when your calculus test has only one problem

Рет қаралды 270,912

Жыл бұрын

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.creator-spring...
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Пікірлер: 346

@AdasiekkkTrzeci Жыл бұрын

Ah yes, a new episode of blackpenredpenbluepenpurplepengreenpen, my favourite!

@blackpenredpen Жыл бұрын

😂

@ItsPungpond98 Жыл бұрын

New episode of bprpbpppgp

@aranyakpati1079 Жыл бұрын

Find indefinite integral of xsinx/(1+(cosx)^2)

@aranyakpati1079 Жыл бұрын

Please help

@DreadTeamLeader Жыл бұрын

Ahh yes, I remember it going down like this in the anime.

@Peter_1986 Жыл бұрын

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.

@jimjim3979 Жыл бұрын

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

@DokterrDanger Жыл бұрын

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*

@Ninja20704 Жыл бұрын

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.

@ES-qe1nh Жыл бұрын

@@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

@landsgevaer Жыл бұрын

@@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.

@megumiasaoka9562 Жыл бұрын

society youtubebu.com/watch?v=udZddgY5Cea n the question as a one million subscriber special

@cristofer6806 Жыл бұрын

@@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

@andreaspatsios9041 Жыл бұрын

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!!!

@AlerGeekVR Жыл бұрын

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!

@andycavanaugh1219 Жыл бұрын

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.

@humzakhan3962 Жыл бұрын

Try learning logarithms, matrices and algebraic whole square solutions

@extreme4180 Жыл бұрын

@@humzakhan3962 bro just passed 3rd grade and is studying high school maths,, i want dedication like him

@awsomeguy3291 11 ай бұрын

GANBAREEEEEE

@doomsdaycookie7034 10 ай бұрын

@@extreme4180 he didnt just pass 3rd grade, his education didnt go past 3rd grade, read the comment

@redtoxic8701 8 ай бұрын

@@doomsdaycookie7034they were joking lol

@rutcimmusic Жыл бұрын

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

@BnSadiq Жыл бұрын

Seeing this I've really understood the meaning of: Don't eat the whole cake in one turn, a slice by slice is good 🍰

@megumiasaoka9562 Жыл бұрын

a thing, y youtubebu.com/watch?v=qTt8Efn8KoU whole cake in one turn, a slice by slice is goo

@MichaelPennMath Жыл бұрын

Congrats on the 1M subs!! Well deserved!!

@blackpenredpen Жыл бұрын

Thanks!

@poopslappa1661 Жыл бұрын

Hey blackpenredpen! You left an outtake in toward the end. That sigh really got me ):

@stickgamer7261 Жыл бұрын

Congrulations for 1 million subscribers !!! Keep it up !

@november666 Жыл бұрын

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

@rotomflux8416 Жыл бұрын

You should make a playlist about generating function, specail functions and Sturm Liouville Systems

@de_oScar Жыл бұрын

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.

@MessedUpSystem Жыл бұрын

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"

@JordHaj Жыл бұрын

@@MessedUpSystem lmao I just revised Laplace transform and this came by, immediately noticed it's L{sqrt(t)}(1) which is sqrt(pi)/2

@mayelonrajanathan9631 Жыл бұрын

Congratulations on reaching 1 Million Subscribers!

@Fraud_watch Жыл бұрын

Congratulations on 999k subs!

@blackpenredpen Жыл бұрын

Thank you!

@Fraud_watch Жыл бұрын

Aww welcome

@wintc Жыл бұрын

You should do a video on infinite sum renormalization techniques. Been watching some statistical physics lectures and they're always popping up. Very interesting stuff and not at all obvious! (to me at least)

@yaleng4597 Жыл бұрын

(turn over the paper) heart attack

@cleanwater524 Жыл бұрын

Congratulations on 1 million!

@Luftwaffle236 8 ай бұрын

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 KZfaq and make more people understand calculus - or even give birth to a whole new generation of masters. God bless you

@tambuwalmathsclass Жыл бұрын

Congratulations for attaining 1M subs. Keep moving 👍

@Maths_3.1415 Жыл бұрын

Congratulations for 1 million subscribers :)

@MicheleeiRettili Жыл бұрын

gotta love those ones!!

@AdrienLegendre Жыл бұрын

Thank you for promoting interest in mathematics!

@YTBRSosyalEmre Жыл бұрын

CONGRATS FOR 1 MILLION BRO

@leonardobarrera2816 Жыл бұрын

Wow, thanks!!! That video is very, very amazing

@activatewindows7415 Жыл бұрын

1 MILLION SUBS!!!! CONGRATS

@adarah00 Жыл бұрын

Seriously I love you guys 😊

@DrinkmoWater. Жыл бұрын

Lots of formulas kudos to all your successes and videos !!

@yisahak Жыл бұрын

Congratulations 🎉 100 subscribes

@jyotiprakashmondal8111 Жыл бұрын

Can you find the radius of a circle which touches Latus rectum , axis and circumference of the parabola Y²=4aX

@TheJara123 Жыл бұрын

Man congrats a Million!!

@josephb5417 Жыл бұрын

ONLY 1K LEFT UNTIL 1MIL

@atinmankotia48 7 ай бұрын

this was a great problem. loved it

@aymantimjicht173 Ай бұрын

Honestly, your contante is Awesome.

@AJ-et3vf Жыл бұрын

Awesome video! Thank you!

@MessedUpSystem Жыл бұрын

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

@AnakinSkywalker-zq6lm Жыл бұрын

I was able to solve everything but the tanh^-1 (x^2) bc my last course never covered that 😮

@Christian_Martel 8 ай бұрын

I love this stuff!

@VinhWins Жыл бұрын

Your videos have saved my grade during my Differential Equations course! I was wondering if you could do a video on how to solve Boundary Value Problems for 4th Order DEs related to deflection of a beam? Thanks again!

@mydali5573 Жыл бұрын

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...

@sanaya9565 Жыл бұрын

Pretty good question for exploding my head but amazing result 👏

@ronin4923 Жыл бұрын

Calculus exam in a couple days, just what i needed!

@HRHKingAaron Жыл бұрын

When he smiles and scoughs it is so cute

@TonyStark-30001 Жыл бұрын

Congratulations for 1M sir . Edit-Love from India❤️

@user-ni6pc4wj6e Жыл бұрын

Congrats 1 million subscriptions！

@123christiansong Жыл бұрын

1 Million subs. Soon!

@puceno Жыл бұрын

congrats for 1M subs , im here since 327 k subs

@blackpenredpen Жыл бұрын

Aww thank you!!

@jusjerm Жыл бұрын

this just makes me realize how much math I forgot over the last 25 years

@s.a.chord4879 Жыл бұрын

I'm in my final 4 weeks of calc 2. I have never once been taught hyperbolic trig functions so that last part flew right over my head

@Fraud_watch Жыл бұрын

Lesgooooo 1 million!!

@somastien9976 Жыл бұрын

That was so funny. I really enjoy it 🥺❤️‍🔥

@bluemashedpotatoes3924 6 ай бұрын

i havent taken any calculus classes so i dont understand anything at all, but i still like to watch

@shivratanyadav8307 Жыл бұрын

Plz pick up more this type problem mix all math concepts

@janda1258 Жыл бұрын

Nailed it!

@tommasotiberi5666 Жыл бұрын

Appreciating the stock of markers in that shelf 😂😂😂

@joshuaallgood7030 Жыл бұрын

You can technically multiply the 2u with the u and use Feynmann integration.

@michaeldoerr5810 Жыл бұрын

Hello can you please show how t integrate x^5/(1+x^7)?

@kh5163 Жыл бұрын

Thanks ✨

@AMANDALOCAL209 Жыл бұрын

Youmare very good at calculus,pls teach basic of calculus

@pepesob9929 Жыл бұрын

When I saw this thumbnail I immediately remembered my Analysis 1 and Analysis 2 classes, ahhh so nostalgic 😌, it only missed some double integrals here

@theimmortalphysicsmaths3962 Жыл бұрын

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

@neanderthal9621 Жыл бұрын

Zero

@anshugupta793 Жыл бұрын

Sir I've seen some of your videos I'm interested in learning calculus where can I find your calculus lessons by you in the channel

@redirir09 Жыл бұрын

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]

@hhhh82user Жыл бұрын

i like watching these even though i have no idea what is even happening

@karryy01 Жыл бұрын

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

@QuiescentPilot 8 ай бұрын

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…?

@nvapisces7011 Жыл бұрын

I think that you can also express the summation as 2tanh¯¹(x)

@justushinkelmann8020 Жыл бұрын

Given is the function f(x) = -x² + 5. Find the tangent, that crosses the point P(3|10), of that function.

@reyespena6324 Жыл бұрын

Y-10=-2x(x-3)

@stevenfallinge7149 Жыл бұрын

For the last part I didn't know the power series of inverse hyperbolic functions so I took the derivative again to get 2x/(1-x^4) and then integrated to get (1/2)(log(x^2+1)-log(1-x^2)). Probably could have also gotten this just remembering the power series for log(1+x).

@abhradeepdas3889 Жыл бұрын

It's just 1/2Y(1/2) the result of integral. Gamma functions

@elismirzali9862 Жыл бұрын

Can you find the integral of sin(e^(-x^2)) from negative infinity to positive infinity??🤔🤔

@andreas5719 Жыл бұрын

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

@youngmathematician9154 Жыл бұрын

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

@lucidreconalt3229 Жыл бұрын

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??

@yiyoungliu8604 Жыл бұрын

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.

@johnporter7915 8 ай бұрын

I would love to see your reaction of one of your students finishing this problem (the exam) in ten minutes

@user-ui7hk6zm1r Жыл бұрын

I'm curious as to what the inverse function to f(x) = x^(1/x) is. I can't solve it at all

@youngmathematician9154 Жыл бұрын

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.)

@gigantopithecus8254 Жыл бұрын

@@youngmathematician9154 its the best function of all time

@user-ui7hk6zm1r Жыл бұрын

@@youngmathematician9154 Thanks a lot!!!

@Mikel08ll8 Жыл бұрын

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

@audibox2605 6 ай бұрын

That integral can be solved by using the gamma function as well....ig it's easier to solve by using it..... great one btw!

@maroinemaro2787 Жыл бұрын

Hey man i have a question is that really important to understand all math proofs or practical problem like that is better ?

@ericfang Жыл бұрын

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.

@amgamer66 8 ай бұрын

I may be 10 months late but...... Did u understand it ?

@shogakun7395 Жыл бұрын

Girl I'm still just on pre-cal and when he started talking i fking died💀

@joyis9638 Жыл бұрын

So this problem was actually one of the easier ones we had to solve back in the day. Thank you for such a great step by step explanation!

@mayankchaudhary8921 Жыл бұрын

broo what did u do back in the days damn💀💀

@LevelUpGA Жыл бұрын

@@mayankchaudhary8921 probably something with differentials equations 🤣

@SuperYoonHo Жыл бұрын

SinQ/CosQ a 10^6

@samuelhawksworth1923 Жыл бұрын

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?

@bjmiftahul_ulum Жыл бұрын

As long as i watched this video, my brain more hot than my phone

@suzziee 5 ай бұрын

im from biology majors and yet i love watching these videos 💀💀

@andtherefore8076 Жыл бұрын

Now we are 1M family 🎉

@hyj7240 Жыл бұрын

How do you find the indefinite integral of e^sinx and e^cosx?

@hyj7240 Жыл бұрын

@@sybrenvandenakker9064 Oh.. But the indefinite integral of e^tanx exists.

@bobingstern4448 Жыл бұрын

Can you try the integral from -infinity to infinity of e^(-x)^2 * sin(x-t)dx and solve for a function of t? I have no idea of wolfram alpha is able to do this

@youngmathematician9154 Жыл бұрын

Try using the Leibniz rule. It states that if you have an Integral(a to b)(f(x,t))dx, where the integral is with respect to x, you get the following: d/dt(Integral(a to b)(f(x,t))dx)=Integral(a to b)(d/dt(f(x,t)))dx. In other words, if you differentiate the integral with respect to t, you can bring the derivative inside of the integral and differentiate the integrand with respect to t. This trick is also called Feynman's trick thanks to Richard Feynman who popularized it.

@garrettdimaano6253 Жыл бұрын

yeah, of course

@sungod9797 Жыл бұрын

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).