

[@ritujithmanoj2133]

@@fivestar5855 Don't they show up while studying kinetics of organic reactions

[@g.reaper7946]

Yo that marker switch thing is certified Mathematician skill.

[@Peter_1986]

This one example is probably quite overkill, however I do generally enjoy problems that include several different concepts, since they feel more rewarding. It's especially nice when math books give problems of that type at regular intervals, since this constantly reminds you of old concepts.

[@26-dimesional_Cube]

Define a limit operator: lim(f(x),n) = a Where f(x) is the function we need to take the limit, n is the critical point (where if f(x) is underfined at that point) and a is the result after the limit approach That means: f(x) < a for every number that is x < n or x > n

[@julianbruns7459]

I dont see how that makes any sense. 1. Why does the function need to be undefined at n? It shouldnt matter if it is defined or undefined if we want to know the limit as x aproaches n. 2. Why does the limit need to be the maximum value of the function? I dont see any use-cases for this defenition of a limit

[@davidblauyoutube]

I can't say I felt satisfied at the end of this video... More like I had been put through a meat grinder lol

[@alberteinstein3612]

Way back when you posted this on FAST, I just used logarithmic differentiation to differentiate the monster. Here’s what I got: ((-3x^2 ln(1-x))/(xlnx- x)) all multiplied to {(1/x)+[(1/(1-x))/-ln(1-x)]-[(1/x)/(ln x -1)]} Thanks for the problem! 😁

[@sumankalyannaik3113]

Hats off to your dedication only for writing that monstrous solution. Lol😆

[@bogdant.8599]

@@sumankalyannaik3113 He's Einstein! He's our king😃

[@Pacvalham]

13:48

[@sumankalyannaik3113]

@@Pacvalham XD

[@bertrandbrodeau2372]

Excellent all-in-one. I love this.

[@stumbling]

"Okay class, I'm going to set only one question for your test." *cheering* The question:

[@blueshell6048]

I put the solution in my graphing calculator expecting some ridiculous graph (assuming I put it in accurately) and I was wonderfully surprised how simple the graph was. This year is my first time taking a calculus class. I’m not the best at it but it is certainly interesting and it was nice that I can understand certain parts of the video.

[@SeeTv.]

To me after figuring out the parts, calculating the derivative isn't that hard. It just requires a lot of concentration and time and space but isn't fundamentally more difficult than some friendlier looking derivatives.

[@liamwelsh5565]

I agree. Derivatives are easy. The only tricky part is not making little mistakes while doing multiple rules at once.

[@Kanbei11]

@@liamwelsh5565 to be honest a function like that is calling out for logarithmic differentiation

[@liamw6976]

After doing calculus for like 2 whole years in college and being in vector calculus rn, this problem is not as bad as it definitely once would have been

[@MrHotBagel]

This is a good problem; definitely going to use! I like how it combines our limit definition of derivatives, power series, and other integration methods and derivative methods all into one. Definitely could improve by making it multiple parts and adding a bit more application based there (like observing flowrates in piping or particle moving in 2-d).

[@liamw6976]

You could also integrate some imaginary parts with complex conjugates into it

[@pontus1055]

I have no idea What you are talking about, but it was very interesting to watch.

[@anastasiskanidis1925]

This is my favorite video of all time

[@derrickblack8632]

The real fun happens when you have to check all student solutions (those students that finished the problem, of course) to make sure their solutions are equivalent to your solution

[@eitansegev]

exactly what i was looking for!

[@stevenschilizzi4104]

Whew! I was expecting a smaller, simpler looking final answer!

[@thatspanishphysicist]

dude, i’m so excited to get into calculus

[@dr.rahulgupta7573]

Thanks , black pen red pen blue pen green pen , for presenting a nice question . Excellent !!

[@papycoima]

I just started disequations at school. I don't know why I open this video, but I don't regret it

[@shinobu9537]

I love your videos so much!! If you haven’t yet, do you think you can do a video on the difference between a function being nondifferentiable and discontinuous at a certain point?

[@Andrux0821]

That was just part A, part B of the exam would ask, “Using the answer you got from part A, draw a slope field.”

[@LuisLascanoValarezo]

Is interesting to see how this makes so much sense in comparison to Línear Algebra

[@cxpKSip]

My Adv. Calc book defined dervatives as taking the limit as x approaches x0 of the function (f(x)-f(x0))/(x-x0). This has the nice property that you can prove that a function f mapping a domain D to the reals is defferentiable at a point x0 in D, despite it not being differentiable everywhere.

[@Gillespie28]

I have my degree but if I took my integral calc exam and saw this question I’d just giggle and skip it.

[@Developer_Max]

0:57 Let h be 0, so (x+h) = x, so (x+h)^3 = x^3. so (x+h)^3 - x^3 = 0. 0/x = 0, so 0 divided by integral is 0

[@alejandrovillegas6754]

i love how you switch markers

[@gb.03]

professor pumar told me about your channel. 100% going to pick you as my next math proffesor

[@ILoveMaths07]

What's his name?

[@aayush1091]

I am kinda proud that i did all this just under 5 min, all in my head, no pen and paper, except for the simplification of the derivative part(cuz that's tedious)

[@jackwheeler609]

Have you ever thought about teaching some of these concepts rather than just implementing them. I'm in Calc 1 right now and have no idea how integrals work yet but I'd love to learn so I can actually participate. Just a thought

[@maxpercer7119]

4:24 for the 0^+, x can be negative though, since -1 < x < 1 so you should do another limit, the case 0^- The limit turns out to be zero anyway.

[@not_vinkami]

Theoretically yes but actually 0- is not needed because the original question has a ln(x) to bound x ∈ (0, 1) instead of (-1, 1)

[@AquaticDot]

Alternative answer: As x is not defined to be between -1 < x < 1, the sum diverges and thus the derivative does not exist.

[@CrittingOut]

Or the derivative is only defined for those bounds

[@AquaticDot]

@@CrittingOut It's defined for even less bounds when you consider that the integral of natural log is divergent from -1 to 0 for x, so it's more that 0 < x < 1.

[@plasmakitten4261]

By checking the domain of his answer it becomes also clear that the derivative does not exist at x=1 (or e, but we knew that already)

[@AquaticDot]

@@plasmakitten4261 True. We also know that from the series being converted into the power series, since x = 1 causes the series to be the divergent harmonic series. I will say, on a slightly irrelevant note, I'm not sure if the natural log integral diverging matters, since I think that would make it the derivative of 0 (at least from -1 < x < 0). So maybe the domain is more (-1, 0)u(0, 1) since x = 0 gives a zero in the denominator for the integral (assuming it even converges in the first place).

[@plasmakitten4261]

@@AquaticDot Is the lower bound of that included though? We know the upper bound is not.

[@7thdayfallout]

Thanks, BPRP. I'll be sure to try putting this as an extra credit question on one of my tests. I teach biology but it will probably be fine.

[@aaronmiller8056]

so relatable - "I have to roll up my sleeves because... ya"!!!

[@RecklessBlaze]

I’m so glad I’m done with calc.

[@thehyperbolicintegral]

I didn't understand the blue summation part because I haven't learned that yet, but being able to understand and do the rest of the math was oddly satisfying

[@alexandrianova6298]

“I have to roll up my sleeves because…yeah. Yeah.” Know that feel all too well.

[@BREAD-kc4zp]

It was pretty easy and direct ngl

[@varun1702]

We love your explanation 😘😘

[@Mrtomix98]

Before swapping integrals and sums, or integrating by parts on objects you do not know yet wether they exist or not (integral of lnt between 0 and x), you might want first to prove it, uniform convergence for the first one and regular proof using limits. If there is no proof of their existence you cannot rigorously use these mathematical items into your calculus…

[@ioangauss]

And now let's study the sign for variation :) (Greetings from France)

[@ILoveMaths07]

Really beautiful question!

[@blackpenredpen]

Glad you think so!

[@luckylucas8596]

The last thing I would add is evaluate the derivative at x = some number. You can’t pick x

[@vijaykulhari_IITB]

Sir I watch your videos sometimes but I have subscribed you because sir your smile is pretty nice

[@vinitoledo1377]

I'm almost sure that the final answer could be written as (3((1-x)*(2-ln(x))*ln(1-x)-x*(1-ln(x)))/((1-x)*(1-ln(x))^2) It's not a huge difference, but it's simple than the answer you boxed in the end

[@Herfuage]

Factoring the last two terms of the numerator we end with 3.(1-x).ln(1-x).(2 - lnx) - 3.x.(1 - lnx)

[@GaryFerrao]

14:09 "i pick derivative on the outside because we can always differentiate" in my test, i will swap them. lol

[@xiangqu]

como dirian en mexico "se mamo" estuvo bueno profe se re entendio gracias desde Argentina!

[@AarushLanjharia]

I feel bad for your students. Love your channel and content though!! I'm an AP calculus student right now and your videos and methods of solving problems actually give me more insight on how these problems could be solved.

[@blackpenredpen]

No need to feel bad for my students. I buy In-N-Out for my students. (Well, two of them this semester. Of course, they had to earn it!)

[@philipeer5772]

Coming from Germany I wonder at what grade level this would be thought. In my Grundkurs (comparable to a normal paced class) we only did Integrals and the ones with the sideways M (don’t remember the English names) but the Leistungskurs (comparable to ap classes) did a little more. Both happened last year at 12th grade (basically jr year) and now we’re doing vectors and layers which is a lot easier imo. Also, physics is really interesting rn as we are working on special relativity theory

[@nathan7726]

most of this was covered in my calc 2 class in uni

[@General12th]

All this was covered in my Calculus 2 class in 11th grade (basically ages 16-17), but that doesn't mean all of this was _understood._ Like anything else, calculus takes time and patience to understand fully.

[@plislegalineu3005]

1:37 I thought you said x to the x power, I double checked and proved myself wrong

[@vicallday3325]

Great video , intro was heart - warming !

[@Rishith198]

This was a lengthy one but quite interesting!

[@pritamsur1926]

Please solve my one integration:- integral of (32-x^5)^(1/5) dx🙂

[@RazorM97]

Thanks for making this!

[@cbayoo9405]

Before I saw this video, I was wishing you were my Calc teacher. After seeing this, Im good😂 All jokes aside, you are helping me out so much with your content. Thank you very much😎

[@shakirullah9434]

Please make a vedio on ROLL, CAUCHY, LAGRANGE THEORAMES.

[@gdmathguy]

This was very nice to do

[@michaferenc6112]

The only thing more satisfying would be if the result was simplified to something short like ln(x) or something like that. I remember that my physics teacher always wanted us to get the result by deriving the formula first and then putting all the numbers in at the end and they were chosen in such way that most of them cancelled out an the result was an integer as opposed to calculating all values on the way and getting some weird fractions that could easily lead to mistakes

[@FIRST_FLOP_PLAYER]

Whenever i see sigma sign in question , I leave the question 🤣🤣😂😂

[@xiang-yue-fung]

Excellent!

[@Randysplaylist]

In the final answer, you can factor out (1-lnx) from the numerator and cancel from the the denominator to make it look a little more clean

[@luisaleman9512]

No, you can't. The third term of the numerator doesn't have a factor of (1 - ln x)

[@paprikaa117]

if i walked into class and saw this question on my desk I'd walk right back out

[@ethanthan3115]

i love the fact that i can understand like 15% of this

[@frozenpeak1524]

watching these videos when i’m about to pass out from sleep deprivation is… i already can’t understand this, adding the sleep just makes me feel as if tilting my head would make my brain flow out

[@valhalla6266]

Imagine having only 5 minutes left in a calculus test and this is the final -boss- question.

[@professorcalculus7885]

Really good videos. I appreciate your efforts

[@jameszhang9326]

Steve, you have indeed evolved with the questions 😂😂

[@comeycallate9959]

You could factorize that last part getting 1 -lnx + 1 = 2-lnx

[@ArenWitsuki]

Thank you for grabbing my attention using that pokeball

[@hannakrupowiecka6311]

Can you really integrate a series like you did for the blue part? I'm intrigued to learn more about this!

[@lassikokkonen5618]

I think you actually can simplify it just a bit further. By turning 3(1-x)(1-lnx)ln(1-x)+3(1-x)ln(1-x) into 3(1-x)(2-lnx)ln(1-x)

[@gregsarnecki7581]

Why not add a Feynman integral trick (partial derivative under the integral sign) while you're at it!😁

[@blackpenredpen]

hahaha, great idea!

[@user-wd8zb9ge5n]

The question the teacher solves in class: 1+1 The question that comes up in the exam :

[@firefightermike7246]

This one question was my entire AP Calc BC final

[@ioangauss]

WHAT THE ... !!! WOOOOW

[@IvyANguyen]

Very cool problem. It is so easy to miss things. I usually use square brackets [ ] if I have 2 levels of ( ).

[@YoYo-ls6pp]

u should do one woth ln and exp

[@user-iy6dt4xp5o]

I feel for the people that saw "lnt" and was confused until they realized it's just ln(t)