Don’t think so
#help-49
lost sphinx
ur goal is to find the shortest c^2, apply pyth theorem by finding the shortest a^2 + b^2
lost sphinx
Do I need to use calculus for this question
u can but no
grand gust
ur goal is to find the shortest c^2, apply pyth theorem by finding the shortest ...
this was what I was trying to get at, OP
gaunt jetty
Do I need to use calculus for this question
Unless you don't know the formula for the vertex of a quadratic...
formal terrace
ur goal is to find the shortest c^2, apply pyth theorem by finding the shortest ...
Where do I apply it
Omg I’m so lost
lost sphinx
a^2 + b^2 is the application
gaunt jetty
Omg I’m so lost
What's the pyth theorem? Write it down
grand gust
it'd probably help if you state the theorem first
gaunt jetty
Ok and you have a + b = 10
formal terrace
But not how to proceed
formal terrace
Ok and you have a + b = 10
Yes
gaunt jetty
This is a system of equations
gaunt jetty
Ok and you have a + b = 10
Try to isolate one variable in this
formal terrace
a=10-b
gaunt jetty
a^2+b^2=c^2
And chug it in this
formal terrace
And chug it in this
(10-b)^2+b^2=c^2
gaunt jetty
Expand
gaunt jetty
no
grand gust
expand the bracket
gaunt jetty
Expand the binomial
formal terrace
100-b^2
grand gust
freshman's dream!
remember that (10-b)^2 = (10 - b)(10 - b), then apply the distributive law
formal terrace
freshman's dream!
remember that (10-b)^2 = (10 - b)(10 - b), then apply the dist...
Isn’t it one plus and one minus
near cliff
what do you have so far?
formal terrace
What has been said here
grand gust
group like terms and simplify
formal terrace
Alr one sec
grand gust
but first you might wanna share what you've gotten after expansion if you wanna check your expansion itself
formal terrace
but first you might wanna share what you've gotten after expansion if you wanna ...
B^2+100-20x+b^2
grand gust
20x?
gaunt jetty
Ye that's fine
formal terrace
Didn’t know what you guys were talking about at first
formal terrace
you mean a^2 **-2ab** + b^2
Ye
gaunt jetty
Simplify like terms now
formal terrace
2b^2
gaunt jetty
Ok awesome
So u want the value that results in the minimum c^2
This corresponds to minimising the quadratic
formal terrace
So u want the value that results in the minimum c^2
The minimumpoint right
gaunt jetty
When is a quadratic at a minimum?
formal terrace
When y”(x)>0 right 😭 I might be wrong
grand gust
I don't think we need to bring out a flamethrower to kill a fly on the wall, but technically wrong way around
formal terrace
When y'(x) = 0
Isn’t that just an extremepoint
gaunt jetty
Itd be a critical point tbat corresponds to the minimum point
formal terrace
I don't think we need to bring out a flamethrower to kill a fly on the wall, but...
I’m studying calculus rn idk why this question is in here 😭
grand pondBOT
formal terrace
Oh okay you should differentiate then
Derive you mean?
gaunt jetty
Differentiate is the proper verb i believe
grand gust
differentiate is the better term
derive has a different meaning in math
though I suppose you could say derivate...? but I'd still stick to differentiate as a verb
formal terrace
Oh okay
near cliff
(Note that since the coefficient of b^2, namely 2, is positive, the quadratic has a minimum point. If this were negative, it would have a maximum point)
gaunt jetty
I've never heard derivate in my life
formal terrace
I study maths in a different language 😅
formal terrace
(Note that since the coefficient of b^2, namely 2, is positive, the quadratic ha...
That’s due to the parable facing up or down right
grand gust
parabola*
formal terrace
4b-20
grand gust
there we go
near cliff
you can simplify that
formal terrace
7
.07
Thank you guys so much 😭 and sorry for the incov
I understand it now
.close
midnight plankBOT
near cliff
@formal terrace there's another, arguably simpler way to do it
Notice that a^2 + b^2 = (a + b)^2 - 2ab = 100 - 2ab.
ab is maximised when a = b (let me know if you want me to elaborate)
Hence, we have c^2 = 100 - 2 * 5 * 5 = 50.
formal terrace
<@807613690093371422> there's another, arguably simpler way to do it
Notice tha...
Elaborate pls
near cliff
.reopen
midnight plankBOT
✅ Original question: #help-49 message
near cliff
Elaborate pls
a + b = 10 means that a and b can be re-written as 5 + k and 5 - k.
ab is thus (5 - k)(5 + k).
using the difference of squares identity, this is 5^2 - k^2, which is of course maximised when k = 0
formal terrace
a + b = 10 means that a and b can be re-written as 5 + k and 5 - k.
ab is thus ...
I see
Thanks!
midnight plankBOT
midnight plankBOT
Available help channel!
graceful ferry
Given b=[-2 -1 -2 1]^T, show that there is no answer to Ax=b using QR decomposition.
can someone help me piece the solution together I kinda don't understand pic 2 (which explains how to use QR decomp to solve the LS problem, even if it's economy QR).
translation for second pic:
The residue vector can be divided into two parts and the total residue is the sum of both the parts.
The first part can be brought to 0 using solution of reverse substitution (?).
The second part can't be changed - the sum of squares of the last part of Q^Tb decides the minimum height of the LS problem.
Note that we could use economy QR to compute solution.
midnight plankBOT
Given b=[-2 -1 -2 1]^T, show that there is no answer to Ax=b using QR decomposi...
graceful ferry
My work: I found economy Q, and for R resulting from Q^TA
How do I continue to show that Ax=b which is Rx=Q^tb has no solution ?
$
Q = \begin{bmatrix}
0.5 & -0.5 & 0.5 \
0.5 & -0.5 & -0.5 \
0.5 & 0.5 & 0.5 \
0.5 & 0.5 & -0.5
\end{bmatrix}$
$R =
\begin{bmatrix}
8 & 5 & 4 \
0 & 5 & 9 \
0 & 0 & 9
\end{bmatrix}$
grand pondBOT
prograce
graceful ferry
What I dont understand mainly: how do I link the second pic with my problem
midnight plankBOT
@graceful ferry Has your question been resolved?
midnight plankBOT
@graceful ferry Has your question been resolved?
graceful ferry
I'm going to sleep but i wanna keep it open what do i do :(
fallow scarab
I'm going to sleep but i wanna keep it open what do i do :(
you just copy paste it another time. help channels aren't meant to be camped
midnight plankBOT
midnight plankBOT
Available help channel!
tidal fulcrum
guys can someone explai this
midnight plankBOT
guys can someone explai this
tidal fulcrum
i dont undertsand the last 2 rows
if f'(x) goes from positive to negative its relative max on f(x)
when im looking for that on a graph..
do i look for when f'(x) slope is positive/negitive or when its above/below the x axis
dawn dagger
the latter
tidal fulcrum
the latter
wdym?
dawn dagger
when its above/below the x axis
the slope of f(x) is indicated by whether f'(x) is negative/positive aka when its above/below the x axis
tidal fulcrum
the slope of f(x) is indicated by whether f'(x) is negative/positive aka when it...
hmm okay i think i understand
i hate this part of calc
😭
dawn dagger
hmm okay i think i understand
one concrete example: do you know the function e^x?
tidal fulcrum
one concrete example: do you know the function e^x?
yea
dawn dagger
so we know e^x is increasing strictly
we can see that by its derivative
,w plot e^x
grand pondBOT
dawn dagger
since the derivative of e^x is aways positive it follows that it is strictly increasing
midnight plankBOT
@tidal fulcrum Has your question been resolved?
midnight plankBOT
tidal fulcrum
guys can someone explai this
i still dont get it tbh
midnight plankBOT
Available help channel!
tidal fulcrum
so when i have something like this
midnight plankBOT
so when i have something like this
tidal fulcrum
lets say question 1
f'(x) = 0 when there's a horizantle tangent on f at x= 4, 1, and 5 where f has relative extrema
is that how i justify first of all?
wait i see how this page works
but then this is what has me lost
question 1, what values of the graph f(x) is increasing
tidal fulcrum
guys can someone explai this
from this table i know that f(x) is increasig when f'(x)>0
oh omg i just look where f'(x) is above the x axis dont i
😭
would this be a valid response:
f(x) is increasing on (-3,-2) where f'(x)>0
cerulean oyster
would this be a valid response:
f(x) is increasing on (-3,-2) where f'(x)>0
yeah
As quick reference, positives and negatives are (obviously) the positives and negatives of f(x);
roots are f(x) = 0;
increasing and decreasing are f'(x) > 0 and f'(x) < 0
f'(x) = 0 or f'(x) doesnt exist are (sometimes) local extrema.
and concativity is determined by f''(x) in the same manner. and the "extrema" are now called inflection points.
midnight plankBOT
@tidal fulcrum Has your question been resolved?
cerulean oyster
f'(x) is clearly the derivative of f(x), right?
tidal fulcrum
yeah
cerulean oyster
So, if they are asking for the slope of f(x) at a point, by knowing f'(x), you can always give an answer even if you dont know what f(x) looks like.
if you recall, the derivative, apart from being the rate of change, also encodes the slope at a point.
cerulean oyster
yep
tidal fulcrum
ohhhh i understand
would this be good justification for 6:
the slope of the line tangent to f(x) is f'(4) which is -2
tidal fulcrum
in 6) they are asking for x = 0 btw
wait so id just look for the point for questions like that?
(0,y)
and y is the answer?
cerulean oyster
for 7), youd consult for f'(4) and then find the negative of its multiplicative inverse.
tidal fulcrum
ah alright so questions 6 and 7 are stright forrward
cerulean oyster
yeah, most of them really are, its just knowing how to read the problem and the graph
cerulean oyster
np
midnight plankBOT
@tidal fulcrum Has your question been resolved?
tidal fulcrum
when i do evt
should 2 values be the same?
midnight plankBOT
when i do evt
should 2 values be the same?
tidal fulcrum
or can they be different
my teacher said something about two of the f(x) being the same
for example here the abs max has 2 of the same values twice
is that just a coincidence or does that must happen
rose trout
It need not happen.
junior flower
coincidence i think
midnight plankBOT
midnight plankBOT
Available help channel!
gaunt jetty
I have a random function $x(t)$ and a deterministic function $h(t)$.
\medskip
Let $y(t)$ be the convolution of those two functions:
[
y(t) = (x\ast h)(t) = \int_{-\infty}^\infty h(\alpha)x(t-\alpha)\dd\alpha.
]
I define the autocorrelation function $R_{yy}(\tau)$ as the expectation value:
[
R_{yy}(\tau) = \vb E[y(t)\2y(t+\tau)].
]
I need to take the Fourier transform of $R_{yy}(\tau)$ with respect to $\tau$
\medskip
\textbf{My Question:} I am trying to prove how the autocorrelation of $y$ is equal to
[
R_{yy}(\tau) = \abs{H(f)}^2\2\FFF{R_{xx}(\tau)}
]
...where $H(f)$ is the Fourier transform of $h(t)$
midnight plankBOT
I have a random function $x(t)$ and a deterministic function $h(t)$.
\medskip
L...
grand pondBOT
gaunt jetty
I keep getting lost in the process because the algebra is pretty nasty. Any ideas on how to approach this?
silent dock
so
,,R_{yy}(tau) = mbb E \left{ int map h{alpha} x(t-alpha) dd alpha int map h{beta} x(t+tau-beta) dd beta \right}
gaunt jetty
Yuh
grand pondBOT
クーリー
Compile Error! Click the 
reaction for more information.
(You may edit your message to recompile.)
silent dock
Bruh what the heck
silent dock
There’s literally no error
Wait wait wait
,,R_{yy}(tau) = mbb E \p[int map h{alpha} x(t-alpha) dd alpha int map h{beta} x(t+tau-beta) dd beta ]
The plane on the WiFi isn’t very good, Lex.
grand pondBOT
クーリー
sudden yacht
sudden yacht
!occupied
midnight plankBOT
Someone else is already using this help channel. If you need help with a question, please open your own help channel/thread (see #❓how-to-get-help for instructions).
sudden yacht
! discussion
gaunt jetty
The plane on the WiFi isn’t very good, Lex.
#humbleflex
sudden yacht
Nope
<@&268886789983436800>
silent dock
,tex>>textgradientglow<< [ iint map h{alpha} map h{beta} mbb E \p[ x(t-alpha)x(t+tau-beta) ] dd alpha dd beta]
grand pondBOT
クーリー
silent dock
I assume x(t) is WSS?
silent dock
Lex, I’m just about to land uh
gaunt jetty
[
\FFF{R_{yy}(\tau)} = \abs{H(f)}^2\2\FFF{R_{xx}(\tau)}
]
grand pondBOT
lethal path
Lex, I’m just about to land uh
you're on a plane and you're Discording? bruh
gaunt jetty
I messed up I meant to write this
The Fourier transform of the autocorrelation is the power spectral density but I ain't throwing jargon in a maths server 
gaunt jetty
Lex, I’m just about to land uh
Aight 
silent dock
Yeah that’s what it’s meant to b
You were kinda yapping
Anyhow
So we have
,tex>>textgradientglow<< [ int int map h{alpha} map h{beta} R_{xx} (tau + alpha + beta) dd alpha dd beta]
grand pondBOT
クーリー
silent dock
textgradientglow<<
I guess you define
[S_{yy}(f) = FFF_{tau to f}{R_{yy}(T)} = int R_{yy}(tau) e^{ -j2pi f tau } dd tau]
grand pondBOT
クーリー
silent dock
Yes
One sec blud I just landed
So that is equal to
textgradientglow<<
[S_{yy}(f) =int! int map h{alpha} map h{beta}\p[int R_{xx}(tau + alpha - beta)e^{-j2pi f tau} dd tau] dd alpha dd beta
]
grand pondBOT
クーリー
silent dock
For the inside integral, let u = tau + alpha - beta so tau = u - (alpha - beta).
textgradientglow<<
[int R_{xx}(u)e^{-j2pi f(u-(alpha-beta))} dd u
]
grand pondBOT
クーリー
silent dock
textgradientglow<<
[e^{+j2pi f(alpha-beta)}\unb{int R_{xx}(u)e^{-j2pi fu} dd u}{S_{xx}(f)}
]
grand pondBOT
クーリー
silent dock
textgradientglow<<
[
S_{yy}(f) = S_{xx}(f) int! int map h{alpha} map h{beta} e^{j2pi f(alpha-beta)} dd alpha dd beta
]
grand pondBOT
クーリー
silent dock
textgradientglow<<
[
S_{yy}(f) = S_{xx}(f) H(-f) map Hf
]
grand pondBOT
クーリー
midnight plankBOT
@gaunt jetty Has your question been resolved?
silent dock
midnight plankBOT
silent dock
midnight plankBOT
Available help channel!
chilly cobalt
Three circles with radii $r_1,r_2,r_3$ where $r_1 < r_2 <r_3$ are mutually and internally tangent at $O$. A point $P$ is external to every circle, and is tangent to each circle at $Q_1,Q_2,Q_3$ respectively. \ Suppose that $PQ_1:PQ_2:PQ_3 = 7:5:3$, and that $r_1=3, r_3=7$. Determine $r_2$.
midnight plankBOT
Three circles with radii $r_1,r_2,r_3$ where $r_1 < r_2 <r_3$ are mutually and i...
grand pondBOT
Copter
chilly cobalt
is my solution correct here?
chilly cobalt
all good?
late rover
chilly cobalt
wtf is man + dad = bomb in sink 😭
late rover
Shouldn't that be (r-3)O3P^2+(7-r2)(49+9k^2)PO1^2
late rover
wtf is man + dad = bomb in sink 😭
human creativity at its peak 
uneven ridge
god damn it
midnight plankBOT
god damn it
uneven ridge
idek how to do that
lyric charm
immediate idea is to maybe cook up something by induction 
dawn dagger
try induction
lyric charm
ain't that what i just said
uneven ridge
immediate idea is to maybe cook up something by induction <:thonk:40657573256390...
why not conduction
im going to radiate thru the vacuum into the sun
what will my two thingies be like
for chain rule
gonna do like
y1, y2, y3 then yn
or smth
cerulean oyster
i was sort of thinking of the general multiple angle formula for cos(nx) to try and clean up a bit, but i think that will just make it messier.
uneven ridge
i kept thinking and then eventually gave up
cant clean it only but
can clean it further into the differentiation it self
ignoring this abomination
atp
lyric charm
why not conduction
im going to radiate thru the vacuum into the sun
???
runic hamlet
have you tried induction?
runic hamlet
ok you could have said that
lyric charm
you could've said that the first time that the word came up 💀
uneven ridge
maybe i do know and its just never been referred to induction for me
lyric charm
what is your native language
lyric charm
in english "induction" is the standard term
anyway induction is a method of proving statements that begin with "for all n ∈ N, ..."
uneven ridge
in english "induction" is the standard term
i meant like they never taught it to us as induction
clever sedge
May I guess japanese
lyric charm
calling said statement P(n), you do it in two steps:
- prove P(1)
- prove that if P(n) is true then P(n+1) is true
uneven ridge
May I guess japanese
very wrong
clever sedge
🙁
lyric charm
P(1) may depending on context actually be P(2) or P(0) or P(-17) or whatever. it's the "lowest" of the statements you care about
uneven ridge
god damn it
assume that entire thing as P(n)
uneven ridge
P(1) may depending on context actually be P(2) or P(0) or P(-17) or whatever. it...
trial and error ??
lyric charm
no
uneven ridge
when you say prove p(1)
i need to prove p(1)=0?
lyric charm
P(n) is the statement that
(1-x^2) y_{n+2} - (2n+1)x * y_{n+1} + (m^2-n^2) y_n = 0
P(0) reads:
(1-x^2) y'' - xy' + m^2 y = 0
makes sense here to start with n=0 instead of n=1 imo
some crunch will be needed to work out y''
cerulean oyster
P() is a mathematical way to refer to a statement that is indexed by cases.
P(1) is usually the way we refer to the base case
P(n) is the nth case.
Induction is usually explained as falling dominoes because, if you can prove that the first case holds, and that the current case is proved true by the previous, you can always sustain yourself in the first case as a chain.
uneven ridge
fuck
lyric charm
idk if you can even avoid anything here that would boil down to an inductive argument tbh
chilly cobalt
idk if you can even avoid anything here that would boil down to an inductive arg...
i mean, you can find the "base case" and then take the n-th derivative using leibniz rule
lyric charm
is leibniz rule going to work
i feel like it's going to be symbolic explosive diarrhea
uneven ridge
is leibniz rule going to work
it does
chilly cobalt
i mean, its not too many terms
uneven ridge
lowkey given up on this coz the diffrentiation and simplification is aids
dawn dagger
ain't that what i just said
u said maybe
midnight plankBOT
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Available help channel!
woeful turret
how to integrate 1/(1+t^4)
midnight plankBOT
how to integrate 1/(1+t^4)
slender walrus
hmm, maybe factorise the 1+t^4
then partial fractions
woeful turret
limits are from 0 to 1
woeful turret
hmm, maybe factorise the 1+t^4
then partial fractions
o ok
thats kinda lengthy right
twilit field
how to integrate 1/(1+t^4)
is this a !xy?
woeful turret
is this a !xy?
help
twilit field
oof
midnight plankBOT
@woeful turret Has your question been resolved?
slender walrus
yeh, seems like you'll have to go through with that partial fractions
woeful turret
oh man
slender walrus
assuming the first part you did was some kings rule stuff
midnight plankBOT
midnight plankBOT
Available help channel!
twilit field
midnight plankBOT
twilit field
Let $d(.,.)$ be the metric on $A$ and thus $B$. Then $\delta(A)= \sup {d(x,y) \mid x,y \in A }$. $\delta(B)= \sup{ d(x,y) \mid x,y \in B }$. But $A \subseteq B$. So ${d(x,y) \mid x,y \in A } \subseteq { d(x,y) \mid x,y \in B }$.
grand pondBOT
wai
twilit field
Is this right so far?
visual tiger
Yes
twilit field
Let $d(.,.)$ be the metric on $A$ and thus $B$. Then $\delta(A)= \sup {d(x,y) \mid x,y \in A }$. $\delta(B)= \sup{ d(x,y) \mid x,y \in B }$. But $A \subseteq B$. So ${d(x,y) \mid x,y \in A } \subseteq { d(x,y) \mid x,y \in B }$. It then follows that $\delta(B)$ is an upper bound of ${ d(x,y) \mid x,y \in B }$. But $\delta(A)$ is it's least upper bound. So $\delta(A) ≤ \delta(B)$
grand pondBOT
wai
lavish venture
yep
lavish venture
i mean there isn't much to shave off other than maybe not needing to fully mention the definitions?
twilit field
i mean there isn't much to shave off other than maybe not needing to fully menti...
Fair, I just mentioned them as I'm just starting off with metric spaces.
lavish venture
yea i don't have any problems with it
i generally use shorter notations
like $\sup_{x, y \in A} d(x, y)$
grand pondBOT
knief
last slate
how tf do u even integrate this
lavish venture
yea its much cleaner
twilit field
I'll close this in like 2 minutes, creating a TeX doc with all solns
Thanks!
.close
midnight plankBOT
woeful turret
how tf do u even integrate this
Sin x = t then kings rule then it's am even function so apply property
midnight plankBOT
Available help channel!
gentle grove
Okay so
Im doing physics and I really cannot figure out how to do this
I have a problem where— in terms of M, R, h, and g— I figured out the linear acceleration and angular velocity of a yoyo (cylinder) falling a specific distance
I did that, that was easy
Now the cylinder has fallen to the ground, broke free from the cord and has not bounced, and has continued spinning. Its now slipping and rolling on a flat surface
Im also assuming a constant friction of mu
I need to figure out the angular velocity as a function of time (t)
How do i do this ;-; if anyone can help
I can also provide any further info needed o7
midnight plankBOT
Okay so
Im doing physics and I really cannot figure out how to do this
I have ...
compact copper
you reckon we can do something with conservation of angular momentum?
gentle grove
you reckon we can do something with conservation of angular momentum?
Oooooooooo
Hmm
Im going to try that
How would I factor in the friction and the slipping to angular momentum ?
compact copper
yeah valid
gentle grove
Hmmmmm mmmmmmm
There probably is an obvious answer here and im just not seeing it
Maybe
compact copper
the friction could maybe be converted into torque
gentle grove
Thats true
midnight plankBOT
@gentle grove Has your question been resolved?
gentle grove
Mmmmmmmmm sure
hoary hamlet
i understand that m_1 = 1, m_2 = sin(1), m_3 = 1
midnight plankBOT
i understand that m_1 = 1, m_2 = sin(1), m_3 = 1
hoary hamlet
and that m_4 = sin(1)
but i dont understand why m_5 is any different to m_1, m_3
since the answer is E
lyric charm
what's the range of f_2?
hoary hamlet
[sin(-1), sin(1)]?
lyric charm
sure is.
or more elegantly: [-sin(1), sin(1)]
what's the maximum value of cos(t) as t goes over that range? or better yet, what's the range of f_3?
crimson rivet
[sin(-1), sin(1)]?
Can u use calculator?
hoary hamlet
nope
crimson rivet
f1=cos x max = 1
lyric charm
[cos(sin(1)), 1]
yes correct
crimson rivet
f2= sin(cos x)
F2<1
lyric charm
[cos(sin(1)), 1]
now find the range of f_4 from this
hoary hamlet
i think i understand it
its because since the max of f_4 is still
1
it never actually touches 0
so the max of f_5 cannot be 1
midnight plankBOT
@hoary hamlet Has your question been resolved?
hoary hamlet
.close
umbral tusk
im not sure what to do for part b
midnight plankBOT
im not sure what to do for part b
buoyant fjord
calculate the cdf of the RV |X-80|
umbral tusk
calculate the cdf of the RV |X-80|
what does cdf and rv mean?
buoyant fjord
what does cdf and rv mean?
cumulative distribution and random variable
gaunt jetty
Cumulative distribution function and random variable
umbral tusk
im not sure where to start icl
buoyant fjord
try to find the density of |X-80|
gaunt jetty
im not sure where to start icl
How would you rewrite |X-80| < b as an inequality involving X
umbral tusk
try to find the density of |X-80|
i dont think im supposed to know into that much depth for this question... (and when i was taught normal distribution, there was no mention of integration with it at my level)
buoyant fjord
How would you rewrite |X-80| < b as an inequality involving X
yeah do this
gaunt jetty
Yeah you dont need integrals here
umbral tusk
How would you rewrite |X-80| < b as an inequality involving X
X < b+80 and 80-b < X
gaunt jetty
Ok so
That means you are looking at a region in the centre of the normal distribution that contains __% of the area
Fill in the blank
umbral tusk
80?
gaunt jetty
Yeah
umbral tusk
hmm lemme try and see what i can do next with that
umbral tusk
is b=4.96 (3sf)?
grand pondBOT
gaunt jetty
,calc 1.282*sqrt(15)
grand pondBOT
Result:
4.9651646498379
I guess it is 4.97 if you are using 3 s.f. but yeah its fine lol
midnight plankBOT
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edgy crater
6 cups, 4 with dose x, 2 with 2x. dose 4x is fatal. fred drinks 2 cups and survives. probability he survives third. are you meant to approach this using conditional probability (find P(drinks 1 cup dose 2x, 1 cup dose x | given he survives) x 0 (since he will drink 3x and hence die next drink) + P(drinks 2 cups dose x | he survives) x 2/4 (he can drink a dose x on the third and survive)
midnight plankBOT
6 cups, 4 with dose x, 2 with 2x. dose 4x is fatal. fred drinks 2 cups and survi...
edgy crater
actually would this just be the probability he drinks (dose x, dose x) x 2/4
onyx tide
How to factor x^2 + 2x -3 now?
midnight plankBOT
How to factor x^2 + 2x -3 now?
cerulean oyster
hey
cerulean oyster
Have you thought of trying some of these yourself?
This one is actually really easy.
cerulean oyster
gang asks for help and gets mad when tasked with use of the prefrontal cortex
clever sedge
😭
cerulean oyster
We will go again, have you heard about Quadratic Formula?
sterile bluff
No don't do that. Its easier to factor
onyx tide
i don’t want to fucking use quadric formula
sterile bluff
Let's work backwards first
onyx tide
I want to use factoring
onyx tide
And what
sterile bluff
So it's a² + a(b+c) + bc
So ultimately, when factoring, were looking for b and c
What adds to 2 and multiplies to -3?
onyx tide
3 and -1
onyx tide
Ok
And when can I use it ?
This method
Only when it’s quadric formula
@sterile bluff
sterile bluff
Only when it's x² + bx + c
And you can find a way for 2 numbers to add to get b and multiply to get c
Which isn't always possible!
So just look for factoring first, and if you can't find those 2 numbers, just use the quadratic formula because they may not be whole numbers or there may not be 2 numbers etc.
@onyx tide
tidal zodiac
onyx tide
@sterile bluff
onyx tide
Two numbers?
sterile bluff
Yes
onyx tide
-1 and 5
onyx tide
Yes
sterile bluff
So what's the equation?
onyx tide
(x-1)(x+5)?
sterile bluff
Yes
You can multiply it out again to check your work but I'll just confirm this one's correct
onyx tide
And the lower number is in the first bracket?
sterile bluff
It doesn't matter
onyx tide
It’ll be the same?
sterile bluff
4 × 5 is the same as 5 × 4
onyx tide
Okay
sterile bluff
With multiplication, it doesn't matter which is first
Anything else you need help with?
onyx tide
Does this method work with x^4?
keen sinew
i mean it works but the equation is way harder
onyx tide
Is this for example correct
keen sinew
yes
onyx tide
Ty baby
sterile bluff
That's not x^4 tho...???
cerulean oyster
im pretty sure it doesnt
onyx tide
What the issue
cerulean oyster
Is this for example correct
note, on step 2. -> 3.
$$-x^2 - 4x$$
$$x(-x-4)$$
grand pondBOT
cerulean oyster
which simplifies to -x(x+4)
night gyro
hello is this steps right
midnight plankBOT
hello is this steps right
night gyro
because when i solve the qudratic i get decimals where did i go wrong
tidal widget
because when i solve the qudratic i get decimals where did i go wrong
Wut
junior flower
yea you should get a positive integer solution
night gyro
yea you should get a positive integer solution
i need to get whole numbers?
night gyro
i would hope the problem is arranged that way
i spotted the mistake the second probability should be 7/x-1
junior flower
if the “correct” answer is not a positive integer then the problem is not designed correctly
night gyro
?
night gyro
if the “correct” answer is not a positive integer then the problem is not design...
what
junior flower
i spotted the mistake the second probability should be 7/x-1
yea
junior flower
the idea is right. what’s the latest expression you came up with?
night gyro
wait let try solve it now
night gyro
the idea is right. what’s the latest expression you came up with?
is giving me an error
i got 21x^2-560x+3899=0
tacit rose
Why did you do x - 8 here?
tacit rose
Oh my bad
tacit rose
Yes looks better
junior flower
yea i think so
night gyro
Yes looks better
now if i solve the qudratic ill get whole niumber?
tacit rose
See if you do
tacit rose
And when you solve for x?
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onyx tide
Factor This x^2 - 12x + 32 = 0
midnight plankBOT
Factor This x^2 - 12x + 32 = 0
jaunty wren
yes
onyx tide
And what are next steps
bold escarp
Try to make the generic rectangles it helps a lot
jaunty wren
And what are next steps
next steps to what
onyx tide
next steps to what
To get the brackets
jaunty wren
if you found -4 and -8 that means that x^2 - 12x + 32 = 0 is the same as (x-4)(x-8) = 0
so with (x-4)(x-8) = 0 you can find the xs that make the polynomial = 0, or roots
rose trout
And what are next steps
The general method is you just split the middle term to get like $x^2 -4x -8x + 32$ and factor the pairs by grouping but in this case what ant says holds
grand pondBOT
Azyrashacorki
onyx tide
.close
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kind ledge
A(5,-2) B(3,4) which make a vector how do i find AB (x,y)
midnight plankBOT
A(5,-2) B(3,4) which make a vector how do i find AB (x,y)
blissful pier
kind ledge
i tried doing (5+3, -2,4)
blissful pier
i tried doing (5+3, -2,4)
not quite, AB is not determined by A+B but rather B-A
kind ledge
ohh
blissful pier
also hello adonis 
midnight plankBOT
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limber plaza
I’m a bit confused on this proof. At 1.4, he seems to just state this again without proof.
midnight plankBOT
I’m a bit confused on this proof. At 1.4, he seems to just state this again with...
limber plaza
I thought that we were proving the identity, but it seems he is using the identity to prove it. I assume I am misunderstanding.
junior flower
that looks like (nearly) the end of the proof of 1.3 (though i can't actually see what 1.3 is because it's on the previous page)
limber plaza
that looks like (nearly) the end of the proof of 1.3 (though i can't actually se...
1.3 is on the 2nd picture toward the bottom. Sorry, uploaded in reverse order.
junior flower
ah
limber plaza
But 1.3 is just stating the identity
junior flower
yea, and the proof ends on the line just below I.4
tall lintel
I’m a bit confused on this proof. At 1.4, he seems to just state this again with...
What identity are you referring to by the way?
limber plaza
Maybe I need to fill in the gaps to see it. I’ll work on it.
The identity in the second picture (1.3)
tall lintel
1^2 + 2^2 + ... + n^2 = n^3 + n^2/2 + n/6?
limber plaza
Yes, that’s the one
wraith dirge
I’m a bit confused on this proof. At 1.4, he seems to just state this again with...
(1.3) states the identity, then it is proven in a slightly different form, then (1.4) states the slightly different form that was proven, no?
tall lintel
Would you agree that this is the proof of that identity?
limber plaza
So, I see how the right simplifies to n^3 - 1^3 by telescoping (or at least I think so), the arithmetic progression for the left feels like it should be explained.
In any way, I now see it. I was getting hung up on the arithmetic progression
And didn’t set them equal to actually see it
Thank you folks 🙂
.close
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tough sapphire
how do i solve b,c, and d? solving with the radians is throwing me off 🥲
midnight plankBOT
how do i solve b,c, and d? solving with the radians is throwing me off 🥲
gaunt jetty
how do i solve b,c, and d? solving with the radians is throwing me off 🥲
Do you knoe the formula for the atc of a circle
For part b
Oh wait u wrote it down
tough sapphire
uuuuh i wrote smth down but idk if it's right
gaunt jetty
Yeah so just put the values in s = r*theta
gaunt jetty
uuuuh i wrote smth down but idk if it's right
Yeah it is
If you are talking about s = r*theta
But evaluate it now
tough sapphire
i plug radian value into theta right
gaunt jetty
Ye
tough sapphire
so just 2.5x12?
tough sapphire
i think my teacher had the wrong answer on her study guide, i got 30
gaunt jetty
Yeah thays correct
gaunt jetty
i think my teacher had the wrong answer on her study guide, i got 30
What did she have
tough sapphire
25 i think it was just a multiplication error at the end cuz she also had 12x2.5
tough sapphire
😔
gaunt jetty
Wanna move on to c now?
tough sapphire
yesss
idk if my formula is right i kinda just sat and thought through it
or.. al least attempted to
gaunt jetty
Ur logic is correct actually
Yeah good job!
But like
You can simplify it
Do you notice something you can eliminate in what you wrote
tough sapphire
pi can go right
tough sapphire
theta/2 x r squared
gaunt jetty
Uh huh
And thats the standard formula for the area of a sector!
So, rinse and repeat, plug and chug and tell me what you get :D
gaunt jetty
Mhm mhm
tough sapphire
so 1/2 / 2 x 36
1/4x36 =9?
gaunt jetty
Nailed it
tough sapphire
my teacher had 36 i think the one mistake is causing the others to get messed up 😭
okay so i'm not tweaking
gaunt jetty
So part d is a bit trickier than that. It needs you to reuse both formulas you nust used for part b and c
Wanna gice it a shot first before I help you? :3
tough sapphire
i'll try to get the formula really fast
carmine sigil
don't troll help channels
tough sapphire
150 cm squared?
gaunt jetty
tough sapphire
thank you so muchhhh
gaunt jetty
Ofc, good luck!
tough sapphire
gaunt jetty
You can close your channel by typing .close if you dont have any more questions :3
tough sapphire
.close
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balmy quartz
midnight plankBOT
lyric charm
what's the domain of y=6-(x-3)^2 as a function of x
balmy quartz
what's the domain of y=6-(x-3)^2 as a function of x
f(x) <= 6
lyric charm
don't overthink it
balmy quartz
x E R??
lyric charm
x in R, yes
or just R tbh
anyway
with x = 3+2sin(t) do we cover all possible values of x in that domain?
lyric charm
bad notation but yes, we do indeed not cover all possible x's
that's why your parametric things DON'T give you the entire curve C
balmy quartz
that's why your parametric things DON'T give you the entire curve C
okay thank you
this makes sense
.closer
.close
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Available help channel!
wind oxide
https://math.stackexchange.com/a/5114780/1126252
so i asked a question
midnight plankBOT
https://math.stackexchange.com/a/5114780/1126252
so i asked a question
wind oxide
i got an answer
but my brain simply refuses to comprehend all the nt at once
and group theory
lets start with this
so first of all
is S a group or a set
because we cant be sure an id elem exists, right?
and also surely there exists x large enough such that e-s_x < 0 and negative nums are not in f's domain
so S is a set
question, by the way (the answer first solves i)
wind oxide
lets start with this
assuming S is a set, then <S> is a 'subgroup generated by S'. WHAT IS <S> A SUBGROUP OF
i will assume the integers (since <S>equal to dZ)
now i need to prove <S> = dZ
where d= gcd(S)
well first of all can i conclude every element of S is in <S>
and so every linear combination of them is also in <S>
right
solid iris
$S=\cbr{f(x)+2020x:x\in\dom(f)}$
grand pondBOT
ロケット・ジャンプ
solid iris
S is a subset of codomain of f, not necessarily a subgroup
wind oxide
S is a subset of codomain of f, not necessarily a subgroup
yes okay thank u
wind oxide
now i need to prove <S> = dZ
now how to deal with this
solid iris
oh this answer is only for i
wind oxide
yes
i need to first learn how to deal with i
though i think the other answer is more suitable
in an olympiad setting
solid iris
in that case S is a subset of Z when we consider <S>
wind oxide
in that case S is a subset of Z when we consider <S>
yes nice i am on the right track
wind oxide
well first of all can i conclude every element of S is in <S>
this also seems right
wind oxide
and so every linear combination of them is also in <S>
but this isnt true now that i think about it
but it should be
since m \in dZ = <S>
solid iris
dZ=<S> comes from d=gcd(S)
wind oxide
dZ=<S> comes from d=gcd(S)
yes but how exactly
solid iris
every element of S is a multiple of d which gives the backward inclusion
wind oxide
every element of S is a multiple of d which gives the backward inclusion
wait okay we need to first know what <S> is no
showing S \subset dZ does show <S> \subset dZ
solid iris
showing S \subset dZ does show <S> \subset dZ
yes thats what i used
so whats the issue?
wind oxide
okay now forward inclusion
wind oxide
so whats the issue?
well i am not sure if we can show forward inclusion using S
wtever u continue dont mind me
solid iris
well i am not sure if we can show forward inclusion using S
use bezouts identity
wind oxide
use bezouts identity
thats the hint i have in the comments
let me try once more
dk = s_m \cdot x+s_n \cdot y
now i need to pick m and n such that the RHS always belongs to <S>
well i can simply pick x=0, y=1 since d divides all s_i
wind oxide
use bezouts identity
right?
midnight plankBOT
@wind oxide Has your question been resolved?
wind oxide
<@&286206848099549185>
woven laurel
hey
wind oxide
hii
woven laurel
sup
solid iris
rather, what do we immediately get after applying bezout?
solid iris
why did u write only two summands?
wind oxide
why did u write only two summands?
well there could easily be more
any finite combination will have a solution
solid iris
write out that form and carefully state what set each object comes from
wind oxide
write out that form and carefully state what set each object comes from
they all come from S
right
solid iris
pls fully write out the form given by bezout, if d=gcd(S) then d=?
wind oxide
pls fully write out the form given by bezout, if d=gcd(S) then d=?
well it would help if u told me what you are talking about because the only bezout i know is ax+by=dk
solid iris
that is bezout for two integers
so now u should be wondering and looking up bezout for ANY set of integers
wind oxide
so now u should be wondering and looking up bezout for ANY set of integers
i honestly cant find it'
do i just repeatedly multiply bezout pairwise to create a eqn in multiply integers?
or is it something more special
solid iris
when u wrote down the case of two integers, did u not realize that it might not apply since S doesnt necessarily have size 2?
wind oxide
when u wrote down the case of two integers, did u not realize that it might not ...
?? S is infinite
i am sure about that
so it should apply to any 2 elements of S
solid iris
thats the issue. the form of bezout u wrote only applies to when S has size 2
wind oxide
thats the issue. the form of bezout u wrote only applies to when S has size 2
wait what?
bezouts applies to Z freely and that doesn't have size 2
solid iris
the form of bezout u wrote only applies to two integers
wind oxide
the form of bezout u wrote only applies to two integers
yes
solid iris
my question is why did u write that form if S is possibly infinite?
wind oxide
my question is why did u write that form if S is possibly infinite?
because it doesnt matter? even if S is infinite i can apply it to any 2 elements so it holds for all the elements
why should i apply it to all the elements of S?
wind oxide
im very confused
i am confused more
wind oxide
what does bezout look like if |S|=3?
do i apply it pairwise? (so for a,b, then b,c, then a,c)
and then add?
solid iris
now that u realize ur form of bezout doesnt apply, u should be curious if generalizations exist
try finding some
wind oxide
any finite combination will have a solution
(this is what i meant by finite combination)
wind oxide
try finding some
is it something else
solid iris
"any finite combination will have a solution" is very vague which is why i asked u to fully write out that form
anyway now u should be wondering even more. this is bezout for finite sets but S is possibly infinite
wind oxide
uhh
i m not sure
i cant just let n approach infinity in this
and i dont think its related to polynomials
solid iris
theres an earlier part of the SE post u mightve taken for granted
d=gcd(S). if S is possibly infinite what does gcd(S) even mean?
wind oxide
d=gcd(S). if S is possibly infinite what does gcd(S) even mean?
oh wait
wait does that mean d=1
no ok probably not
well gcd(S) means a number that divides every number in S
so the gcd of (f(1) + 2020, f(2) + 4040, ...) for all natural numbers
am i missing something here?
solid iris
now u should be asking urself more questions
does gcd of infinite set always exist? why or why not?
wind oxide
does gcd of infinite set always exist? why or why not?
well it should always exist if the set is a subset of integers (which it is since we're talking about gcds)
since every integer is divisible by 1
for nZ its n
o
solid iris
all u said is gcd>=1
wind oxide
all u said is gcd>=1
well if nothing divides all the numbers we can let it be 1
solid iris
the gcd may or may not blow up as we take more numbers
wind oxide
for nZ its n
but like in this case we can be sure its n
solid iris
try to answer
does gcd of infinite set always exist? if yes prove it. if no give a counterexample
orchid fossil
https://math.stackexchange.com/a/5114780/1126252
so i asked a question
oh you are also on MSE? nice.
i am too on it
wind oxide
does gcd of infinite set always exist? if yes prove it. if no give a counterexam...
let us assume for the sake of contradiction that the gcd DNE, i.e. there exists no integer such that divides evey element of our infinite set.
but every integer is divisible by 1, so the set always has 1 as a common divisor of every element. So there exists an integer that divides every element of our infinite set. This is a contradiction. BWOC, GCD must exist
hopefully this is good enough
solid iris
not at all
wind oxide
not at all
solid iris
1 is a common divisor but that doesnt say anything about the GREATEST common divisor
wind oxide
1 is a common divisor but that doesnt say anything about the GREATEST common div...
well we had assumed no integer exists
but we just proved an integer exists which is a contradiction
so our assumption must be wrong
therefore the complement must be true
no?
wind oxide
1 is a common divisor but that doesnt say anything about the GREATEST common div...
i mean this does seem right though
but BWOC this proof is correct no?
solid iris
assuming the lack of greatest common divisor does not imply lack of common divisors
thats ur fatal flaw
scenic wyvern
well we had assumed no integer exists
what if there are infinitely many of them?
I believe that's what roketto is implying here?
wind oxide
assuming the lack of greatest common divisor does not imply lack of common divis...
huh??
we always have a greatest element
if we have common divisors
if we dont have a greatest element u possibly cant have common divisors
wind oxide
what if there are infinitely many of them?
infinitely equal integers still have a well defined gcd
wait
infinitely many unequal integers dont
o
solid iris
mhm
wind oxide
mhm
so if we had infinitely many uneqal common divisors
we wont have a gcd
but what set can even satisfy that
solid iris
i suggest u do a direct proof instead of contradiction
empty scroll
ok
wind oxide
i suggest u do a direct proof instead of contradiction
ok i give up
hand me the solution
solid iris
that sounds a bit more forceful than u probably intended
wind oxide
that sounds a bit more forceful than u probably intended
oh ok sorry
but yes pls give the solution
solid iris
$\gcd S=\max\br{\bigcap_{i=1}^\infty F_i}$ where $F_i$ is the set of factors of the $i$th number
grand pondBOT
ロケット・ジャンプ
solid iris
agree?
midnight plankBOT
@wind oxide Has your question been resolved?
solid iris
$\bigcap_iF_i\ss F_1$ and $F_1$ is finite, so $\bigcap_iF_i$ is finite, so the max exists
grand pondBOT
ロケット・ジャンプ
solid iris
recall the def of intersection
wind oxide
recall the def of intersection
oh wait rt
OH
THATS SO EASY
HOW COME DID I NOT SEE THIS
🤦♂️
well why did we even prove this
solid iris
u were probably overwhelmed by the amount of detail in the SE post and took many things for granted
wind oxide
u were probably overwhelmed by the amount of detail in the SE post and took many...
maybe
wind oxide
What course is this for
great question, idk
remote flame
looks terrifying
remote flame
great question, idk
Im assuming you are also in college?
solid iris
an unexplained gap in the SE post: S is possibly infinite but bezout only applies to finite sets
theres a fact that bridges the gap. try to explore and make a conjecture. google if u give up
i have to sleep now. good luck
midnight plankBOT
@wind oxide Has your question been resolved?
inland patio
Let $X$ be distributed according to a Poisson distribution with parameter $M$, which itself is a random variable. The author writes $X\mid M=m\in\text{Po}(m)$, which I guess means $P(X=k\mid M=m)=\frac{m^k e^{-m}}{k!}$. First I thought $P(X=k\mid M=m)$ is a regular conditional probability, but I'm confused; $X$ is discrete whereas $M$ is continuous it seems. What is then $P(X=k\mid M=m)$ supposed to mean?
midnight plankBOT
Let $X$ be distributed according to a Poisson distribution with parameter $M$, w...
grand pondBOT
psie
frank swift
pearl hull
Don’t troll here 
twilit jetty
Let $X$ be distributed according to a Poisson distribution with parameter $M$, w...
should just be the chance that X, with M being set to m, is the integer k and not some other integer
P(X = k | M = m) with m fixed is a pmf
compare this to if you set M = lambda in advance as a given, itd be the same ordeal
inland patio
should just be the chance that X, with M being set to m, is the integer k and no...
So in measure theory language, $P(X\in A\mid M=m)=\nu(m,A)$, where $\nu$ is a Markov kernel I think.
grand pondBOT
psie
midnight plankBOT
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hoary hamlet
how do i go about solving this?
midnight plankBOT
how do i go about solving this?
umbral laurel
how do i go about solving this?
Can we use the property of a triangle that one side is always lesser than the sum of other 2 and use some trigo to rule the about out ( sorry that's the thing i can think it may be stupid)
midnight plankBOT
@hoary hamlet Has your question been resolved?
shell wigeon
It's not very hard to construct counterexamples for the first two