For this example
#help-27
supple knot
violet vault
Yeah
Like you're saying my junk in the middle is all of my alpha/2, or am I wrong?
I'm looking at the tutorial and they didn't really explain how they got their columns
violet vault
Each row is my z-score, then, right?
I don't understand the first row, no
And they don't go over that in the tutorial
They were like "we're looking up this row", but there's not an explanation for how they chose that column
"Example 2
The height of plants in a certain garden are normally distributed with a mean of μ = 26.5 inches and a standard deviation of σ = 2.5 inches. Approximately what percentage of plants are greater than 26 inches tall?
Step 1: Find the z-score.
First, we will find the z-score associated with a height of 26 inches.
z-score = (x – μ) / σ = (26 – 26.5) / 2.5 = -0.5 / 2.5 = -0.2
Step 2: Use the z-table to find the percentage that corresponds to the z-score.
Next, we will look up the value -0.2 in the z-table:"
To me they don't explain the column thing at all
supple knot
-0.2 = -0.20
violet vault
Oohh
Wait
So for my z value here, I have 0.025, so you're saying I'll find that value at where my first decimal place is 1.9 and my second decimal place is 0.06 to get 1.96 when you add them?
supple knot
Yes 1.9+0.06=1.96
1.9 being from the first column
And 0.06 is from the first row
Then 0.025 is the area to the right
So 1-0.025 is the area to the left
violet vault
Got it! Fuck they should seriously just put a legend on there
That was the biggest reverse engineering of the century
Thank you so much
That makes so much sense
supple knot
Glad to help. We've all been where you are
violet vault
Yeah, I mean I'm a junior so it's not like I don't know how to study. I'm just stressed with finals and work and exams and projects. Everything is due all at once
Thank you
How do I give you karma on here?
!done
devout snowBOT
If you are done with this channel, please mark your problem as solved by typing .close
violet vault
.close
grand warren
Let q be a definite quadratic form, and I want to show that it is either positive definite or negative definite.
\ I then assume that q is neither positive nor negative definite, which leads to a contradiction. So $\exists x \neq 0, q(x)>0$ and $y \neq 0$ such that $q(y)<0$.
\ Let's posit $\varphi: t \mapsto q(tx+(1-t)y)$ with $x$ and $y$ well chosen.
Thus $\varphi(t)=t^2 q(x)+(1-t)^2 q(y)+2t(1-t)b(x,y)$ where b is the polar form of q.
\
We have $\varphi(0)=q(y)<0$ and $\varphi(1)=q(x)>0$, $\varphi$ being continuous we deduce by the intermediate value theorem that $\exists t_0 \in (0,1) , \varphi(t_0)=0$. q being defined, it is necessary that $t_0 x+(1-t_0)y=0$, and here I'm stuck...
devout snowBOT
Let q be a definite quadratic form, and I want to show that it is either positiv...
grand warren
if x and y are collinear, $\exists \lambda, y=\lambda x$ and $q(y)=\lambda^2 q(x)<0$ which is not possible because $q(x)>0$
woven radishBOT
tm
ancient trellis
hey guys, this limits goes to 0, right?
devout snowBOT
hey guys, this limits goes to 0, right?
ancient trellis
wolfram gave me an error
supple knot
,w lim n to inf (e)^n * n! / n^n
ancient trellis
but, if I'm right, I think that, using the alternate series test, this goes to 0
woven radishBOT
supple knot
,w lim n to inf (e)^n * n! / n^{n+1/2}
woven radishBOT
devout snowBOT
@ancient trellis Has your question been resolved?
ancient trellis
hey guys, if I use n-th term test with x=e, the limit goes to inf
then, the series divergers for x=e
by comparison, it also happens in x=-e
so the radius of converge is (-e,e)
yeah, got it
devout snowBOT
supple knot
yes
devout snowBOT
wicked turtle
what's the context here, looks like they are expecting you to compute a numerical approximation?
yes but i assume this means you're writing a computer program to calculate it?
supple knot
would help if there was more context to the problem like bungo asked
devout snowBOT
@wintry umbra Has your question been resolved?
supple knot
like what topics/methods you've learned recently for this problem
you learned all of those just recently?
devout snowBOT
@wintry umbra Has your question been resolved?
devout snowBOT
@wintry umbra Has your question been resolved?
devout snowBOT
devout snowBOT
Available help channel!
stuck vortex
devout snowBOT
stuck vortex
how
fresh fox
do you know how area and volume scale with length?
if i have a square and i make each side twice as long, how much bigger will the area be?
if i have a cube and i make each side twice as long, how much bigger will the volume be?
stuck vortex
if i have a square and i make each side twice as long, how much bigger will the ...
4 times larger?
fresh fox
4 times larger?
correct
what about the volume question
stuck vortex
and volume becomes 8× larger
stuck vortex
do u see any pattern
no
fresh fox
no
if the length increases by n
the area goes up by n² and the volume by n³
this works for any shape
eg a circle's area is pi r²
if u double the radius u get pi (2r)² = 4 pi r²
so if the length of the statue went from 12 to 15
what factor did it increase by?
and the volume (and therefore the mass) will increase by that factor cubed
devout snowBOT
@stuck vortex Has your question been resolved?
stable flare
@stuck vortex Do you still need help?
solid osprey
im stuck on (j) i dont see any patterns, any hints?
devout snowBOT
im stuck on (j) i dont see any patterns, any hints?
solid osprey
normally when i want to prove an integer is good, i split it into a good integer and a bad integer, then try to prove that they combined is good (ok this doesent make sense lemme write an example)
M=30=5×6
by CRT, n^n=1 mod 5 and n^n=1 mod 6, since M=6 is good then n^n=1 mod 6 => n=1 mod 6, trying out numbers for n^n=1 mod 5 the ones that dont imply n=1 mod 5 are all in the form 4+20k, 8+20k, and 12+20k for all integers k, and these are all even which share a factor with 6 so if you choose one of them to be n, then its impossible for n^n to be 1 mod 6, so the only other possibility is that it implies n=1 mod 5 so M=30 is good
for proving bad i try to find a contradiction
devout snowBOT
@solid osprey Has your question been resolved?
topaz beacon
9 should work
topaz beacon
,w (9^9) mod 14
woven radishBOT
topaz beacon
oh lemme look
solid osprey
smh how did i even mess that up
topaz beacon
youre familiar with fermats little theorem right?
im gonna start spitballing while i figure this out
solid osprey
yea
topaz beacon
ok, so lets think about totients then
whats special about totients of all odds up to 14 that 14 breaks?
solid osprey
i conjecture that a good M cant have a prime factor>3
topaz beacon
all evens whoops
topaz beacon
i conjecture that a good M cant have a prime factor>3
step d shows you 10 is good
solid osprey
whats special about totients of all odds up to 14 that 14 breaks?
honestly no idea
solid osprey
did you just guess and check to get 9 mod 14?
yes
topaz beacon
ok lets look at that a little more
solid osprey
i just checked 3 5 9 and it worked for 9
topaz beacon
by flt if a^b = 1 mod 14, we can reduce a mod 14 and b mod 6
solid osprey
mhm
solid osprey
i think because of that it cycles trough every odd a and b and prolly one of them is bound to be 1 mod 14?
solid osprey
i dont actually understand it well enough
i kinda half wrote each step but its really terribly written
restive river
Is this occupied
topaz beacon
so in order to have an odd number, we must have ord of it be 3
an odd number with an odd ord
proof
odd coincidence
no pun intended
solid osprey
so in order to have an odd number, we must have ord of it be 3
to have an odd number to be good?
topaz beacon
to have an odd number disprove an even number from being good
solid osprey
oh
why do you need the ord of it to be 3? are you taking M=14 for the example and 3|6?
topaz beacon
if you want n^n congruent to 1, you need n odd
since we reduce the power by an even number, we need it to be congruent to 3 mod 6
or 1
but a power of 1 would require a 1 as the base
solid osprey
ok and 5 mod 6 wont work because the ord divides 6 and the ord also has to divide 5 mod 6, and the gcd of them both is 1 but the ord cant be 1 so it has to be 3 mod 6 right?
topaz beacon
yes
and i was looking through cubes that happen to work and 9^3 worked
so that was kinda lucky
i dont really know where to go
solid osprey
hm
topaz beacon
so my thought process was if phi(M) was a power of two this cant happen
but then M=42 is fine
oh
thats because 3|42 and phi(42)=12
so we must have a power of 3 so that we have an odd order
solid osprey
you mean like the order has to be 3 for it to be an odd order?
topaz beacon
but then x^3 = 1 mod 42 cant happen if x is a multiple of 3
topaz beacon
you mean like the order has to be 3 for it to be an odd order?
yes
solid osprey
well if the odd prime factors of phi(M) cant divide M then M is bad right?
topaz beacon
thats what it looks like
solid osprey
but is that what it wants us to find tho cause seems hard to use
topaz beacon
i dont know, but this feels like a step in the right direction
solid osprey
it does, maybe we can change it to be something more general and easier to put in use?
topaz beacon
maybe thats enough for what we need if 2013 acts nice enough
because part k narrows down our search to 30 possibilities
,w factor 2013
woven radishBOT
solid osprey
totient(2013)=1200
yeah so (k) is saying if you add a factor of 2 and 5 (aka multiply by 10) then its good
topaz beacon
well if the odd prime factors of phi(M) cant divide M then M is bad right?
this might just be prime factors in general
2 is a little touchy
solid osprey
ok yeah this looks good rn
solid osprey
this might just be prime factors in general
oh yea, M is always divisible by 2 since M is even
solid osprey
yeah so (k) is saying if you add a factor of 2 and 5 (aka multiply by 10) then i...
but multiplying by 10 changes phi
like m=10k for k from 1 to 30, phi(2013m)=phi(20130k)=
phi(20130)phi(k)gcd(20130,k)/phi(gcd(20130,k)) only changes when k is either a multiple of 3 or 11
actually nvm
topaz beacon
i think at this point, you should just test each case
some are pretty easy like 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 110, 120, 150, 160, 180, 200, 210, 220, 240, 250, 280, 300
thats a good majority of them
if it turns out this was for an aime problem id start panicking
oh yeah also 130 and 140
and 170
almost all of them work
solid osprey
like m=10k for k from 1 to 30, phi(2013m)=phi(20130k)=
phi(20130)phi(k)gcd(20130...
the extra factor of gcd doesent do anything, /phi(gcd(20130,k)) should be fine as it actually doesent increase the prime factors of phi(2013m), the only worrying thing is phi(k) but it seems like 1<=k<=30 the only troublemaker ks k=29 as phi(29)=28=7×4 and 2013×10×29 isnt divisible by 7, so thats prolly (l)
topaz beacon
oh nice
solid osprey
okay thank you!!
solid osprey
indeed
without the walktroughs and you id never figure this out in a million years lol
ty 
.solved
devout snowBOT
topaz beacon
id never figure this out without the walkthroughs either lol
youre welcome
in case you ever want my help with contest math in the future, just ping
devout snowBOT
Available help channel!
tough vigil
!help
devout snowBOT
To ask for mathematics help on this server, please open your own help channel or help thread. See #❓how-to-get-help for instructions.
tough vigil
north roost
so
tough vigil
P is mentioned but not used
it used
north roost
since the sets are distinct (excluding the 1)
isnt it just 1 + number of primes less than 50?
supple knot
How is a union of a set just a number?
north roost
n()
supple knot
What does n() mean
supple knot
Still flawed then
north roost
@tough vigil is n() defined somewhere
fossil locust
P is mentioned but not used
Oh wait
fossil locust
isnt it just 1 + number of primes less than 50?
Yep
Union of {1,2}; {1,3}, {1,5}...
rare kernel
devout snowBOT
rare kernel
can someone help with this
not able to understand the different cases i have to take
like do i have to consider 7+2 and 7+1+1 or just 7+2
patent token
So, till (1+x^9) we can get x^9 after that, we cant so the sequence important to us is (1 + x)(1+x^2)... (1+x^9)
rare kernel
So, till (1+x^9) we can get x^9 after that, we cant so the sequence important to...
yeah got that
patent token
after we just have to form cases
patent token
like do i have to consider 7+2 and 7+1+1 or just 7+2
1+1 isnt possible as there is only one (1+x)
rare kernel
oh.
patent token
so one case is 7+2
patent token
no worries have a great day
rare kernel
no worries have a great day
btw are u a jee aspirant by any chance
patent token
this year
rare kernel
oh atb man 🙏
rare kernel
.close
devout snowBOT
devout snowBOT
Available help channel!
tough vigil
!help
devout snowBOT
To ask for mathematics help on this server, please open your own help channel or help thread. See #❓how-to-get-help for instructions.
tough vigil
autumn girder
!showyourwork
devout snowBOT
Show your work, and if possible, explain where you are stuck.
autumn girder
The arrow just means that the relation goes from integer to integer
$R = \left{(x, y)| y = 2 x + 1, x \in \bZ, y \in \bZ\right}$
woven radishBOT
@autumn girder
autumn girder
It's the same as this if thus makes more sense
Have you studied relations @tough vigil
?
pseudo basin
!showyourwork
it is just !show btw
devout snowBOT
@tough vigil Has your question been resolved?
rare kernel
find n
autumn girder
!status
devout snowBOT
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
potent tusk
this is definitely some jee brainrot
autumn girder
!showyourowkr
devout snowBOT
Show your work, and if possible, explain where you are stuck.
rare kernel
this is definitely some jee brainrot
😂
i just wrote all the fractions as
1-1/2, 1-1/4,1-1/8
and then tried multiplying -1 inside
and got a total of nothing
@autumn girder could u help me w what to do
autumn girder
Have you tried using the identity $(1 - x)^n = \sum_{i = 1}^n(-1)^i{n \choose i}x^i$?
woven radishBOT
@autumn girder
autumn girder
Well, you can use the distributive property of multiplication over addition to take the added terms out
and then apply this
rare kernel
Well, you can use the distributive property of multiplication over addition to t...
huh wait i didnt understand this
how do i take the added terms out
autumn girder
huh wait i didnt understand this
$a(b + c) = ab + ac$
woven radishBOT
@autumn girder
rare kernel
wait so u want me to distribute that over 5 terms?
rare kernel
dude..
autumn girder
<@946379454056386560>
which you can then use this on
stone stump
oh no, 5 terms which look basically the same
autumn girder
I mean doing that you get $\left(\frac 1 2\right)^n + \left(\frac 1 4\right)^n + \left(\frac 1 8\right)^n + \left(\frac 1 {16}\right)^n + \left(\frac 1 {32}\right)^n$
woven radishBOT
@autumn girder
autumn girder
Well try completing the problem first
devout snowBOT
autumn girder
yeah sorry judged too fast
np
devout snowBOT
Available help channel!
north roost
devout snowBOT
north roost
how do we know that ${\bar{A} \neq \emptyset}$?
woven radishBOT
k
sand quarry
how do we know that ${\bar{A} \neq \emptyset}$?
Who says it is
pseudo basin
how do we know that ${\bar{A} \neq \emptyset}$?
we don't
pseudo basin
it exists because it's given a clear definition
it is the set of all points x ∈ A which satisfy x ∉ g(x)
north roost
so g(x) is some function that maps each element of A to p(A), returning a set. we define A bar as a set containing the elements of A not found in any set of the range of g(x). am i understanding this correctly
and A bar must exist because it is clearly defined
?
so we use A bar as a counterexample that g(x) is not surjective
is this a non-constructive proof, im new to proof as well as this
😭
pseudo basin
we define A bar as a set containing the elements of A not found in any set of the range of g(x). am i understanding this correctly
no
you're not understanding it correctly
if you want though i can explain the same idea in more hands-on terms
would you like that? @north roost
north roost
yes plz
pseudo basin
ok rigth
right*
imagine A is a set of people
$\wp(A)$ is then the set of all possible group photos that can be made with members of $A$, including one empty photo'' i.e. $\rien$ as well as portraits'' i.e. singleton subsets of $A$
woven radishBOT
Ann
north roost
mhm
pseudo basin
now $g: A \to \wp(A)$ is an assignment of people to group photos. in other words, you can think about it like \textbf{giving} each person one group photo from the big pile that is $\wp(A)$.
woven radishBOT
Ann
pseudo basin
our goal will be to show that we couldn't possibly have given out the entire pile no matter how we do it.
(which is the translation of "g is not surjective" into this setup)
@north roost with me so far?
north roost
yes
pseudo basin
ok right
now
call a person "happy" if they can see themselves in the group photo they got, and "unhappy" otherwise.
i.e. $x$ is happy iff $x \in g(x)$.
woven radishBOT
Ann
pseudo basin
then $\bar{A}$ is the group photo with all the unhappy people.
woven radishBOT
Ann
pseudo basin
makes sense?
north roost
yes
pseudo basin
now the thing is
nobody could have possibly received that group photo
if the recipient of this photo is happy they would see themselves in it -- but then, because A-bar depicts precisely all of the unhappy people, it'd mean the recipient is also unhappy.
but if the recipient were unhappy, they wouldn't see themselves in this photo -- and yet they would appear there, because of their unhappiness
so assuming that the "unhappy bunch" photo was given out to anybody at all leads to a paradox
north roost
im kinda confused with the conclusion
north roost
if the recipient of this photo is happy they would see themselves in it -- but t...
"but if the recipient were unhappy, they wouldn't see themselves in this photo -- and yet they would appear there, because of their unhappiness"
why would they appear
there if they are not
there
pseudo basin
they'd appear there by virtue of being unhappy
the unhappy-bunch photo by definition shows all the unhappy people
north roost
but that photo is not given to anyone, right?
with all the unhappy people
let me rephrase it. so the unhappy recipient will have the picture that doesnt have themself in there, but they will still appear in the group picture of unhappy people that is not given to anyone
because if it belongs to another unhappy person, that person is happy
so
there is a group photo of unhappy people somewhere, but it is not given to anyone
right?
😭
@pseudo basin 🙏
pseudo basin
but that photo is not given to anyone, right?
that's the point.
pseudo basin
**nobody could have possibly received that group photo**
see here again
pseudo basin
our goal will be to show that we couldn't _possibly_ have given out the entire p...
,
quartz scroll
Why is the angle between two vectors equal to the angle between the corresponding radius vectors, if the vectors are perpendicular to the radius vectors?
devout snowBOT
Why is the angle between two vectors equal to the angle between the correspondin...
quartz scroll
or this picture, if it's clearer
devout snowBOT
@quartz scroll Has your question been resolved?
quartz scroll
<@&286206848099549185>
devout snowBOT
@quartz scroll Has your question been resolved?
devout snowBOT
@quartz scroll Has your question been resolved?
devout snowBOT
@quartz scroll Has your question been resolved?
mental pebble
Hi.
wispy oyster
!xy
devout snowBOT
Please show the original problem, exactly as it was stated to you, with the entire original context. A picture or screenshot is best. If the original problem is not in English, then post it anyway! The additional context might still be helpful. Do your best to provide a translation.
wispy oyster
!da2a
devout snowBOT
No need to ask “Can I ask…?” or “Does anyone know about…?”—it’s faster for everyone if you just ask your question! See https://dontasktoask.com/
potent tusk
No need to ask “Can I ask…?” or “Does anyone know about…?”—it’s faster for every...
erm actually they just said hi
wispy oyster
erm actually they just said hi
!nogpt
devout snowBOT
erm actually they just said hi
Please do not trust ChatGPT or similar AI tools for mathematical tasks, as they often generate output which "sounds correct" but has numerous factual or logical errors. Use of these AI tools to answer other people's help questions is strictly against server rules (see #rules).
devout snowBOT
@mental pebble Has your question been resolved?
granite bough
hi im going crazy
devout snowBOT
hi im going crazy
granite bough
so i have a like-sided triangle with side 3
and i was wondering what its height was
oh wait
i just figured it out lmao
nevermind im not feeling crazy anymore
.close
devout snowBOT
devout snowBOT
Available help channel!
thorny cape
Hello
blazing jetty
where are you stuck at?
blazing jetty
okay umm first we'll factor out the -2 so that the coefficient of x^3 becomes 1
making it -2(x^3-5x^2+8x-4)
thorny cape
ok what happens after that
blazing jetty
well there are 2 ways you can do this, either find one root by trial-error or factorise
and by looking at this i could see that x=1 would satisfy the equation (this is the trial-error method where you put values like 0,1, etc and find a root)
devout snowBOT
devout snowBOT
Available help channel!
elfin drift
how prove that log(x^x) > x for x >= 10
devout snowBOT
how prove that log(x^x) > x for x >= 10
elfin drift
yea ln
elfin drift
Wait is it true that ln(x^y) = y ln(x) for x >= 10?
severe bolt
ln(a^b) = b ln(a)
elfin drift
Wait is it true that ln(x^y) = y ln(x) for x >= 10?
I feel like this might follow from calculus , but i cant prove it yet
elfin drift
ln(a^b) = b ln(a)
do they have to be integers or can they be real numbers for sufficiently large positive x,y?
severe bolt
do they have to be integers or can they be real numbers for sufficiently large p...
]0;+infty[
Any real positive number
Is it french?
I thought all people using it
elfin drift
alr i think i get it
so once ln(x) becomes > 1 and increasing then ln(x^x) = xln(x) > x
thanks
.close
severe bolt
x,y inside R*+
devout snowBOT
devout snowBOT
Available help channel!
worn plover
devout snowBOT
worn plover
i have a few questions about this proof
how do we get that M_j=j^2/n and m_j=(j-1)^2/n ?
thin fern
how do we get that M_j=j^2/n and m_j=(j-1)^2/n ?
Because f is strictly increasing
and M_j is defined to be sup of f on the interval, and m_j is defined to be the inf
so the sup would be f of the rightmost point, and the inf would be f of the leftmost point
sullen island
it should be (j/n)^2 = j^2/n^2 really
worn plover
but isnt the left and rightmost points 0 and 1?
thin fern
but isnt the left and rightmost points 0 and 1?
Of each interval created by the partition I mean, not [0,1]
worn plover
hm
im still not seeing how they came up with those specific values
each subinterval is length of 1/n right?
acoustic leaf
the partition is {j/n} for 0 <= j <= n, so each subinterval is [(j-1)/n, j/n]
thin fern
devout snowBOT
@worn plover Has your question been resolved?
supple compass
Guys any ideas on this
deft token
PA
Point of inflection and rate of change
Found by defining the graph
Graphing it
And analyzing
With find inflection
To find rate of change at a certain point
Dy/dx(f(x)) | x= point value
What you need to do
Understand how to graph peace wise functions and start the second gragh at the point where the drug is no longer effective.
Common words for the analysis in worded form
And info can be in dot points
PB
Define equation
Graph equation
Menu analysis
Inflection
Is the maximum value higher than the overdose cap yes or no
No
Graph additional equation from point of inflection
And if the next maximum isn't greater that the overdose cap continue.
Yes
Once the next pill passes the overdose threshold delete that graph and start next graph at the point the drug is no longer effective in the bloodstream
Repeat until you reach the specified domain. Of the question
Count the maximum amount of pills allowed to be taken within the domain(for eg 24hs or 12h)
What you need to learn
How to connect and graph the equation in a piece wise function with 5 to 10 graphs
Fastest ways to write them out
Using the define functions
In order to connect each together
Understand how to explain what you've done and the questions in relation to it.
This is the information I know about it .
I need help Graphing drugs on Cas in relation to log and exponential.
With something like this
f(t)=A×e^-bt ×In (bt+2)
And. I was wondering if anyone had some practice questions like it. Or information that could help me
devout snowBOT
PA
Point of inflection and rate of change
Found by defining the graph
Graphing ...
@deft token Has your question been resolved?
devout snowBOT
@deft token Has your question been resolved?
deft token
<@&286206848099549185>
drifting mauve
deft token
The single question is at the bottom
I need help Graphing piece wise functions on Cas
devout snowBOT
@deft token Has your question been resolved?
@deft token Has your question been resolved?
@deft token Has your question been resolved?
@deft token Has your question been resolved?
@deft token Has your question been resolved?
devout snowBOT
devout snowBOT
Available help channel!
wise nacelle
help, looking to confirm my answers
devout snowBOT
help, looking to confirm my answers
devout snowBOT
@wise nacelle Has your question been resolved?
fossil locust
,calc 500 * e^(-0.032 * 3)
woven radishBOT
Result:
454.23200803435
,calc 500 * e^(-0.032 * 8)
woven radishBOT
Result:
387.07098439612
,calc 500 * e^(-0.032 * 20)
woven radishBOT
Result:
263.64621202152
fossil locust
half-life is correct though
fossil locust
no, 454, 387, and 264
fossil locust
i thought it was 264 since u had to round up
yeah it is 264
wise nacelle
oh ok tysm
fossil locust
np!
wise nacelle
.close
devout snowBOT
devout snowBOT
Available help channel!
ornate bluff
Help
ornate bluff
Is it correct?
steady robin
Is it correct?
yeah. you couldve saved some hassle imo if you just completed the squares:
x^2 + 10x + 25 + y^2 - 6y + 9 - 25 = 0
(x+5)^2 + (y-3)^2 = 5^2
centre is (-5, 3)
radius is 5
but thats completely personal preference
devout snowBOT
steady robin
It said "you need not complete the square"
oh lol 😭
devout snowBOT
Available help channel!
sand isle
devout snowBOT
lament kraken
Is this an accurate interpretion of (A union B) int C
Yes
lament kraken
$\cup$
sand isle
Yay
woven radishBOT
Carbonite
lament kraken
its this btw
woven radishBOT
Light
devout snowBOT
devout snowBOT
Available help channel!
eager stone
I learned this just few days ago and im struggling with solving by myself, can I get a guide just for the 1st one
devout snowBOT
I learned this just few days ago and im struggling with solving by myself, can I...
eager stone
Mathematically, are u aware what each option means?
hmm what do u mean
i know i have to use P to find max/ min rate of change
I dont exactly know what to do with 2,1 vector
sand isle
you have to normalise that vector
normalise your vector, find the directional derivative and see what result u get
on the basis of that u should get ur answer
eager stone
you have to normalise that vector
what does normalizing the vector mean, cant i use it for directional derivative equation directly?
sand isle
what does normalizing the vector mean, cant i use it for directional derivative ...
normalising the vector means converting your vector into a unit vector
sand isle
the derivative equation requires u to find the unit vector because converting it into a unit vector limits its length such that the speed of change only depends upon the direction of the vector and not how long it is
if u get what i mean
eager stone
the derivative equation requires u to find the unit vector because converting it...
that makes sense
sand isle
yup yup
eager stone
so I find the direction of 2,1 vector, i find the change of direction of the equation? and compare
like max, min rate of changes
sand isle
sorry i soned out
eager stone
lol np
eager stone
yes
sand isle
yeah, follow it by normalising direction vec and the rest of the steps
u should be fine
sand isle
what are integers
devout snowBOT
what are integers
kind tiger
whole numbers, positive, negative or 0
..., -2, -1, 0, 1, 2, ....
or do you mean more philosophically?
unreal forge
integers
smoky gyro
1
kind tiger
..., -2, -1, 0, 1, 2, ....
these
sand isle
oh
unreal forge
anything without a decimal
sand isle
So whole numbers and integers same thing?
kind tiger
sometimes people say whole numbers to mean just positive (and maytbe 0)
kind tiger
its not really a technical term
sand isle
So no fractions and decimals = whole number...
unreal forge
yes
sand isle
but every integer is a whole number
Wait huh
kind tiger
dont stress it tbh
sand isle
But do integers include fractions
kind tiger
no
unreal forge
nope
smoky gyro
But do integers include fractions
2/2 
unreal forge
real question
wispy oyster
whole numbers are nonnegative integers
unreal forge
yeah, follow it by normalising direction vec and the rest of the steps
how did u answer this
but dont know what an integer is
sand isle
how did u answer this
listen
winter patrol
but every integer is a whole number
you have those mixed around
unreal forge
BUDDY IS COOKED
sand isle
you have those mixed around
Eh?
smoky gyro
listen
wait that is a good question how do u know what a directional derivative is and not an integer we havent even learned this in class 😭
sand isle
wait that is a good question how do u know what a directional derivative is and ...
to answer it simply
I studied ahead because I wanted to
😭🙏
unreal forge
he so deep in the rabbit hole he forgot the entry
hollow ice
to answer it simply
to answer it simply, bro used mathcord instead of google
sand isle
he so deep in the rabbit hole he forgot the entry
REAL
smoky gyro
I studied ahead because I wanted to
and u called me the nerd thats crazy
unreal forge
so im working with 3d geometry
sand isle
and u called me the nerd thats crazy
😔
unreal forge
and i dont know what ts called but imma write it out
sand isle
What
winter patrol
vauge description and don't include negatives
unreal forge
ah damn whole numbers are positives?
hollow ice
Natural = 1,2,3,...
Whole = 0, 1, 2, ....
Integers = ..., -1, 0, 1, ...
unreal forge
shi mb translation error
sand isle
Natural = 1,2,3,...
Whole = 0, 1, 2, ....
Integers = ..., -1, 0, 1, ...
O
What about rational then
unreal forge
Natural = 1,2,3,...
Whole = 0, 1, 2, ....
Integers = ..., -1, 0, 1, ...
yep what he said
sand isle
to answer it simply, bro used mathcord instead of google
We love mathcord 🔥
solid osprey
What about rational then
any number that can be represented as a/b for integer a and b
sand isle
any number that can be represented as a/b for integer a and b
Technically that's any number
solid osprey
basically combining integers and fractions
solid osprey
Technically that's any number
you cant write sqrt(2) as a/b
sand isle
How not
sand isle
whole numbers are integers
unreal forge
guys how do i go from a Parametric equation to a Cartesian equation of the line
unreal forge
is that right had to ask deepseek for translation?
smoky gyro
whole numbers are integers
sqrt2 is not an integer like skissue said
smoky gyro
what's the problem with it
cuz.... its not rational??
solid osprey
ok how about this
sand isle
what
solid osprey
rational numbers are fractions
unreal forge
and check if its rational
sand isle
and check if its rational
I might as well do my every question on calculators and not bother solving them 😭
sand isle
rational numbers are fractions
yes
smoky gyro
light maybe instead of studying ahead of the class and looking at directional derivatives which arent even gonna be in our final u should touch up on basic set knowledge 😭
sand isle
light maybe instead of studying ahead of the class and looking at directional de...
I'm doing it today man </3
unreal forge
light how old r u
sand isle
light how old r u
nineteen
unreal forge
dayumn
eager stone
light maybe instead of studying ahead of the class and looking at directional de...
noo light learn all directional derivatives and solve my questions ☺️
smoky gyro
I remember sqrt 2 as a/b proof
proof by contradiction
sand isle
I did it in grade 10
wispy oyster
integers do not have any fractional or decimal part to them
sand isle
proof by contradiction
YEAH
sand isle
integers do not have any fractional or decimal part to them
so fractions aren't integers?
wispy oyster
no
well
2/2 is an integer
because it is 1
2/3 is not an integer
because it has a decimal part to it
sand isle
so fractions where a/b are different and b isn't a factor or multiple of a?
wispy oyster
no
sand isle
because it has a decimal part to it
oh right
sand isle
Technically every number as a decimal point
wispy oyster
2.00 is still an integer
if you are a cs student, u would call this a double
but whatever
unreal forge
mia u do cs?
wispy oyster
no
unreal forge
oh ok
sand isle
I see 🤔
unreal forge
hes just ragebaiting atp
sand isle
and furthermore
sand isle
hes just ragebaiting atp
no
I'm not
Mathematics takes understanding
Sybau
And and
smoky gyro
Mathematics takes understanding
common this is like standard math level
sand isle
Rational numbers are real
unreal forge
jk
sand isle
yes?
wispy oyster
yes
unreal forge
Rational numbers are real
so are integers?
sand isle
are irrational real?
unreal forge
yes
wispy oyster
yes
sand isle
alr so lemme make a graph of this
unreal forge
we reinventing xy axis with this one🔥
devout snowBOT
smoky gyro
very interesting
noo its not interesdting 😭 icl pmo 🥀
devout snowBOT
Available help channel!
chilly quest
devout snowBOT
chilly quest
AD = AB
What’s length for AC?
I am stuck at this question idk how to start
<@&286206848099549185>
hollow ice
do you know cosine rule?
chilly quest
Yes
hollow ice
then you can use that in the triangle ADC
chilly quest
How
hollow ice
You know the angle ADC = ADB+BDC
chilly quest
Yes
hollow ice
you also know CD and can calculate AD
chilly quest
How
hollow ice
pythagoras' theorem???
chilly quest
Okay
What do I do after finding AD and ADC
Oh
Cosine therome
So many steps
Is there easier way
Thanks thou
hollow ice
Is there easier way
idk cant think of one on the fly rn sry
cos rule is the first think that came to mind seeing this
chilly quest
cos rule is the first think that came to mind seeing this
So pro
Okay thank you so much for helping me
hollow ice
np
chilly quest
Have fun on the fly
devout snowBOT
@chilly quest Has your question been resolved?
pearl grotto
hey
pearl grotto
how would I do b
I can't use the discriminant
I did dy/dx and got 4x^3 + 6x^2
then set it to 10
turned that into 4x^3 + 6x^2 - 10 = 0
but got 2 solutions
timber pebble
,w D[x^3 * (x+2), x]
woven radishBOT
timber pebble
,w graph 2x^3(2x+3)-10
woven radishBOT
timber pebble
timber pebble
sounds like the question is wrong
pearl grotto
sounds like the question is wrong
i dont think so lol
wait
this is the correct working
pearl grotto
,w graph 4x^3 + 6x^2 - 10
woven radishBOT
pearl grotto
does this look correct
timber pebble
3x^2(x+2) + x^3 = 3x^3+6x^2+x^3 = 4x^3+6x^2
vital sedge
pearl grotto
i still dont get why i have 2 solutions
devout snowBOT
@pearl grotto Has your question been resolved?
royal laurel
I would like it to be as general as possible. Say, I got a matrix and a function.
devout snowBOT
I would like it to be as general as possible. Say, I got a matrix and a function...
royal laurel
How should I compute the resulting function after applying the matrix?
Let's say I got y=1/x, and the matrix to be:
autumn girder
wdym by applying the matrix?
sand quarry
the expression 1/X is not well-defined for a matrix X
You need to define it yourself if you want to give it some sort of sense
royal laurel
Like considering the matrix as a transformation. I wonder what should I exercise to obtain the function after transformation.
royal laurel
You need to define it yourself if you want to give it some sort of sense
Let's say x belongs to real number but not 0
royal laurel
How should I compute the resulting function after applying the matrix?
I just wonder what is the common method to gain the function after transformation.
sand quarry
Confusion
royal laurel
Also, I would like to learn the rationale behind the method as well.
potent tusk
how do you transform a function by a matrix lmao
royal laurel
how do you transform a function by a matrix lmao
Function is a graph. Transformation alters the graph.
restive river
From what I'm getting, you seek to obtain a set of transformed points after applying that matrix on the points (x,y) lying on the graph of y=1/x, correct?
potent tusk
wait nvm i get it now
royal laurel
From what I'm getting, you seek to obtain a set of transformed points after appl...
Exactly, sir.
restive river
Just multiply the matrix M with the column vector (x,y)
And put y=1/x
You'll get the set of transformed points, but not a function per se
royal laurel
Just multiply the matrix M with the column vector (x,y)
I will get a transformed column vector of (x,y) after multiplying it by matrix M.
restive river
Yeah
royal laurel
By putting the transformed vector, let's say, (x',y') into y=1/x. I should get the transformed function.
restive river
Computing 1/v where v is a vector is not well defined in general
And the latter part of your argument doesn't make sense
royal laurel
And the latter part of your argument doesn't make sense
By the latter part, you mean the result will not be a function?
restive river
There will be no result if you put vector v = (x',y') in f(x) = 1/x unless you resort to exclusive definitions for 1/v
What is your original complete question?
royal laurel
What is your original complete question?
I will send it immediately, sir. Give me one second
I WILL TRANSLATE THEM. IT IS A QUESTION FROM COMPETITIVE MATH.
restive river
No it's not translation
restive river
Just multiply the matrix M with the column vector (x,y)
It's this
And you can see after applying the matrix, you get a point (and to be exact, a set of points for non-zero Real x)
royal laurel
And you can see after applying the matrix, you get a point (and to be exact, a s...
set of points, as a whole , should be a function.
restive river
A set is not a function
A function is a mapping or relationship between sets (with more conditions)
royal laurel
A set is not a function
A function is a mapping or relationship between sets (w...
May I know what elicits the qualification?
restive river
What you are doing here is applying M on every point (x,y) lying on y=1/x
You apply a matrix on a point (specifically, the position vector)
You're doing this
$$\begin{bmatrix} 1 & -1 \ 1 & 1 \end{bmatrix} \begin{bmatrix} x \ y \end{bmatrix}$$
woven radishBOT
à뜜
royal laurel
The matrix should convert the graph. The resulting (x',y') should be considered the graph after the transformation. That is, (x',y') display the transformed function on the xy plane.
@helper
west birch
I WILL TRANSLATE THEM. IT IS A QUESTION FROM COMPETITIVE MATH.
我看得懂:)，so you mean MN(x, y) where y=1/x
royal laurel
How should I compute the resulting function after applying the matrix?
<@&286206848099549185>
west birch
that is acting on R2 and adding a constraint
tawny pewter
The matrix should convert the graph. The resulting (x',y') should be considered ...
you consider the graph of 1/x as a subset of the plane is that it?
royal laurel
我看得懂:)，so you mean MN(x, y) where y=1/x
Glad you understand it, and yes, that’s exactly what I mean
ornate bluff
Matrices got remove in my syllabus this year
west birch
you consider the graph of 1/x as a subset of the plane is that it?
yes or algebraic variety as you may
tawny pewter
you could apply the matrice to this vector
west birch
I WILL TRANSLATE THEM. IT IS A QUESTION FROM COMPETITIVE MATH.
so what theyre saying is that you rotate the whole plane, and then you draw out the rotated curve
royal laurel
so what theyre saying is that you rotate the whole plane, and then you draw out ...
Exactly
Since it’s a rotation matrix in this case
west birch
you could apply the matrice to this vector
this is correct
tawny pewter
but then figuring out the resulting curve could be tricky
west birch
the point is that imagine (x', y') is the rotated set of points(curve), so every single original point (x, y) obeys y=1/x, and you rotate each point to the resulting point
restive river
but then figuring out the resulting curve could be tricky
One way is to express (x - 1/x , x + 1/x) in the form (t,g(t)) if we are finding the closed form of rotated graph
west birch
is that the justification youre asking?
royal laurel
is that the justification youre asking?
Yes. Thank you sir
frank agate
this rotate
tawny pewter
tawny pewter
this rotate
yep and stretch a bit
west birch
One way is to express (x - 1/x , x + 1/x) in the form (t,g(t)) if we are finding...
the general solution is like this, in the video you posted the solution assumes the form x'^2+ay'^2=b since we know y=1/x is hyperbola
the answer follows since b does not depend on x' or y', so a is set to be -1
restive river
yep and stretch a bit
wait nvm
restive river
Yeah
devout snowBOT
@royal laurel Has your question been resolved?
unique vortex
Linear approximation
devout snowBOT
Linear approximation
supple knot
Should be cos(0) in the gradient
unique vortex
What is this supposed to represent
linear approximation?
supple knot
y^2 is not linear
unique vortex
Linear approximation
here's whole exercise
unique vortex
Show the full page where you got this from
there's nothing but this line
supple knot
Then there is only a partial answer
supple knot
Linear approximation
This is really two questions
unique vortex
This is really two questions
I already solved (a)
Only (b) left.
Everything else on these pages is just other exercises.
supple knot
Yea so like I said above, they only provided a partial answer
foggy ermine
Use Taylor Theorem?
devout snowBOT
@unique vortex Has your question been resolved?
west birch
Seems like you're not wrong, it's just the second order translation messing with you
*grad f dot (x-a)
@unique vortex
unique vortex
Seems like you're not wrong, it's just the second order translation messing with...
Ty, got the same answer. Messed it up due to two dot products once though.
grad(x,y,z) really helped with hessian matrix
How to calculate the Taylor polynomial
of degree two?
waxen crescent
The nth term of a Taylor polynomial is $\frac{f^{(n)}(z)}{n!}(x-z)^n$ where z is where you're centered at
woven radishBOT
Parsec
unique vortex
The nth term of a Taylor polynomial is $\frac{f^{(n)}(z)}{n!}(x-z)^n$ where z is...
So, (1,0,0) is center?
waxen crescent
Oh nevermind that's only for single variable
unique vortex
essentially the same thing
So, linear approximation and second degree tp are the same thing?
west birch
no i mean multivariate takes similar form as single variate
i think you're looking for frechet derivative
i saw that before but not familiar sorry
pearl grotto
yo guys