An IMO Bronze Medallist Stood No Chance - How a Calculus Exam made nearly all First Years Fail
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- Опубліковано 16 чер 2024
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Today we take a look at an Australian advanced multivariable calculus and Ordinary Differential Equation exam, that made nearly all the first year's fail... It got everything, from contours, diffy q's over to logistic equations with harvesting and vector calculus. Enjoy! =D
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0:00 Intro
2:37 Background Information
4:14 Exercise 1
7:40 Exercise 2
11:49 Exercise 3
19:12 Exercise 4
25:16 Exercise 5
29:08 Exercise 6
35:19 Exercise 7
39:44 Exercise 8
44:19 Exercise 9
48:10 Exercise 10
52:21 Why did they all Fail though?
53:09 How did some still pass?
I love the fact that youre collaborating with tom! Next is michael penn
^^
Yes please collab with Michael Penn
I don’t think Michael Penn has any sort of sense of humor at all though.
@@PapaFlammy69 please collab with him (if you think the collab has value). If so, can you make him laugh? It would be so cool to *finally* see him smile :)
@@Reliquancy he does, ever heard of the good place to stop?
Who is the IMO Bronze Medallist?? Certainly not me...
Hot
Me neither
Subbed to your channel 😀
:D
Mocking mind your decisions perhaps
I remember that exam so clearly - funniest exam I’ve ever taken because everyone was so clearly defeated and had given up 😂
You looked around afterwards and there was this look of pure death in everyone's eyes. No one even knew what to feel afterwards because their capacity for emotion was gone
Same with 1071 hey
Pretty sure I was just looking around and laughing
The beers at the red room tasted pretty good after that
this was the same with my MATH2001 exam in 2019. so insanely hard. by 1 hour in i hadn't answered any questions yet. looks of pure defeat in peoples eyes as they were leaving. i was on 80% across the assignments and ended up barely getting a 4 in the course
I'm amazed that I've never seen the 42-69-69 triangle from the intro before.
I once made it to the Iberoamerican math olympiads and it was so hard I became an engineer.
fellow Ibero olympic
F
t+c stands for tom cowford or something like that
P+f stands for papa flammy
Tom Crawford :)
@@TomRocksMaths ohh I'm extremely sorry ☹️☹️
for the first question, you could go further and get an explicit equation of y in terms of t. If you multiply both sides by 1-y, you get a quadratic in y. Then plug the coefficients into the quadratic equation and you would get the two branches of the solution curve.
For the Boat Problem: you should probably calculate the directional derivative along the direction of the vector (7,12), but then you need to evaluate it at the point (2,1). This means as much as that the directional's derivative along the direction of vector (7,12) has, generally speaking, a different value over the lake, you then think at the other piece of information no longer as a vector but rather as a point (2,1) where you evaluate the derivative.
Brilliant is hooking my boy out with yet another video 😁
Exercise 4 has a really interesting result. It says that entropy is maximized when all elements are the same, or when all your things are uniform ( xj = 1/N for all j ).
It's so cool that you'd get an answer that makes sense after a convoluted math problem
ya ikr I enjoyed watching that question the most
I think it's because -xlnx is always concave, so Jansen solves it nicely
For the second exercise what you need is dT/dt along the path of the boat which can be writen as ∇T ⋅ v using the chain rule, where v is the velocity vector. To evaluate it at that point and at that velocity you just do 13*∇T(2,1)⋅(7,12)/sqrt(7²+12²), note that v=13*(7,12)/sqrt(7²+12²) to get the correct magnitude.
I wasn't careful with the units, the magnitude of the velocity that they give you is in m/s so it should be changed to km/s
Rly nice discussion at the end flammabro, those poor first years didn't deserve such brutality D:
:c
Shoutout to so many of my friends, RIP
Great stuff!
The third exc. can be solved by rotor or rotational. The ♡×F=0, vectorial product of nabla and F is null vector
Its also written as curl(F) or in some places, rot(F). For his problem, we need that to be different from the null vector so that we can say it is not conservative
This exam is the exam from my university, I did the MATH1052 course which was the normal course, because MATH1072 was the optional advanced course, kudos to you for going through the exam! Perhaps I'll send you some exams that I've experienced❤
Thanks for this video
As a time traveller, this was very useful for the exam. Thank you!
7:49 oo nice view🔥
:D
Hello! I'm the one who sent in this exam and there were a few things that I think I forgot to mention in my emails.
For exercise 1, you actually had to solve explicitly for y (leaving the answer in the form y = ...). This led to a lot more computation, a bifurcation, and selecting a branch based on the initial condition. This made the question even more ridiculously complicated. This wasn't stated in the exam itself, but rather in the lectures by the professor.
For exercise 2, the differentiation was absolutely atrocious and having no calculators meant that trying to simplify our expression was horrible. And yes, the direction vector is just (7, 12) and not (5, 11).
For exercise 3, the curve is actually different to the one Tom parametrised. The curve in the question is actually easier, as it's just three-quarters of a circle (starting at (2, 0) and ending at (0, -2)).
Exercise 4 was also my favourite one, and was actually the only one I enjoyed solving as it didn't require stupid computations and had a nice answer at the end.
In the lectures, homework assignments, and class notes, we were never actually taught what a bifurcation was or how to identify them. This made exercise 9 almost impossible for us.
We unfortunately couldn't leave the derivatives as exercises for the marker in exercise 10 :)
The fact that you took almost an hour to do the exam without even actually doing some of the computations just shows how bloody hard it was for us.
Hey! I also go to UQ and took 1072 in semester 2 2018. The course is designed for students with a keen interest in maths and is an extension on top of the easier MATH1052, obviously you know this, but others may not. In saying that this exam seemed pretty reasonable in terms of difficulty, but too harsh in terms of time required. I think it's really cool that you sent it in and this will be a great resource for students trying to study for this course in the coming years!
Solving y^2/(1-y) = .5e^t for y sounds incredibly nasty lol
@@benoplustee Um, take 0.5e^t as C and now it simply becomes a quadratic equation y^2+Cy-C=0. Or y= (-C +/- sqrt(C^2+4C))/2 and substitute C=0.5e^t in that. Just three to four extra steps. I don't understand what's so nasty about it.
@@purnabhishek hey, thanks, I hadn't immediately seen that. You're not wrong, but I think framing it in a "wow, cant believe you didnt figure this out lol" kinda tone is pretty unnecessary
@@benoplustee wait, I assumed you thought solving quadratic equations were nasty. My bad.
For exercise 8, just by subtracting sqrt(1+z) from both sides and multiplying by 2, one gets 0≤(sqrt(1+z)-1)^2, which trivially holds.
Try doing a STEP paper! They’re the entrance exam used by Cambridge for Mathematics. There are plenty of interesting questions :)
Love you and Tom
0:04 I would totally see Andrew Wile stamping his proof of Fermat's Last Theorem with the EZ stamp.
The first question is similar to the deriving the logistic growth function’s formula
For the second problem you had to be careful with the 13 m/s information, i.e you had to calculate the velocity. And it’s in the direction of (7,12) so the velocity is proportional to (7,12). For the third one you just had to check mixed partial derivatives (Poincare Lemma) instead of solving PDEs.
Thanks Alvaro! =)
For Question 3 it works either way, If you see that you cannot work out a scalar field, Phi, for which F is the gradient of Phi, then it's not conservative, the method done in the video essentially amounts to the Poincare Lemma, as the Poincare Lemma is derived from this.
bet you cant do an A-level further maths paper speedrun
As someone taking A-level further maths, that would be really helpful lol.
Then probably try STEP speedrun lol
Eric I need to see this
@@Eric-dt7bt ye that's better
I did when I started doing university math, I regretted why they created such a thing like further math. I think the French system where they customize their math depending on the series you take is more helpful. I observed that many students who passed through the French baccalaureate (terminale C especially) were more fluent in advanced math concepts in university. It took me the whole of first year to figure out what the hell mathematical analysis was.
Excercise 4 can also be solved without Lagrange multipliers by expressing a fixed x_k by the other x_j, and partially differentiating
Happy new year flammy😊
🎊🎉😂
brilliant video bro
:)
Why am I watching this, I don’t even like math
lol
Because u like math
idk, why?
woosh now you like it, you can thank me later ;)
give it a try
Very cool video.
In #4 (25min) you don't need to find lambda and substitute to get 1/N, you can skip this only need x_j = const(j) & Sigma(x_j) =1.
It's probably (7,12) as a direction vector imo. The magnitude is 13, her speed is 13m/s, so you know, profs tend to make the numbers not too overly complicated
This seems like a mix of calculus 3, differential equations, and discrete math, which are classes you can't even take first year. I think it's a bit of an unreasonable test as you need to have knowledge on all 3 subjects for this test and these classes aren't taught until 3rd- 4th year at most universities. Considering that most people failed this test, it seems like students should appeal this test.
You think so probably because you're in the US. Other systems are little faster than that. They have 3 years for a bachelor degree in math. They also take single differential and integral calculus in high school and even go little further in further mathematics.
@@michaeltamajong4659 I'm in Canada, and you can learn calculus 1 as an option. You could also opt to take the advanced placement classes and probably have a deeper understanding of the topic. I've never personally took these classes, but from what I've seen they cover super basics of differentials and not multivariable calculus. I'm not sure what high school curriculum in Europe is like, but if the class average is not passing, that can't really be good. Also, some of the derivatives he took, probably knowing Mclauren series would of helped as your derivatives wouldn't get out of control. I would also be surprised if they learned something like gradient in high school, but maybe they learn it.
@@michaeljjan6344 in most european countries, since they take single variable calculus in high school, they just start with multi-variable calculus in he university as one of the first year courses. I was unfortunate that in my first year, we were introduced to multivariable calculus in the physics courses. I did not understand what was going on, how the operators worked it would have been better to take MV calculus as a course. Instead we started with mathematical analysis which included differential equations, abstract algebra, linear algebra, and affine/euclidean geometry.
@@michaeljjan6344calculus 1 and 2 are required for first year to even apply to faculties like Engineering, Physics, Chemistry, etc.
Manitoba btw!
@@michaeltamajong4659 I'm in the US, and by 12th grade I'm covering around the same level, but switching discrete for complex analysis. I think this course is meant for very advanced first years. US system is very customizable to be as easy or brutal as you want
I am a first year, and except for the ODE part, i wouldn't really be surprised if this was given as our midsem exam on the 3rd
Holy crap I’ve taken this course at UQ as a first year maths student. I took it back in 2017 and it was still pretty hard. Maths exam at UQ are typically rough.
Definitely man
dunno about that second question. You didn't consider the speed 13m/s. It gave an unclear direction, but i would have guessed they wanted (5,11) as a "direction", but normed and then multiplied by the speed 13m/s to get the actual direction (together with the speed). Else the whole second part of the question makes no sense to me. And then ofc to obtain the answer to plug in (2,1) into the scalar product.
In Australia I believe they do SV calculus in high school just like the other European systems. The time should be the most significant factor that caused many to fail as you propose.
yup!
This exam is insanely long! for the fourth exercise, once you have got that all the xj were equal, the only solution that satisfies the constraint is 1/N, and you were done. and I agree, this is the only cool exercise
Ex 3, i usually prefer to verify if curl(F) =0 which is of course equivalent to the method shown
Bro please make a detailed video on binomial distribution
Whichever it is going to be (true that, the wording is not clear but it should be 7,12) to get the rate of change per unit distance v must be normalized
I love and respect you brother.
I always watch your video and you helped me a lot. I am also mathematics student at National University, Bangladesh. B.sc 3rd year.
10:46 (7,12) as that is the direction along which to compute the derivative.
Small correction- it should be normalised..
jens: this is quite a simple equation...
me who likes biology and does not know any maths but still likes watching your channel: ya sure 👁👄👁
Hello comrade
Hey, regarding P2, I think you missed out the velocity. Idk if I'm right but this is what I would do, If by any chance someone finds a mistake let me know it.
I'm still not sure If the direction is either (7,12) or (5,11) but it doesnt matter. I will use (5,11)
The line that meets points (2,1) and (7,12) can be defined as R = (2,1) + V*t
Here V is our velocity, so we will just need to normalize our (5,11) vector and then multiply it by 13 wich gives us (5,11) * 13/Sqrt(146)
This new vector has the (5,11) direction but its module is 13
Then I would evaluate the Directional Derivative at (2,1) using our vector V
39:07 quite sure that setting h to be zero is not what we are looking for - you are changing the initial conditions to the end result, you are solving for one special case
colloquially speaking, the time of the fall as well as the end velocity is obviously dependant on the starting height, and obviously if you set the starting height h to be zero then the time of the fall is also zero, as is the velocity at that point
I believe there is a better way to solve question 6 (and the solution in the video actually seems to be wrong)
Rewrite the function as (x+1)(y-1)/e^(xy). It's easy to see the function is never positive, so the maximum is 0 at the triangle edges.
Now, to actually solve for the minimum:
The value of -f(x, y) is equal to the area A of the rectangle (-1, 1) to (x, y) divided by the exp of the area B of the rectangle (x, y) to (0, 0). Clearly you want to maximize A and minimize B, so the minimum point is on the hypotenuse. You'll find that A = B = x(1-x)
Now let x(1-x) = t. Since t/e^t is increasing for 0
Tom writing closed intervals with parenthesis pains me
xD
Yeah. It does.
I like the French system that uses square brackets everywhere. Open intervals are written as ]a,b[ it is really interesting
@@michaeltamajong4659 I was taught this notation at highschool. But I was also taught the parenthesis notation a little later.
@@suurlu01 yeah. I think it keeps the distinction between a point and an interval. Using parentheses can sometimes cause confusion with beginner mathematicians who are also studying coordinate geometry (well, not to professionals like you)
@@suurlu01 please come clearer
32:23 I like how he can write two pefect parallel line of equation but write it in a tilted way.
Brother plz make video on shell method using polar coordinates to find volume and using double integral. Plz i have problem on this.
And he seriously calls an OXFORD university professor 'my boi' 🤯
For #3, the circle goes from (2,0) to (0,-2), NOT (-2,0). Should be 3/4 of a circle. My answer came out to 3pi + 2/3
Aufgabe 6 kam bei uns eins zu eins in Analysis 2 für angehende Lehrkräfte dran 😂
Wow, the second part of the second question is incorrect, you need a unit vector in the direction of the differentiated vector.
Second Problem: Directional derivative, shouldnt you use the direction vector of length 1 for this?
Hope you bring Grant sanderson soon
For the 3. exercise you could also just show that there exists a point for which rot(F) != 0 because rot(F) = 0 and F is conservative are equivalent statements.
Are you sure about this? I don’t think the rot operator is defined for R2. Moreover rot(F)=0 and F conservative are equivalent statements only under the hypothesis of simply connected domain. Feel free to correct me if I’m wrong.
@@Alberto-tq7yv rot F is defined in 2d.
The entropy problem is a lot easier without Lagrange multipliers! Assuming there is a unique solution, by symmetry all the x_j must be equal, and since they sum to 1, we have x_j = 1/N.
I think that in an optimization problem with symmetry, there are two scenarios. One is where all the variables are equal and the other is when there are N solutions that are symmetric wrt each other. So this problem, you could have had N solutions with one being (N, 0, .....), (0,N, 0,.....), and so on. If you could prove that there is only one solution or even maybe that there are less than N solutions, then I think you could conclude that all the variables must be equal to each other.
For the second part of exercise 2, can’t we form a line between points (7,12) and (2,1) where our starting point is (2,1) ie L = (2,1) + t((7,12)-(2,1)) and then the x and y components of the line can be substituted into our temperature function? That way, temperature is only a function of t and the rate of change at point (2,1) is just the derivative of the temperature evaluated at t=0?
ooof I go the university, the paper is from and he's correct. it was a hard exam and most people failed it.
At 2:07 when u said ad revenue, I got an ad! XD
oh xD
Use a AdBlocker
no
@@PapaFlammy69 we won't use an ad blocker! Never :)
As someone who had to that class this year, the fact that there was so much calculation to do was not foreign (at least for me) and I don't think it was the problem. At my high school, we had similarly ludicrous exams where no one in a class of 15 would finish all of the ~12 questions within the 2.5 hour limit*. Because of covid, we had a 10 question, 12 hour, open book, take-home exam (basically a mini assignment) which questions of similar difficulty and I (someone who had gotten 7's in all of my previous subjects) still had a lot of trouble with it (along with many other people I talked to about it). So I don't think time was (at least completely) the problem. Furthermore, that class (MATH1072) is taken is 2nd semester so students would have done uni exams before.
*At high school, your grade would be determined by criteria, not the pure amount of question you got right so you could still get an A, even is you didn't finish a couple of low level questions.
Yeah, this was the 2019 exam, and I was in that class as a fifth course at least for the first half before I realised it would've been a terrible idea to see it through. Zoltan just isn't a very good lecturer, and the rest had Jon Links for the second half. The content is just way out of whack following 1071, it's too broad, and isn't covered well enough. It wasn't until math2001 that stuff made sense. If you're doing that this summer you will enjoy it :)
This exam gave me 0th hand PTSD.
Is that Tales of Berseria music at about 40 mins?
May I know, in Exercise 4, if each x_j is the not the same then how do you proceed? (24:08 timestamp) Can you still substitute directly into the equation G?
What I understand is that the x_j = e^(-1-lambda) is substituted into the restriction G (sum from j=1 to N), but the e^(-1-lambda) is independent of j, so it's allowed to be factored out of the summation symbol, leaving e^(-1-lambda) * (sum(j=1 to N) 1) 1 , which is e^(-1-lambda) * N
Andd on 24:50 i think a negative sign was left out accidentally (lambda = - ln(N) - 1 ) :D
Hey what’s the music in the background you’re using? It’s super relaxing and I’d like to study with it.
The solution of the 6th exercise looks a bit sketchy. In the interior of the triangle you can just differentiate but using this to find a critical point on the boundary? I think you should check the interior first, see if there are critical points and then parameterize the boundary
For #3, could you just do part 2 first to save time? (Once you get a nonzero answer, you can use that to say that the field is nonconservative, right?
why is it the perfect triangle?
7:52 .. hehe booii... man of culture!!
The uschi's boat question in the second part there is a bit of stuff to do. It is asking for the instantaneous rate of change at (2,1) so I think you would take the directional derivative in the direction (7,12) and then evaluate it AT the point (2,1) cause usually when you need to calculate the max rate of change at some point you take the gradient evaluated at that point and then just take the magnitude, but that's for max rate of change so I kinda tried to transpose it here, I m not certain. Also this is actually a doubt, the problems I did required me to normalise the vector before taking the directional derivative but ik that isn't applicable everywhere so I was wondering is anyone could clarify that.
well i knew i was mentally dumb
but after watching this video i now know im dumb in coolness also
For 3), why not just show that the mixed second partial derivatives are not equal, thus the field cannot be conservative by Clairaut's theorem?
I guess he wanted to use the definition instead. Or maybe he just forgot
Math professor used super hard math test.
First year students: no protective shields left
Math professor wins
Wow lmao the EZ. stamp sure looks nice for my maths exam lmao
:D
Do u think you could try the indefinite integral of sin(x^x)? I cant find anything about it which tells me its either impossible or pappa flammable just hasn't seen it yet
Absolutely insane to only have 2 hours for these 10 exercises. Even with a more normal 5 hours exam time this would be a tough one.
In those systems, they have been used to calculus already for almost four years before entering the university. I had calculus for 4 years (effectively 2) years in the secondary school. It is effectively 2 because the lower secondary calculus is not really advanced. It is just the intro.
Systems like that of Australia do their university in 3 years as opposed to 4 in the 🇺🇸. Well, it may differ sometimes. Looking at that exam the only thing I saw i guess as you did that it is the time.
As a first year, I think I could do that exam in 2.5 hrs. Five hours is too much. The 30 minutes it thinking and planning time.
For question 3, could you also show that the curl of the function is non-zero, so it can't be a conservative function?
We hadn't learnt any calc 3 stuff at this point
I‘s love to send some NEXT LEVEL spicy math exam that actually is not optional and also held around the 2nd semester but sadly all our materials are strictly prohibited to be shared as per our universities guidelines and obviously i‘d only be a practice exam only showcasing a possible option for the actual exam that we as students obviously only see once while writing it & another time if you wish to review it 😂
Can u do some STEP questions plz dad
Well, I am a 2-time IMO bronze medalist and would certainly not pass without having studied anything related.
hey papa, i have a challenge for you
Let ξ_n be independent random variables that take values 1 and -1 with probability p and q=1-p respectively. Prove that the series from n=1 to infinity of ξ_n /n always converges if p=q=1/2 and always diverges otherwise
have fun :))
This is a really cool challenge! To be accurate shouldn't we say the series almost always converges? If its a 50/50 chance there is still a possibility that all the ξ_n land as 1 and the series diverges. But this would just be one in uncountably infinite or more possibilities.
@@hybmnzz2658 if it was a finite sum, sure. But it's an infinite sum.
Epic
17:00 Tom is getting 3)(ii) wrong. The question says "the arc of the circle...from (2, 0) to (0, -2)", but Tom is evaluating it from (2, 0) to (-2, 0) and then back along the x-axis from (-2, 0) to (0, 2).
It was asked to calculate the line integral anticlockwise. It's absolutely correct.
Dutta says you are wrong -- but you are absolutely correct and he is absolutely wrong! The curve is along the circle counterclockwise from theta = 0 to theta = 3 pi/2. Tom was wrong and mistaken that he thought it was a closed curve.
Brother, i would like to request you that plz give me a calculus pdf book which you most love and read.
Plz.
I don't read books, sry ^^
Learn calculus through manga
Watch some organic chemistry tutor, do practice problems on Khan Academy, and crunch it out
James Stewart single and multivariable calculus is a good one
@@PapaFlammy69 How did you learn then?
I was in this cohort. I did the semester one advanced calc course and felt so miserable I dropped mathematics altogether 😍 Glad to know I wasn’t just dumb
I’m right there with you brother
Same!!! Everyone was complaining that it was hard but I didn't think it was hard enough to be featured in a video like this omg.
Oh hey Jason
the answer in the last question is t e^-2t^2 and not cost
sorry the question is for the non homogeneous equation. .
I see, you are a man of culture as well...
On par with my diffy q exam, prof literally gave problems we’ve not done bc Covid meant we couldn’t cover it and said we should be able to figure it out ffs... that and random calc questions which weren’t covered in the course lol
Naturally the average was disastrously low, such that the grade was never released and they were prompted to curve when they didn’t plan to
:/
We'd covered all the content briefly, but there was just no way to do all the algebra required to finish the test in two hours
When I first started watching this video, I knew what the problem would be as soon as you held up the wad of paper that was the exam. While the content wasn't particularly horrific (tough and challenging but fair) it just seems like far too much to ask of students in their first year when university style exams are totally new. What I do find worse is that calculation crunching doesn't prove you're a good mathematician in the same way that following a recipe from a cookbook doesn't show that you're a good chef. Poor quality exam 😂
nobody who knows any geometry would ever consider the "diference vector"
1072 was a shitty course. Everything is not rigorous. I regret signing up for it, and so do my friends who followed through and failed this exam. Poorly run, bad content
This whole exam was much easier than I initially thought, just standart Calc. II problems. But damn, the exercise with the triangle along would take a newbie student at least 15-20 mins. No wonder noone passed..
ye 3:
It was not a difficult question but a moderate one.
My calculus lecturer would set the threshold at 60%😖
Soo Papa>IMO medalist? Now that's what I like to see! Good luck in the future as well man!
hehe :D
Kyaaaa! Papa replied and hearted!!! This must be a sign that I will pass all the math courses in my phys major🤧🥳
In the first exercise, you must explicite y as a function of t.... it must not be a problem for you, i guess... greetings and happy new year.
I bought the stamps. They are so hilarious 😂
My memory is a bit hazy, but don't you need to normalize your direction vector before taking the directional derivative (re: ex2)?
You're right! I actually said that part but it seems like I accidentally cut it out ^^' We only want the direction, so that makes sense :)
This looks like calculus III.
Except it is combined with differential equations into one course
@@michaeltamajong4659 differential equations is not commonly a stand-alone course unless they are PDEs. I'm not arguing with you just saying.
@@patricksalo2425 I had a basic ODE course, then more advanced non linear ODE course and then PDE. First, second and third year respectively.
0:22-0:25 : The most ungerman thing from a German.
:D
I want the link of the calclus book of first year students of this college
Its just multivariable calculus together with differential equations done as a single course. I think They do single variable calculus in high school.
@@michaeltamajong4659 Yes we do. HS student here, in my mathematics course we cover pretty much single variable Differentiation and Integration. Higher levels touch on Vectors and differential equations