my all-in-one calculus question (uncut)
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Күн бұрынLearn more calculus here 👉 brilliant.org/blackpenredpen/ (20% off with this sponsored link)
Here's another all-in-one ultimate calculus question that I came up with! We have the first principle limit definition of derivative of 1/x^2, then the power rule, followed by an infinite geometric series and we will end with an integral with partial fraction decompositions. This is an integral of an infinite series of the derivative of the limit of a difference quotient! I would say this is an ultimate calculus question for the final exam! Free feel to use this for your class if you are also a calculus teacher! #calculus #blackpenredpen
If you like this video, then you should watch this video next: "when d/dx(x^2) becomes the hardest question on your calculus final" 👉 • calc 1 final be like (...
0:00 limit definition of derivative of 1/x^2
2:20 derivative by using the power rule
2:58 infinite geometric series
5:17 partial fraction decomposition with quadratic denominators
10:54 the actual integration
14:12 integral of 1/(1-x^2) is coth^-1(x) when abs(x) is greater than 1
*How to integrate 1/(1-x^2): • the integral of 1/(1-x... *
16:16 WOW!
17:18 Check out Brilliant
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@blackpenredpen 2 жыл бұрын
Do you like my videos edited or unedited?
@jamesh7505 2 жыл бұрын
unedited
@28sarthakdwivedi44 2 жыл бұрын
Sir do you conduct zoom meetings also.. please share to me also 😞😞😢😋😋
@decCoulissante 2 жыл бұрын
Un
@mohamedibrahim1023 2 жыл бұрын
Unedited
@ArdenChew271 2 жыл бұрын
Unedited but u should speed up erasing and writing way too long things
@Luka_D_Snots 2 жыл бұрын
You know things go serious if you see bprp using 5 colors on a question
@Claymore_II 2 жыл бұрын
Now he’s BPRPBPRPPP
@rayane--21pilots8 2 жыл бұрын
@@Claymore_II i had a stroke guessing it xD
@ils7103 2 жыл бұрын
1. that was actually funny 2. you somehow have about twice as many likes as THE PINNED COMMENT BPRP MADE HOW
@Luka_D_Snots 2 жыл бұрын
@@ils7103 As you can see, many other comments are hearted by bprp. I exchanged the heart for 500 likes. jk lol 😂
@patolol3091 2 жыл бұрын
@@Claymore_II bprpbpppgp bp¹=black pen rp=red pen bp²=blue pen pp=purple pen gp=green pen
@HDitzzDH 2 жыл бұрын
Imagine this question being on your final exam. You go through all of these steps and calculations just to forget the +C at the end which single-handedly causes you to fail the exam.
@ranjitsarkar3126 2 жыл бұрын
Bruh you forgor the dx
@aadilashraf1597 2 жыл бұрын
Bro if I am forgetting only the c part in this question...... Then I am sure.... I will be able to solve other questions easily 🤣🤣🤣🤣
@marvin5729 2 жыл бұрын
It's called stuying ET (Elektrotechnic) and IT (Informationtechnic)...
@user-yr3uj6go8i 2 жыл бұрын
Then you have a garbage math teacher/professor if your only mistake of forgetting the +C at the end causes you to fail the entire exam.
@memebaltan Жыл бұрын
@@user-yr3uj6go8i all or nothing
@xissel 2 жыл бұрын
Oh yeah it's just the inverse hyperbolic cotangent of x over the fourth root of 6, I could have guessed that
@gusjak3454 2 жыл бұрын
For sure🤣🤣❤️
@temujinartem2375 2 жыл бұрын
It’s crazy the shit they make you remember in calc but then in any science they give formula sheets longer than the test
@MusicalInquisit 2 жыл бұрын
You forgot the plus c
@RobinD0903 2 жыл бұрын
13:19 LMFAO I FELT THAT “why am I doing this” 💀
@jonahrankin6978 2 жыл бұрын
yeah it sounds so genuine lol
@SpaceWithSam 2 жыл бұрын
He’s always energetic and I love it.
@morsxsx 2 жыл бұрын
you can see the passion through his eyes
@baptiste5216 2 жыл бұрын
Fermat : "I have discovered a truly marvelous proof of this, which this board is too narrow to contain." blackpenredpen : "hold my beer"
@christopherdyson1158 2 жыл бұрын
This is completely off topic, but I think I came up with an interesting problem. Given the recursive functions: f(n) = af(n-1) + bg(n-1) g(n) = ag(n-1) + bf(n-1) Where a=0.5 and a+b=φ Show that both f and g are fibbonacci sequences. I'll put my solution at the end. Stumbled upon this fact while trying to solve for g for any a or b... idk how others may attempt to solve it but I wrote them as a discrete time convolution. I got this equation for g(n): g(n) = (b^2-a^2)g(n-2)+2ag(n-1) Was trying to solve for f or g to write sin(nt) in terms of sin(t) and cos(t)... but accidentally made a mistake, one of the equations were supposed to add and the other was supposed to subtract the b term.... was doing this in the first place because, I was hoping the equation for sin(nt) would help solve the integral for (sin(Nt)cot(t/2)+cos(Nt))cos(nt) where N and n are two non-negative integers. Which I plugged into a couple integral calculators... and it didn't give me a solution. When you evaluate that integral in respect to t from -π to π, and divide by 2π, its supposed to be an inverse discrete-time fourier transform that should equal 1 for all |n|
@kingshiva5427 2 жыл бұрын
Challenge in Maths 🏆🏆🔥🔥 kzfaq.info/get/bejne/jNWKjcilqJurlok.html
@dizzyd7315 2 жыл бұрын
All that effort for 12 likes lol
@franceskotambonop5687 2 жыл бұрын
@@dizzyd7315 lol
@vaibhavsrivastva1253 Жыл бұрын
@@dizzyd7315 Other people are illiterate.
@toastedbred2961 2 жыл бұрын
You know it's gonna be hard when he doesn't recommend calc teachers to use this in a test at the start of the video
@blackpenredpen 2 жыл бұрын
lol
@TheRealUSArmy 2 жыл бұрын
@blackpenredpen ngl I'd pass out if I saw that question 😳😳
@kingshiva5427 2 жыл бұрын
@@blackpenredpen Challenge in Maths 🏆🏆🔥🔥 kzfaq.info/get/bejne/jNWKjcilqJurlok.html
@bavanindhra2733 2 жыл бұрын
"blackpenredpen"->"blackpenredpenbluepenpurplepengreenpen" Peak character development ladies and gentlemen
@trusiant1729 2 жыл бұрын
then comes penbrow
@sfglunywand8042 2 жыл бұрын
I’m a sophomore in high school and have no idea what he’s doing but I still watch a bunch of his videos because they are interesting and relaxing. Keep up the good work👍
@1upGAMeR200 2 жыл бұрын
Same, I really enjoy these kind of videos :D
@aidendoty7711 8 ай бұрын
i was in your position a couple years ago. i’m now in calc 2, watched this whole video and actually understood all of it, and i’m very proud of myself. keep studying and you’ll get there💪
@ranjitsarkar3126 2 жыл бұрын
Wait. Where's the dx? That's -100 score for you.
@xinpingdonohoe3978 2 жыл бұрын
My first thought also.
@leoofficial527 2 жыл бұрын
@@xinpingdonohoe3978 same
@DarshanShah10 2 жыл бұрын
5 minutes for the limit, differentiation and series; and 13 minutes for integration... Well...
@comma_thingy 2 жыл бұрын
more like 5 minutes to do partial fractions and 3 minutes to do the integral
@mohammadmahmood8255 2 жыл бұрын
Limits, derivatives and integration in one question. It should be called fundamental question of the calculus...Complete package.. Amazing
@EyeSooGuy 8 ай бұрын
Don’t forget the sigma!
@3ckitani 2 жыл бұрын
Ah yes, blackpenredpenbluepengreenpenpurplepen
@gurpreetkhera8266 2 жыл бұрын
Watching your videos, I actually started loving math !
@sdkdoom 2 жыл бұрын
I've always wanted to be a teacher, but after watching you, it made me feel so dumb that I'm no longer sure I'm capable of it. I hope I can be as intelligent as you are one day.
@gusjak3454 2 жыл бұрын
Just keep practicing man. Everything comes on point to the ones who really know what they want. Just giving you some strength ❤️!
@jovianarsenic6893 2 жыл бұрын
Remember that he’s been teaching for about a decade at a massive institution
@yunghollow1529 2 жыл бұрын
A young man as cool as a Shaolin monk with a Kung Fu Master beard and has a Pokemon ball in his hand explaining math problems... This combination made my yung hollow mind literally explode.
@alexanderhjertkvist7537 2 жыл бұрын
It makes me so happy to see someone this exited about maths! Keep up the entertaining content😁👍🏼!
@Chaosdude341 2 жыл бұрын
I like these mega problems! They're great review.
@blackpenredpen 2 жыл бұрын
Glad to hear : )
@anupamadewpura 2 жыл бұрын
Heyy I solved this on my own before I went further into the video. I'm proud of myself 😍
@vasnormandy2 2 жыл бұрын
Idk if it’s easier but I factored out the 1/6 on the top and bottom from the beginning to lessen the burden of roots to get 1/((x^4/6)-1).
@marciochs5523 2 жыл бұрын
Great content man. As always
@joannaford7137 2 жыл бұрын
Love your content helps with my love of mathematics, regards
@danny89620 2 жыл бұрын
Thank you I have tears of joy now
@hyper_phantom_darxteel 2 жыл бұрын
Partial fractions could be used in a different way by adding and subtracting x^2 (x squared) and 6^(1/2) (square root of 6)
@nicollerodriguez4486 2 жыл бұрын
Your videos are everything
@ShebastianReyes 2 жыл бұрын
This made me realize how much I've forgotten about partial fraction decomposition 🥲 time for a study montage
@kruksog 2 жыл бұрын
I got all the way to the integral on my own and said "I do math for fun, but I draw the line at PFD."
@stumbling 2 жыл бұрын
STUDYING HARD NOW, HE'S GOTTA RECLAIM ALL OF THE MATH THAT ESCAPED FROM HIS BRAIN INTEGRALS, DERIVATIVES AND PARTIAL FRACTIONS HE'S MASTERED THEM ALL NOW HE'S READY FOR ACTIONNNN MATH-E-MATIC! AUT-O-MATIC! FEELS LIKE MAGIC WHEN YOU DERIVE JUST TO STAY ALIVEEEEE! MATH-E-MATIC! AUT-O-MATIC! MATH-E-MATIC! AUT-O-MATIC! *music fades out as you ride off on a motorcycle*
@armanavagyan1876 Жыл бұрын
More all one calculus questions thanks)
@alberteinstein3612 2 жыл бұрын
Can you please make a video that introduces hyperbolic functions and some sample problems with them? I’ve never known anything about them and would love to learn more about them!
@blackpenredpen 2 жыл бұрын
An old video here kzfaq.info/get/bejne/l6lllryUnL7Meac.html
@alberteinstein3612 2 жыл бұрын
@@blackpenredpen thanks!!!
@aubertducharmont 7 ай бұрын
Wait until you hear about elliptic functions.
@radheyshyamsharma1975 2 жыл бұрын
its amazing sir.. your videos always enlighten me.....
@jibster5903 2 жыл бұрын
I love this!
@evancamus3322 2 жыл бұрын
just learned what's the derivative of coth^-1. Great video !
@dudono1744 2 жыл бұрын
here is a challenge: find f(x) such that f'(x) = f(x+1) oh and i forgot something: there is a x such that f(x) = 1 (just multiply the solution by some number to get f(x) = 1 if you don't have it already). This is just to exclude f(x) = 0
@abhirajranjan1518 2 жыл бұрын
Brilliant!!
@WinterSoldier42 2 жыл бұрын
bro hes getting serious with these problems. I love your videos BPRP. Keep up these amazing videos. Im a high school student in AP Calculus AB. You help me alot with studying! Thanks man!
@blackpenredpen 2 жыл бұрын
Glad to hear it, thank you : )
@WinterSoldier42 2 жыл бұрын
@@blackpenredpen yessirrrr
@user-nl7mn4rr6s 2 жыл бұрын
coefficients can be simplify to -(6)^(1/4)
@adityaadhana5126 2 жыл бұрын
This guy is gonna change my career!
@armanavagyan1876 Жыл бұрын
Your awsome Professor)
@mostafakhaled9702 2 жыл бұрын
hello bprp. there's a limit that's been puzzling me for a while: lim(x,y)--->(0,0) (xy^3)/(x^2+y^6) if you try many directions like polynomail functions or y=sinx or y=e^x-1 the limit = 0 but if you saw the graph for that two variable function you see that the limit does not exist at (0,0) so from what direction is the limit not equal to 0?
@polynoodle8186 2 жыл бұрын
Just took my Calculus CLEP exam and got the score I needed; 64 ain't great, but i get my credit so it's cool. I've never been confident in Math, and quite frankly I went into the exam expecting to fail- I'm quite happy that wasn't the case. Im very grateful for all the videos you make, and passing this exam was definitely thanks to you lol. Keep it up!
@matthewcaimbeul8722 2 жыл бұрын
Never heard of it until looking it up. Seems very silly they wouldn’t just give you the AP Exam.
@Vladimir_Pavlov 2 жыл бұрын
It is necessary to replace the variable in time to avoid radicals. ∫6dx/(x^4-6)=6^(1/4)∫ dt/(t^4-1). t=x/6^(1/4). Here the fraction under the integral sign is so simple that it can be decomposed into elementary fractions without the method of indefinite coefficients. 1/(t^4-1)=(1/2) *{1/(t^2-1)-1/(t^2+1)}=(1/2)*{(1/2)[1/(t-1)-1/(t+1)]-1/(t^2+1)}= = (1/4)*1/(t-1)- (1/4)*1/(t+1)- (1/2)*1/(t^2+1)
@Vladimir_Pavlov 2 жыл бұрын
6^(1/4)* ∫ [(1/4)*1/(t-1)- (1/4)*1/(t+1)- (1/2)*1/(t^2+1)]dx= = 6^(1/4)* {(1/4)*ln|t-1|-(1/4)*ln|t+1|-(1/2)* atan(t)} +C= =(6^(1/4)/4)* ln|(t-1)/(t+1)|-(6^(1/4)/2)*atan (t) +C = =(a/4)* ln|(x-a)/(x+a)|-(a/2)*atan (x/a) +C . a= 6^(1/4).
@alberteinstein3612 2 жыл бұрын
This is beautiful Definitely a surprise to be sure, but a welcome one to wake up to
@mighty8357 2 жыл бұрын
You are a madman!
@dr_ned_flanders 2 жыл бұрын
What is the minimun number of pen colours a bprp needs to write an equation such that none of adjacent terms have the same colour? Is it like the 4 colour map problem?
@spaghetti1383 2 жыл бұрын
The partial fraction decomposition can be done without a system of equations if you consider that 1 = 1/(2*sqrt(6))*[(x^2+sqrt(6))-(x^2-sqrt(6))]. This improved memory efficiency allowed me do the problem in my head reasonably fast. Also, BPRP was running out of room and the video was already long so he used hyperbolic trig functions. But you can decompose 1/(x^2-sqrt(6)) with the same trick or Heaviside cover up.
@ernestschoenmakers8181 Жыл бұрын
My thoughts exactly.
@fotclana2363 2 жыл бұрын
You are amazing good job💪
@Mystery_Biscuits 2 жыл бұрын
There's no dx at the end of the integral at the top :( xD
@victorgalvez2547 2 жыл бұрын
You're right.
@Hyperdrive 2 жыл бұрын
It is incomplete
@fivestar5855 2 жыл бұрын
This looks ultimate
@TEGPS3 2 жыл бұрын
gonna need this for when i retake calc 2… but on a side note, where did you get the pokeball cover for your mic?
@nuranichandra2177 2 жыл бұрын
An absolute monster
@eugeniodiazbarriga8140 2 жыл бұрын
Excelente !!. Y su libro muy bueno ..
@user-nv4id1hq2t 2 жыл бұрын
the very first limit: isn't the definition of derivative? derivative of x^(-2) is -2/x^3
@someguy650 2 жыл бұрын
Yes u are right. He mentioned it in the video
@xinpingdonohoe3978 2 жыл бұрын
But when it's in first principles form, you should do it as an actual algebra question.
@Freddy-pp4nc 2 жыл бұрын
@@xinpingdonohoe3978 why, unless they say you have to why not save time
@xinpingdonohoe3978 2 жыл бұрын
@@Freddy-pp4nc because they are saying to. That is the structure of the question. If it is in first principles form, you need to do it as such. If it's in d/dx form, you can just derive it. With first principles at least you know where you're working to, so it isn't completely abstract algebra.
@nico-kryonix3095 Жыл бұрын
For the coth^-1, i would instead have used another partial fraction and then 2 times the Ln fonction, is it right ?
@TitaniumRailgun 2 жыл бұрын
I just realized how much more I have to learn
@eliteteamkiller319 2 жыл бұрын
These are the kinds of questions my General Chemistry I and II professor was fond of - the entire final exam in one question.
@dieforyou0077 2 жыл бұрын
Hello which one is biger e^x and x^n when x->+∞
@michaeltimpanaro5622 9 ай бұрын
In which level of calculus would this problem be found? Thanks
@Leo-ws3bp 2 жыл бұрын
If you have 1/(0/1), would that be undefined, or would it be 0 because (a/b)/(c/d)=(a/b)×(d/c) meaning 1×(0/1)=0
@renatotafaj1507 2 жыл бұрын
The students: "don't worry he said he would put only one ex. on the exam" The exercise: are you sure about that 😂😂🤣🤣
@blackpenredpen 2 жыл бұрын
Hahaha!
@buqet 2 жыл бұрын
i never thought i would be so satisfied by an math equation :D
@blackpenredpen 2 жыл бұрын
I am glad to hear that : )
@dianeweiss4562 2 жыл бұрын
You could have added the dx in the original problem. It was probably written in white. Excellent problem. I can solve it today, thanks to what I’ve learned from you over the past few years. I may have told you before that as a physicist, my first year of calculus was an “honors” class where all we did was proofs. The only think I really loved about that class was that we used Apostle and I fell in love with proofs by induction.
@EyeSooGuy 8 ай бұрын
I’m 41 and just learning calculus. I just sat there with wide eyes and all shocked at how much work it’s needed. Damn. I think my brain almost literally exploded. Lol.
@ariffiyatno1167 2 жыл бұрын
what we can use it in daily life?
@DoomDG 2 жыл бұрын
When h approaches zero doesn't you 2xh become 0 as well? How do you get 2x?
@yehogwon5667 2 жыл бұрын
06:32 It also can be nice way to use Heavyside cover up method, I guess. :)
@higherhighermath 2 жыл бұрын
Amazing
@akshatzz_iitb6052 2 жыл бұрын
At first, this question seemed pretty daunting but when I started solving I thought this question was really simple. I solved inside equations pretty quickly but when I got to the integration I was like "bruh f*ck this shit". In my opinion, the integration isn't difficult, it's just very lengthy.
@thethoughtrobot5486 2 жыл бұрын
Very good
@28sarthakdwivedi44 2 жыл бұрын
Which is greater x^y ! Or y^x ! Given that x and y are real numbers? Pls answer this
@danielkovacs6809 2 жыл бұрын
What value does f(x)=cos(cos(cos(cos(...(cos(x)))))))...) (infinitely many cos) approach in a kind of nice form? I am really interested in it, but I cannot find it.
@ilpaghynonpaga 2 жыл бұрын
Cool stuff
@bruh-sy9kc 2 жыл бұрын
I lost it at the integration part Until then i was really happy that things were going smooth
@sudeepdalbanjan8595 2 жыл бұрын
Blackpenredpen Collab with purplepen greenpen and bluepen
@liamryden558 2 жыл бұрын
This is highly epic
@marcelfernandez8626 2 жыл бұрын
Broo whats de result of integral of 1 power x?? im so confused thanks
@zhiyuanliu9533 2 жыл бұрын
The final answer can be further simplified into: -[(∜6)/2]coth⁻¹[x/(∜6)] - [(∜6)/2]tan⁻¹[x/(∜6)] + C however, what if the question is: (Without the restrictions/without the previous infinite series question) ∫[6/(x⁴-6)]dx for -∜6 < x < ∜6 Since coth⁻¹[x/(∜6)] is not defined for -∜6 < x < ∜6, what expression would it be? !
@theimmux3034 2 жыл бұрын
In Euclid's proof of infinite primes it is stated that the product of the first n primes plus one is not divisible by any of those primes. Seems intuitive right? Wouldn't it then make sense that also the product of the first n primes MINUS one would also not be divisible by - again - any of those primes? However this isn't true at all. Could you _please_ shed some light onto this problem I am facing?
@raphschru9865 2 жыл бұрын
I think there is an error in the last expression. The red part in factor of coth-1 and tan-1 should be (root4(6))^2 instead of just root4(6). So after simplification you get (-1/2)coth-1(x/root4(6)) - (1/2)tan-1(x/root4(6)) + c
@willbishop1355 2 жыл бұрын
Erase part of the board!!! Also at the end, you could simplify square root of 6 / (2 * 4th root of 6) to just (4th root of 6) / 2.
@Mathcambo 2 жыл бұрын
Solve Equation|3^x=cos(√x) in KZfaq: Mathcambo
@luccaventurini1011 2 жыл бұрын
"Why am I doing this 😞" was the best part 😂😂😂
@enjoylifewithme 2 жыл бұрын
Integration of 1/sin^7x + cos^7x possible
@eliaspapa4415 2 жыл бұрын
I really felt the "Why am I doing this" at 13:20 on deeper levels than is comprehensible.
@Audjem1 8 ай бұрын
13:26 “why am i doing this”. that’s how i feel when i do implicit differentiation
@chorizoenmascarado4976 2 жыл бұрын
Hello there, big fan of you. I have a cuestion, aren´t you missing the x part on the partial fractions? Greetings from Mexico
@lolek_m573 2 жыл бұрын
I just think, that's awesome
@ariaelidove 2 жыл бұрын
13:19 - My reaction when it's 4 am and I'm only watching these videos
@mrjazz2570 2 жыл бұрын
Hey blackpenredpen, i have a problem ive tried to solve, but no answer yet. It goes like this LOGa(x) = a^x , you need to solve for an a value so tangency occurs (both equations kiss eachother) Ive got the value to be about 1.45, but its not satisfying to estimate it lol Thanks:)
@Tank.Dripsey 2 жыл бұрын
It's nearly 2am and I've been up for over an hour trying to do this. I lay restless in bed thinking about this question
@Feministsiken3169 2 жыл бұрын
Awosome
@eliteteamkiller319 2 жыл бұрын
I love how you look at it so proudly when you're done, like it's your child. 😂
@blackpenredpen 2 жыл бұрын
😆
@PlutoTheSecond 2 жыл бұрын
When you reduce the fraction at 9:47, couldn't you have canceled out factors of sqrt(6) instead? Then you would have gotten -sqrt(6)/2 with the denominator already rationalized.
@dqrksun 2 жыл бұрын
why didn't you just change (x+h)^-2 and just ue the limit definition to evaluate it at first?
@umeshpandit7215 2 жыл бұрын
Best video
@mitochondriaman4798 2 жыл бұрын
Wait, when solving the Σ, how did x⁴ leave since the fraction is (6/x⁴)'? Shouldn't it be something else?
@user-jh3kz7dp2z 2 жыл бұрын
he says it's raised to the FIRST POWER, so it's just ^1
@xinpingdonohoe3978 2 жыл бұрын
I don't understand your issue with the series. What's wrong with him answer?
@tornadot1182 2 жыл бұрын
I don't really know what your issue is, but I hope this helps. The series is a geometric series (of the form a^r, where |a| should be less than 1 for the series to converge. If the series starts from n=1, then the value of the series is a/(1-a). If you're confused about the inequality, then it's 6/|x^4|< 1, 61). Hope this helps.
@sambhusharma1436 2 жыл бұрын
You are awesome
@naeemalqise8978 2 жыл бұрын
A technician takes X hours to visit the stores in a couple of streets in one shift, and when he finishes his tour, another technician revisits the same stores again, needing other X hours to finish the second shift, and so on. For example, if the first technician started his shift at 1:00pm, he finishes it at 7:00pm and the second technician starts his shift at 7:00pm and finishes it at 1:00am, and so on. The supervisors used to make a quick meeting with all technicians twice a day at 1:00 am and 1:00 pm, so they need to finish their tours exactly 1:00. Identify the mathematical notation for the number of shifts should be made by the technicians in order to achieve this, write the name of this mathematical value, and find it for two tours, one with X=7 and another with X=11 how I can solve this Question
@tacticalcacti7613 2 жыл бұрын
Somebody get this man a bigger whiteboard! xD
@lebreizhdu29 2 жыл бұрын
Congratulations from a French student. 🇫🇷❤️
@aashsyed1277 2 жыл бұрын
Where are higher derivatives and tailor series