Entrance examination to Stanford University
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- เผยแพร่เมื่อ 9 ก.ย. 2024
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@higher_mathematics
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1. Stanford does not have an entrance exam. 2. Stephen Hawking did not go to Stanford, nor did he teach there. 3. If you're going to use that much machinery for such an obvious question, at least prove that there are no other solutions.
He made the nearly exact same video with the same thumbnail with Harvard instead of Stanford.
Easily one of the worst math channels on TH-cam.
Clickbait should be a capital offense
How can a university not have an entrance exam?
Stanford and Harvard both have entrance exams which consist of one question as seen above.
@@haggismcbaggis9485 🤣😂
Assuming that the examination must be completed in a given time period, it is possible that the point of this question is to see the intuitively obvious solution = 4. I remember (67 years ago) a NY State Science & Engineering Scholarship exam where there was a multiple choice for the product of two huge integers. I noticed that the product ended in 6 and only one set of integers had last digits whose product ended in 6. Solution in a few seconds and onto the next problem!
It’s Stanford university. Think they’d care about the time you’re given?
@@Ron_DeForestNope, they only care how much money your old man gives them, take as many semesters as you like 💵💸💸💸
@@joecaves6235
That may be true of the Admissions office, Joe. But no graduate student TA gives a little brown squishy one about what anybody's old man gives to get his name on a building.
Consider yourself one more data point for "Cynicism is the wisdom of the dolt."
After the stage where you multiply by 2^20, split the right side into 2^4x2^16. This becomes 16x2^16.
Both sides are of the form Ax2^A=Bx2^B so A=B can be equated giving 20-X=16 at a much earlier stage and no Lambert W function required!
Friggin' brilliant! Thank you.
Hey what's the logic with which you can say that A = B from the equation you provided.
I think this very nice idea can be clarified additionally for example this way.
Consider equality Ax2^A=Bx2^B.
We have value B as positive (equals to 16). So if A is negative, then equality can't be acheived (Ax2xA is negative, Bx2^2B is positive). Suppose both A and B is positive. In this case if they are different the equality is also cannot be true because the function X*2^X is monotonic for positive X (you can take first derivative and check its sign).
wow
@DukeofEarl1961 sure, but that reduces the opportunity for adsense.
Sixteen lines of maths too many. It was blindingly obvious the answer was x=4 three seconds after looking at the question.
There's a reason why you are always supposed to show how you arrived at the answer...
@@fredpagniello3267
We could just notice that 2^x + x is an ever increasing function and clearly has no solution for x < 0.
Next notice that when x = 0, then 2^x + x = 1. This fact coupled with the fact that 2^x + x is an ever increasing function means that there is one and only one real x, such that 2^x + x = 20.
By observation, we see that 2⁴ + 4 = 20.
So x = 4 is the one and only real solution.
@@davidbrisbane7206Your explanation can be reduced to a very simple "abracadabra here's my answer." Yours works for simple problems with an obvious answer only. The purpose of working through a simple problem is to develop the acuity and facility to tackle those of greater difficulty. Essentially, you are building up skill from the inside out, not the reverse. The latter, by the way, is a standard for the approach to languages; the student learns the equivalent in their native tongue and therefore thinks in it and translates into the target language, exactly the opposite from how the human brain operates (that is, direct form function relationship, association of the basic structure that allows for acquiring more complex ones).
No credit without showing work
@@abefroman7393 How do you show the working if there is not to be done?
You can guess 4, and since 2^x + x is a strictly increasing function, it has at most one solution.
wgat about complex numbers?
You can guess because x happens to be a integer
If it were = 21 it wouldn't work
@@pierrekilgoretrout3143 2^x isn't well defined if x isn't real, it has infinite values (not every number, but infinite numbers)
Assumption: X must be a real number. 2^x+x=20 2^x=20-x analysis: 2^x is strictly increasing, 20-x is strictly decreasing. Conclusion: there can only be one solution. Now you can use newtons approximation methode or just start with small whole numbers as that is always a good start. one, nope, two, getting closer, three, almost there, four. OHH it's four how convinient.
I try 1, then, 2, then 3, then 4 and I find that 4 is the solution. It takes 10 seconds
Same😂😂😂🎉
Didn't need to go this far
I did what you did. Seemed too easy. I was expecting the video was going to say it was a trick question.
Painful to watch. Distortedly complex
Same. The only thing I can think of to make this make sense is that the question must say, "Using the Lambert W function, show that....". This makes no sense otherwise...
I'd never trust anyone who made x's like that
It made the equation more surreal than it should have been. :)
What's unusual about those X?
Look at his e and l. Try not to be confused whenever he writes e^ln2.
Yeah, 2 c's do not equal x.
@@BoundaryElephant That's literally how people write x in cursive. It's the norm.
As an engineer, all i did was plot a graph of 20-x and 2^x. Only one intersection shows the existence of only one solution.
After that, a middle school kid can guess the answer is 4.
That is to say, if getting to the answer is your ultimate goal instead of proving it rigorously.
2ˣ = 20 - x
(1/2)⁻ˣ = 20 - x
(1/2)²⁰(1/2)⁻ˣ = (1/2)²⁰(20 - x)
(1/2)²⁰⁻ˣ = (1/2)²⁰(20 - x)
(20 - x)2²⁰⁻ˣ = 2²⁰ = 2⁴2¹⁶ = (16)2¹⁶
20 - x = 16 => *x = 4*
a faster way
2ˣ = 20 - x
2⁻ˣ(20 - x) = 1
(20 - x)2²⁰⁻ˣ = 2²⁰ = (16)2¹⁶
hi...exellent❤
Exactly
Can you justify that A*(2^A) = B*(2^B) implies A=B?
That is why the person in this video clip get paid more , and you don’t get paid.
@@rhodexlx1659 implies A = B as a possible solution
This show how unpopular mathematic can be present...
Well. Without wanting to belittle W for LAMBERT, I did it head on. First: "x" must be even, because the power of 2 will always be even and, consequently, the value of "x" too, since the sum of the two is equal to 20, which is even. So "x " € {0, 2, 4, 6....}. Therefore, when we substitute the options from this set, on the third attempt, we find the answer.
1)knowing the answer is 4, we can always quickly architect an identity to reason the answer (trying to beat the author with time).
2)In actual fact, the quickest method starting from scratch is to a)sketch y =2^x and y=20-x and we can easily explain that there is one and only one intersecting point at x=4
How do you do it when it’s a^x + x = b?
I would say you do it using the Lambert W function. Won’t be quite as clean because you can’t break a or b, but you’ll get an answer better than the xth root of a function of x.
@@eliteteamkiller319 I mean sure, but is that really the point of the question? To see if you know about the lambert W function? The answer would be obvious for those who do, and completely impossible for those who don't. That isn't a good question.
I want to thank the author of this video for making this. This was a useful and clear introduction to how repeated use of the Lambert Function W, which I’d never heard of before, can help solve a problem of A^x -x = B.
25 years ago while in my engineering math classes I sounded like a lot of these commenters, that I’d never need all this math wiz-bang hand-waving stuff, that profs were making something simple hard. I certainly heard that from working engineers that they never used the math they learned in school. That may explain a lot of problems we now have in building things that our fathers and grandfather seem to do with much less trouble.
Over the years I’ve learned that the tools in a contrived problem can mask their usefulness in solving a badass problem later down the line. In the coding work I do, I’m using more of what I learned nearly three decades ago than ever before and am actually going back and relearning what I’ve forgotten in multi variable calculus, DiffEq, Fourier and Laplace Transforms.
So, again, thanks!
Elementary x=4
2*2*2*2+4=20
Thank you you are the only person in 52 years to explain a math problem to me in a way I can understand 😊
@@br4524me? Thank you
“Don’t ask me why.” This is the problem with math instruction. Maybe the mathematicians don’t know either. “Just do this. Then do that. Don’t ask me why.”
Ranks right up there with PFM (Pure F*ing Magic) and ATAMO (And Then A Miracle Occurs), which is essentially his explanation for why he uses the Lambert W function.
@silverhammer7779 : You can find the proof, but it's either a book or a college course of its own.
This channel loves the Lambert W function.
😊 2⁴ + 4 = 16 + 4 = 20 🎉
x = 4
Looked at a couple of videos. One of the worst math channels on TH-cam. He make several times the same video with every time the same stupid answer. He only changes the title of the video (one is Harvard, the other is Stanford, in both thumbnails there is Hawking who went to neither). And noone learns anything because the bloody Lambert W function is utterly useless.
And yes, you can find x = 4 in 10 seconds.
This channel exists purely for clicks
@@lolilollolilol7773 To be honest, I did feel like I learned something here. Have a minor in math, but that was from back in the 1980s, roughly the bronze age or something. But still, I did feel like I learned something, so I felt like my time spent on this was ok.
@@lolilollolilol7773but you get to see one math trick which may become useful in your next life so I give a thumb up 😊
A useful way to look at this problem might be to appreciate the structure that led to the approach. Sure, you can solve this specific problem quickly by inspection, but what if the solution turned out to be something less obvious like √3? Some students may not know that problems with the variable in both polynomial and an exponent form require the Lambert W function. I thought there was secondary value in the way he had to repeatedly manipulate the equation to apply the W() function. That too is a skill worth learning for more difficult questions, IMHO.
Yeah I don't really get the people here in comments going “why even bother, it's obviously 4 by inspection”. Like it's there to provide you with something more tangible than just a^x+x=b, to teach a complex method. It's actually really good video showing how to solve the LambertW part without a calculator, I didn't even realize that ofc you could do that by further manipulating the expression inside the W(…).
@@arttu3232 Definitely some big egos in the channel who don't even post their own solutions. 🙄
And yet, there's no explanation for the reasoning that brought in that 2^20. I don't see how a method that relies on an intuitive leap from out of nowhere is a better method.
@@nathanialharris3300the W function relies on the symmetry of multiplication and exponent. Since you can't get rid of the 20 you bring it into tge exponent.
To the people saying this was a bunch of lines too many and a waste of time, go change 20 to “a” and try again. The point is not to solve one problem. It is to explore a technique.
This method cannot be used for “a” . For example cannot be solved for 21
It is much faster just to try a few numbers, 1 to 5. The answer is 4.
Finding no structured way to solve this problem, I used guess and check I found an immediate solution. x=4, then I checked 2^4 + 4 = 20.
Solved less than 30 secs just by trying numbers. Even it was a test, you can still guess the "in between" numbers for x.
you actually need 3 seconds. We have an increasing function + increasing function = constant. We know about increasing functions that they are injective, meaning that for an x there exists only one f(x). So when we have 2 increasing functions added, we get one increasing function, which means an injective function => there is only one solution, which you can solve by trial and error. x=4 works obviously, and this is the only solution
Meta AI: To solve the equation 2**x + x = 20 for x, we can start by isolating the term with the exponent:
2**x = 20 - x
Next, we can recognize that 2**x is an exponential function, and to solve for x, we can use logarithms. However, since the base is 2, we can also try to find the solution by inspection or trial and error.
Let's try some values of x:
- x = 1: 2**1 + 1 = 3 (too low)
- x = 2: 2**2 + 2 = 6 (too low)
- x = 3: 2**3 + 3 = 11 (still too low)
- x = 4: 2**4 + 4 = 20 (matches the right-hand side)
So, the solution is x = 4.
Note that this equation has only one solution, and it can be found using logarithms or numerical methods if needed.
Puts the democratisation of mathematical teaching back 30 years. Come back NancyPi! We miss you more than ever.
Mathematics is beautiful. But, in a test, most of the time we don't have this rich and precious time.
X=4 2^4 = 16 x= 4 2^4+4= 20
Thanks, never heard of Lambert W function, so had to dig into. Learned something new, always nice!
A binary search gave me the answer.
Since a binary number can be written as a power of 2... You write 20 in binary and then you see the 4 being the only available response.
Stanford does not have an entrance exam. Stop lying.
Chatgpt, Clickbait!
Lighten up, Francis!
They used to have in 19th century.
@@abefroman7393Stripes
Still a sexy solution.
Bro really went the longest possible way around solving a 30-second equation.
There's this thing called "Trial and Error" and in Singapore they call it "Guess and Check". I was never taught this when i was young. I was happy when my daughter had homeworks about it when she was in primary 4. Not sure about other countries but i think Singapore is right in having this included in the primary/elementary school Maths
2^x is a rising exponential function... and 20-x is linear... the point of intersection is one. The answer is 4, which can be guess within 3 seconds.
easiest way by correlation: 2^×+x=16+4, 2^×=16and x=4, 2^4=16, then 16+4=20
Knew Lambert W could be used. But instead I did my own proof. So x>0 for a solution. Rewrite as x= 4*(5-2^(x-2)) we know the paranthesis must be positive so x
Είναι συνάρτηση 1-1 και βρίσκεις τη μοναδική λύση. Όλο αυτό μπορεί να λυθεί σε 3 γραμμές
Pretty sure Will Hunting was mumbling 'there's a better way' after the first 20 seconds...
Ridiculous, I solved the equation in 10 seconds just in my head while the mathematician took 12 minutes!
And way detoured for that
Why is that better? The other way is way hotter. Literally got turned on seeing Lambert W twice.
Seriously though, replace the number 2 by a and the number 20 by b and try to solve it by inspection. Exactly. The point is not to solve one problem. It is to look at a technique to solve a set of problems.
@@eliteteamkiller319 So use more complex numbers as an example.
And if you did that on an entrance exam you’d get a zero because you didn’t actually demonstrate an understanding of mathematical principles
The purpose of the question wasn't the answer.
The purpose was the proof: or display of knowledge.
the majority of comments not understanding this is frightening
Having found x=4 by inspection and observing that 2^x + x is strictly increasing, there can be no other solutions.
I'm not a fan of math, but correct me if I'm wrong here.
The point of this problem isn't to come up with the right answer (Even I knew the answer at a glance). The point is to show that you understand the mathematical principles needed to arrive at the answer. The reason would be that when you come across a problem you can't solve at a glance, you will have a way to get to the correct answer.
Is that a fair summation?
Disclaimer: I watched the end to verify my answer was correct. I watched the beginning until the 5th line and my eyes started to cross and I had to stop. :)
The hard part, IMO, is that there are non intuitive steps. Possible that someone who watched this, would be able to solve some future problem, using similar methods though.
x = 4 "by inspection" - that is all that is sufficient and necessary to answer the question.
Off to find learn about the Lambert W function. Thanks for the question.
Or ... We could just notice that 2^x + x is an ever increasing function and clearly has no solution for x < 0.
Next notice that when x = 0, then 2^x + x = 1. This fact coupled by the fact that 2^x + x is an ever increasing function means that there is one and only one real x, such that 2^x + x = 20.
By observation, we see that 2⁴ + 4 = 20.
So x = 4 is the one and only real solution.
2^x+x=20 --- solve this expression
2^x=20-x --- 2^x is exponentiation function (with base 2) wich means that expression is above 0 then 20-x too is above 0: 20-x>0 / 20>x / x=0 --- for negative x function 2^x is lower then 1 but 20-1=19 that very bigger than 1 and not equal wich means that x not negative
Log2(2^x)=Log2(20-x) --- set logarithm on left and right expression
x=Log2(20-x) --- logarithm function correct only for expression is above 0: (20-x)>0 / 20>x / x
Short description of solution without or short comments:
2^x+x=20
2^x=20-x / x>0
Log2(2^x)=Log2(20-x)
x=Log2(20-x) / x
My approach was: 4 is clearly a solution. now show that the function f with expression f(x)=2^x + x - 20 admits a single solution on R, so the the only solution is 4.
f(x) is the increasing function on R (because its first derivative is positive on R)=> the equation f(x)=0 has unique solution.
Решение графически :построение экспоненты 2 в степени х и линии 20-х.Пересечение ответ 4.вариант легче метода логар и формул подстановок. Постановки перебор большой. Степень понизить к уравнению.возможно и здесь 2 варианта.
To solve the equation \(2^x + x = 20\), we can use trial and error or numerical methods, as there isn’t a straightforward algebraic solution. Here’s a quick way to find an approximate solution:
1. **Try different values for \(x\)**:
- For \(x = 3\): \(2^3 + 3 = 8 + 3 = 11\)
- For \(x = 4\): \(2^4 + 4 = 16 + 4 = 20\)
We see that when \(x = 4\), the equation holds true.
So, \(x = 4\) is the solution to the equation \(2^x + x = 20\).
In what course is the Lambert W function taught. It's not taught in Precalculus and it"s not a calculus problem. Would my high school Algebra teacher know this function?
Probability less than 1% with 95% confidence. (i.e. p
If this was a multi-choice test, then substitution would work, however if essay type test, you would need to show your work. Being Stanford, most likely the later. My math Instructors always said - show your work. If you know how to solve the problem but make a stupid calcultion mistake, you can still get partial credit instead of no credit. I’m 76 and still remember that advise from high school.
They should kick you out of Stanford for writing your x's like that
It can be resolved by geometric method: functions y1=20-x and y2=2^x have just 1 intersection. So we can obviously say that x=4 and its 1 single root.
Это устная задача. Угадываем корень x = 4, монотонно возрастающая функция y = 2^x + x пересекается с y = 20 только в одной точке.
If you got the answer in 3 seconds, That is jee not maths🗿
He used an obscure trick and reasoning that I, quite frankly could not follow. The video teaches us nothing other than how "clever" its author is. Much easier to find it by examination in less than a minute.
Inspection*, I agree with you
Maybe this problem is not meant to teach, but to titillate. I was certainly titilated.
Dont blame others if you didnt understand something. This is completely understandable and its only you who doesnt want to understand and work hard on this question, atleast try to understand. P.S the W is a function which is used to solve these type of equations although he showd how this function worked you can atleast try to understand it.
x-1, 2^x=2, 2^x+x=3
x-2, 2^x=4, 2^x+x=6
x-3, 2^x=8, 2^x+x=11
x-4, 2^x=16, 2^x+x=20
It's so easy sum
2^x + x = 2 × 2 × 5
2^x + x = 2² × ( 2² + 2⁰)
2^x + x = 2^4 + 4
Therefore x = 4
This is genius 👍🏽
Clever and rather simple!
can you walk me through your thought process for this?
@@hahaureadmyname 2^x + x is the same as 2^4 + 4 . I have only made factors of 20 rather than using functions .
A lovely question, even a 10 year old can spot that x=4 is a solution but showing it is the only solution is a lovely use of the Lambert W function
Note that this solution is not a general solution for the initial formula. It works just exactly for a few numbers like 20 because this is one of the solutions of the W formula. Or said otherwise: The initial formula is a nearly exponentional graph to base 2 if you would plot it. The W function is another exponential graph. Those graphs only match on a few locations, maybe even only on y=20. That's the reason why a simplification using W can even be done. Without that, no easy solution. Just try it with another number than 20.
The same as: 20-x/2^2=x
If a+x/b=x
Then x = a/b-1 [*TM]
Notice:
20+x/4 = x
x = 20 / 3
x = 6.666
26.666 / 6.666 = 4
Not convinced?
If 2^x+x=11
So 11/2=5.5
5.5+11 = 16.5
16.5/5.5= 3
2^3+3=11
Not 'helpful', but as a rule, take 1 less than the power and the answer or solution pops out. Another case of 'if you know you know'! :D watch this space.
20-x/20-x = *clueless*
(i)x=4 is a solve.
(ii)2^x + x is a monotonically increasing function.
(iii)Then, x=4 is only one solve.
This and all similar problems are created specifically to have a solution only with the W function. They're trying to make W be considered a fundamental function, like logs and trig functions.
I never learned about this W lambda function at univerisity. Weird...
I have minor in math, albeit decades ago. And it didn't ring a bell.
I think complicated methods to show how to apply something like the W function is fine, but it would be simpler (on a test) to just do x=4 by inspection and prove that the derivative of 2^x+x is always positive, hence there is just that one solution.
Modern education has failed our younger generation.
One glance and you can recognize it's 4, but showing work for proof slightly more complex.
I’d found the solution x=4 without making all your calculs in less than a minute 4+2^4=4+16=20
X=4 is an evident solution.
2^x+x=20
x≡0 (mod2) → x=2k (k>0)
2^x
I went: I know 2x2=4 [2^2], and 4x2=8 [2^3], and 8x2=16 [2^4], and 16+4=20 [2^4 +4], thus if 2^X +X = 20, then X = 4. Simple, and I did this in my head in a few seconds.
Started with 3. That didn’t work, 4 did. Took 10 seconds.
2^x+x=20
2^x+ x=16+4
2^x+x=2^4 + 4
x=4 Done!
The complicated solution was an interesting display of mathematical entertainment, but it is totally unnecessary
2^x+x=70; instead of the solution, it was more interesting to make another similar equation
x=4 is correct
The derivative is always positive, so it can't have 2 solutions
Done! It's that easy!
I figured it was 4 at first, but I wanted to see the real solution rather than rely on a trial and error solution.
20=2² . 5
=2² . (2²+1)
=2² . 2² + 2²
=2^4 + 4
=> x=4
Maybe I’m missing something, but when 2^20 was split into 2^16.2^4, it looks like the reason to do that rather than say 2^8.2^12 is that you already know the answer is 4 and are trying to prove it, rather than trying to find the answer, otherwise how would you know to choose 4 and 16?
I know almost nothing of math, but got it in about 5 minutes. It's really pretty basic. I find that I appreciate math as I get older.
I know people who can figure this out just by looking at it, without having to show how. Those are the people that academic mathematicians are jealous of.
OK. Now I remember why I sat in algebra class and looked out the window.
Maths was my weak point at school, and even i got the answer in about 20 seconds . Simply guessed it would be around 4, checked it and it was right.
2x2=4 4x2=8 8x2 =16.
16+4=20
X=4
2^x + x = 20
2^x = 20 - x
Log(2^x)=Log(20-x)
xLog2=Log(20-x)
x= Log(20-x)/Log2
=Log(20-x)/0.301
=3.322(Log(20-x))
20 is a small number. There are only 4 integer numbers that make 2 to the power of that number less than 20. It takes up to 3 iterations to get the right answer, which can easily be done without use of pen and paper (in mind). It takes about a few seconds to come up with x=4. The algebra of the solution presented here is very curious though. I watched with interest and it was entertaining. Thank you!
By inspection, X=4. About four seconds.
One of two real root was easy (natural number), but try to solve another root (real irrational number).
In order to do the step when you split 2^20 into 2^4 * 2^16, you needed to be able to see ahead to the next step when you replaced 2^4 with 16 to make the multiple equal the power. Therefore you needed to have solved the original equation in your head to be able to perform the massively complicated alternative... 🤔🤷
Damn thats one fascinating piece of maths 😮👍👍
I did it in my head in seconds before clicking the video.
Who's on first, What's on second!
X is 4 and we don't need a 13minute explaining it. It's equally solvable by asking "show your work for a difference of two squares." Ideology. It just matches 4... Its not hard nor complex.
As an Indian aspirant I have calculated the answer orally within seconds
~2,84. Использованы свойства монотонных функций y=2^x; y=20-x. Одна возрастает, другая убывает, поэтому других решений нет.
The beauty of Lambert's W is what you can see on the graph.
This can't be on any entrance exam. It's a preliminary question.
Or you just brute force it given that 2 to the power x is really simple e.g 2 to the power 2 is 4, to the power 3 is 8 and to the power 4 is 16. You already know that it has to be less than 20 so you only need to know which combination equals 20 e.g 4+2 or 8+3 or 16+4. It takes like 5 seconds
Engineers:hmmmmmm.....oh the answer is 4
Mathematicians: I need to make a general method of solving this sum.
Start putting values for x from 0 and stop when you get your answer. This equation do not have any relevance anywhere in engineering or science
CORRECT. 2 times 2 times 2 times 2 is 16 ( That's 2 to the power of 4 ), plus 4 is 20. Utterly trivial by simple observation and calculation.
Ridiculous. This can be GUESSED almost immediately. The exponent values are 2,4,8,16. So 4 gives 16, and if you add the four you get 20.
On a first look one tries integers to get a feel for the question, and 4 clearly works. The next step is to think about the function to see if there may be other solutions. But y = 2^x + x is clearly an exponential-like function that rises consistently with values bounded below by 0. So only one real solution. As a uni entrance exam, I think going into the lambda function is a bit extreme... especially as the manipulations to get there are more extreme than the trial and error to find 4.
It is easier to make 20 into 2 to power of 4 plus 4 and then connect that x is 4