My all-in-one calculus problem
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День томуLearn more calculus on Brilliant: 👉brilliant.org/blackpenredpen/ (now with a 30-day free trial plus 20% off with this link!)
I made this all-in-one style calculus problem for you as an early Christmas gift. We will find the derivative of sin^2(x^2), which requires the chain rule twice, then we need to find a closed form for the infinite power series 1+x^2+x^4/2+x^6/6+..., then we have the limit of sqrt(x)/ln(x) and the limit of ln(x)/sqrt(x) as x goes to infinity. Finally, we will put everyone together and integrate it!
#calculus #math #challenge #blackpenredpen
🛍 Shop my math t-shirt & hoodies: amzn.to/3qBeuw6
0:00 Christmas is coming, so I made an all-in-one calc 2 problem or you
0:20 limit of ln(x)/sqrt(x) as x goes to infinity
1:45 derivative of sin^2(x^2), chain rule twice!
2:57 Power series for 1+x^2+x^4/2+x^6/6+...
4:00 solving the integral
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Thank you all!
@blackpenredpen 6 місяців тому
Learn more calculus on Brilliant: 👉brilliant.org/blackpenredpen/ (now with a 30-day free trial plus 20% off with this link!)
@maxvangulik1988 6 місяців тому
i like how the limits of integration are actual limits
@atripathi6349 6 місяців тому
nothings better than solving an integral on Christmas's
@hanckNCR 6 місяців тому
its christmas?
@anadishrivastava4804 6 місяців тому
Agreed
@michalkrawczak 6 місяців тому
@@hanckNCRit's always Christmas if you have integrals to solve
@Aaron_1112 6 місяців тому
@@michalkrawczak😔
@aninditabasak7694 6 місяців тому
And Christmas also happens to be the birthday of Newton, who invented calculus.
@7yamkr 6 місяців тому
Every scary problem is not necessarily tough & Every tough problem isn't scary😊
@EyeSooGuy 6 місяців тому
😱(lol)
@llawliet7163 6 місяців тому
Only thing scary is his face in the thumbnail 😂😂 but fr tho great video
@AdityaMishra-nd7cq 6 місяців тому
Is this UKpostsr from China if yes then the china is my favorite country 😂
@d3generate804 5 місяців тому
@@AdityaMishra-nd7cq he is a Taiwanese living in america
@lunam7249 Місяць тому
chuck norris says ..."hold my beer"
@trelosyiaellinika 6 місяців тому
I've graduated from a mathematical school and even went to Mathematics faculty at the university for a year before changing my mind and becoming a general surgeon... It was a very tough decision as there was no scientific material that didn't interest me at the time... But maths has always remained my love and mania and I've always benefited from the knowledge while creating various complex macros for my work... However, I had almost forgotten most of its juicy parts... It's been more than 36 years after all! Now, I am retired and very much enjoy your videos, remembering and solving them in parallel... It charges my batteries and gives me a sense of satisfaction like winning a chess match! Thank you very much! You are doing a great job!
@blackpenredpen 6 місяців тому
Thank you so much for the comment!
@andripula8986 6 місяців тому
to end with a repeating integral, brilliant problem!
@loulephille 6 місяців тому
Imagine checking your socks at early morning and finding a paper with this integral written and a message from Santa : "Integrate the above to receive gift"
@mchanc 15 днів тому
well worry no longer my friend.
@juxx9628 6 місяців тому
Ok. Trying first before seeing the video. Step 1: Evaluate limits. On the bottom one, use L'Hopital rule and get (1/x)/(1/2√x). Simplify and get 0. The top one use L'Hopital rule to get (1/2√x)/(1/x). Simplify and it diverges. Step 2: Derivative. Just use the chain rule twice. f(y)= y² y(t)= sint t(x)= t² df/dx = df/dy • dy/dt • dt/dx = 2y • cost • 2t Recall the definitions of the variables: 2•2x•sinx•cosx Step 3: Power series. Recall the Maclaurin series for e^x, then put x² as the input. That easy. e^x². Step 4: The monster. The integral looks like 0-inf∫ 2•2x•sinx•cosx• e^-x² dx. Use substitution j=x², dj=2xdx (bounds of integration stays the same and we already have dj in the integral) =0-inf∫ 2•sinx•cosx•e^-j dj Recall doble angle formula for sinx and name the integral I: 0-inf∫ sin(2j)•e^-j dj = I Use IBP or DI method, just the same: D: + sin(2j) - 2cos(2j) + -4sin(2j) I: e^-j -e^-j e^-j After the setup, this ends like: I = (sin(2j)•e^-j)]0-inf + (2cos(2j)•e^-j)]inf-0 - 4I Notice that first term goes to 0 and in the second term I changed the bounds thanks to the minus sign. Now, in the second term, the limit as j approaches 0 is 2 and when j approaches infinity is just 0 thanks to the exponential and the squeeze theorem. So, finally: I = 2 - 4I 5I = 2 I = 2/5 Thanks for reading, love you.
@samueljehanno 5 місяців тому
Interesting
@cemsaglam9241 5 місяців тому
i've just realized by reading your comment that IBP is short for "integration by parts"
@juxx9628 5 місяців тому
@@cemsaglam9241 Yeah, it's a confusing way to write it. I first got confused because in spanish it is just simply despicted as integration by parts or "the cow" (la vaca) because of some mnemotecnic to remember IBP.
@PRIYANSH_SUTHAR 6 місяців тому
This guy can intimidate you with full innocence.
@sergeygaevoy6422 6 місяців тому
And it is a Laplace transform in the end.
@dinokiller9186 6 місяців тому
The numerator was easy but I couldn't guess the denominator part 👍👍
@MokshitArora. 6 місяців тому
That e^x² at the denominator was great . I was thinking it to be some different series and was thinking to use limit as a sum (converting an infinite sum to definite integral)
@M7RAA 5 місяців тому
how did he get that though?
@MokshitArora. 5 місяців тому
@@M7RAA use tailor series expansion on e^x you will get the series or if you know series of sine and cosine then also you can get that After that replace x with x² and you will get the mentioned series We can reverse it also by finding function with series by writing it as a limit on summation and then converting into Reimann sums then integrating
@cheerio662 6 місяців тому
Been watching you for 2-3 years now as a highschool student and could finally solve on of your all-in-one questions by myself! Feels great to go from knowing nothing and just liking the magic numbers to solving something that looks scary (but really wasnt) all by my lonesome. Thank you for the content you provide!
@tambuwalmathsclass 6 місяців тому
Wow, incredible. 💪 But isn't the final answer supposed to be -2/5 ?
@ABHIGAMING-yo9my 5 місяців тому
Bro function is always positive so answer should be positive
@joshhh___ 5 місяців тому
@@ABHIGAMING-yo9myThe function f(x) = sin(2x)e^(-x) is not always positive on [0, inf), but ∫₀^∞ f(x)dx is still equal to 2/5.
@valentinvanhees8690 6 місяців тому
i really liked this!! my first really hard integral that i solved first try! would love to see more power series-integrals
@aimgaming4744 6 місяців тому
Love these kind of questions, keep it up!
@o_s-24 6 місяців тому
All of calculus 2 summarized in 11mins. Awsome!
@xum0007 6 місяців тому
I’m only a freshman so I’m taking algebra 2 honors right now. I must say this looks way harder than what I do in class right now (which is a pretty low standard) but if you’re interested in the subject it shouldn’t be too bad.
@matheusdossantos9252 5 місяців тому
@@xum0007Algebra II also called "Linear Algebra"? After the diagonalization content it can get a little more complicated depending on your teacher.
@michellekagansbu 13 днів тому
@@matheusdossantos9252 I don't think he means linear algebra
@hsod0 5 місяців тому
You are really awesome!!! Actually, thank you for what you are doing, I'm into mathematics even more because of your videos and I'm really having fun watching them. Please, keep it up. These videos really make my day
@pedri_meet 6 місяців тому
That was great!! It's like quick revision
@pekorasfuturehusband 5 місяців тому
I’ve been wanting another all in one problem for a while now, thanks for the early present!
@jonny8448 6 місяців тому
Thanks professor!!! Christmas is coming and I have to find a crazy Christmas problem for my channel!!!🎄🧑🎄🤶 PS. Not as crazy as yours!!! I wouldn't be able to come up with something like this!!!🤩🤗
@armanavagyan1876 6 місяців тому
Thanks PROF 👍
@phillipalter6499 6 місяців тому
My calc professor will love this, thanks
@softllamaspajamas 6 місяців тому
What a thrilling problem! I’ll give it a go myself closer to Christmas!
@TypoKnig 6 місяців тому
Merry Calcu-mas!
@PhysicalScienceInSinhala 6 місяців тому
It's amazing 😃❤️
@scottleung9587 6 місяців тому
Yay - the answer is 2/5 for the 25th (of December)!
@yoniziv 6 місяців тому
Loved it
@Peter_1986 6 місяців тому
I once saw an integral that had integrals as limits of integration, lol.
@TsukkiSenpai727 6 місяців тому
So what’s the answer to 1/5 + 1/5 ? BlackPenRedPen: sooo actually
@igorhaladjian5718 5 місяців тому
Love the Christmas T-shirt !
@stapler942 6 місяців тому
"Two limits, a derivative, a power series, and an integral wander onto a board..."
@Jadamhodges 6 місяців тому
Wonderful!!!😊
@myththelegendtyson 6 місяців тому
We should have an advent of integration. Each day a new integral problem
@joen_enjoyer 6 місяців тому
ty much appreciated
@MichaelZankel 6 місяців тому
It’s not Christmas without integration!
@nickfleiwer5272 6 місяців тому
Thanks a lot for this years Christmas present 😂😂😂 but I might return it later haha
@hidden_leafy 6 місяців тому
Best Christmas gift I've ever received lol
@diptenduchatterjee950 5 місяців тому
Yess!! Done in the first attempt. Good question
@AlejandroGD17 5 місяців тому
First time I'm actually able to solve one of these!!
@nikko2505 6 місяців тому
This is simply Laplace Transform
@aubertducharmont 6 місяців тому
When you got to the final form of the integral, I would just use contour integration to get the answer. I dont like doing that much integration by parts. And also that series in the numerator arent necesserily described by the e to -x squared formula. As you wrote only a finite number of parts, in this case four, there is an infitnite amount of formulas for these four parts of the series. One could pick that after x^2/6 would come 69 and find a formula for this, with use of the Gregory-Newton formula.
@istvanszabo3275 5 місяців тому
Merry Christmas 2u 😃
@cristofer6806 6 місяців тому
do you have any plans on doing calc 3 stuff, would love to see more of that
@mauriziomorales5303 10 днів тому
Qué EJERCIZASO!!!! I LIKE IT, THANK YOU!!!!!
@DC_EDITS 5 місяців тому
Great christmas present
@herbie_the_hillbillie_goat 5 місяців тому
Tis the season.
@thebeardman7533 6 місяців тому
It is to early for I still have calc lectures but when Christmas comes be assured that I will solve it
@AlumniQuad 6 місяців тому
IT'S A CHRISTMAS MIRACLE!
@fwelling2703 5 місяців тому
gonna come back to this video in a year to see if I understand yet.
@catss2949 4 місяці тому
done!
@akgamer4215 6 місяців тому
Solve this without denominator
@TomMarAlem1987 6 місяців тому
My boy's giving us a surprise in the denominator.
@gaariwala 5 місяців тому
Sir do a Fourier transform of e power x
@coyotestarrk2632 6 місяців тому
Thank you so much for this BRILLIANT vid and explanation!!
@longlong10203 5 місяців тому
i thought you are gonna talk about the Gaussian Integral when i saw e^x^2, it's almost, phew
@stevencarr4002 6 місяців тому
To get the limit why not put u = ln(x), then we have e^0.5u in the denominator and u in the numerator as u goes to infinity. This is obviously zero.
@xwf1335 6 місяців тому
Nice bro
@hotlatte1222 6 місяців тому
Great work!! But i think it is more likely for Halloween, not Christmas.
@blackpenredpen 6 місяців тому
lol, it should really be for Thanksgiving since it's just next week! haha
@hotlatte1222 5 місяців тому
@@blackpenredpen Maybe this question fits all 3 festivals. When seeing it in the beginning, it is so horrible for Halloween. When solving it, it is like the games of finding eggs in Thanksgiving. And finally you reveal the solution with clear steps; which is just a Christmas gift. So cool.
@Passersby98 5 місяців тому
I'm expecting that Mr Tsao could demonstrate how to solve ODE
@user-pm1kf9ko4v 6 місяців тому
Since it's my bday, i'll take this as my bday gift
@Ashaiksameer 5 місяців тому
Can u make a roadmap of mathematics and concepts in it😢
@atishthatei8842 6 місяців тому
make me fun as i do in cristmas . thanks bro . but quite a easy one
@yencheonglee5940 Місяць тому
This question is simple. The limits can be found easily, next I replace t=x^2 and come out with \int e^{-t}sin(2t) dt, and then I solve lim_{s -> 1} Laplace transform of sin(2t) by subtracting s=1 in the result.
@Siddhartha.Chatterjee 6 місяців тому
I have not watched it yet... But please tell me it's 2/5 Edit: Ok, I messed up somewhere at plugging infinity at the last part (for some reason I forgot that even with infinity, the sin & cos function would be finite, and applied L'Hopital, somehow ended up having I=-4I, allowing me to say I=0 at x->infinity), but anyways the answer still ended up the same....
@namename7000 6 місяців тому
Hello, how to solve factorial equations like this: 3x!-x^x-2=0 do you have a video about this?
@richardfredlund8846 6 місяців тому
0,1,2 are trivial solutions, but for different numbers that looks really hard... interesting looking problem type.
@migueldomingos4570 5 місяців тому
If x's domain is positive integers: You can just do some bounding. Rearrange to 3x! = x^x + 2 and notice that the RHS grows much faster than the LHS, to formalize it you can prove by induction that for x>= 3 x^x > 3x! and thus all solutions will be smaller than 3 and you can easily check that 0,1 and 2 works as richard stated
@MichaelZankel 6 місяців тому
Isn’t it -2/5?? Because it was (-sin2u + 2cos2u )/(5e^u), so (-) ALL of that is (-2*1)/5 at the end!! No?
@saadansari1757 6 місяців тому
Even I think the same
@MichaelZankel 6 місяців тому
@@saadansari1757yeah, Idk why he didn’t put a (-) on the cos at the end.
@Anmol_Sinha 6 місяців тому
It is actually -(sin2u + 2cos2u)/(5e^u) , here -ve is in the outside. During the application or the upper and lower limit of integral, we got -(-(2/5)). I don't think in any part of the video it showed the -ve only on sin(as your comment suggests)
@Anmol_Sinha 6 місяців тому
@@MichaelZankelthe minus never got distributed in the expression. Look at the brackets carefully
@saadansari1757 6 місяців тому
@@Anmol_Sinha okay thanks
@CrushOfSiel 6 місяців тому
Ah damn, I was close. Been a while since I did calculus. I got the limits and the numerator right but I thought the denominator was cos(x) and then I was stuck, it is similar.
@mickelsantiagoquispenamuch4961 6 місяців тому
Happy X-mass
@natrok 6 місяців тому
Bro just made calculus final boss 💀💀
@user-kh3mo5dn4y 6 місяців тому
nice one
@jakehu 6 місяців тому
The kid who just guesses 2/5😂
@brucekritt7036 Місяць тому
Strange.. The answer I'm getting is -(2/5). Based on (d/du)[e^(-u)*(sin(2u)+2*cos(2u))] = -5*e^(-u)*sin(2u). I checked that derivative carefully.
@anticlashers2617 6 місяців тому
I likes your videos ❤. Love from india🇮🇳
@evansaschow 3 місяці тому
I hate doing IBP, so I’d much rather decompose sin(2u) into its exponential form
@omerzaferdundar7586 6 місяців тому
the answer is -2/5 10:39 you mismultiplied - and - (the second - is just for sin0 which is 0)
@dylanogden9337 5 місяців тому
I would like to try this before watching, but I don't understand the series in the denominator. Could you provide the next two terms, please?
@DravenFNM 3 місяці тому
i think its -2/5, you overlooked the last fraction
@knowledge90s93 Місяць тому
Which of the following sequences could represent the impulse response of a stable discrete-time system? k^2 (-0.65)^k 2^k ksin(k)
@jbucata 6 місяців тому
Instead of 2/5, for Christmas you should have made the answer be 12/25
@user-bm6xz6pq5z 6 місяців тому
SLOW DOWN ONE HOLIDAY at a time! We haven't even made it past Thanksgiving yet!
@blackpenredpen 6 місяців тому
My bad 😆
@user-xd2dj1qt2e Місяць тому
we can solve it by gama function
@doug2855 5 місяців тому
Can you explain the math behind cos, sin , tan etc. Like how did cos(45°)=1/sqrt(2).
@doug2855 5 місяців тому
More generically how would you hand calculate the value of cos(x). X being a random value
@aryanjoshi3342 5 місяців тому
@@doug2855 power series
@rufusmafija8674 6 місяців тому
hey there i have an incredibly hard question for you: try to find the integral of sqrt(3x²+x) do you know to solve that?
@umertaiyab5500 6 місяців тому
i wanted to know how does trigonometric substitution work when you substitute sinx or cosx as they can only have the value from -1 to 1.
@conanedojawa4538 6 місяців тому
i think that the limit of sinx /e^x when x goes to infinity the sine function goes to a finite value 1 or -1 but e^x goes to infinity then the limit will be zero but I don't know it will be 0 plus or 0 minus
@A_Random_Ghost 6 місяців тому
If you're talking about the final limit. When you have a bounded numerator and a denominator that goes to infinity. You can just conclude the limit goes to zero. And the reverse goes to infinity.
@A_Random_Ghost 6 місяців тому
@@abcd-ug8tj Yeah, I forgot that was a thing 😅.
@aimlessideas1165 6 місяців тому
2/5 for the 25th👀
@pritamsur1926 6 місяців тому
Please solve this integration.. integral of (32-x^5)^(1/5)🙂
@TozzaYT 6 місяців тому
u sub?
@pritamsur1926 5 місяців тому
@@TozzaYT mathematics🙂
@Integralsouls Місяць тому
N o i c e
@user-yx4yi3wv3s 6 місяців тому
Hey blackpenredpen is there in the complex numbers a function thats inverse equals it's derivative? Thank you
@samueljehanno 5 місяців тому
Interesting
@EmpyreanLightASMR 5 місяців тому
You should have ended the video by saying the answer is that Christmas is on December 2/5! This was a blast!
@skltfz4997 5 місяців тому
a journey for warrior
@hearteyedgirl 5 місяців тому
Although I do know 1=0!,1=1!, 2=2!,6=3! and know the intention of the question but the intention itself remains ambiguous There's no way to know if the series really is x^n/n!
@darcash1738 6 місяців тому
Very nice. Now let’s see Paul Allen’s integral… Nah I’m just joking Paul Allen couldn’t top this one 😂
@JSSTyger 6 місяців тому
Oh what a wonderful Christmas gift!! (Murmur murmur...)
@PaawanS 6 місяців тому
When evaluating the numberator for u=inf, you say it's finite so its precise value doesn't matter. However, how do you account for the fact that sin(2u)+2cos(2u) can sometimes equal 0? Why is it okay to assume it's non-zero in the limit?
@carultch 6 місяців тому
Sine and cosine are both functions of exponential order. This means that an exponential decay function as its input goes to infinity, will shrink to zero either faster than these functions, or as fast as these functions. This is one of the criteria for a Laplace transform to exist, is that the function has to be of exponential order, which is why sine and cosine have Laplace transforms, but secant and tangent do not.
@mathmachine4266 6 місяців тому
Your thumbnail makes it look like you're being held against your will
@jakehu8733 6 місяців тому
I calculated -2 on my first try.
@mainsera4407 6 місяців тому
I was close except for the power series because I started at 1 instead of zero, which is where I got lost. I got x^(2n-2)/(n-1)! for the series, starting at 1 which fits. Does anyone know if you could still solve it this was this series or does a power series have to start at 0? (Power series is my weakest topic I don’t understand them well)
@cormalan9894 5 місяців тому
Where does that sum definition of e^x^2 come from?
@adamburt7200 5 місяців тому
Hi, i was wondering if: lim n->∞ ((n^2)/(x^n))=1 has any solutions for x. And if not, is there any value, this could be equal to, so that it would have a solution? Im still in highschool and dont know how would i solve it. Love your channel.
@oxydoreduction2483 5 місяців тому
It doesn’t have any solutions. If x1, the limit is 0 (because n^2=o(x^n) if x>1).So no solutions (you could have found this by yourself honestly…)
blackpenredpen
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blackpenredpen
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