my all-in-one calculus question
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- Äas pĆidĂĄn 6. 08. 2024
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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: đ âą my all-in-one calculus...
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Another all-in-one calculus question (uncut): đ czcams.com/video/3s1WYUWYEKU/video.html
I love that heâs just holding a PokeBall while crushing my brain
He gotta crush'em all
Facts
Think that it is his mike or something like that.
@@adrienleblancpiette6239 it is his microphone
@@adrienleblancpiette6239 his âmikeâ
"If you're a calculus teacher, feel free to use this on your calculus test"
Me: NOOOOOOOOOOOOOOOOOOOOOOOOOOOO
đ
At least you have the answer now xD
Well, you know the solution now
@@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. â€ïžâ€ïžâ€ïžâ€ïž u unended.
@@aviralkumarbarnwal83 look up questions online and do them a lot that's how you engrave it into ur memory
Calc teachers DO NOT feel free to use this on a test
Lol why
@@blackpenredpen it is a little bit difficult ahahahha, the limiti and the power series are not impossible but the rest is quite hard
@@blackpenredpen its hard
@@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.
@@Geo25rey good
My suggestion:
Let each student have the choice. Either they take a normal calculus test or they only have to do this problem
đ
Or take a test of problems like these (bonus points)
and if you get it incorrect you get a 0
@@cjxchess17 Funny seeing you here haha
@@cjxchess17did not expect you here
calculus teachers would be : "you didn't close the parentheses at 6:44 , B-".
jokes aside, i love your channel!
Noooo, I can't unsee itđđ!
I noticed that too and it was bugggggggging me! BPRP is still awesome!!
*anger sounds*
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!"
I also had the same doubt
If you also like filling the whole board with math
đ
I bought my own whiteboard because of you : D
Ok, so now you use green pen as well
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..
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)
@@kushaldey3003 bule pen too đ
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)
I was about to comment that.
i was about to post this very same comment too ^^
Screw that I'll just say "assuming all the conditions are met we have" and proceed.
Now convince me that this is indeed the derivative using an epsilon-delta proof.
NO ILL LITERALLY DROP OUT
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
Definitely for exam season. đđ
Totally not complicated đłđł
Not doing calculus every day, this was a refreshing way to quickly review many topics. Thanks!
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.
As a 13 year old, I find this very interesting and now I wanna learn calculus.
In case you haven't yet, you should watch 3Blue1Brown's "introduction to calculus" series here on CZcams. It'll give you a wonderful and qualitative understanding of the subject.
@@General12th ok
@@armgord thats cool!
@@armgord Yh me too.
@@armgord Yh me too. (no)
I've always likes your small clarifications as you go along. You always have unusual math content which is refreshing and interesting.
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)
Exactly what I thought, but my conundrum is still not resolved since it's not useful.
@@ViguLiviu?
Well, I am not only impressed about you coming up with the question, but also with the fact that I could understand you everything
I open youtube after a 4 hour calc 2 test and this is the first suggestion đ
đ
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
This is actually crazy and great at the same time, great idea, video, and performing! :D
Thanks!
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
Yo that marker switch thing is certified Mathematician skill.
C=O
d/dt
so you even managed to smuggle some organic chemistry in
respect!
d/dt in organic chemistry? I've never heard about it (I study a chemical Engineering)
@@fivestar5855 Dichlorodiphenyltrichloroethane (DDT) is an insecticide
yeah, i know, lame joke.. but he literally said it as "dee dee tee"
I thought organic chemistry too when i saw the C=O
@@manuelgandaras7871 carboxyl bond
@@fivestar5855 Don't they show up while studying kinetics of organic reactions
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.
"Okay class, I'm going to set only one question for your test."
*cheering*
The question:
I love your videos so much!! If you havenât yet, do you think you can do a video on the difference between a function being nondifferentiable and discontinuous at a certain point?
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. Iâm not the best at it but it is certainly interesting and it was nice that I can understand certain parts of the video.
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.
Excellent all-in-one. I love this.
Thanks for making this!
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).
You could also integrate some imaginary parts with complex conjugates into it
Great video , intro was heart - warming !
I can't say I felt satisfied at the end of this video... More like I had been put through a meat grinder lol
exactly what i was looking for!
Really good videos. I appreciate your efforts
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.
I agree. Derivatives are easy. The only tricky part is not making little mistakes while doing multiple rules at once.
@@liamwelsh5565 to be honest a function like that is calling out for logarithmic differentiation
Excellent!
Way back when you posted this on FAST, I just used logarithmic differentiation to differentiate the monster. Hereâs 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! đ
Hats off to your dedication only for writing that monstrous solution. Lolđ
@@sumankalyannaik3113 He's Einstein! He's our kingđ
13:48
@@Pacvalham XD
Fab!
Super enjoyable as always.
i am loving this
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
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.
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
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
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
Thank you so much â«đđŽđ, finaly was easy. I wish you the best
Really beautiful question!
Glad you think so!
We love your explanation đđ
amazing !
Is interesting to see how this makes so much sense in comparison to LĂnear Algebra
I have my degree but if I took my integral calc exam and saw this question Iâd just giggle and skip it.
This is my favorite video of all time
That was fun. I actually kept up with you this time.
Wondering if you could 'further tweak' the initial question in some inspiring way s.t. further simplification of the outcome is nicely achievable ?
Very cool problem. It is so easy to miss things. I usually use square brackets [ ] if I have 2 levels of ( ).
This was very nice to do
Woww great !!
dude, iâm so excited to get into calculus
Thanks , black pen red pen blue pen green pen , for presenting a nice question . Excellent !!
That was just part A, part B of the exam would ask, âUsing the answer you got from part A, draw a slope field.â
Fabulous explanations. .. congratulations from đđŠđ·
professor pumar told me about your channel. 100% going to pick you as my next math proffesor
What's his name?
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.
Whew! I was expecting a smaller, simpler looking final answer!
Sir I watch your videos sometimes but I have subscribed you because sir your smile is pretty nice
I have no idea What you are talking about, but it was very interesting to watch.
Thanks!
đ
Can you really integrate a series like you did for the blue part? I'm intrigued to learn more about this!
Iâm so glad Iâm done with calc.
It was easy to understand
âI have to roll up my sleeves becauseâŠyeah. Yeah.â Know that feel all too well.
This is amazing lolll
I just started disequations at school. I don't know why I open this video, but I don't regret it
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.
Theoretically yes but actually 0- is not needed because the original question has a ln(x) to bound x â (0, 1) instead of (-1, 1)
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.
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!)
Steve, you have indeed evolved with the questions đđ
I had this really hard antiderivative question at my uni. It was to find the antiderivative to absoluta value of sinx
Imagine having only 5 minutes left in a calculus test and this is the final -boss- question.
Please make a vedio on ROLL, CAUCHY, LAGRANGE THEORAMES.
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
No, you can't. The third term of the numerator doesn't have a factor of (1 - ln x)
Pls Mr. How do I become this fluid and great in math, I've seen many of your videos and I'm absolutely amazed the way you explain and solve the problems. I really admire you.
Truth be told I think itâs a lot of practice
Practise......
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
Can your explain the motivation for using 1/(t)^2 in the LâHopitalâs calculation?
i love how you switch markers
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)
Btw - could you do a video on infinity factorial? I have seen some videos which say that it is equal to pi*Square root of 2 but this makes no sense so I am sure youn could do a video on this that would clear this up.
Alternative answer: As x is not defined to be between -1 < x < 1, the sum diverges and thus the derivative does not exist.
Or the derivative is only defined for those bounds
@@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.
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)
@@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).
@@AquaticDot Is the lower bound of that included though? We know the upper bound is not.
Man youâre crazy xD
How?đ
Well no matter i still love your videos
Hello sir, will you also be covering multivariable calculus?
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 (donât 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 weâre doing vectors and layers which is a lot easier imo. Also, physics is really interesting rn as we are working on special relativity theory
most of this was covered in my calc 2 class in uni
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.
Whenever i see sigma sign in question , I leave the question
đ€Łđ€Łđđ
I have a question , how to plot graph of integration of e^x logx dx. Feel free to answer and thank you for your videos
Can you kindly share any easy approach to solve:
y=x-1 & y=cos(x)
TIA
if i walked into class and saw this question on my desk I'd walk right back out
If you didn't bring the negative to the denominator before you differentiate, would it give you the same result?
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 calculusâŠ
Thx
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
watching these videos when iâm about to pass out from sleep deprivation is⊠i already canât understand this, adding the sleep just makes me feel as if tilting my head would make my brain flow out
This was a lengthy one but quite interesting!
so relatable - "I have to roll up my sleeves because... ya"!!!
And now let's study the sign for variation :) (Greetings from France)
Bro i lov eyour bvideso bro omg so good