Function and integral inequality - Viewer Submission
ฝัง
- เผยแพร่เมื่อ 9 ก.ค. 2024
- Submit your math problems to me at mathoutloud40@gmail.com and I'll attempt a solution as I see it for the first time.
Follow me on Twitter: / mathoutloud40
Follow me on Mastodon: mathstodon.xyz/@MathOutLoud
#math #maths #mathematics
This is a very simple application of the fact that I = int [ f ( x ) , a to b satisfies (b- a) * Min (f(x) ) < I < max (f (x ) .
Man, for each k=1,2,..., you have
log(k) < int_{k}^{k+1} log(x) dx < log(k+1)
For some n, add all the inequalities for k=1,2,...,n-1,
sum log(k) < int_{k=1}^{n} log(x) < sum log(k+1)
log((n-1)!) < (log(x^x)-x)|_{1}^{n} < log(n!)
log((n-1)!) < log(n^n)-n+1 < log(n!)
Now you take the power of e,
(n-1)! < n^ne^{1-n} < n!
In other words, instead of opening each integral, taking the e power of each inequality, multiplying everything and using telescoping reasoning for the product,
it is better to add all inequalities, so the sum of integrals is the total integral, take the e power and it is done. That's clean. And intuitive.
And if I actually did this test, I would leave the following message for the person that made this question:
"Really? "n^ne^{1-n}"? (n/e)^ne is way better!"
That’s a fantastic alternate and essentially equivalent solution! Thanks for sharing!
BTW that was a question from the University of Cambridge entrance exam for maths in 1957. Brutal question for 18 year olds!!
Probably a bit on the difficult side, but whether it’s an appropriate level for an entrance exam is open for debate. But what I find interesting is how the style of question that’s asked is essentially the same now as it was about 70 years ago. You could have told me this was on the exam last year and I wouldn’t have questioned you!
This is definitely NOT brutal. It is an easy question and formulated in a great way to test the student's proof reasoning and style.
The only thing I would change is writing
(n/e)^ne
instead of
n^ne^{1-n}
Again, it’s all relative to your background. This is most certainly a rather advanced question for typical high-school students.
@@mathoutloud humm background? I don't think so. The only different thing the student needs to know for this question is
Int log(x) = (-1+log(x))x
That's pretty much the only different thing, isn't it? Strictly increasing function, log, fatorial, exponentiation, basic integral theory, there is nothing hard about those.
What percentage of high-school students do you think are able to find an antiderivative of the logarithm?
Thank you so much for doing that question, I thought it was really challenging. I got to the last inequality you got to, but tried multiplying by the expression by (n-1)^(n-1) but did not help. Thank you again.
This is not challenging. It is pretty straight forward.
@samueldeandrade8535 it’s all relative.
@@mathoutloud it is all relative. But this doesn't mean we can't classify things as "brutal" or "easy". I don't understand why after someone says "brutal/challenging" you kinda agree with it, and if someone else says "easy" you reply with "it's all relative".
This clearly shows you don't think "it's all relative". You actually think "it is probably a bit on the difficult side" ...
I never understood such ... I don't know, reaction? It is like the "issue" some people have with
"It's easy to show that ..."
For some reason, a lot of people don't like when the "easy" label is used. Why? Actually I see such people trying to prevent or accuse arrogance coming from "easy" label users. Something that makes NO sense.
Anyway ... just some random social philosophy narrative, I guess ... I don't know why I write those, no one cares ...
@@mathoutloud what I can say is:
the question and its suggestion were GREAT, the thumbnails with black background and white font is AMAZING, it is CLASSIC, it is MODEST, it is CLEAR, I freaking love the thumbnails ... I guess the titles are great too. AND something GREAR about your videos is that they are GENUINE. It is you genuinely solving an exercise. Not the result of a bunch of takes for you to look good.
Just some positive opinions so I don't look that much of an a..h...
Oh, I forgot the conclusion: and all this is NOT relative. It is objectively true. Haha.
@samueldeandrade8535 because context matters? This question is obviously intended for high-school students seeing as how it’s on a university entrance examination. This question is without a doubt very difficult for a typical high-school student, but if you have studied calculus and analysis at a university level for a couple years then it’s essentially trivial. Hence, relative.
Perhaps my response to you takes its form due to your dismissal of someone else’s struggle with the problem. There are a lot of things that the world doesn’t need, but I’d like to include in that list condescending and elitist attitudes regarding mathematics education. If Raj thought the problem was brutal, then the problem was brutal for him. Saying “it’s not brutal” in the face of their comment is hardly a way of encouraging someone that’s excited to learn and attempt these problems.