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@cicik572 жыл бұрын
my i ask, why following is not working: if f(2x) = 3f(x) then integral from a to b (f2x) = F(2b)- F(2a) = 3(F(b) - F(a)) so integral from 0 to 2 f(x) = F(2) - F(0) = 3(F(1) - F(0))=3*1 = 3 something must be wrong in this allroach, can you point to it, pleace?
@ivandebiasi66572 жыл бұрын
The integral from a to b of f(2x) is not F(2b) - F(2a). Try just with f(x) = 1: the integral is b-a but F(2b) - F(2a) is 2b-2a = 2(b-a). The correct formula is 1/2 (F(2b) - F(2a)). Also f(2x) = 3f(x) not F(2x) = 3F(x)...
@emmanuelbalogun34042 жыл бұрын
This looks wrong. You stated that u = 2x, so how can you then leapfrog into stating that u is equivalent to x? not convinced
@samuelmoss24802 жыл бұрын
there’s actually an easier solution: just integrate the identity they give you from 0 to 1 and substitute to find an expression for integral from 0 to 2 then subtract
@hmiscoordinator27512 жыл бұрын
Apologies for horrible shorthand, using my phone. Int(f,x,1:2) = Int(f,x,0:2) - Int(f,x,0:1) = Int(f,2x,0:1) - Int(f,x,0:1) = 3*Int(f,x,0:1) - Int(f,x,0:1) = 3*1 -1 = 2. #
@drpeyam2 жыл бұрын
Your videos are an integral part of my day 😁
@chsh96862 жыл бұрын
🤣🤣
@naturelover41482 жыл бұрын
Hello Sir!
@muratkaradag37032 жыл бұрын
Yours and BlackPenRedPen, I love to watch at the end of the day 🤩🥰
@bryantwiltrout54922 жыл бұрын
I see what you did there sir
@amirmahdypayrovi93162 жыл бұрын
@@muratkaradag3703 also me♥
@Macieks3002 жыл бұрын
the first solution is like if someone forgot that 1+1=2 and instead used the equation 1+1/2+1/4+...=2
@blackpenredpen2 жыл бұрын
😂
@RB-ew6lo2 жыл бұрын
perfect summary :-)
@MrMusic551232 жыл бұрын
too easy for Harvard-MIT
@siobhangraham72802 жыл бұрын
1+1 may equal 2, but f(1)+f(1) doesn't necessarily equal 2f(1) :P
@devyanshatharv18612 жыл бұрын
@@siobhangraham7280 it doesn't?
@jessehammer1232 жыл бұрын
Wow, both solutions are so neat! I appreciate how yours doesn’t require anything infinite.
@xinpingdonohoe39782 жыл бұрын
Except the geometric series.
@lih33912 жыл бұрын
@@xinpingdonohoe3978 what do you mean?
@xinpingdonohoe39782 жыл бұрын
@@lih3391 it says "doesn't require anything infinite" but there's a geometric series, which is an infinite series.
@gamerdio25032 жыл бұрын
@@xinpingdonohoe3978 His solution doesn't use a geometric series though
@stephenbeck72222 жыл бұрын
Xinping Donohoe watch to the end of the video, he has an alternate solution.
@vladimirkhazinski37252 жыл бұрын
When I first saw the question, I thought it was ridiculously easy. When I saw the official solution I thought it was ridiculously hard lol
@blackpenredpen2 жыл бұрын
And then ridiculously easy again, right? 😆
@saruarahemadrakib51872 жыл бұрын
@@blackpenredpen 😏
@nanamacapagal83422 жыл бұрын
@@roxannemackinnon2213 it turns out the function IS cx^r (it's (log_2(3)+1)x^(log_2(3)))
@Willtedwards5 ай бұрын
Did you know how to solve it when you first thought it was easy, or just assume it was easier to solve?
@Atlas_Enderium2 жыл бұрын
The official solution was so needlessly complicated 💀 substitution is such a great tool
@vitalsbat23102 жыл бұрын
skull substitution
@miantony64932 жыл бұрын
kzbin.info/www/bejne/sGOyqol_f9-hftk This question is super difficult
@btwtempest2 жыл бұрын
your solution made this look ridiculously easy
@noahtaul2 жыл бұрын
To solve the question at the end: move the 1 to the other side and multiply by e^x. You can pull the whole thing together to get (e^x*f(x)-e^x)’
@omaromy87222 жыл бұрын
That is brilliant 👏 👌
@DRoo952 жыл бұрын
Brilliant solution, I found a more extensive/less neat way to prove it (first proving that f(x) + f'(x) = 1 is the optimal solution, then solving the ODE), but I found the same answer: Question: how did you go from e^x*f(x) + e^x*f'(x) - e^x
@omaromy87222 жыл бұрын
@@DRoo95 He didn't go anywhere, it's still the same, it's just written in the derivative form
@DRoo952 жыл бұрын
@@omaromy8722 I must say I haven't heard the derivative form in my uni years. Enlighten me. Why are we allowed to remove the e^x * f'(x) term by taking the derivative over the remaining terms? EDIT: Ah never mind. Reverse multiplication rule. Whoops 😂 brilliant way of finding it
@manizarin22332 жыл бұрын
wow I solved it with ODEs but this one is way to better and more elegant👌👌
@kobethebeefinmathworld9532 жыл бұрын
I think there's a faster way to do the integral: by FToC, I = I_1 - I_2 where I_1 is from 0 to 2 and I_2 is from 0 to 1 which is already given to be 1, then for I_1 we can do u-sub by letting u = x/2 to change the bounds into 0 to 1 and the expression inside of the integral becomes f(2u) * 2du to apply the given formula f(2x) = 3f(x) and obtain f(2u)* 2du = 6 f(u) du. So it yields to I_1 = 6*1 = 6 and we get the answer I = 6 - 1 = 5
@kobethebeefinmathworld9532 жыл бұрын
Oh, never mind. I didn't finish the video and just realized you used the same method.
@walaaaaaa34562 жыл бұрын
@@kobethebeefinmathworld953 xD
@learningmathswithconcepts72032 жыл бұрын
Yeah exactly,i did same
@sushrut43282 жыл бұрын
I did the same, and this way is so easy. Why to complicate things? duh!
@hongshengzhu44112 жыл бұрын
I use the same method as yours :D
@UberHummus2 жыл бұрын
Your insight is always phenomenal
@opstall2 жыл бұрын
I'm terrible at all contest problems (for a math professor, anyway), and this one took me under a minute. Thanks for sharing.
@blackpenredpen2 жыл бұрын
I was actually very confused at first when I saw their official solution. But then after I decided to try it on my own, I solved in within minutes.
@davidebic2 жыл бұрын
The first method was mind-boggling but uselessly complicated.
@3ckitani2 жыл бұрын
Harvard MIT do be flexing their integral skill
@dbro12052 жыл бұрын
Agreed. It was just fancy steps
@georgefan29772 жыл бұрын
Basically a race of whoever comes up with a more uselessly sophisticated solution
@bilalmalik50022 жыл бұрын
@@georgefan2977 For a uselessly complicated question
@gabrielroberto58192 жыл бұрын
when he was doing the first method I thought he would do it in an easier way, I mean, after he made the integral from 1/2 to 1 he could do the integral from 0 to 1/2 and it wold be enough to draw a conclusion by making an equation, but it wanted to be unnecessarily difficult, seem that MIT tried to use all calculus features kkkkkkkkkkk
@Re-lx1md2 жыл бұрын
The MIT Tournament problems you gave at 0:18 are proving to be fun! Problem 3 has me stumped currently
@Notthatkindofdr2 жыл бұрын
In Problem 3 I think you first need to solve for f'(0), g'(0), and h'(0). But Problem 7 is really bizarre. It does not even make sense the way it is written (since A(M) is a random variable and cannot just have a "limit value"), but even when I interpret the problem to mean what I think they intended, it is not a calculus problem at all (more of a number theory problem). 🤷♂
@sinecurve99992 жыл бұрын
Consider integral[f(x), {x, 0, 2}]. We can make the substitution 2u = x transforming our integral into integral[2*f(2*u), {u, 0, 1}]. Applying the rule f(2x) = 3f(x), our integral becomes integral[2*3*f(u), {u, 0, 1}] = 6. integral[f(x), {x, 1, 2}] = integral[f(x), {x, 0, 2}] - integral[f(x), {x, 0, 1}] = 6 - 1 = 5.
@wowZhenek2 жыл бұрын
Man, I feel proud of myself for literally instantly figuring out the solution, which ended up similar to your approach
@blackpenredpen2 жыл бұрын
👍
@cosmicgxming88082 жыл бұрын
I'm currently taking AP Calculus BC in high school and I understood everything up to 2:41 but I lost track about everything after that lmao. The hard part about this type of math isn't necessarily the execution process itself but about connecting all the information in the question to formulate all the steps needed to arrive at a conclusion.
@adamschroeder39462 жыл бұрын
Man said in the description "good challenge for calc 1 students"... umm nah calc 2 maybe? (im in BC too lol)
@willo13452 жыл бұрын
Yes, takes a lot of practice. There are a lot of rules to remember and you often make little mistakes that screw you in the end. Practice is the only way to really get it because you won't know how to solve a problem until you do it.
@martinrosol77197 ай бұрын
@@willo1345Actually, you should understand the rules, not "remember" them.
@willo13457 ай бұрын
@@martinrosol7719 You do not have time to derive everything from scratch on a test and, even if you did, that increases the chances of human error. So yes. You need to remember the rules.
@xninja23696 ай бұрын
Bro this wasn't even hard to understand , specially limit part was easiest some type of limit can't be solved by L'HÔpitals rule or infinity GP or anything and you need to solve it and it takes 2-3 pages to solve. ..
@andrewkarsten52682 жыл бұрын
I immediately did the first step of the first method, and realized it would repeat, but it gave me the idea for the second method, so I did the second method instead. That was much simpler in my opinion
@ankitbasera84702 жыл бұрын
This is how I did it f(2x) = 3 f(x) Integrating both sides from x=0 to x=1 Integration 0 to 1 of f(2x) = 3 Put 2x = t I get Integration 0 to 2 of f(t) = 6 Divide the range of integration from 0-1 and then 1-2 Integration 1 to 2 f(t) = 5
@KewlWIS6 ай бұрын
this is also cool
@IdeesDePhysique2 жыл бұрын
The solution from the MIT was so convoluted. I naturally approached the problem with your solution in 1 min. lol.
@alexdotdash77312 жыл бұрын
Such a great teacher! Amazing video🔥
@miantony64932 жыл бұрын
kzbin.info/www/bejne/sGOyqol_f9-hftk
@ayaanpatel96672 жыл бұрын
again an amazing video from you
@soundest87682 жыл бұрын
That’s ridiculously satisfying. Thanks for the step-by-step.
@VinTheFox2 жыл бұрын
When I plotted the curve on a graph to visualize the problem, I intuitively came up with the first method as the way to solve it. That might be the approach they took as well.
@byronvega82982 жыл бұрын
I did it waaaay differently. Take the integral from 1 to 2 of f(x) and set it equal to y Then add 1 on both sides but on the LHS as the integral from 0 to 1 of f(x) Then make the substitution x->2v Now use the identity 3f(v)=f(2v) Resulting in six times the integral of f(v) from 0 to 1 which implies 6 = y+1
@titiyop72862 жыл бұрын
It's the same thing that he did but the other way around
@nqnqnq2 жыл бұрын
the second method (yours) is so neat and simple!
@KurtColville2 жыл бұрын
Your solution is SO much better. Well done!
@wavingbuddy35352 жыл бұрын
Everyones saying the official solution was really complicated, but it wasn't really, it was mathematically beautiful and utilizes A level/ basic year 1 university algebra.
@adnanhussain99062 жыл бұрын
Maybe not necessarily “complicated” but people say it is because of the other more efficient and easier way to answer the question.
@omerbar75182 жыл бұрын
When I saw your solution I just flopped in my chair. I love how you take solutions to MIT test and be like: "yeah but I can do it 14 times better". Keep the good content up! (although I'm 4 months late lol)
@dabmanplz14512 жыл бұрын
Nobody seems to calculate f(x) and is the simplest way for me. Knowing that f(2x)=2x for the usual f(x)=x, and f(2x)=4x for the usual f(x)=x^2, then our f(x) must be something like f(x)=x^k, with k between 1 and 2. Just substituting in the condition of the problem gives us that f(x) = x^log2(3). Then you just need to do the integrals. With the first you realize our function actually needs a constant: f(x) = a * x^log2(3), with a = log2(3)+1, because that's the value that makes the integral 1. And them you just do the second integral, and its 5. I believe this method may be a little longer that the fastest one proposed, but you need to know practically nothing about integrals to solve it: no changes of variable, no nothing. And also you learn what is happening with f.
@abhimanyus16702 жыл бұрын
You don't get a calculator. But that's a method for sure
@PotmosHetoimos2 жыл бұрын
Thank you, I was wondering what f was!
@MarkMcDaniel2 жыл бұрын
You're simplified technique to solve this was great!
@ideatronzvvvicunit947610 ай бұрын
Both solutions are realy great . What is wonderful in the first solution is the idea, the approach using infinite sum series
@adamtedd2000er2 жыл бұрын
Since f is Riemann integrable on (0,2), it is continuous a.e. on (0,2) and therefore it can be shown the function agrees a.e. with the formula: f(x) = x^( log_2(3) ) * (log_2(3)+1). Interesting to note you can find an explicit formula from the vague assumptions about f(x).
@artemisnimrod29489 ай бұрын
My solution: start calculating points on the line that satisfies f(2x)=3f(x) starting with (1,1). (1,1), (2,3), (3,9), (4,27), (5,81). The pattern is pretty obvious. It's just (2^n,3^n). So, the equation f(x)=3^[log{2}(x)] goes through these points. The integral of this function from 0 to 1 is less than 1, though, so we need to add a scalar, "a." a*int{0 to 1}(3^[log{2}(x)])dx ==> a = log(6)/log(2). So, [log(6)/log(2)]*int{0 to 1}(3^[log{2}(x)])dx =1. Change the bounds of integration: [log(6)/log(2)]*int{1 to 2}(3^[log{2}(x)])dx = 5. I have no idea if this is a sound way of doing this, but it's how I got the right answer.
@furiousfajitaa23672 жыл бұрын
This restored my passion for maths
@laprankster32642 жыл бұрын
I literally just guessed f(x)=cx^(n) where (c,n) are real numbers. I solved for n to get log2(3). And just integrated to get cx^(log2(6))/log2(6). I then plugged in the F(1)-F(0) to find c=log2(6) (yay for cancellation). I finally plugged in F(2)-F(1) to get 5 as the final answer.
@hemantbhosale30602 жыл бұрын
We can substitute f(x)=f(2x)/3 In given integral which gives integral from 0 to 2 is equal to 6 but we know integral 0 to 2 is integral 0 to 1 plus integral 1 to 2 we know integral 0 to 1 is 1 and integral 0 to 2 is 6 therefore integral 1 to 2 is 5
@sajedtabbabi2 жыл бұрын
Some day I'll be a mathematician
@ΑνναΜαρια-ψ7σ2 жыл бұрын
idk if its right but F’(x)=f(x) so in this case we have F(2x)/2=3F(x) . The integral from 0 to 1 for f(x) is F(1) -F(0) and the one from 1 to 2 is F(2)-F(1) . By putting 1 in the first you get that F(2)=6F(1) so the second integral is 5F(1) . Also by putting 0 you get that F(0) =0 so from the first integral F(1)= 1 so 5F(1)= 5
@kostasl1808 Жыл бұрын
Είσαι ο μοναδική στα σχόλια που έχει ακριβώς την ίδια λύση με τη δική μου και η μοναδική που δεν το έλυσε μέσω Αλάσκας. Ή εμάς στα σχολεία μας τα κάνουν πολύ απλά ή σε όλους τους άλλους πολύ περίπλοκα.
@F_A_F12310 ай бұрын
F(2x)/2 = F(x) is generally wrong. Because if you increase the F by a constant, the thing you get still is the antiderivative of f(x), but it doesn't satisfy F(2x)/2 = 3F(x). (F(2x) + C)/2 = 3(F(x) + C) F(2x)/2 + C/2 = 3F(x) + 3C C/2 = 3C Which is wrong for C ≠ 0
@zerglingsking2 жыл бұрын
That first method was pretty cool! I did something more similar to your answer: I take integral of f(2x) between 0 and 1, which is equal to 3 times the integral of f(x) between 0 and 1. But it is also 1/2 * integral of f(x) between 0 and 2. So we get 1/2 * ( integral of f(x) between 0 and 2) = 3 * (integral of f(x) between 0 and 1) The right side is equal to 3, and the integral between 0 and 2 of f(x) can be separated between 0 and 1, and 1 and 2. The integral betwene 0 and 1 is known so we have 0.5 * (integral of f(x) between 1 and 2 + 1) = 3 which gives integral of f(x) between 1 and 2 = 2 * ( 3 - 1/2) = 5!
@JBOboe7202 жыл бұрын
f(x)=log_2(6)x^log_2(3) totally works despite x
@eja17312 жыл бұрын
When you realize he’s holding a Pokeball the whole time
@stefanopetrone9900 Жыл бұрын
Using the identity f(2x)=3f(x) and some logic i found the function f(x) = c * x ^ (ln(3)/ln(2)) Found c with known integral and finally calculate the other integral
@TheDannyAwesome2 жыл бұрын
My initial reaction was that f(2x)=3f(x) looks like a functional equation for some Ax^n. Under the assumption that there is a unique correct answer to this problem, we can find one function satisfying it, and take that as f(x). Use the functional equation to find n, and then use the fact that the area between 0 and 1 is 1 to determine A.
@adayah2933 Жыл бұрын
Nah, there is a ton of such functions.
@ATL457 ай бұрын
I did this out explicitly by realizing the given information describes a function for which a vertical dilation can be equivalent to a horizontal dilation. The family of power functions, f(x)=ax^n, have this property, and for these values (2 and 3) we end up with n=log_2(3) and a=n+1=log_2(6). Evaluate F(2)-F(1) and you get 5.
@Maxence1402a2 жыл бұрын
About the last question, let g(x) = f(x)-1+exp(-x), which gives g(x)+g'(x) = f(x)+f'(x)-1 0, g(x) > 0, then let a = max {y in [0,x), g(y)=0} (which exists since g is continuous and g(0)=0), and mean value theorem yields b in [a,x] such that g'(b)=g(x)/(x-a) > 0, therefore g(b)+g'(b) > 0 (x >= b > a so g(b) > 0) which contradicts g(x)+g'(x) = 0, g(x)
@paxtonjk2 жыл бұрын
I literally learned about geometric series today In my ap calc bc class. So this was a fun example of what I learned today.
@yourmom44762 жыл бұрын
I solved it by assuming a monomial solution ax^n. Using the first information gives a(2x)^n=3ax^n -> 2^nx^n=3x^n -> 2^n=3 n=log2(3) Our function is ax^log2(3). Putting this into the integral from 0 to 1 and taking the antiderivative gives (a/log2(3))x^log2(3)+1 evaluated from 0 to 1. log2(3)+1 can be simplified to log2(6) by making 1 into log2(2) and using log rules. (a/log2(6))(x^log2(6)) evaluated from 0 to 1. 1 to any power is 1 and 0 to any power is 0 so a/log2(6) which must be equal to 1, so a=log2(6). Our final function is log2(6)x^log2(3). Putting this into the integral from 1 to 2 and taking the antiderivative gives (log2(6)/log2(6))x^log2(6) evaluated from 1 to 2. The logs cancel, 2^log2(6)=6 1 to any power is 1 6-1 is 5. It doesnt prove that any function with these requirements has an integral from 1 to 2 of 5 but it solves the problem :)
@jaredwoolsey56922 жыл бұрын
Dang you just beat me to it. I made the same assumptions and got: f(x) = x^(log2(3))*(log2(3)+1) Nice job!
@Jeff-zc6rr2 жыл бұрын
Justt integrate the identity f(2x)=3f(x) from 0 to 1. Then right side is just 3. Do a change of variable u=2x and you will get an integral from 0 to 2 on the left side. but split up into two integrals from 0 to 1 and then 1 to 2. That second integral is what you need and you will get the answer 5.
@dbro12052 жыл бұрын
I found the second solution as more efficient
@cwcarson2 жыл бұрын
Used infinitely fewer terms 👍
@dbro12052 жыл бұрын
And the second substitution for the second method was just way easier
@farklegriffen26242 жыл бұрын
For your challenge at the end: the answer is infinity. For instance: Assume f(x) = lim [ a→1, 1/(3x-3a) + 1/(3a) ], as 'a' gets arbitrarily close to 1, f(x) approaches infinity. This function also satisfies the requirements that f(x) + f'(x) ≤ 1 and f(0)=0.
@MathIsFun1372 жыл бұрын
Unfortunately, your function is not continuous everywhere. From the conditions in the problem, f’(x) exists for all x, which means f(x) is continuous for all x. But your function does not exist when x = a. In fact, since your function does not take an actual value at x = 1 by what you yourself showed, you know that it’s not continuous. I’m also unsure about being able to define a single-variable function like that in the first place, as one could argue that there’s technically two variables there.
@Biggyweezer692 жыл бұрын
That limit evaluates to 1/3(x/(x-1)) and is undefined at 1 as the limit from the left is -infinity and the right is +infinity. If the function were to increase going from left to right it would be bounded by f(x)
@DRoo952 жыл бұрын
You make the mistake of trying to "move" asymptotic behaviour to infinity. -For any of the approaching functions for a -> 1, the function isn't continuous, thus not fulfilling the constraints. You can't just make a "half limit", taking one property from the real function for a=1 (continuity on [0,1) ) and take another property from the limit for a -> 1 (the value for f(1) ). In general, even a limit of continuous functions does not always lead to a continuous functions. -If you fill in a = 1, you do get an asymptote at x = 1, which is -inf when approaching from the lower side of x. -If you do do a -> 1, you're making a limit of non continuous functions, thereby not proving your function is continuous. In fact, if you're interested. You could check my reply, in which I proved 1-1/e to be the maximum answer.
@DRoo952 жыл бұрын
@@Biggyweezer69 if you're interested, you could look at my reply. I think I've got a prove why 1-1/e is the maximum answer.
@DRoo952 жыл бұрын
@@MathIsFun137 if you're interested, you could look at my reply. I think I've got a prove why 1-1/e is the maximum answer.
@stephenbeck72222 жыл бұрын
Yeah, your approach is what I would do. I don’t even know who would come up with splitting the original integral into sub intervals between 2^n-1 and 2^n in order to solve when there’s a much more intuitive way.
@floydmaseda2 жыл бұрын
A real analysis professor would come up with the geometric series method. Real analysis professors are in their own weird world.
@bjao86192 жыл бұрын
I think the original solution was also quite nice though, yes not the most efficient but it was easy to follow and there wasn’t as much confusion with the switching variables. Idk I just started learning analysis and it was really cool seeing a lot of the things I learnt being applied in a solution like that.
@spazticdrummer72 жыл бұрын
OMG I did it the way you did before watching any of the solutions. Time to take the rest of the day off.
@theeternalwanderer1902 жыл бұрын
Interesting method, thank you for sharing! I did it by looking at the pattern to get f(x) = k 3^log_2(x) for some constant k, did the given integral to find k= ln(6)/ln(2) and then integrating to get 5. Not as elegant though 😅
@hrvojedjurdjevic21232 жыл бұрын
I also thought that finding solution without finding f(x) would be more elegant, but I did it the same way as you did, anyway. Since f(0)=0, I assumed f(x) = c x^a for some constants c and a, from f(2x)=3f(x) it follows a=log_2(3), and from the value of the known integral it follows c=a+1. As the resulting function is same, it yields the same result when integrated from 1 to 2.
@F_A_F12310 ай бұрын
Did you prove that that's the only function satisfying the criteria?
@michaelwarnecke34742 жыл бұрын
At the end, we just want the function for which f(x) + f'(x) = 1, which is: f(x) = 1-e^(-x) f'(x) = e^(-x) f(x) + f'(x) = 1-e^(-x) + e^(-x) = 1 f(1) = 1-e^(-1) = 1-1/e Not a proof, but the way I got the solution.
@kqnrqdtqqtttel17782 жыл бұрын
Yeah you need to prove that requirement f(x) + f’(x) = 1 yields the maximum value in the first place
@MrHotBagel2 жыл бұрын
So why do you set f(x)dx = f(u)du in the last portion of your problem. I am having trouble conceptualizing that since f(u)du=f(2x)*2dx, when you perform substitution back. I understand that the "x" or "u" is simply a dummy variable; I do not understand how the dummy variable "x" is similar to the x previously. This may have been why this method was not chosen; since it seems to coincidentally lead to your integral, from 1 to 2, being equal to 5.
@pedroribeiro15362 жыл бұрын
The sponsor of this video is Brilliant and this solution was *super brilliant*
@CalculusIsFun12 ай бұрын
Did it in my head. Nice warm up for the day.
@jakehawks10902 жыл бұрын
That entire problem felt like a backwards knight move in chess, going backwards to go forwards later. Incredibly smart solution 👍
@maaikevreugdemaker92102 жыл бұрын
Loved it. Thanks.
@dlmcnamara2 жыл бұрын
I'd start with the decomposition: \int_1^2 f(x) dx = \int_0^2 f(x) dx - \int_0^1 f(x)dx ; substitute 2u=x into first integral; second integral =1 (given); easy peasy.
@methodiconion85232 жыл бұрын
I got my answer in a very different way. I decided to find a specific function which satisfied the first requirement. After a little thinking, I came up with g(x)=3^log2(x)=x^(ln(3)/ln(2)). I integrated this and found something which doesn't have an integral from 0 to 1 of 1, so I multiplied it by its denominator to get F(x)=x^(ln(3)/ln(2)+1). Putting in x=2 boils down to F(2)=3*2=6. Edit: Added a single closing parenthesis. P.S. I feel like my function obviously doesn't work for the domain from minus infinity to 0. Anyone know if there's a valid solution sub zero that my answer fails to get, but which this substitution method can produce? P.P.S. Just watched both of the video's explanations. I don't believe either is able to find a solution for x=0, -((-x)^(log2(3))*(log2(3)+1))|x
@vvthanh2 жыл бұрын
you can define f(x)=0 for any negative x, it doesn't matter what the negative part of the function looks like I like your approach, very brute force, but not really the spirit of the question 🤪
@seansean17282 жыл бұрын
It’s perfectly well defined as a complex function
@Infinium2 жыл бұрын
I’ve always loved your channel, another great video! 😊
@punstress11 ай бұрын
LOVE THIS! Can't believe how they complicated such a simple solution ... but it was kind of cool. Darn showoffs!
@chaosredefined38342 жыл бұрын
I can see an approach to do the f(x) + f'(x) inf, this becomes the derivative. Step 2: Find the maximum value of f(x + 1/n) with this strategy. Step 3: Find the maximum value of f(x + 2/n) with this strategy. Step 4: Find the general pattern to get the maximum value of f(x + k/n) with this strategy. Step 5: Use induction to prove that it is the maximum. Step 6: Use this to find the maximum value of f(x + n/n). Step 7: Take the limit as n -> inf of the above, and set x = 0, and f(x) = 0. This will give you lim n-> inf f(n/n) which is f(1) This is way too much work for me to do with my day off, but if anyone wants to try it, feel free to tell me how you go.
@Jeremy-jj4nj2 жыл бұрын
Your solution is the effective one. The other one is nice to flex muscles though :)
@laxminarayanbhandari8552 жыл бұрын
I have a super easy way. Integral from 1 to 2 f(x) dx= integral from 0 to 2 f(x) dx- integral from 0 to 1 f(x) dx = 2*integral from 0 to 1 f(2x) dx-integral from 0 to 1 f(x) dx =5*integral from 0 to 1f(x) dx=5
@ruchirkadam85102 жыл бұрын
NICE.
@laxminarayanbhandari8552 жыл бұрын
@@ruchirkadam8510 thanks
@srijanagarwal9642 жыл бұрын
Try breaking the given integral into (0,2) and (2,1) and substitute x=2u in the first one....some may find that easier.
@schizoframia4874 Жыл бұрын
That’s so cool!!
@Alex-ud6kv2 жыл бұрын
Interesting solutions! Since f(x)=x^2 follows a pattern of f(2x)=4f(x) i realized f(x)=ax^n (actually absolute value of x since it should work for all x, but that doesn't matter for this question). Then plugging that into f(2x)=3f(x) yielded n=log_2(3). Plugging into the given integral then yielded a=1+log_2(3), and we then have f(x)=(1+log_2(3))x^(log_2(3)) which we can use to solve the final integral. *"log_2(3)" meaning "log base 2 of 3"
@albertobernal25372 жыл бұрын
It took me at least 30 mins to watch the video in full due to pausing and backtracking; it's been many years since my last math course at uni. But halfway through, it became very clear. I'm looking forward to watching your earlier uploads on a regular basis and refresh my calculus. Thx a lot sir!
@ftbex92242 жыл бұрын
I like your video,they not only train my English but also give me a demonstration to teach calculus
@sbunny82 жыл бұрын
The way I did it is f(2x)=3f(x) told to me it's an exponential function, i.e. f(x)= ax^n and I just had to find a and n. Since f(2x)=3f(x), we get a(2x)^n=3ax^n, hence (2^n)=3 so n=log2(3). To get a, use the integral (a/(n+1))x^(n+1) from 0 to 1, so a/(n+1)=1, hence a=n+1. Now we know the mystery function is *f(x)=(n+1)x^n where n=log2(3)* . The integral of that is x^(n+1). Evaluate from 1 to 2 and we get 2^(n+1)-1, which is 2^(log2(3)+1)-1. That evaluates as (3)(2)-1, which is 5. Took about the same length of time, and I not only found the integral but I know the function itself, so I'm ready for follow-up questions like what's f(2). I suppose my answer isn't as clever but it works.
@richardvidil69942 жыл бұрын
Why not just integrate both sides of f(2x) = 3f(x) from 0 to 1? Since we know f(x) integrated from 0 to 1 is 1, the right hand side is just 3. Now sub u = 2x, on the left hand side, changing the limits to from 0 to 2 and introducing the 1/2 factor. We now have integral of f(u) du from 0 to 2 = 6 (clearing the fraction). Therefore integral of f(u) du from 0 to 1 plus integral of f(u) du from 1 to 2 = 6, but we already know integral f(u) du from 0 to 1 is 1, therefore integral of f(u) du from 1 to 2 = 6 - 1 = 5.
@DRoo952 жыл бұрын
At the question in the end: The optimal strategy will be to always maximize f'(x). (So f'(x) = 1 - f(x)). If some other function fulfilling this constraint g is better (so g(1) > f(1) ), then by midpoint theorem, there has to be a point x1 in [0, 1) where g(x1) = f(x1) and g(x) > g(x) for all x > x1. However, since g(1) > f(1), we can again use the midpoint theorem to find that for some x2 in (x1, 1) we have g'(x2) > f'(x2). Since g(x2) > f(x2) we have g(x2) + g'(x2) > 1. Thus such function cannot exist. Now we've established that f' = 1 - f, we have to solve this differential equation. We have: df/dx = 1 - f df/(f-1) = -dx (multiply both sides by -dx/(1-f) ) Int(df/(f-1)) = -Int(dx) Ln(f-1) = c1 - x f-1 = e^c1 * e^-x f = c2 * e^-x +1 Now we need to fill in f(0)=0 to find that c2 = -1 and we find: f(x) = 1 - e^-x With f(1) = 1-1/e = 0.63... the maximum value. This one was a lot harder than the one in the video 😅
@violintegral2 жыл бұрын
I have an alternate, albeit similar solution to this problem. Since working with a differential equation is much easier than working with a differential inequality, I set f(x) + f'(x) = a =< 1. This first order equation is separable, but I think a nicer solution involves treating it as a first order linear equation. Multiplying both sides by e^x, we have e^x·f(x) + e^x·f'(x) = a·e^x. The left hand side is clearly d/dx(e^x·f(x)), so we can integrate both sides, giving e^x·f(x) = a·e^x + C. Dividing both sides by f(x), we have found f(x) = a + Ce^-x. Using the initial condition f(0) = 0, we can see that C = -a, so f(x) = a(1 - e^-x). At this stage we can evaluate f(1), giving f(1) = a(1 - e^-1). Since 1 - e^-1 > 0, f(1) is trivially maximized when a = 1, given the restrictions on a, so f(1) = 1 - e^-1, as you showed.
@DRoo952 жыл бұрын
@@violintegral nice solution! And a neater way of solving the DE. But if I'm not mistaken, you're only proving the maximum for functions with f(x)+f'(x) = a with a being a constant. Via this proof, there could technically be some other function with f(x)+f'(x) non constant but smaller than 1 that still produces better results, right? I know it seems trivial that a function with f + f' < 1 shouldn't score better than one which always had f + f' = 1, but doing that via the analytical way was kinda the first part of my proof
@DRoo952 жыл бұрын
@@violintegral in fact, @noahtaul had an even neater solution. Start with f + f'
@carstenmeyer77862 жыл бұрын
@@DRoo95 If you assume *f(x) + f'(x)* to be integrable, you can set *f(x) + f'(x) =: u(x) = 0* In the above *H(t)* is Heavyside's step-function that jumps from zero to one at *t = 0* . We use the inequality *u(t)
@gustavoespinoza79402 жыл бұрын
The second solution was literally the first approach that popped into my head
@misterrivs94572 жыл бұрын
Nice video! I did the question in a matter of a couple of minutes in a few simple steps: I let F'(x)=f(x) so integrating f(x) would give you F(x). I then used the first identity (integrating f(x) from 0 to 1 equals 1-----------call this identity {1}) to obtain F(1)-F(0)=1. I then subbed f(x)=f(2x)/3 into the first identity as well Iintegrating f(2x)/3 from 0 to 1 equals 1) to obtain 1/6[F(2x)] bounded by 0 to 1 is equal to 1. Eventually expanding out you get F(2)-F(0)=6-------call this identity {2}). Subtracting identity {2} by identity {1}: F(2)-F(0) - [F(1) - F(0)] = 6 - 1 ---> F(2)-F(1)=5 Which is the same as the integral of f(x) from 1 to 2 :))
@adayah2933 Жыл бұрын
But you don't know if there exists F satisfying F'(x) = f(x).
@mathboy81882 жыл бұрын
Easier way still: You can do this in your head. On an exam, you can bypass the implicit math requirement to justify that the answer exists, and just ASSUME that there IS an answer. So ASSUMING this question has an answer, it's the same answer for ANY function that satisfies f(2x) = 3 f(x) and INT{ x=0 to x = 1 of f(x) dx } = 1. So you only need to find ANY function that satisfies that, integrate it, and you're done. For the students I tutored for the SATs, I called this this the *"Make it Concrete"* trick. Work: First, it ain't gonna be linear ( f(2x) = 3 f(x) ), and have 2 constraints, so will need 2 unknowns on a general form of function. The easiest choice to try is f(x) = a x^n, and it works: Constraint #1: f(2x) = a 2^n x^n = 3 f(x) = 3 a x^n implies 2^n = 3 which gives n ( n = ln(3)/ln(2), but don't even need to do that!). Constraint #2: 1 = INT{ x=0 to x = 1 of f(x) dx } = INT{ x=0 to x = 1 of a x^n dx } = a x^(n+1)/(n+1) ]_{ x = 0 to x = 1} = a/(n+1), which gives a ( a = n+1 = 1 + ln(3)/ln(2), but again, don't even need that!). Thus INT{ x=1 to x = 2 of f(x) dx } = INT{ x=1 to x = 2 of a x^n dx } = a x^(n+1)/(n+1) ]_{ x = 1 to x = 2} = ( a/(n+1) ) { 2^(n+1) - 1^(n+1) } = (1) { (2)(2^n) - 1 } = 2(3) - 1 = 5.
@mehrdadmohajer38472 жыл бұрын
thx. alot for the posting. well done🙂
@mayhemistic601911 ай бұрын
I also solved this originally in a similar fashion. Took me two minutes to figure out. Integral 0->2 f(x) = Integral f(x) 0->1 + Integral f(x) 1->2 (Let's call this M). Now we do te substituiton x = 2t in the integral. So 2f(2t)dt from 0->1 which is 6f(t)dt from 0->1. So the final equation becomes 6 = 1 + M. So M = 5 🙂
@raja28509 ай бұрын
Yes bro.
@leickrobinson51862 жыл бұрын
Yep, that’s how I did it! Easy enough to do in your head. :-D (Their solution WAY overcomplicated the problem, like walking around the block to go next door.)
@gaspardh.76432 жыл бұрын
There's a easier solution : Take the integral between 0 and 1 of f(x)dx =1, with u=2x and f(x) = f(2x)/3 you get 1 = 1/6 times the integral from 0 to 2 of f(u)du, and with Chasles relation you get 1= 1/6*(1+S) with S the integral to find, and that makes S=5
@Nathan-cz8uk2 жыл бұрын
Your way is definitely cleaner than theirs. Another, more hack-y way that happens to work: if we assume the function f(x) is of the form f(x)=Ax^n (motivated by the fact that f(ax)=bf(x) works for functions of that form), then the two given conditions give n = log base 2 of 3, and A = log base 2 of 6. Then you can directly compute the desired integral with the power rule.
@li__on64032 жыл бұрын
I recently did something similar. The integral of t^a*(1 mod t) dt from 0 to 1 = (1/a+1)-(riemann-zeta-function(a+2)/a+2). The way you do it is similar. (Ik that x mod y doesn't really make sense for real variables, but I just used what desmos used for that expression)
@UNIverso.Matematica11 ай бұрын
The solution of the question of the end: f+f' f'=af => af+f=0 => f(a+1)=0 => suppose f ≠ 0, a=-1. Part. associated: f(x)=1 Complete: f(x)= Ae^(-x)+1 f(0)=0 => Ae^(0)+1=0 => A+1=0 => A=-1 => f(x)=1-e^(-x) => f(1)=1-e^(-1) If f(hom.) =0, f(hom.)(1)=0. So f(0)=1, but it is not possible, because f(0)=0 So the sol is 1-(1/e) Ansatz technique is so cool!
@chaulamyu73282 жыл бұрын
Consider int(f(u)*du) from 0 to 2. If we let u = 2x, then we have int(f(2x)*d2x) = int(f(2x)*2*dx) from 0 to 1. Since f(2x) = 3(fx), so it also equals to int(6f(x)*dx) from 0 to1 = 6*1 = 6. Note that int(f(u)*du) from 0 to 2 = int(f(u)*du) from 0 to 1 + int(f(u)*du) from 1 to 2. Therefore, we have 6 = 1+ int(f(u)*du) from 1 to 2. Therefore, int(f(u)*du) from 1 to 2 = 6-1 = 5
@chaulamyu73282 жыл бұрын
Haha, just notice the second half of the video after typing the comment
@milanh13492 жыл бұрын
This is great. Thank you :)
@keinemaschine30382 жыл бұрын
My approach was simliar to yours, except I got rid of the integral equation by using the antiderivative. Then I got an easy functional equation to solve. Nicely selected Question!
@maelhostettler10042 жыл бұрын
Nice solution but there is a way simpler approach : f is continuous so let : F(x) = integral from 0 to x of f(y)dy thus F(0) = 0 and by definition F(1) = 1 then take the substitution : y = 2t thus dy = 2dt and F(x) = 2 * (integral from 0 to x/2 of f(2t) dt) then by definition F(x) = 2 * (integral from 0 to x/2 of 3*f(t)dt) so F(x) = 6 * (integral from 0 to x/2 of f(t)dt) so F(2) = 6 * F(1) = 6 and integral from 1 to 2 of f(x)dx = F(2) - F(1) = 6 - 1 = 5
@ajbiffl46952 жыл бұрын
I think finding the first, complicated solution comes down to which integral you start u-substituting in first. From 1 to 2, you go to 1/2 to 1, and so on, but if you start with 0 to 1, you get immediately to 0 to 2
@rata40052 жыл бұрын
I intregrated f(2x)=3*f(x) which gave me F(2x)/2=3*F(x) +C F(2x)=6*F(x)+C and also noticed that the integral 0 to 1 of f(x)dx=F(1)-F(0)=1 F(1)=1+F(0). F(2*0)=6*F(0)+C => F(0)=-C/5. Then is used the informations to determine the value of the integral 1 to 2 f(x)dx=F(2)-F(1)=6*F(1)+C-F(1)=5*F(1)+C=5*(1+F(0))+C=5*(1-C/5)+C=5-C+C=5.
@saroshadenwalla3982 жыл бұрын
I did something similar but integrated the identity between 0 and x so I got 1/2( F(2x)-F(0))=3(F(x)-F(0)) which gives F(2x)=6F(x)-5F(0). Setting x=1 gives F(2)=6F(1)-5F(0) and so F(2)-F(1)=5F(1)-5F(0)=5(F(1)-F(0))=5
@chimero38062 жыл бұрын
very good question! I enjoyed it!
@selimders54732 жыл бұрын
Realy nice question keep it going !!! :))
@OnionKing-cm4qh2 жыл бұрын
I love it when 2 methods (one which has a limit) are used to get the same result.
@danielontheedge2 жыл бұрын
Integral from a to b is the same as sum of integral from a to c and c to b. Is there a mathematical theorem that supports the case where we add an infinitely countable set of intervals to form the integral from a to b? This question also reminds me of the Riemann sum of area of rectangles that approaches the area under curve as n goes to infinity. What supports the leap from adding area of infinitely countable number of rectangles that allows us to make the conclusion that it becomes equal to the area under the curve
@mathcanbeeasy2 жыл бұрын
I=Int from 0 to 2 of f(x) dx x=2u then dx= 2du I=2*integral from 0 to 1 f(2u)du =6* integral from 0 to 1 of f(u)du=6 So I tegral from 1 to 2...=6-integral from 0 to 1...=6-1=5. Took me 1 min and 2 lines. Very nice problem to se how students think substitutions, variable changes and new limits of integration.
@antonioprovolone28152 жыл бұрын
The MIT given solution is what my Math teacher from high school would define as Complicating Easy Problems Office
@johncalculusmathsclass59982 жыл бұрын
Wonderful step, brilliant
@mathevengers11312 жыл бұрын
First method was really creative.
@marcelohenrique51012 жыл бұрын
Amazing, professor. Greetings from Brazil
@chazzbunn78112 жыл бұрын
I solved it using nothing more than the identity given and substitution, but a little differently. First, I did this: Int_0to1_{f(x)}dx = int_0-(1/2)_{f(x)}dx + int_(1/2)to1_{f(x)}dx = 1 int_(1/2)to1_{f(x)} = 1 - int_0to(1/2)_{f(x)}dx This will be important later. Next, I took the integral to be solved for and substituted 2u = x and then used the given identity: Int_1to2_{f(x)}dx = 2*int_(1/2)to1_{f(2u)}du = 6*int_(1/2)to1_{f(u)}du Next, I substitute the identity I listed above: 6*int_(1/2)to1_{f(u)}du = 6*[1-int_0to(1/2)_{f(u)}du] I use another substitution, u = 1/2w, and using the given identity once more: 6*[1-int_0to(1/2)_{f(u)}du] = 6*[1-(1/2)*int_0to1_{f(w/2)}dw] = 6*[1-(1/6)*int_0to1_{f(w)}dw] = 6*[1-(1/6)] = 6 - 1 = 5