my all-in-one calculus question (uncut)
HTML-код
- Опубликовано: 1 авг 2024
- Learn 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
For more math for fun videos like this, be sure to subscribe 👉 bit.ly/3o2fMNo
My just calculus tutorial channel: 👉 / justcalculus
💪 Join our channel membership to unlock special perks: bit.ly/34PGH4h
📸 Math notes on my Instagram: / blackpenredpen
🛍 Shop math t-shirt & hoodies: bit.ly/bprpmerch
10% off with the code "WELCOME10"
Equipment:
👉 Expo Markers (black, red, blue): amzn.to/3yYLqOf
👉 The whiteboard: amzn.to/2R38KX7
👉 Ultimate Integrals On Your Wall: teespring.com/calc-2-integral...
---------------------------------------------------------------------------------------------------
**Thanks to ALL my lovely patrons for supporting my channel and believing in what I do**
AP-IP Ben Delo Marcelo Silva Ehud Ezra 3blue1brown Joseph DeStefano
Mark Mann Philippe Zivan Sussholz AlkanKondo89 Adam Quentin Colley
Gary Tugan Stephen Stofka Alex Dodge Gary Huntress Alison Hansel
Delton Ding Klemens Christopher Ursich buda Vincent Poirier Toma Kolev
Tibees Bob Maxell A.B.C Cristian Navarro Jan Bormans Galios Theorist
Robert Sundling Stuart Wurtman Nick S William O'Corrigan Ron Jensen
Patapom Daniel Kahn Lea Denise James Steven Ridgway Jason Bucata
Mirko Schultz xeioex Jean-Manuel Izaret Jason Clement robert huff
Julian Moik Hiu Fung Lam Ronald Bryant Jan Řehák Robert Toltowicz
Angel Marchev, Jr. Antonio Luiz Brandao SquadriWilliam Laderer Natasha Caron Yevonnael Andrew Angel Marchev Sam Padilla ScienceBro Ryan Bingham
Papa Fassi Hoang Nguyen Arun Iyengar Michael Miller Sandun Panthangi
Skorj Olafsen Riley Faison Rolf Waefler Andrew Jack Ingham P Dwag Jason Kevin Davis Franco Tejero Klasseh Khornate Richard Payne Witek Mozga Brandon Smith Jan Lukas Kiermeyer Ralph Sato Kischel Nair Carsten Milkau Keith Kevelson Christoph Hipp Witness Forest Roberts Abd-alijaleel Laraki Anthony Bruent-Bessette Samuel Gronwold Tyler Bennett christopher careta Troy R Katy Lap C Niltiac, Stealer of Souls
---------------------------------------------------------------------------------------------------
💪 If you would also like to support this channel and have your name in the video description, then you could become my patron here / blackpenredpen
Do you like my videos edited or unedited?
unedited
Sir do you conduct zoom meetings also.. please share to me also 😞😞😢😋😋
Un
Unedited
Unedited but u should speed up erasing and writing way too long things
You know things go serious if you see bprp using 5 colors on a question
Now he’s BPRPBPRPPP
@@Claymore_II i had a stroke guessing it xD
1. that was actually funny
2. you somehow have about twice as many likes as THE PINNED COMMENT BPRP MADE
HOW
@@ananas-nevermind As you can see, many other comments are hearted by bprp. I exchanged the heart for 500 likes.
jk lol 😂
@@Claymore_II bprpbpppgp
bp¹=black pen
rp=red pen
bp²=blue pen
pp=purple pen
gp=green pen
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.
Bruh you forgor the dx
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 🤣🤣🤣🤣
It's called stuying ET (Elektrotechnic) and IT (Informationtechnic)...
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.
@@user-yr3uj6go8i all or nothing
Oh yeah it's just the inverse hyperbolic cotangent of x over the fourth root of 6, I could have guessed that
For sure🤣🤣❤️
It’s crazy the shit they make you remember in calc but then in any science they give formula sheets longer than the test
You forgot the plus c
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|
Challenge in Maths 🏆🏆🔥🔥
ruclips.net/video/VoZZetC5GfU/видео.html
All that effort for 12 likes lol
@@dizzyd7315 lol
@@dizzyd7315 Other people are illiterate.
He’s always energetic and I love it.
you can see the passion through his eyes
13:19 LMFAO I FELT THAT “why am I doing this” 💀
yeah it sounds so genuine lol
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
lol
@blackpenredpen ngl I'd pass out if I saw that question 😳😳
@@blackpenredpen Challenge in Maths 🏆🏆🔥🔥
ruclips.net/video/VoZZetC5GfU/видео.html
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👍
Same, I really enjoy these kind of videos :D
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💪
Wait. Where's the dx?
That's -100 score for you.
My first thought also.
@@xinpingdonohoe3978 same
Fermat : "I have discovered a truly marvelous proof of this, which this board is too narrow to contain."
blackpenredpen : "hold my beer"
"blackpenredpen"->"blackpenredpenbluepenpurplepengreenpen"
Peak character development ladies and gentlemen
then comes penbrow
5 minutes for the limit, differentiation and series; and 13 minutes for integration... Well...
more like 5 minutes to do partial fractions and 3 minutes to do the integral
Limits, derivatives and integration in one question. It should be called fundamental question of the calculus...Complete package.. Amazing
Don’t forget the sigma!
Ah yes, blackpenredpenbluepengreenpenpurplepen
Watching your videos, I actually started loving math !
Heyy I solved this on my own before I went further into the video. I'm proud of myself 😍
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.
Just keep practicing man. Everything comes on point to the ones who really know what they want. Just giving you some strength ❤️!
Remember that he’s been teaching for about a decade at a massive institution
It makes me so happy to see someone this exited about maths! Keep up the entertaining content😁👍🏼!
I like these mega problems! They're great review.
Glad to hear : )
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).
Great content man. As always
Love your content helps with my love of mathematics, regards
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)
Thank you I have tears of joy now
Your videos are everything
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.
This made me realize how much I've forgotten about partial fraction decomposition 🥲 time for a study montage
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."
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*
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!
An old video here ruclips.net/video/aC5cYc7XhIs/видео.html
@@blackpenredpen thanks!!!
Wait until you hear about elliptic functions.
its amazing sir.. your videos always enlighten me.....
just learned what's the derivative of coth^-1. Great video !
More all one calculus questions thanks)
I love this!
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!
Glad to hear it, thank you : )
@@blackpenredpen yessirrrr
Brilliant!!
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
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!
Never heard of it until looking it up. Seems very silly they wouldn’t just give you the AP Exam.
This guy is gonna change my career!
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)
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).
This is beautiful
Definitely a surprise to be sure, but a welcome one to wake up to
You are a madman!
This looks ultimate
These are the kinds of questions my General Chemistry I and II professor was fond of - the entire final exam in one question.
Your awsome Professor)
coefficients can be simplify to -(6)^(1/4)
I just realized how much more I have to learn
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.
My thoughts exactly.
You are amazing good job💪
An absolute monster
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?
This is highly epic
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.
There's no dx at the end of the integral at the top :( xD
You're right.
It is incomplete
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?
Excelente !!. Y su libro muy bueno ..
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.
gonna need this for when i retake calc 2… but on a side note, where did you get the pokeball cover for your mic?
Cool stuff
I lost it at the integration part
Until then i was really happy that things were going smooth
i never thought i would be so satisfied by an math equation :D
I am glad to hear that : )
Hello which one is biger e^x and x^n when x->+∞
Amazing
Very good
For the coth^-1, i would instead have used another partial fraction and then 2 times the Ln fonction, is it right ?
Hello there, big fan of you.
I have a cuestion, aren´t you missing the x part on the partial fractions?
Greetings from Mexico
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
I just think, that's awesome
06:32 It also can be nice way to use Heavyside cover up method, I guess. :)
Best video
In which level of calculus would this problem be found? Thanks
You are awesome
Congratulations from a French student. 🇫🇷❤️
the very first limit: isn't the definition of derivative?
derivative of x^(-2) is -2/x^3
Yes u are right. He mentioned it in the video
But when it's in first principles form, you should do it as an actual algebra question.
@@xinpingdonohoe3978 why, unless they say you have to why not save time
@@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.
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.
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.
Which is greater x^y ! Or y^x ! Given that x and y are real numbers? Pls answer this
I really felt the "Why am I doing this" at 13:20 on deeper levels than is comprehensible.
what we can use it in daily life?
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
Awosome
Wait, when solving the Σ, how did x⁴ leave since the fraction is (6/x⁴)'? Shouldn't it be something else?
he says it's raised to the FIRST POWER, so it's just ^1
I don't understand your issue with the series. What's wrong with him answer?
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.
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
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?
When h approaches zero doesn't you 2xh become 0 as well? How do you get 2x?
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
"Why am I doing this 😞" was the best part 😂😂😂
The students: "don't worry he said he would put only one ex. on the exam"
The exercise: are you sure about that 😂😂🤣🤣
Hahaha!
I love how you look at it so proudly when you're done, like it's your child. 😂
😆
Integration of 1/sin^7x + cos^7x possible
Blackpenredpen Collab with purplepen greenpen and bluepen
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.
Broo whats de result of integral of 1 power x?? im so confused
thanks
13:26 “why am i doing this”. that’s how i feel when i do implicit differentiation
I'm gonna learn calculus and come back here to see if I can understand more than 20% of this video
the "why am i doing this" is so real
why didn't you just change (x+h)^-2 and just ue the limit definition to evaluate it at first?
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:)
Is this ASMR? I get the feels just by watching this
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.
I may be wrong with the following thought, but here goes: if inside a fraction, both numerator and denominator have the same sign - in our case, both are positive - its absolute value is the fraction itself. Which means that |6/(x^4)|sqrt_4(6), and not |x|>sqrt_4(6).
Great