when your calculus test has only one problem
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- Опубликовано: 30 сен 2024
- Here's a beautiful all-in-one calculus question for you guys! Of course, it includes a limit, an integral, a power series, and a second derivative! Here's Laplace's way of solving the Gaussian integral: • how Laplace solved the...
tanh^-1(x), inverse hyperbolic tangent in terms of logarithm • Inverse tanh(x)
Previously:
all-in-one calculus question ep1. • my all-in-one calculus...
all-in-one calculus question ep2. • my all-in-one calculus...
🛍 Get a Taylor series t-shirt (as seen in the video): blackpenredpen...
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Ah yes, a new episode of blackpenredpenbluepenpurplepengreenpen, my favourite!
😂
New episode of bprpbpppgp
Find indefinite integral of xsinx/(1+(cosx)^2)
Please help
Ahh yes, I remember it going down like this in the anime.
now the society wants you to make even *harder* question including some deadly *definite triple integrals* along with *laplace transforms* and *partial derivatives* just casually floating around in the question as a one million subscriber special
Edit: alrighty here u are, at a million subs
Congrats for that👍
.
.
.
now gimme my question
*pweeease*
Ive always wondered, whats the difference between regular derivatives and partial derivatives. I’ve seen it quite a number of times when he does differential equation and Feyman’s technique, but no idea what it really means. Thank you in advance.
@@Ninja20704 With sufficient degrees of freedom, like say some function plotted on x, y and z it may sometimes be practical to keep one variable constant such that we can "slice" the plane and examine a regular 2d plane for derivations or such
@@ES-qe1nh Agreed.
In addition to that, a partial derivative implicitly depends on what other variables you have, since they are to be kept constant.
For instance, suppose
f(x,y) = x+y
Then the partial derivative (I write D because of my keyboard, but I mean the partial-d) D/Dx equals
Df/Dx = 1
If I reparametrize, or transform my coordinate system to new variables x and z, where x remains the same but z = x-y, then f(x,z) = 2x-z, so now suddenly
Df/Dx = 2
even though we changed nothing essentially about either f or x!
Alternatively, suppose that y itself is a function of x, say y(x) = x², then f(x) = x+x², then we can compute the normal full derivative as
df/dx = 1+2x
I guess the thing to note is that all of the derivatives of f with regard to x are different. So they are actually different beasts, not just different notations.
society
youtubebu.com/watch?v=udZddgY5Cea
n the question as a one million subscriber special
@@Ninja20704 they appear more frequently in physics than maths
but to simplify the definition, it’s basically the derivative except it gives more pain than normal derivative
With this all in one calculus becoming a thing, you are showing again why you are among the top mathematicians in the platform, if not the best
This is genuinely a very powerful test of someone's calculus skills.
I guess it might be a bit overwhelming to use it during an actual test, but it can definitely be used by students as a self-check.
Even dough I didn’t understand most of the video, I find your channel really interesting and I love to watch your videos. I can clearly see your passion to maths and your happiness during all videos. Keep going man! I really admire people like you!
I cant express how much I love this channel. I am currently studying Soil science and agricultural chemistry and surprisingly enough the math needed for it is extremely advanced.I unfortunately lost some time and almost dropped out but right now I am determined to graduate.I started learning math by myself from the fundamentals to calculus and now I'm trying to study complex analysis by myself,and this channel just keeps me motivated.Thank you Mr. bprp!!!
As someone who’s education didn’t go past 3rd grade. Thank you for your videos, I’m doing my best to learn all the things I missed out on.
Try learning logarithms, matrices and algebraic whole square solutions
@@humzakhan3962 bro just passed 3rd grade and is studying high school maths,, i want dedication like him
GANBAREEEEEE
@@extreme4180 he didnt just pass 3rd grade, his education didnt go past 3rd grade, read the comment
@@doomsdaycookie7034they were joking lol
That little backtrack at 7:46 was funny XD I thought I accidentally rewound the video cuz I spaced out for a single second the first time
You should make a playlist about generating function, specail functions and Sturm Liouville Systems
Seeing this I've really understood the meaning of:
Don't eat the whole cake in one turn, a slice by slice is good 🍰
a thing, y
youtubebu.com/watch?v=qTt8Efn8KoU
whole cake in one turn, a slice by slice is goo
For the first bit, I just noticed that it’s the same as the evaluating (1/2)! Via the gamma function, which is sqrt(pi)/2
ONLY 1K LEFT UNTIL 1MIL
I may not know what you are doing right now and may get frustrated while trying to understand it, but I'm telling you, I WILL be back in a month and I WILL get it. Cya in 1 month, or 4 weeks, or 30 days, or 1800 hours, or 108000 minutes, or 6480000 seconds. I'll be back.
I may be 10 months late but......
Did u understand it ?
Congrats on the 1M subs!! Well deserved!!
Thanks!
With the gaussian integral you can take advantage of the fact that the integrand is an even function and the integral is bounded symmetrically, so you can change the lower bound to zero and double the result of that. That shows right away that our "half-gaussian" integral in the 'u' world is sqrt(π)/2, no worries about convergence.
I have been dealing with gamma function so much lately that as soon as I saw the limit I instantly realized "ok the limit is just x going to sqrt(pi)/2"
@@MessedUpSystem lmao I just revised Laplace transform and this came by, immediately noticed it's L{sqrt(t)}(1) which is sqrt(pi)/2
Can you find the radius of a circle which touches Latus rectum , axis and circumference of the parabola Y²=4aX
(turn over the paper)
heart attack
I was able to solve everything but the tanh^-1 (x^2) bc my last course never covered that 😮
As a kid that doesnt understand calculus entirely. My honest reaction was:
WHAT THE FUC-
Who came here from instagram?
me
Congratulations for attaining 1M subs. Keep moving 👍
Congratulations for 1M sir .
Edit-Love from India❤️
If you expect your viewers to just know the Gaussian Integral, you should expect them to know the Gamma function.
Anyone else just watch the video even though they have no idea what he did or how to do the problem? LMAO
I'm curious as to what the inverse function to f(x) = x^(1/x) is. I can't solve it at all
To do this, we will need the Lambert W function, which blackpenredpen just seems to be obsessed with for some reason (watch his older videos). This function is defined as the function W(x) for which W(xe^x)=W(x)e^W(x)=x. In other words, W(x) is the inverse function of xe^x.
Let's start with y=x^(1/x) and swap x and y to get x=y^(1/y). We want to solve for y.
Taking the reciprocal of both sides gives 1/x=1/(y^(1/y)). But since 1=1^(1/y), we have 1/x=(1^(1/y))/(y^(1/y)). Using exponent properties, we get 1/x=(1/y)^(1/y).
Taking the natural log on both sides, we get ln(1/x)=ln((1/y)^(1/y)). Using log properties, we get ln(1/x)=(1/y)ln(1/y).
Since 1/y=e^ln(1/y), we get ln(1/x)=ln(1/y)e^ln(1/y). We are now in a good place to use the Lambert W function since the RHS is of the form ke^k.
Doing that gives W(ln(1/x))=W(ln(1/y)e^ln(1/y))=ln(1/y), by the definition of the Lambert W function.
We can now solve for y easily:
ln(1/y)=W(ln(1/x))
1/y=e^W(ln(1/x))
y=1/e^W(ln(1/x))=e^(-W(-ln(x)) (using log and exponent properties)
Hence, the inverse function to f(x)=x^(1/x) is e^(-W(-ln(x)). QED
(As an exercise, try to check that this is the correct answer using the definition of the Lambert W function.)
@@youngmathematician9154 its the best function of all time
@@youngmathematician9154 Thanks a lot!!!
Congratulations on 999k subs!
Thank you!
Aww welcome
Hey !!!!!!!
Namaste🙏
I'm challenging you
Solve the integral without using any property....
I = 2/π ∫ dx/( 1+e^sinx)(2+cos2x)
Limits from -π/4 to π/4
Zero
I didn't recognize the power series so I took it to be the integral of x^[2(2n+1)] and turned into a geometric series, then integrated to get the log version hahaha
May i request you to make more of these all-in-one questions? I find it very amusing to solve and it was incredibly satisfying when i got the question right. This may just be the right tool for me to do brain exercises during leisure times. I love your work very much. I hope you gain an even greater reach on RUclips and make more people understand calculus - or even give birth to a whole new generation of masters. God bless you
There's a way to find exactly (1+(2+(3+(4+..)^1/4)^1/3)^1/2)^1/1 [The sum of n n-roots of n plus the next root]
What bullshit. I wouldn't even bother, I'd transfer to another place or class where, hopefully, memorizing and practicing is enough, and some complex riddle they just made up one night isn't determining pass/fail on the tests. You can tell that someone just constructed this one day like hmm what if I put 3 functions and 3 tricks and combine them just to make it harder (us people not scamming kids call this obfuscation, the art of taking a known direct readable value and making it appear complex or different; usually used for security or by malicious software as a sort of easily breakable, but deterrence strategy). This is ironic that aimless obfuscation is being used to scam students. At least unraveling like real obfuscation might work in engineering, especially reverse engineering, but yet this looks to be just some pass/fail (thus badly designed) riddle which would impact only if someone passes a class, thus, quite bad. "Just do steps 1 thru 9 in that order, but it's way easier for us who wrote the steps than to unravel the steps and ordering yourself " and at the end you aren't even doing it for like a purpose. Again what a beautiful example of how scams work nowdays. He knows to make it look easy just applying known steps in minutes, while the scammed ones have to spend like an hour or more. I would simply pass this or avoid it.
前幾個步驟我有抓出來，不過後續的一些特殊函數值，需要記的，就算不了了
bruh wtf is that
Hey mr Blackpenredpen, can you help me with this integral, it's very hard 😭😭
The integral of: [(e)^(e)^x]/(x^(n+1))
Pls help me my calc prof will check my homeworks the day after tomorrow
🙏🙏🙏🙏
Can you find the integral of sin(e^(-x^2)) from negative infinity to positive infinity??🤔🤔
Very interesting problem, with a lot of concepts rolled into one! The only gripe I have, though, is that this seems to rely very heavily on the student remembering the solutions to past problems. Recognizing the Gaussian integral is pretty reasonable, but would the student be screwed if they didn’t have the Taylor series for the inverse hyperbolic tangent memorized and be able to recognize it…?
congrats for 1M subs , im here since 327 k subs
Aww thank you!!
Why didn't you take the laplace transform of the integral to find the answer, it was much easier....
I forgot 😆
"when calculus test only has one problem"
you cant solve any of 100 questions
I am doing IB math aa hl, will I learn how to do this amazing math, I'm in year 12 ?
Couldn't we just use gamma function for the limit integral?
Gamma Functions of the integral whuch x is aproaching wuld make things easier for those who have done advanced calculus. It's just sqrt(pi)/2...
1k for 1M
Sir please solve this question √9-4√5
oh hell no dude if this was on my calc 1 final i would have just left wtf
oh hell naw
Thank you for promoting interest in mathematics!
2:29 I think that should be a minus sign. It will give you the same answer at the end tho cause you square it
Can show the solution for the integral from 0 to 1 of ((x^2)-1)/(ln(x))? Somehow the answer equals ln(3), but any online source gives the answer in terms of the Exponential Integral, and uses numerical approximation to get a value that visually looks equal to ln(3), but it doesn’t show how to plug in the bounds to get that answer. I get that you can substitute u = ln(x) and dx = (e^u)du and so the expression becomes integral from -infinity to 0 of (e^3u - e^u)/u du. This seems to not be directly solvable in terms of real valued closed form/elementary functions. The question was on our advanced calculus quiz, and somehow the correct answer (multiple choice) was ln(3).
Can you solve, what sin(cos(-sin(-cos(sin(cos(...)))) is approaching to if that sin(cos(-sin(-cos(...)))) part is repeating itself infinitely times? So sin(cos(-sin(-cos(sin(cos(-sin(-cos(sin(cos(-sin(-cos(...)))))))))))).
What am i even watching i'm drunk and i don't think that being sober would help me understand that
LMAO
Congrulations for 1 million subscribers !!! Keep it up !
Only 6k for 1M
Bprp is my favourite channel on RUclips :)
Could you perhaps try to solve lim x -> infinity of x/(tan((pi/2)-pi/x)) in one of your upcoming videos, I think the result will be surprising to you but I wouldn't know how to solve this using classical calculus techniques
Here's how I did it:
The denominator in the limit is tan(pi/2-pi/x)=cot(pi/x)=1/tan(pi/x), by trigonometric identities. Hence, the function inside the limit is x/(1/tan(pi/x))=xtan(pi/x). Our limit is now lim(x->inf)(xtan(pi/x)).
Now, we will introduce a substitution. Let t=pi/x, meaning x=pi/t. As x->inf, t->0+. Our limit becomes lim(t->0+)((pi/t)tan(t)).
Taking the pi out of the limit since it's a constant gives pi*lim(t->0+)(tan(t)/t).
We can rewrite tan(t)/t as (sin(t)/cos(t))/t=(sin(t)/t)*(1/cos(t)), using trigonometric identities. Our limit becomes pi*lim(t->0+)((sin(t)/t)*(1/(cos(t)))=pi*lim(t->0+)(sin(t)/t)*lim(t->0+)(1/cos(t)). We can do this because both of the resulting limits exist.
The first limit, lim(t->0+)(sin(t)/t), is famously equal to 1 and blackpenredpen definitely made a video on it already.
The second limit can be evaluated using direct substitution: 1/cos(0)=1.
Our limit is hence equal to pi*1*1=pi. QED
This challenge is for you!🔥
Solve:
a³ + b² = 1 ;
a² + b³ = -1
Note: The solutions of these equations are real integers.
Fermat's Last Theorem. No thanks.
Trivial solution: (a, b) = (0, -1)
isnt there some stuff from ramanujan to solve this? think i have seen something similar
a = 0
b = -1
Integrats of log2dx?
Answer please
Thank you:-)
hey could you or anyone in the comments show why when you find the area between the two curves y=x⅔ + √(1-x²) and y=x⅔ - √(1-x²) [the two curves which gives a heart shape when you graph them together] the area is = to pi??
The two curves go from -1 to 1, so just using area between two curves, you get
integral(-1, 1) (x^2/3+sqrt(1-x^2)-(x^2/3-sqrt(1-x^2)) dx
the x^2/3 cancels, and you get
integral(-1, 1) (2sqrt(1-x^2)) dx
and you can notice that this is the area of two semicircles with radius 1, so the area would be pi*1^2 = pi.
We need only 1K left to 1M subscribers!
Who was here before the thumbnail changed?
👇
Not enough greens functions
gotta love those ones!!
Congratulations for 1 million subscribers :)
As i can see, d²/dx² of the whole thing inside is just equal to (x^2(2n+1))/(2n+1).
For the limit we know that x approaches √π/2, don't ask why cuz it's too easy. And also the limit is not undefined when x=√π/2 so we just put x=√π/2 and the thing left is the sum series of 1/4*π^(2n+1)/(2n+1)
Well, i think from here u guys can solve this on your own
Calculus exam in a couple days, just what i needed!
Suggestion: differentiate the general term of the series first. Lots of junk gets cancelled.
but why (sqrt(pi))/2 while the gaussian integral is just sqrt(pi)
I like your vids so much
bprp i have found this differential equation that nothing can solve, you know way more than me in maths, could you solve it ?
f(x+1)=f'(x)
Now prove that using limit def of derivative
Bro shut up 😢
Beautiful
this just makes me realize how much math I forgot over the last 25 years
alright then, prove it using the epsilon-delta definition
😆
Imagine this problem in a world without latex or similar software...
please help me, I found an equation that make me exhausted, I'm hardly to solve it =(((. tan(x)*x=1. Some equations as have the same type like this are also equivalent.
Whoooossshhh!!!!! Well, That went right over my head. Maybe next year I can do better.
I guess math is not my cup of tea. Thanks for the video.
take care.
snwer=one
This is why I hate exams from the Math department. They tends to make exam about obscure properties and not so much about what was taught in class like how to do Limit or integration of the infinite series. Like, if you are not deep into the subject and knowledgeable about how some history about this type of setup, you would totally fail this exam. As an engineer, I find this kind of question elitist.
So I’ve just finished real analysis and we didn’t learn about the hyperbolic inverse tangent being that sum. What module would that be taught in?
That integral can be solved by using the gamma function as well....ig it's easier to solve by using it..... great one btw!
Yeah maybe watching Calc 3 videos when I only know Calc 2 was me marginally overestimating my skills
Wat
Solve the Rubik's cube using algebra🤓🤓🤓
Pretty good question for exploding my head but amazing result 👏
freaking nerd
wait I'm taking Calculus in high school
Bro do some other math. Your channel is pretty good but it's just calculus 1 and 2 forever
Make a very complicated question that ends in a funny number :)
If this is your test, I'm glad you weren't my calculus professor in college.
s.v.p je veux la solution de cette équation 8-16y^2=dy/dx et merci beaucoup
lim 1/pow(n, n) = 1
n -> inf
can you tell me why it equal 1
thanks
Bro you are genius.
I am a students of Mathematics from Bangladesh 🇧🇩
What level of calculus is this, is this normal for highschool students to know
P.S Not from the US, so i have no idea
why the hell am i watching this until finish even though i don't understand even a single things in this video (except differentiating 2 u to 2) is beyond me....
im from biology majors and yet i love watching these videos 💀💀
What is that part with Sum from 0 to infinity?
And the function
1 mil 👍
bruh. i still have to look up how to factor when my brain takes a dump. one day il atleast understand what your saying. i hope.
I trust every word the math guy says when he wears a calculus t-shirt and uses 2 pens with the same hand (and has the 'e' number framed on the wall)...
Congratulations 🎉 100 subscribes
why we are using that identity?.... what is that identity about?... is there any source to know something deep about that identity?
Hello can you please show how t integrate x^5/(1+x^7)?
Anyone also notice the boxes of whiteboard markers?