0, 1, 2
#help-43
noble fossil
uhh idk
spark wigeon
i already spoiled the hard part
tired willow
oops ig
spark wigeon
(1/6)(5/6) + (5/6)(1/6) is how often you get one 3, do you see why?
noble fossil
i fucking hate probability
noble fossil
(1/6)(5/6) + (5/6)(1/6) is how often you get one 3, do you see why?
no i geniuenly dont im sorry
1/6 yes bcs
tired willow
(1/6)(5/6) + (5/6)(1/6) is how often you get one 3, do you see why?
wait yes what
noble fossil
theres 1 3 in 6
tired willow
wait no what
spark wigeon
because we add up mutually exclusive cases
noble fossil
but why 5/6
spark wigeon
because we don;t want to roll 3 again
tired willow
aren't there two orders, (3, not 3) or (not 3, 3)
uhh am i stupid how does that get that
wait
tired willow
for x = 1 wouldnt that be like 10/36
because you multiply by 2 and not 1/6
uh
yeah?
noble fossil
ok wait
spark wigeon
@tired willowwe both get 10/36
noble fossil
noble fossil
yo i do not understand wtf happened in the solution
tired willow
yo i do not understand wtf happened in the solution
let $X$ = number of times a 3 appears in 2 rolls
each roll is independent (doesn't depend on previous outcomes, basically):
$P(3)=\frac{1}{6}, \quad P(\text{not }3)=\frac{5}{6}$
boreal girderBOT
esker
tired willow
i hope atleast that is clear
noble fossil
why is the chance to get 3 in both throws 1/36
tired willow
why is the chance to get 3 in both throws 1/36
because 1/6 for first roll's 3, and 1/6 for second roll's 3 and we're supposed to get the case where its SIMULTANEOUS, not individual
noble fossil
wait that makes sense
tired willow
so we multiply
noble fossil
mb
tired willow
yuh
tired willow
nono
noble fossil
1/6
noble fossil
omg
tired willow
let $X$ = number of times a 3 appears in 2 rolls
each roll is independent (does...
can u now try computing the probabilities for possible values of $X$: $X=0,1,2$
boreal girderBOT
esker
tired willow
and x is just the number of 3s in 2 rolls, remember
just replace x in your head with that phrase and it shouldnt be that hard
noble fossil
5/6, 5/6
1/6,5/6
1/6,1/6
tired willow
subtracted
no twin
the stake isnt subtracted from the payouts in the table
the payouts 0, 5, x are the amounts you receive depending on X = number of 3s in 2 rolls
dont confuse small x with capital X lol mb
tired willow
because you get 5 for one 3 rolled
not quite u cant just multiply i think
because the cases dont happen equally often
remember
X = 1 happens with a diff probability than X = 2 or X = 0
we dont scale payouts directly in such cases
we use expected value
tired willow
X = 1 happens with a diff probability than X = 2 or X = 0
this is the reason we first calculated the probability for each case and now we use the expected value for each case and set it equal to our stake
tired willow
yeah so how tf do i do this
the expected value is 2 which is our stake
$E = 0\cdot P(X=0) + 5\cdot P(X=1) + x\cdot P(X=2)$
boreal girderBOT
esker
noble fossil
yes
tired willow
is that somewhat more understandable?
noble fossil
expected value is 2
noble fossil
yeah ngl i dont get it sorry
tired willow
remember this is because theres a long run balance condition
noble fossil
why do we do 10/36 times 5
spark wigeon
we're just getting average win
noble fossil
why times 5 tho why 5
tired willow
yeah ngl i dont get it sorry
so the situation is:
you roll a die twice
you count how many 3s you get
that number is $X$
so $X$ can be:
$0, 1, 2$
and the payouts are:
$X=0 \to 0$ euros
$X=1 \to 5$ euros
$X=2 \to x$ euros
i think u understand the question upto this part, which is good and u also calculated the probabilities for each possibility of X
boreal girderBOT
esker
tired willow
“long run” means if you play this game like 10,000 times, you don’t care about single games anymore
you care about the average money per game
like frowny said
noble fossil
“long run” means if you play this game like 10,000 times, you don’t care about s...
hm okay
yo oh my god
noble fossil
if i get 5 euros for one 3 rolled
spark wigeon
our goal is average win equals 2, because it costs 2 to play
noble fossil
then i get fucking 10 for two 3s rolled
tired willow
if i get 5 euros for one 3 rolled
yess twin but one 3 rolled doesn't happen all the time too
spark wigeon
the hardest part is what i said
tired willow
and also the other two cases
spark wigeon
the rest makes sense
spark wigeon
then i get fucking 10 for two 3s rolled
you get x
spark wigeon
you need to set x to somerthing
maybe 10 is correct
maybe you need to make it less or more
noble fossil
yess twin but one 3 rolled doesn't happen all the time too
why
if X = 2
then two 3s were roled
tired willow
then two 3s were roled
twin, i think u mean: “1 three gives 5, so 2 threes gives 10”
but that assumes both outcomes are equally likely, which is false
we just found out that each case doesn't happen as often as the other, so we need to accoutn for that too
like X = 1 happens a lot more times than X = 2 so you might end up getting 5 euros more than 10 euros
the thing is, payouts dont SCALE linearly with the number of 3s
they scale with the probability factor
tired willow
yeah i mean
it could be
but
like
in this case if u solve for x u get 22
which is way off
so its best to assume you dont
spark wigeon
ping me if you want, can't imagine why lol
tired willow
noble fossil
twin, i think u mean: “1 three gives 5, so 2 threes gives 10”
but that assumes ...
ohhhhhhhohhhh
ohhhhhhhhhhh
ohhhhhhhhhh
noble fossil
nah ngl i dont understand this
tired willow
oof
tired willow
what part do u think is very iffy or weird
tired willow
why is it times 5
times 5? where? u mean in the payout?
spark wigeon
it's just average win
tired willow
$5\cdot P(X=1)$
boreal girderBOT
esker
spark wigeon
you should know "expected value" for irl purposes lol
noble fossil
i just got confused
tired willow
yuh
noble fossil
no i got it
tired willow
fact check status true
noble fossil
like
tired willow
aight
noble fossil
expected value
yea
but yk
the thing is in probability
you can’t prepare yourself
there could be so much variation
valid ore
Help me solve 11 th q
compact pewterBOT
Help me solve 11 th q
strange spear
Help me solve 11 th q
Uhh its cropped out
There is no 11th question in your picture
midnight basin
also, we dont solve it for you, we guide you to the answer! what have you tried and where are you stuck on?
lavish gull
,rccw
boreal girderBOT
lavish gull
,rccw
strange spear
Holy image quality
boreal girderBOT
valid ore
also, we dont solve it for you, we guide you to the answer! what have you tried ...
I have no idea how to start
calm stirrup
tired willow
I have no idea how to start
try constructing a triangle where the sum of two sides is indeed not bigger than the third, which is its smaller or equal to it
can u notice what happens to the angle opposite to the third side which is fixed?
or you can even fix the two sides and try drawing a third side smaller than the sum of those two
valid ore
try constructing a triangle where the sum of two sides is indeed not bigger than...
But Don don't i need to prove bigger ?
tired willow
But Don don't i need to prove bigger ?
yes, but if we're able to prove that a smaller or equal one can't exist, the opposite is automatically true
if the sum can't be smaller or equal to the third side, it must be greater
valid ore
What abt this method ?
tired willow
What abt this method ?
yuh this is legit
placid whale
What abt this method ?
,rcw
boreal girderBOT
compact pewterBOT
@valid ore Has your question been resolved?
visual flint
compact pewterBOT
visual flint
how do I subtract n1(M)-n0(M)? And find minimum
Both matrices are invertible
Total 2^25-1 invertible matrices
ornate sable
compact pewterBOT
ornate sable
Can someone help with 7 I know we will be using a Taylor approximatation and I can bound the error but then we have an arbitrary n for our sum
How would I show it for all those n? Would I have to use induction?
midnight basin
ah wait you dont even need to use induction or prove for an arbitrary n
applying the taylor series you get an alternating series yes
so all you need to do is use the alternating series estimation theorem which states that the value will always be trapped between any two consecutive partial sums
compact pewterBOT
@ornate sable Has your question been resolved?
ornate sable
so all you need to do is use the alternating series estimation theorem which sta...
I don't know the alternating series estimation
ornate sable
Is there no other way without it?
midnight basin
I don't know the alternating series estimation
midnight basin
Is there no other way without it?
well, i doubt it
ah, wait, hold on there is an algebraic method
ornate sable
?
midnight basin
the top expression is obtained thru polynomial long division
for the bottom, since the expression is strictly positive, and youre integrating from 0 to 1/2 which is also positive, that term is also positive (lower bound)
and since 1+x^4 will be always larger than 1, we can find the upper bound of the bottom term by x^16/(1+x^4) < x^16
so you can bound the integral below, evaluate the integral above, and see if it therefore fulfils the question
ornate sable
Isn't that the error?. Also your Taylor polynomial has three terms. But how can that help if we want for n terms
midnight basin
Isn't that the error?. Also your Taylor polynomial has three terms. But how can ...
but why might you want it for n terms?
and yes, it is the error
ornate sable
Because the function is equal to $\sum_{k=0}^n(-1)^k x^{4k} +E_{4n}(x)$. But we can just choose the n since that proves it for only one n
boreal girderBOT
BigBen
midnight basin
hold on, like, $\frac{1}{1+x^4} = \sum_{k=0}^n (-1)^k x^{4k} + E_{4n}(x)$ is a universally true identity which is true for all n though
boreal girderBOT
wjs
ornate sable
Ye so then we intergate both sides and want to show that our right side is in the bounds of question 7. But how can I compute the sum if it an arbitrary n.
midnight basin
which is why the question gives us the liberty to choose an n and not an arbitrary n lol
(and any reasonable question would give us such liberty)
ornate sable
Ye but how can you know which is n is the right choice
midnight basin
Ye but how can you know which is n is the right choice
pick one that is small enough such that you dont have to compute insane number of fractions
and pick one that is boundable
ornate sable
No but why is that valid. Are you just saying that they will all be the same since lowering the n will increase the error proportionally
So could I not just choose n=1
midnight basin
how would you bound in that case
ornate sable
Bound what?
midnight basin
No but why is that valid. Are you just saying that they will all be the same sin...
what changes with n is our window of uncertainty and the more n is the lesser this uncertainty window is so yes i guess sorta?
im confused
midnight basin
So could I not just choose n=1
lets say you did the taylors expansion on this but used the first term and the remainder term right
ornate sable
$\sum_{k=0}^1 (-1)^kx^{4k} +E_1(x)$
midnight basin
if you pick too small the bounds of your remainder term might be too large such that when you have the bound for the actual integral itself, then
boreal girderBOT
BigBen
midnight basin
if you pick too small the bounds of your remainder term might be too large such ...
you might not be able to prove it
ornate sable
if you pick too small the bounds of your remainder term might be too large such ...
What's the issue with the error term being bigger? It is simply because our approximatation is worse but either they will add up to the function
midnight basin
theres nothing wrong with the error term bigger you just must make sure it fulfils the question's bound thingy
like the problem with n = 1 and n = 2 is that you cannot bound them properly such that you prove the question
midnight basin
**BigBen**
like in the case of this you would have 1/(1+x) ~ 1 right and thats no way to prove the question's statement
so i guess when approaching this question it's really just find the target error and make sure your integral gets bounded, try to not use an n that is too large or else the resulting fraction would be too annoying to evaluate
ornate sable
like the problem with n = 1 and n = 2 is that you cannot bound them properly suc...
Ye so I'm confused about why
No matter what n we choose the Taylor polynomial plus the error will be the same
Then if we integrate them it should be the same no matter what n
midnight basin
yes but the problem is we dont know what the integral of the error term is specifically
our best shot is to bound the error term and integrate the upper and lower bounds
ornate sable
But I tried on desmos and that doesn't hold. What is wrong with my reasoning?
midnight basin
so that we get
a < integral of non-error terms + lower bound of the integral of error term, integral of non-error terms + upper bound of the integral of error term < b
midnight basin
But I tried on desmos and that doesn't hold. What is wrong with my reasoning?
wdym
ornate sable
ornate sable
yes but the problem is we dont know what the integral of the error term is speci...
But isn't the integral of (a+b) = integral (a)+integral (b)
midnight basin
yeah
ornate sable
Ok so f(x)= Taylor polynomial plus error
midnight basin
but we dont know the integral of the error term so our best shot is to bound it is what ive been trying to say 😭
midnight basin
Ok so f(x)= Taylor polynomial plus error
yes
ornate sable
So no matter what n we have this sum is the same
midnight basin
but we dont know the integral of the error term so our best shot is to bound it ...
we dont know (conventionally) how to integral x^16/(1+x^4)
ornate sable
But what I'm saying is if the sum never changes. The the integral can never change
midnight basin
So no matter what n we have this sum is the same
yes, agree, if we pick n = 1 however, we will get an error term we dont know how to integrate
and if we bound this n = 1 error term, we would not meet the condition of
a < integral of non-error terms + lower bound of the integral of error term, integral of non-error terms + upper bound of the integral of error term < b
so you must pick an n that is reasonable to compute and meets this target error
ornate sable
But what I'm saying is if the sum never changes. The the integral can never chan...
Soemthing is wrong we this since it doesn't hold up when I have tried different n
ornate sable
? The highest degree in the polynomial would be 20
midnight basin
huh?
ornate sable
midnight basin
so therefore the remainder term must jump up to 24 no?
every single term in this specific series jumps by a power of 4 not by a power of 1
midnight basin
like if
$$\frac{1}{1+u} = 1 - u + u^2 - u^3 + u^4 - u^5 + \frac{u^6}{1+u}$$
sub u = x^4
$$\frac{1}{1+x^4} = (1 - x^4 + x^8 - x^{12} + x^{16} - x^{20}) + \frac{x^{24}}{1+x^4}$$
boreal girderBOT
wjs
Compile Error! Click the 
reaction for more information.
(You may edit your message to recompile.)
midnight basin
so instead of 21 it should be x^24 here
midnight basin
What is accounting for these differences?
cell 13 is wrong
you have to subtract the remainder
it is an alternating series
ornate sable
Ok so then I just need to find an n where my error is possible to integrate
ornate sable
Or could I just do this
So we see that any n doesn't matter so could I choose an say n=1 to find the bounds of that derivative and then uses the bounds for the error because I don't see can integrate the error
Also there's no issue right for choosing some n? Since it just says it needs to hold for n+1
midnight basin
yes so essentially youd be using lagrange error
ornate sable
or possible to bound within the confines of the question
Well the all seem to be of 0.493958
midnight basin
but i dont think this would be recommended because like
this is a nightmare
for doing all the derivatives
midnight basin
So we see that any n doesn't matter so could I choose an say n=1 to find the bou...
if n here is the order of the derivative then no i think like realistically you might need much more
ornate sable
I'm not following?
What other way do we have? Im not sure how to integrate it we represent it as one of those fractions
ornate sable
Well the all seem to be of 0.493958
Also is this not an issue? How can it fit in the bound if it is the bound
midnight basin
I'm not following?
suppose you pick n = 1, you need to find the maximum bound M of the 2nd derivative on [0, 1/2]
f''(x) according to mathway is $f''(x) = \frac{20x^6 - 12x^2}{(1+x^4)^3}$.
even if you calculate that error perfectly this only proves the integral is less than only proves the integral is less than 0.493967 when the question demands it to be less than 0.493958
boreal girderBOT
wjs
midnight basin
with increasingly large n the derivative gets even more difficult to evaluate and more notorious to compute
midnight basin
Also is this not an issue? How can it fit in the bound if it is the bound
??
im really confused
midnight basin
What other way do we have? Im not sure how to integrate it we represent it as on...
you can use the alternating series estimation theorem, you can use the algebraic trick with n = 4 in taylor's polynomial
ornate sable
??
These are all 0.493958...
So they all must be the same
And they are all greater than 0.493958
ornate sable
So for the trick your saying that instead fo integrating the term we simply bound them
midnight basin
So for the trick your saying that instead fo integrating the term we simply boun...
mhm
ornate sable
So why have it for n=4 though we could do that for any of the errors
ornate sable
why would there be an issue
midnight basin
So why have it for n=4 though we could do that for any of the errors
because i found it convenient to bound
ornate sable
They are all greater than the upper bound for 7
midnight basin
?
ornate sable
0.493958051077>0.493958 no?
ornate sable
So he's truncating it
Ok then so then let's say the upper bound is ≤0.493958051077
For any error isn't it simply bound by the numerator
For n=4 the error is bound by x^16
midnight basin
For any error isn't it simply bound by the numerator
the denominator also bounds it too
x^16/(1+x^4) <= x^16
ornate sable
Ye I mean it is always ≤ the numerator
midnight basin
ah, yeah thats right
ornate sable
But either way if we integrate that bound from 0 to 1/2 and then integrate the Taylor polynomial for n= 4 from 0 to 1/2 we break our bound
midnight basin
which means the question had an error
midnight basin
where did the denominator go to here?
also please state what the purpose of a sent image is and what the underlying thought process behind it is since us helpers wont be able to know what youre trying to accomplish
ornate sable
Well we used the upper bound since we can integrate it
midnight basin
Well we used the upper bound since we can integrate it
right
ornate sable
also please state what the purpose of a sent image is and what the underlying th...
The purpose of the image was to show that if we integrate this upper bound and the sum that with the intergral of the Taylor polynomial we break the upper bound for the question
midnight basin
but the question is wrong in the first place, like, huh??
why are we flogging a dead horse
we break the upper bound for the question, yes, because the question is in and of itself incorrect
ornate sable
Because supposedly it has a solution but I can't understand it
midnight basin
you shouldve sent this long before..
ornate sable
you shouldve sent this long before..
I didn't want to look at the answer
midnight basin
Because supposedly it has a solution but I can't understand it
either ways, this still ignores the fact that this is wrong
ornate sable
His second step seems to be of the form 7.11
midnight basin
the integral in itself is over the bounds
ornate sable
So what of his solution? How did he bound the n+1 derivative?
midnight basin
Because supposedly it has a solution but I can't understand it
hold on
do u see the problem
how is 0.493852 < 0.49375 😭
midnight basin
the second line where they define the error bound they divide everything by (n+1)! this only works when you are using the general Taylor formula but in the first line the author chooses to use the geometric expansion which does not have any factorials
so this solution is also wrong
the factorial came out of nowhere
midnight basin
Well. This has been a disappointing question
in any case... type .close if done, commiserations for your moral luck in meeting this question
ornate sable
Thanks for trying to help me through the problem
midnight basin
anytime
ornate sable
.close
compact pewterBOT
compact pewterBOT
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noble fossil
compact pewterBOT
noble fossil
“Check the relative position of gk and h depending on k.”
so im kind of confused
in the II equation
do you just pretend that if mü is also -2
ancient raft
Lets check if the lines are parralel first
What condition would they hsve to satisfy
ancient raft
Are you the german guy
noble fossil
and the other vector cant be on the other line
wait ur the german guy?
yes
due
im literally rushing linear algebra rn
ancient raft
Im the one that helped you with the grpahs I and II
ancient raft
Yeah
ancient raft
im literally rushing linear algebra rn
Allright
Also hier musst du dir die richtungsvektoren anschauen
ancient raft
kolinear
Ja
Sie müssen Vielfache voneinander sein
Damit die geraden parralel sind
noble fossil
ja genau
aber problem ist
bei der zweiten gleichung tut man also sozusagen einfach so
das mü auch -2 ist?
ancient raft
Nein
noble fossil
weil
noble fossil
k muss -8 sein
nen sorry
ich meine
k muss -4 sein
accsoo
aber k muss ja nicht immer -4 sein
ancient raft
k muss -4 sein
Ja
ancient raft
aber k muss ja nicht immer -4 sein
Korrekt
Ja also nur für k=4 sind sie parallel
-4
Dann müsstest du untersuchen ob die geraden für k=-4 identisch oder echt parallel sind
noble fossil
Dann müsstest du untersuchen ob die geraden für k=-4 identisch oder echt paralle...
ja
alsodann einfach k einsetzen
oder ne
ancient raft
Ja
noble fossil
brauch ich nicht mal oder
ancient raft
Doch
noble fossil
kann ich nicht einfach den stützvektor von gk in h einsetzen
ancient raft
Ok ja
noble fossil
und dann sagt mirdas ergebnis ob es identisch oder echt parallelja
ancient raft
Ja
noble fossil
hä und dann
falls k nicht -4 ist
dafür gucke ich dann ob sie dann windschief sind oder ein schnittpunkt haven?
ancient raft
Ja
noble fossil
scheiße okay
compact pewterBOT
Available help channel!
round plank
Question: Is there a specific reason why primes leave a gap approx equal to n/ln(n)?
compact pewterBOT
Question: Is there a specific reason why primes leave a gap approx equal to n/ln...
silver canopy
Thats not what that is
Its not the gap between primes it's the total number of primes up to n
inner terrace
pi(n) ~ n/ logn
short ferry
Question: Is there a specific reason why primes leave a gap approx equal to n/ln...
but even though its not simple, it is something that can be proved. It's not just some guess based on observation
inner terrace
among first n integers, the proportion of prime is approx 1/logn
round plank
And if we take the limit as n approaches infinity?
round plank
but even though its not simple, it is something that can be proved. It's not jus...
I'm also interested in the assumption of a higher order term that is more efficient and precise than it, if possible.
short ferry
I'm also interested in the assumption of a higher order term that is more effici...
you can check the "more precise estimates"
round plank
I find it interesting how the error term fluctuates between -8 and +13 in the very early iterations. If I may ask; why that wobble?
Though the information you sent it quite sufficient for deep analysis. Thanks.
compact pewterBOT
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kind crane
Why do we have to use the normal approximation ?
how else would oyu solve the problem
pearl elk
how else would oyu solve the problem
Can’t you use binomial?
kind crane
sure you can try
kind crane
sure if you can use a calculator go for it
pearl elk
sure if you can use a calculator go for it
The makexheme doesn’t accept
My method but I Don’t understand why
kind crane
no idea what makexheme is
if they don't let you use calculators then that's why you need to use normal approximation
pearl elk
if they don't let you use calculators then that's why you need to use normal app...
They do that’s the thing my calculator is acceptable
kind crane
where does it say you can't use binomial?
compact pewterBOT
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wintry sleet
Hiya! i have finals comming up in a few weeks, and was woundering if there is some practice i could get a recomendation for? (I am in 9th grade geometery btw)
compact pewterBOT
Hiya! i have finals comming up in a few weeks, and was woundering if there is so...
compact pewterBOT
compact pewterBOT
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winged lion
<@&268886789983436800>
plucky jewel
7th Problem.
compact pewterBOT
7th Problem.
What step are you on?
1. I don't know where to begin.
2. I have begun but got stuck midway.
3. I got an answer but I was told that it's wrong.
4. I got an answer and would like my work checked.
5. I have a question about someone else's work/solution.
6. I have completed the problem and don't need help anymore. Thank you.
7. None of the above
My progress:
rotund sphinx
My progress:
keep going with this
plucky jewel
```
What step are you on?
1. I don't know where to begin.
2. I have begun but go...
2nd step
rotund sphinx
keep going with this
consider how you can factor it
spoiler alert: ||Simon's Favorite Factoring Trick||
plucky jewel
consider how you can factor it
Okay
plucky jewel
consider how you can factor it
I ddint look.the spoiler
But
Aft that blue shade equn I did the possible factorisation right?
rotund sphinx
Aft that blue shade equn I did the possible factorisation right?
factorisation normally refers to product of stuff
here you're adding two quantities in x,y together
so it's not super helpful (unless I'm missing something)
plucky jewel
factorisation normally refers to product of stuff
Hmm so leme think
plucky jewel
so it's not *super* helpful (unless I'm missing something)
Imma check out the spoiler
Sfft
rotund sphinx
so it's not *super* helpful (unless I'm missing something)
see but now you have a $y^2$ term here, so that basically gives you the same "problem" as here
boreal girderBOT
Civil Service Pigeon
rotund sphinx
ideally you'd want to tack on a constant term instead of something in x and/or y
perhaps it would help if you
,texsp ||rewrote it as $x-xy+y=1$||
boreal girderBOT
Civil Service Pigeon
plucky jewel
Hmm
plucky jewel
ideally you'd want to tack on a constant term instead of something in x and/or y
This?
rotund sphinx
here you're adding two quantities in x,y together
again, same issue.
Hint: ||-(-y)=y||
plucky jewel
Hint: ||-(-y)=y||
Woah
plucky jewel
Hint: ||-(-y)=y||
Sry man I ain't that good at this manipulation and all but yeh I got.it
rotund sphinx
good
bright orchid
which question
rotund sphinx
which question
See pins.
bright orchid
nice
compact pewterBOT
I can solve the problem now.
If you are done with this channel, please mark your problem as solved by typing .close
plucky jewel
<:catthumbsup:614540188747563008>
Hey But I'm getting wrong answer.
rotund sphinx
rip
rotund sphinx
!show
compact pewterBOT
Show your work, and if possible, explain where you are stuck.
rotund sphinx
wrong picture?
rotund sphinx
well you got x=1 or y=1
plucky jewel
Hmm
rotund sphinx
I'm asking how you got from that to your final answer
plucky jewel
So
plucky jewel
I'm asking how you got from that to your final answer
Got that?
rotund sphinx
By x=y=1
$(x-1)(y-1)=0$ does not imply both $x=1$ and $y=1$. In general,
$$pq=0 \implies p=0 \textbf {or } q=0.$$
So you have $x-1=0$ \textbf{or} $y-1=0$. Aka $x=1$ \textbf{or} $y=1$.
boreal girderBOT
Civil Service Pigeon
rotund sphinx
yeah there's some kinda casework
plucky jewel
and see what happens
rotund sphinx
yeah there's some kinda casework
tbf ||the cases for x=1, y=1 are basically the same up til labeling (why?) so it's not too bad||
but it's just donkey work from here, so imma go now
plucky jewel
but it's just donkey work from here, so imma go now
Hmm bye
Can some1 tell how to make the case
<@&286206848099549185>
How to make cases for x=1 or y=1
compact pewterBOT
@plucky jewel Has your question been resolved?
dull moon
How to make cases for x=1 or y=1
I mean you already have $1= x+y-xy$, now rewrite it as $x+y-xy-1=0$ and factorize it.
boreal girderBOT
Annie Maqionde
hazy obsidian
Suppose that $N = \overline_{a_na_{n-1}...a_0}$ is an $n$ digit number in base 10, for which $N = 2^k$ for some $k$. Show that any other permutation of $N$ with $n$ digits is not a power of 2.
compact pewterBOT
Suppose that $N = \overline_{a_na_{n-1}...a_0}$ is an $n$ digit number in base 1...
boreal girderBOT
Copter
Compile Error! Click the reaction for more information.
(You may edit your message to recompile.)
hazy obsidian
how do i do this?
elfin finch
Wait isn't n to 0 n+1 digits?
compact pewterBOT
@hazy obsidian Has your question been resolved?
hazy obsidian
Wait isn't n to 0 n+1 digits?
oh right mb
compact pewterBOT
@hazy obsidian Has your question been resolved?
violet magnet
Can you do modular arithmetic?
dull moon
Suppose that $N = \overline_{a_na_{n-1}...a_0}$ is an $n$ digit number in base 1...
This may or may not work, but consider a contradiction, by considering that a number is equal to the sum of its digits modulo 9, and that under the permutation, the sum of digits is invariant
when is $2^m \equiv 2^k \pmod 9$?
boreal girderBOT
Annie Maqionde
violet magnet
I don't think if I remember it right but I think any numbers with base 10 is congruent the sum of its digits in mod 9.
Ah then just test the cycle for 2^k mod 9, perfect.
hazy obsidian
hmm
compact pewterBOT
@hazy obsidian Has your question been resolved?
boreal girderBOT
Coder decoder
marble adder
compact pewterBOT
marble adder
how to do this?
fallow fractal
have you tried anything?
marble adder
im not sure where to start
grand tundra
It is 22√155
hearty pike
I don't think there is a particular approach, it's just calculation I think.
fading flicker
how to do this?
sqrt(a) x sqrt(b)=sqrt(a x b)
fading flicker
On squaring we get 484*155
don't give the answer out immediately
marble adder
sqrt(a) x sqrt(b)=sqrt(a x b)
ill try this
fallow fractal
It is 22√155
this is not true i believe 
marble adder
i looked at the answer sheet i think it says 75020 but im not sure how to get there
fading flicker
ill try this
note also that ab x cd=abcd so 11 x sqrt5 x 2 x sqrt31 = 11x2 x the square roots multiplied
grand tundra
don't give the answer out immediately
I didn't give the final answer
fading flicker
I didn't give the final answer
you basically just did
they don't know how to get there, you were unclear.
grand tundra
Sorry.
hearty pike
i looked at the answer sheet i think it says 75020 but im not sure how to get th...
Since it's of the form [a×sqrt(b) × c×sqrt(d)]² it just becomes [ac×sqrt(bd)]² which is just a²c²bd
grand tundra
i looked at the answer sheet i think it says 75020 but im not sure how to get th...
Sqrt(2)*Sqrt(3) = Sqrt(6) . Try that for any two numbers whose product is the third one.
marble adder
this is apparently the answer
fading flicker
this is apparently the answer
it is shown, could you clarify what you may be confused about?
hearty pike
this is apparently the answer
Yeah since (a×b)^c = a^c × b^c
hearty pike
👍
compact pewterBOT
@marble adder Has your question been resolved?
compact pewterBOT
compact pewterBOT
Available help channel!
wary mulch
What is this exercise asking us to do?
compact pewterBOT
What is this exercise asking us to do?
upper bane
What’s -> R
the codomain of the function
wary mulch
How is the interval the function
upper bane
the function is f though?
wary mulch
the function is f though?
Yeah
Also how do I write mathematic proofs because I have to be writing proofs using thereoums which is pretty intimidating for me
fading flicker
What is this exercise asking us to do?
Note that e^x>1+x => e^x-1-x>0, now, let f(t)=e^t-1-t and f(t)>0 for t values ranging from 0 to x
wary mulch
Any advice would be appreciated
fading flicker
Also how do I write mathematic proofs because I have to be writing proofs using ...
for 1
don't be intimidated
it's just jargon spam and logic
understanding is key
wary mulch
What are the rules
fading flicker
you can watch a video on it they probably have better analogies and stuff
wary mulch
I never wrote a proof in my life
Maybe if I try and do one
I should get the gist of it
fading flicker
What are the rules
wdym rules? in proof? nothing really, just don't say anything unrelated to what you're proving and don't say anything wrong or would not be a logical step in whatever you're doing
I'm saying this as a person who doesn't really write proofs
sum1 else who smarter than me correct me
-# I just spam hence and therefore
wary mulch
What if ur trying to prove something but your not progressing because your just writing in equivalent forms if that makes sense
fading flicker
What if ur trying to prove something but your not progressing because your just ...
most people don't do that for exams
wary mulch
then how do we go about it
proud orchid
Note that e^x>1+x => e^x-1-x>0, now, let f(t)=e^t-1-t and f(t)>0 for t values ra...
how do you know f(t) > 0 for 0 ≤ t ≤ x
thorny kestrel
What is this exercise asking us to do?
the exercise is asking you to show the inequality, nothing more
how would you answer the question if you didn't have to write a "proof" for it
wary mulch
Well x+1 is just a constant line and increases at a constant rate
but e^x increases at a multiplying rate
proud orchid
this only tells you that e^x eventually goes above x + 1
wary mulch
Yea
thorny kestrel
but e^x increases at a multiplying rate
but something like 2^x is also a multiplying rate, yet it goes under the line x + 1
so the argument doesn't work
wary mulch
but something like 2^x is also a multiplying rate, yet it goes under the line x ...
It doesn’t?
thorny kestrel
proud orchid
you can verify that ✓2 < 3/2 algebraically
wary mulch
It is bigger than x+1
proud orchid
(the point is we don't have pictures)
thorny kestrel
It is bigger than x+1
not for all x>0
thorny kestrel
sure but this is not what is asked
proud orchid
it's still a useful thought
thorny kestrel
you said e^x > x+1 for all x > 0 because e^x is multiplicative and x+1 is a line, but if that was really the reason, then that would mean that the same applies for 2^x which is also multiplicative
proud orchid
they did not say this
wary mulch
I said something similar but not when x>0
proud orchid
they don't understand what the proof is doing and so they're thinking about the problem on their own to try to motivate the proof
wary mulch
Yep
wary mulch
What is this exercise asking us to do?
Or you can just explain me the proof
proud orchid
think about how you might use calculus to detect if there is some t where f(t) < 0
when you work with e^x the only two things you know are its power series and the fact that taking a derivative gives itself
so think about derivatives
sonic walrus
Or you can just explain me the proof
the mean value theorem states that if $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$ then there exists some number $c$ in $(a,b)$ such that $$\frac{f(b)-f(a)}{b-a} = f'(c).$$
boreal girderBOT
melon
sonic walrus
the proof is first defining f then stating that it satisfies the hypotheses so we can actually apply the theorem
sonic walrus
How is the interval the function
x is being treated as a real number in the proof, maybe it would make more sense for you to see it as
consider f:[0,a] -> R by f(x) = e^x - 1 - x.
the variable in their function is t and so anything that isn't t is just a constant
proud orchid
i think it's just they haven't really seen the domain codomain notation before
wary mulch
Yeah I did but it really confuses me
proud orchid
let f(x) = e^x - x - 1. the core idea of the proof is that you somehow want to use the intuition that f'(x) > 0 implies f is always increasing
wary mulch
But I havnt seen it being defined on the interval
proud orchid
because you know that f(0) = 0
so firstly why is f'(x) > 0 for all x > 0
this is something you should compute
compact pewterBOT
@wary mulch Has your question been resolved?
dawn elbow
I have to find out the slope but don't know how to approach
compact pewterBOT
I have to find out the slope but don't know how to approach
sonic wyvern
,rccw
boreal girderBOT
sonic wyvern
the line has what condition?
is it tangent to the circle? intersects the circle at two points? does not intersect the circle? something else?
dawn elbow
it wasnt given in the question
sonic wyvern
then for all we know this is just a random line though.
glacial onyx
the q seems incomplete
sonic wyvern
it can be any line.
glacial onyx
because the point of tangency isnt given
sonic wyvern
any of the answers can be right.
dawn elbow
it possibly can't be a tangent though, but is there any way to solve it
sonic wyvern
I don't think an incomplete question can be solved.
dawn elbow
well what if it was a tangent indeed?
sonic wyvern
tangent at where?
glacial onyx
because the point of tangency isnt given
^^
dawn elbow
to the circle given ofc, if only the radius could've been 2 units
glacial onyx
to the circle given ofc, if only the radius could've been 2 units
yes but you need the coords
also the line can basically be anything a diameter, chord, tangent
so the question isnt clear enough
sonic wyvern
I still stand by my conclusion that this seems inconclusive.
dawn elbow
tangent to the circle at where?
well if the circle was of radius 2 u and then the y intercept is already given as 2, so the tangent could be possible at (0,2)
silver canopy
to the circle given ofc, if only the radius could've been 2 units
put simply: your equation can be represented as [
c^2 = 25 \2(1+m^2)
]
As there are two unknowns in the equation, we can not determine a unique value for $m$
boreal girderBOT
dawn elbow
so the question isnt clear enough
i understand that, i was confused too
sonic wyvern
well if the circle was of radius 2 u and then the y intercept is already given a...
but that assumes c is also given. and that also assumes that the circle is really tangent to the circle at (0, 2).
dawn elbow
put simply: your equation can be represented as \[
c^2 = 25 \2(1+m^2)
\]
As the...
isn't c 2?
sonic wyvern
no.
silver canopy
assuming tangency*
dawn elbow
i apologize, really, the c was indeed 2, i couldnt find the exact question so i wrote it down to share
sonic wyvern
but hang on.
silver canopy
but
sonic wyvern
the radius of this circle is 5, isn't it?
silver canopy
i dont think this is doable
glacial onyx
slope is coming negative
sonic wyvern
and it's centered at the origin.
glacial onyx
which isnt possible
silver canopy
because it violates the tangency
sonic wyvern
how can the y-intercept be inside the circle?
silver canopy
yeah it would not work
dawn elbow
thats the issue, i think the question is wrong, but then i didn't want to assume considering i dont know this topic well enough
silver canopy
@dawn elbow your problem is really badly formulated right now. Do you have a reliable source to confirm what the original question is?
sonic wyvern
actually I'll just leave it to the two of you, too many people here right now. all the best OP!
dawn elbow
<@1164565746790183012> your problem is really badly formulated right now. Do you...
Actually no, i was shared this same info by someone else to solve it
and before concluding it wrong, i wanted advice from someone better
glacial onyx
Actually no, i was shared this same info by someone else to solve it
the info is incorrect then
flint cliff
compact pewterBOT
silver canopy
Its the rule when you have a repeated root
Like
,, \41{x(x+1)^3} = \4A{x+1} + \4B{(x+1)^2} + \4C{(x+1)^3}+ \4Dx
As an example
boreal girderBOT
flint cliff
ah okay i see
and here why is the num s+1
shouldn't it be 1
or its been factored out
silver canopy
shouldn't it be 1
As a rule, when doing PFD, the resultant dsnominators will be of degree n and the numerators will be of degree n - 1
So if your denominator is linear, like s - 1, which is degree 1, your numerator is gonna be of degree 0
Which is a constant
Likewise, if your denominator is a quadratic, like in your problem, which is degree 2, your numerator is gonna be of degree 1, meaning a linear equation in the form of As - B
compact pewterBOT
@flint cliff Has your question been resolved?
echo merlin
compact pewterBOT
echo merlin
just wondering, since our prof has abandoned us 2 days before the exam ( 😭) the solution for this exam q is convoluted. wondering if someone can check my solution
$R=f(X)\cup{(0,0)}$ is compact iff it is sequentially compact. let $(y_n)_{n=1}^{\infty}$ be a sequence in $R$. note either $y_n=(0,0)$ or $y_n=f(t_n)$ for some $t_n\in X$, so $$d(y_n, (0,0))\leq \sqrt{\left(\frac{\cos(t_n)}{t_n}\right)^2+\left(\frac{\sin(t_n)}{t_n}\right)^2}=\frac{1}{t_n}\leq 1$$
so the sequence is bounded. by bolzano-weierstrass, a convergent subsequence exists, $Y$ is sequentially compact
boreal girderBOT
lyric
compact pewterBOT
@echo merlin Has your question been resolved?
silver canopy
$R=f(X)\cup\{(0,0)\}$ is compact iff it is sequentially compact. let $(y_n)_{n=1...
Im a bit confused
BW guarantees that you have a convergent subsequence, but it doesnt necessarily guaantee that the limit point is in R does it
echo merlin
oh good point
silver canopy
For R to be sequentially compact, the lmit of the subsequence must belong to R
So you gotta prove that
compact pewterBOT
dull moon
<@&268886789983436800>
upper bane
guess we are back in scam season after a couple of weeks
dull moon
the amount of times i saw the same thing its becoming boring
compact pewterBOT
silver canopy
guess we are back in scam season after a couple of weeks
it stopped?
upper bane
for a couple of weeks
silver canopy
X.E. on that
upper bane
the last spike was roughly on the date of this message
glacial onyx
the amount of times i saw the same thing its becoming boring
fr
compact pewterBOT
Available help channel!
pearl elk
compact pewterBOT
cerulean bramble
Erm, so you know how we can do tan as a fraction of two trig functions? @pearl elk
cyan sluice
Erm, so you know how we can do tan as a fraction of two trig functions? <@107597...
-# sorry to interrupt, you mean tanh right
cerulean bramble
-# sorry to interrupt, you mean tanh right<:cat_thonk:586238610957926430>
-# Well, yes, I do, but I'm trying to intuit that response
cyan sluice
-# Well, yes, I do, but I'm trying to intuit that response
-# oh oh ok, good strategy
cerulean bramble
A similar identity exists w/ hyp. trig. functions
-# Alternatively, use the exponential definition of sinh and cosh, because you then just have a quadratic (admittedly hidden but it is still a quadratic)
-# This would could be more helpful because you can more clearly explain why there aren't any solutions, per what the question asks of you
worn cloud
you can write coshx as (e^x+e^-2)/2 and sinhx as (e^2-e^-2)/2
cyan sluice
you can write coshx as (e^x+e^-2)/2 and sinhx as (e^2-e^-2)/2
-# em, that -2 is sholud be -x
cerulean bramble
-# Alternatively, use the exponential definition of sinh and cosh, because you t...
-# And also that's what I'm alluding to here lol
cyan sluice
-# This ~~would~~ could be more helpful ~~because you can more clearly explain ...
-# iib has a solution?
worn cloud
yes x my bad
cerulean bramble
-# iib has a solution?
oh right mb
compact pewterBOT
@pearl elk Has your question been resolved?
wheat pasture
Is it related to ii though?
(only insofar as to remember the rule, from what I remember 
)
wheat pasture
I’ve done this
I also think you may want to resub u and simplify that? 
wheat pasture
Either way, beat me to it and you have the idea 
wheat pasture
Maybe differentiate what you have again? 
pearl elk
Hm
cyan sluice
or multiply the sqrt(2-x^2) on the other side first /
~~either way either a quotient or product rule
pearl elk
And then. I just show that they’re equal ?
Ohhh i see
Thank you so much everyone
.close
compact pewterBOT
compact pewterBOT
Available help channel!
dapper ravine
compact pewterBOT
dapper ravine
how do i do part d?
reef oasis
how do i do part d?
Find a vector that satisfies both conditions
dapper ravine
what conditions?
reef oasis
for plan 1 and 2
take a vector from plan 2 for example
whaat condition it must verify to be in plan 1 aswell ?
finite terrace
be perpendicular to the normal of plane 1?
Think about it like this. If I have two lines, if they share a point, what has to be true about those lines?
finite terrace
they intersect
Exactly!!
dapper ravine
gotchu
finite terrace
In order for a line on one plane
To be on another plane
What do those planes have to do
finite terrace
to be apart of 2 planes?
Yep
finite terrace
the planes would have to be rotations of each other
I would argue that if the planes intersected (much like our lines example) they would share a line in common
dapper ravine
right right
finite terrace
right right
Do you see how these two planes share a line (often called the line of intersection)
dapper ravine
yeah i get that
im just trying to think about how we get a vector representation of that line 😭
finite terrace
im just trying to think about how we get a vector representation of that line 😭
In 3D, what two things do we need for a line?
dapper ravine
well
depends on which representation
you can do it with just a normal
wait
what am i on about
sorry
you need 2 direction vectors
and a point
😭
no
1 point
1 direction vector
im tripping
finite terrace
How do you think we’re gonna find those two things?
finite terrace
the direction vector? uhhh i have no clue
The direction vector is just going to be the cross product of your normal vectors
dapper ravine
is there a way to do it without the cross product? its not part of our spec 😭
and its one day before the exam
😄
im cooked bro
🥀
finite terrace
is there a way to do it without the cross product? its not part of our spec 😭
You guys didn’t learn how to do the cross product?? Are you sure?
finite terrace
that i dont do
And you don’t need to, lol. I just realized, you can parametrize one plane
compact pewterBOT
compact pewterBOT
Available help channel!
woeful silo
Find the range of values of y = (x^2 - x + 1)/(x^2 + x + 1) if x takes real values.
compact pewterBOT
Find the range of values of y = (x^2 - x + 1)/(x^2 + x + 1) if x takes real valu...
woeful silo
y is defined at all x as x^2 + x + 1 = 0 has no solution.
solemn matrix
wdym ranges of values
like you want us to find like y < 0 or some stuffs like that?
livid crypt
Find the range of values of y = (x^2 - x + 1)/(x^2 + x + 1) if x takes real valu...
I will help you...
solemn matrix
first of all numerator and denominator both are positive
orchid vine
wdym *ranges* of values
range of the function
livid crypt
Find the range of values of y = (x^2 - x + 1)/(x^2 + x + 1) if x takes real valu...
Assume the whole equation as y
solemn matrix
range of the function
hm ic
livid crypt
Now cross multiply
orchid vine
Now cross multiply
yea this is the only method lol
woeful silo
x^2 (y-1) + x(y + 1) + (y-1) = 0
orchid vine
x^2 (y-1) + x(y + 1) + (y-1) = 0
since x takes real values what can you say about the discriminant
livid crypt
Please do like this
solemn matrix
Damn
solemn matrix
yea this is the only method lol
I don't rlly think so
woeful silo
since x takes real values what can you say about the discriminant
I believe the solution interval of D >= 0 is a subinterval of the range of y.
woeful silo
I don't rlly think so
yeah lol
livid crypt
Hope it helps...
compact pewterBOT
Hope it helps...
As a helper, please do not give out answers that could be copied as a homework solution. Have the student work through the problem themselves and guide them along the way.
livid crypt
compact pewterBOT
@woeful silo Has your question been resolved?
compact pewterBOT
compact pewterBOT
@old rivet Has your question been resolved?
fair raft
What does P^(-1) mean when P is a prime ideal?
the inverse of P in K?
old rivet
the inverse of P in K?
So basically the "regular" inverse of the ideal P?
So that P(-1)P = K?
compact pewterBOT
@old rivet Has your question been resolved?
old rivet
.close
compact pewterBOT
fair raft
<@&268886789983436800>