so there are 7 such elements that work in Z_9 right
#help-49
runic hamlet
which elements in Z_9 do not work?
twilit field
I'm so confused
none?
okay, so there are 6 elements of order 9
2 of order 3
and 1 of order 1
.close
midnight plankBOT
midnight plankBOT
Available help channel!
zealous schooner
midnight plankBOT
zealous schooner
How do I approach this question?
ancient ridge
You should start by following the hint and finding an alpha that makes that quantity a linear combination of i and k.
last slate
midnight plankBOT
last slate
How do i show 3?
i did the first using IVT
and second using Vieta
But when i try to show 3
i'm not able to compare x_1 and x_3 using their intervals in 1
near cliff
Could you translate the problem?
last slate
Could you translate the problem?
sure
last slate
Could you translate the problem?
Consider the equation $x^3 -x^2-2x+1=0$
-
Prove that the equation Has 3 roots that verify $-2<x_1<-1 , 0<x_2<1 ,1<x_3<2$
-
Show that $x_1 + x_2 + x_3 = 1$
-
deduce that $\abs{x_2} < \abs{x_1} < \abs{x_3} < 2$
grand pondBOT
Drk
grand pondBOT
Drk
last slate
i think making the x_1 interval smaller using dichotomy method would work
but is that like the only way
steep ledge
x1+x3 = 1-x2 which is between 0 and 1
so x1+x3 > 0 so x1 > -x3, since x1 and -x3 are negative it implies |x1| < |x3|
last slate
x1+x3 = 1-x2 which is between 0 and 1
so x1+x3 > 0 so x1 > -x3, since x1 and -x3...
Oh yeah
thanks
.close
candid night
heyy
candid night
so
can anyone help me abit here?
ill translate
The function's y= f(x) E(f)=[-3;7]. What is the function's g(x)=f(x-4)+2 E(f)?
the answer is D
i have no clue from where to even start
gaunt jetty
What's E
thorn sorrel
what transformations does the function undergo
candid night
probably a coordinate
ig
gaunt jetty
Ok so not a coordinate
candid night
Does the transformation f(x) -> f(x-4) affect the range of the function
idk all ik is that -4 moves the function to the right and +2 makes it go up
candid night
the y axis
gaunt jetty
the y axis
So, if some transformation makes the function go "up or down" it would affect fhe range yes?
candid night
So, if some transformation makes the function go "up or down" it would affect fh...
yes
candid night
the -4 makes it go right
gaunt jetty
Answer the question though
candid night
yes
it does
i'm sorry i don't understand if i didn't give enough info or you're trying to gauge my knowledge on the topic
gaunt jetty
Nah i am just helping you try to solve it
candid night
ahh okk
gaunt jetty
A transformation like f(x-4) doesn't affect the range bcuz this wouldn't change the range of y values
gaunt jetty
There's a + 2 there
candid night
you mean x-4 is a?
candid night
okki
candid night
ohh
undone yacht
Np
undone yacht
have a great day :))
U2
graceful ferry
How do I know for which values of alpha the matrix is positive definite?
midnight plankBOT
How do I know for which values of alpha the matrix is positive definite?
fathom onyx
!status
midnight plankBOT
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
Check the definition of positive definiteness, would be my first port of call
graceful ferry
Progress:
Um so the very long term i got, how do i know for which values of alpha it is positive ?
graceful ferry
Check the definition of positive definiteness, would be my first port of call
I'm trying to do it using the definition of positive definite
fathom onyx
There is another definition I might consider trying:
[Note: I haven't checked how this would work for this question, but I'd take a look at this]
graceful ferry
Alr will try with this
But this doesn't seem like an easy matrix to find its' eigenvalues
midnight plankBOT
@graceful ferry Has your question been resolved?
graceful ferry
I can't find any easy way to find the eigenvalues, do I have to just do det(A-lambdaI) ....?
There must be a better way
obtuse basin
There must be a better way
Another characterization for positive definiteness is that the determinants of all upper-left submatrices are positive
Even though there's still some tedious work needed for this, it's better than using the definition or the characterization using eigenvalues
graceful ferry
I mean I didn't learn this in class so idk if I should be using this
Honestly I'm lost it seems like they just want me to do the hard lifting
midnight plankBOT
@graceful ferry Has your question been resolved?
midnight plankBOT
@graceful ferry Has your question been resolved?
graceful ferry
.close
opal hollow
Hi im kinda confused on where the thing highlighted in purple came from
midnight plankBOT
Hi im kinda confused on where the thing highlighted in purple came from
golden wagon
Hi, it seems like it was omitted that x^(e^x) is inside sin
Just to be sure, e^x is an exponent of x and not sin x?
opal hollow
Yes
opal hollow
Just to be sure, e^x is an exponent of x and not sin x?
^
opal hollow
Hi, it seems like it was omitted that x^(e^x) is inside sin
?
golden wagon
If you were to take the ln of y=x^e^x on both sides you would get what was written (also use ln on both sides, as the derivative of ln x is just 1/x)
opal hollow
If you were to take the ln of y=x^e^x on both sides you would get what was writt...
Sorry is there any way for you to write that out? Its okay if you can’t
golden wagon
$y=x^{e^x} \ \ln y = e^x \ln x$
grand pondBOT
FriedRiceKing
golden wagon
This is where I suspect the purple part came from; sin was omitted
opal hollow
Hm
Oh
Wait
Then why is there a +log x
Also
Would i take the derivative of the green and purple seperately?
So that would just make it 1/1+x
Or would i do the product rule then the derivative
golden wagon
Then why is there a +log x
Frankly I don’t see why it appeared
opal hollow
O
golden wagon
Would i take the derivative of the green and purple seperately?
You would do product rule
opal hollow
Ok
golden wagon
Also, make sure to use take the natural base log of both sides instead of base 10
opal hollow
Ok
golden wagon
This is because $\frac{d}{dx} \log y = \frac{1}{y \ln (10)}\cdot y’$
grand pondBOT
FriedRiceKing
golden wagon
Of course you could use base 10, but it would be more complicated
opal hollow
Mmm
Just to be clear. The derivative of ln is 1/x right? Then how do we get rid of log if we arent doing derivative
golden wagon
At the start, you take the ln of both sides instead of the log
golden wagon
Since we want to use ln instead of log, at the start we take the ln on both sides
This means what you have is almost good, you just have to replace the logs with ln
And you forgot ln for (1+x) (first term)
thorn sorrel
log means ln most of the time
golden wagon
Oh? I was taught log means base 10
opal hollow
Since we want to use ln instead of log, at the start we take the ln on both side...
Ok
.close
wraith notch
This is not making sense to me
midnight plankBOT
This is not making sense to me
wraith notch
So, the answer is c, but assuming that you approach like, infinity, then it would eventually go to the deviation 102
Unless this is a z score answer o
No that makes no sense
Oh there isnt a sqrt I guess
I'm lost
sigma over n.... but not sigma over sqrt(n), why
lethal path
sigma over n.... but not sigma over sqrt(n), why
sigma over sqrt(n) is correct
they've made a typo in the options, so it is 102/sqrt(100) = 102/10 = 10.2
wraith notch
Oh okay
This book had a couple typos that I noticed, and is a bit weirdly formatted, thanks
.close
midnight plankBOT
midnight plankBOT
Available help channel!
wanton steeple
would LCM of pi and pi^2 exist?
midnight plankBOT
would LCM of pi and pi^2 exist?
wanton steeple
i guess thats true, what is the general definition?
wanton steeple
its usually only defined over the integers
yeah in that case none should exist
runic hamlet
a is called a multiple of b if there exists an integer k with a=bk
thats the definition over the integers
you can adapt this to so-called "rings"
a is called a multiple of b if there exists an element k in the ring such that a=bk
the problem is, if we take the real numbers as our ring, then stuff gets very boring
because everything is a multiple of everything
(except zero)
cause you can always choose k=a/b
you can work in the ring Z[pi] which essentially means "all finite expressions involving sums/multiplications with integers and pi"
so all numbers like 3, pi, 18-pi, 19pi^2+pi^17, ...
in that ring pi^2 is a multiple of pi as expected
and then lcm(pi,pi^2)=pi^2
wanton steeple
you can work in the ring Z[pi] which essentially means "all finite expressions i...
what would be a "ring" ?
runic hamlet
ring basically means "set of numbers in which you can add, subtract and multiply but not necessarily divide"
wanton steeple
i see
wanton steeple
and then lcm(pi,pi^2)=pi^2
so the answer would indeed differ depending on your number system;
i think i get it.
thanks!
runic hamlet
yw
wanton steeple
.close
midnight plankBOT
midnight plankBOT
Available help channel!
vital quartz
i am so confused on how to start answering.
midnight plankBOT
i am so confused on how to start answering.
vital quartz
i am confused on what values mean what things in the triangle
twilit jetty
first, what is something you know you can do if you know one of the sides
for example if a is 1, what can you figure out about b and c
lusty python
i am confused on what values mean what things in the triangle
a, b, c are just the sides of the triangle
vital quartz
what does 1, 2, and 3 mean????
granite sandal
i am so confused on how to start answering.
i think these are the sides of the triangle
lusty python
i am so confused on how to start answering.
May i ask, what's the 1. 2. 3. thing on the table mean?
granite sandal
and you can see that the angles are 45 degree
twilit jetty
oh thank god I thought you guys were going to spoil it
granite sandal
tan 45 = perpendicular/hypotenuse
twilit jetty
you dont know either
vital quartz
May i ask, what's the 1. 2. 3. thing on the table mean?
exactly my point
i dont even know too
twilit jetty
what does 1, 2, and 3 mean????
heres a hint
usually if you have a table
one direction means one thing
and the other direction has to mean something different
granite sandal
i dont even know too
nah i think you just need to fill value of sides for the triangles
look on the left a,b,c
lusty python
i am so confused on how to start answering.
I don't even need trig for this
granite sandal
just need to designate the side length
twilit jetty
yea you need to fill in side lengths into the table
now you can see here there are three different side lengths in this 3x3 grid
lusty python
i am so confused on how to start answering.
In this example, try calculating the other angle?
twilit jetty
now taking a look at this picture,
lusty python
PLEASE DON'T SPOIL
vital quartz
In this example, try calculating the other angle?
its a right triangle
twilit jetty
do you think these all belong to the same triangle or to different ttriangles
lusty python
!noans
midnight plankBOT
The purpose of this server is to help you learn, not to hand out answers. Do not ask someone to give you the answer directly.
twilit jetty
PLEASE DON'T SPOIL
thats not what the spoiler is
vital quartz
do you think these all belong to the same triangle or to different ttriangles
its only 1 triangle per given
twilit jetty
its only 1 triangle per given
wdym
wise flicker
ig it's just bout triangles with sides a,b,c fill corresponding value in columns
lusty python
its a right triangle
Yes, try calculating the other angle
twilit jetty
ig it's just bout triangles with sides a,b,c fill corresponding value in columns
youve just spoiled it
goddammit there are too many people int his channel
wise flicker
youve just spoiled it
wdym
twilit jetty
3 of you dont even get the question
1 of you spoiled the core skill that MSG needs to learn
you needed to figure out for yourself what they were even thinking when they made the table
its a critical skill called "reading comprehension"
if you just have someone explain the table for you, youll never get it when the next BS thing rolls around
vital quartz
it does not say anything in the book
twilit jetty
no it doesnt
vital quartz
it just said get the values
twilit jetty
yes it does
that would be exactly what a chart like this would do to you
what you have to do here is combine common sense with some deduction, this is a very strange question to figure out
the question being what the table is supposed to mean
now as before, we established these are all of different triangles, right
they cant all be the same triangle
vital quartz
the question being what the table is supposed to mean
complete the values
vital quartz
they cant all be the same triangle
is it even possible to get 9 values from a triangle?
twilit jetty
potentially if you organize things a particular way, but for now, no
you can get up to three values from the triangle
that is because in the diagram on the right, they only labelled three things
and correspondingly in the table there are the same three labels
a, b, and c
does this make sense?
twilit jetty
now knowing we can only get 3 values at a time,
lets focus on where each number is in the table
take 8√2 for example
what value would this correspond to?
vital quartz
b
vital quartz
but what would i put in 1 and 3?
twilit jetty
lets focus on what Im asking for now
Im leading you to eventually figure out what the 1, 2, 3, 4, 5, 6 mean
vital quartz
aight
twilit jetty
its important you come up with what they mean yourself, that way youre familiar with how you gotta think inside the box
twilit jetty
b
for each value in this table, youre recognizing you can get two other values
lets just have those numbers down for reference for now
first, say c is 7/2 * √2
what would a and b be
vital quartz
they would be equal
twilit jetty
yes theyd be equal, but what would their values both be
vital quartz
yes theyd be equal, but what would their values both be
7/2?
twilit jetty
thats correct
now unless you want to memorize 6 numbers,
ideally youd want to write this down somewhere
keep in your head that both 7/2 and 7/2 need to be written somewhere in this table
lets go on to the next one:
if b is 8√2, what would a and c be?
vital quartz
a - same
c - 16???
twilit jetty
thats correct
so now we have 8√2 and 16 also needing to be written down
now for the final triangle for this table
a = 10√2
what is b and c?
vital quartz
b - same
c - 20
twilit jetty
nice
vital quartz
OH
twilit jetty
so our first step to dealing with this table: just do the problem without it and see what happens
second step it seems youre figuring out right now
vital quartz
I FEEL SO DUMB RN
twilit jetty
dw about it
this table is absolute ass at presenting itself
what youre doing rn is actually pretty smart
you figured out what the table is supposed to do without much of any hints
vital quartz
twilit jetty
yep, thats what you do
now given that the numbers on the top didnt seem to do anything
and in the table below that, the numbers continue 4, 5, 6
what do you think the 1, 2, 3 mean?
vital quartz
same triangle with the values changed
twilit jetty
think less mathy for this one
what would 1, 2, 3, 4, 5, 6 be doing on your assignment?
vital quartz
i think i can do the others now
twilit jetty
lol
twilit jetty
what would 1, 2, 3, 4, 5, 6 be doing on your assignment?
do you know what these numbers are for
vital quartz
what would 1, 2, 3, 4, 5, 6 be doing on your assignment?
different triangles
twilit jetty
you gotta say what theyre for at least if you know it
that isnt exactly what Im looking for
theyre problem numbers
they number that you have six problems to do
1 is the first problem, 2 is the second problem, etc.
they dont have to be different triangles either
they could serve you two of the same triangle in two problems
so close, yet so far
youre gonna have to memorize that instead of think of it next time you see something like this
midnight plankBOT
midnight plankBOT
Available help channel!
hoary drift
I tried-
- substituting the second polynomial = u^m
- substituting x^m = u
midnight plankBOT
I tried-
- substituting the second polynomial = u^m
- substituting x^m = u
fallow scarab
differentiate the answers 
hoary drift
Its not an exam so I want a subjective solution
fallow scarab
differentiating the answer will give you the substitution
hoary drift
Assuming m = 1 will be the better shortcut anyway
hoary drift
differentiating the answer will give you the substitution
True
Bruh the question was printed incorrectly
😭
.close
midnight plankBOT
hoary drift
This was the correct question
midnight plankBOT
Available help channel!
modern lynx
I am trying to integrate $x^2 + 1$ on the interval [0,3] using riemanns sum
midnight plankBOT
I am trying to integrate $x^2 + 1$ on the interval [0,3] using riemanns sum
grand pondBOT
Mystic
modern lynx
why is my answer 9 please spot the mistake i cant find it ive reviewed it multiple times
fallow scarab
this already looks wrong
or that's the last step that was correct
next step after that should be to sub in delta x = 3/n
as delta x goes to 0, n goes to inf
modern lynx
lmao riemann himself 💀
fallow scarab
you have delta x goes to 0, but n has a direct relationship to delta x
modern lynx
yeah that is why i am converting all the delta x's to 3/n
so i can replace lim delta x to 0 with lim n to inf
shouldnt it be an inverse relationship
delta x = 3 / n
fallow scarab
then you did that incorrectly here
f(a) = 1 + a^2
f(a + delta x) = 1 + (a + delta x)^2
and so on
modern lynx
a is 0 so f(a) is 1
alright i understand it now
i missed all those 1's
they sum upto n-1
thanks riemann
i love this formula btw
thanks for inventing it
modern lynx
midnight plankBOT
modern lynx
I am trying to integrate $x^2-x^3$ from -1 to 0 using riemanns sum
grand pondBOT
Mystic
modern lynx
Does this summation only apply to the i's?
I mean can I do this after that step
im getting the wrong answer so i probably cannot
so how do i do it then
placid spoke
I mean can I do this after that step
you didn't divide by 6 for your sum of i^2
modern lynx
you didn't divide by 6 for your sum of i^2
that would give me 4/3 instead of 8 right
well 4/3 -1/4 -3/2 isnt correct either
the answer is 1/12
fallow scarab
where's the original question
fallow scarab
where did this term go
modern lynx
let me correct it
modern lynx
where did this term go
it got even worse now
modern lynx
no i know it is an i
fallow scarab
oh even worse
fallow scarab
1 + 1 + 1 + ... + 1 n times equals n
modern lynx
should it be n*i
fallow scarab
1 + 2 + 3 + ... + n is not n
fallow scarab
did you make more mistakes?
modern lynx
did you make more mistakes?
this will give me 4/3 - 5/2 -1
which is not 1/12
are you sure this method is correct
am i the only problem here 😭
fallow scarab
what "method" are you referring to
modern lynx
what "method" are you referring to
your method
fallow scarab
yea it's right
modern lynx
mistake here
this term gives me 5/2 instead of 0 right
fallow scarab
another mistake here
modern lynx
no 4
sudden yacht
Yeah you forgot a 4
fallow scarab
who says the answer is 1/12 ?
fallow scarab
another mistake here
modern lynx
alright well now it is 4/3 - 5/2 - 1/4
this yields 17/12
fallow scarab
more context
the first line at the top you were just simplifying 1/n * f(-1 + i/n)
but the bottom you plugged that into the sum
fallow scarab
from the beginning
fallow scarab
1 + 1 + 1 + ... + 1 n times equals n
use
modern lynx
sum of wat
grand pondBOT
riemann
modern lynx
uhh
fallow scarab
if 2/n is inside the summation why cant i take it out
since you factored out 2/n
modern lynx
n i guess
fallow scarab
1 + 1 + 1 + ... + 1 n times equals n
right
modern lynx
1 + 1 + 1 + ... + 1 n times equals n
how am i gonna use this on 2/n
fallow scarab
if 2/n is inside the summation why cant i take it out
you factored out 2/n
modern lynx
where
fallow scarab
fallow scarab
from the beginning
.
modern lynx
i took it out
modern lynx
from the beginning
this is incorrect cuz i did not write the 2i/n term
fallow scarab
write what this equals to
modern lynx
write what this equals to
nothing
modern lynx
it just goes outside right
earnest quest
cant believe Bernhard Riemann himself is here helping people
fallow scarab
write out exactly what you mean by "goes outside"
modern lynx
write out exactly what you mean by "goes outside"
wat i mean is that the summation does not apply to it
fallow scarab
nah
modern lynx
if 2/n is inside the summation why cant i take it out
.
fallow scarab
to illustrate, take for example n=3
modern lynx
then why this
fallow scarab
2/3 + 2/3 + 2/3 = ?
modern lynx
2
fallow scarab
i thought you meant "take it out" to mean "factor out"
fallow scarab
2/3 + 2/3 + 2/3 = 2/3 * (1 + 1 + 1) = 2/3 * 3 = 2
modern lynx
so it is sum of 2/n upto n terms
fallow scarab
**riemann**
to be able to use this
modern lynx
that is just 2 right
fallow scarab
correct
modern lynx
so we have a 2
fallow scarab
2/n * n = 2
fallow scarab
at least you got one of the summations correct
modern lynx
whats wrong with you riemann get out of my curriculum
fallow scarab
skill issue
fallow scarab
everyone makes these mistakes their first time
modern lynx
everyone makes these mistakes their first time
my professor told me that formula its his fault 😭
imagine if i started from this dude
.close
tired cedar
What am I missing here? (Polynomial Functions)
midnight plankBOT
What am I missing here? (Polynomial Functions)
tired cedar
It should be 4.7713 T
in the last part of the equation, there is an extra T next to D3, which combines to that T
sudden yacht
Omg I would have never seen it 😅
tired cedar
instead of 1506.8655
could you show me your step by step please?
sudden yacht
i get 1506.8349
Alright then I think this is it
mystic condor
you accidentally added the last bit instead of keeping it as a T term
like you treated that as a constant
tired cedar
I don't understand
tired cedar
yeah
mystic condor
you seem to have added that
tired cedar
yes
-7.139x10^-13 x (3500^3)
= 0.03684625
add the T to that, and then combine like terms with the other T
sudden yacht
That part is correct, I've just checked right now
mystic condor
the constant?
sudden yacht
No, the linear term
mystic condor
yeah its definitely the constant thats wrong
tired cedar
I think I asked before but could you give me a step by step on how you did the constant?
sudden yacht
yeah its definitely the constant thats wrong
Yeah I get 1506.8349
mystic condor
Yeah I get 1506.8349
he got .8655
tired cedar
*she
sudden yacht
he got .8655
Yep that's why
mystic condor
sorry
sudden yacht
The other is fine
mystic condor
i normally say they to avoid that i completely forgot there
tired cedar
I think I see now, I must've rounded it randomly during the calculating
2.051875 turned into 2.0519
sudden yacht
I think I see now, I must've rounded it randomly during the calculating
Oh that's the mistake for sure
sudden yacht
In maths (and also physics, I'd say) you never round up during the steps, only at the end
tired cedar
I think my confusion was rounding the coefficients, I didn't know what that meant at first tbh
sudden yacht
I love how the smallest thing could make the biggest difference
If you've done or will do physics, you'll see this much better
tired cedar
I most likely will with weather science/meteorology
mystic condor
i always just use the store functions on my calculator to avoid rounding errors
im rly stubborn about it
sudden yacht
If you've done or will do physics, you'll see this much better
Especially with extremely tiny things (like atoms) or extremely big ones (astronomy)
tired cedar
i always just use the store functions on my calculator to avoid rounding errors
how do you do that outside of storing it in the y= function?
mystic condor
depends on the calculator ig
tired cedar
Ti-84 Plus
mystic condor
press sto, then one of the letters
on my calculator for some reason you dont use alpha to press the letter so test that out
midnight plankBOT
midnight plankBOT
Available help channel!
tired cedar
I'm overall just lost on how to figure out the asymptote, I've tried 1 and 0 since thats what it looked like on my graphing calculator.
midnight plankBOT
I'm overall just lost on how to figure out the asymptote, I've tried 1 and 0 sin...
fallow scarab
vertical asymptotes can occur when the denominator equals 0
tired cedar
so does that make it undefined?
fallow scarab
yes typically vertical asymptotes mean the function is undefined at that x value
fallow scarab
the function is undefined, but the x value exists
a simpler example is $g(x) = \frac{1}{x}$
grand pondBOT
riemann
fallow scarab
the vertical asymptote occurs when the denominator is zero. do you know what x value that is?
tired cedar
if it is zero then x is zero?
fallow scarab
correct x = 0 is the vertical asymptote. g(0) is undefined, because 1/0 is undefined.
fallow scarab
vertical asymptotes can occur when the denominator equals 0
yes
tired cedar
yes
sorry when it was said that way it didn't really make much sense
thanks tho, I get it now
.close
midnight plankBOT
midnight plankBOT
Available help channel!
pliant sorrel
midnight plankBOT
pliant sorrel
SO i already did the algebra got simlpified it to (n^2 + n + 1)/(n(n+1))
then i wrote out the sum
and its 3/2 + 7/6 + 13/12 + ... + 4086463/4086462
and i wrote it then as 1+1/2+1+1/6+1+1/12+...+1+1/4086462
and i summed all the 1's
and its 2021 + 1/2+1/6+1/12+...+1/4086462
but then how do i sum the other part
idk
small jasper
,w true or false sqrt(1+1/n^2+1/(n+1)^2)=(n^2+n+1)/(n(n+1))
grand pondBOT
small jasper
But anyway, finding the sum of $\frac{1}{n(n+1)}$ is the textbook example of partial fractions $\to$ telescoping
grand pondBOT
Civil Service Pigeon
pliant sorrel
so what do i do with 1/n - 1/(n+1)
small jasper
so what do i do with 1/n - 1/(n+1)
Write out the first few terms
fair frigate
A circle is given with two distinct secant lines drawn from an external point P. One secant intersects the circle at points A and B in that order as one moves toward the circle from P. The other secant intersects the circle at points C and D in that order from P. All points A,B,C,D lie on the circle. You are told that the measure of ∠APC is equal to the measure of ∠APD. Using only geometric principles that hold for all circles—such as inscribed angle relationships, similar triangles, and properties of intersecting chords—prove a relationship that must always hold between the directed lengths PA,PB,PC, and PD. Your proof should: Establish at least one pair of similar triangles created by intersections of the secants with the circle. Explain why the angle relationships guaranteeing similarity must hold for any pair of secants drawn from the same external point. Use the similarity to derive a proportionality involving the products of segments formed along each secant. Conclude with a general statement describing the invariant quantity determined by the external point P and the circle. You must justify each step using circle theorems, triangle similarity criteria, or other accepted geometric principles.
midnight plankBOT
A circle is given with two distinct secant lines drawn from an external point P....
shut elk
lavish venture
why are you opening a second channel
lavish venture
.close
midnight plankBOT
midnight plankBOT
Available help channel!
honest briar
how do i calculate the closed form
midnight plankBOT
how do i calculate the closed form
honest briar
is there a formula
night axle
is there a formula
What do you think it could be
night axle
Have you done part a
honest briar
yep
night axle
That might give an idea
night axle
Ok, and does that give any idea
Does it look like anything you recognize
What about expanding out a_4 in terms of a_1
honest briar
uhuh
night axle
So, what about a4? a5?
night axle
4(2a1+1)+2
Doesn’t quite seem right
honest briar
I'm really confused 😭
night axle
But you’ll notice, you keep stacking things, like
2+1
2(2+1)+1
2(2(2+1)+1)+1
honest briar
uhuh
true
Wait I'll brb don't close this channel I gotta go do something
If it gets closed I can make another one actually I'll be like a good hour
.close
midnight plankBOT
midnight plankBOT
Available help channel!
rapid talon
Help me
rapid talon
Ok
rapid talon
Okok
modern sapphire
flat tartan
guys
flat tartan
whats the pythagrean theorum
vague ember
$a^2 + b^2 = c^2$
grand pondBOT
_rayn123_
vague ember
@flat tartan
flat tartan
just math in general
vague ember
make sure u understand
concepts
very well
do practice questions
try to extend yourself and do harder ones
simple as that
vague ember
uhm it depends on the topic tbh
hexed barn
if possible, rederive certain key formulas, like the quadratic formula, yourself. this gets you into the habit of asking why formulas are like that, what they represent, when you can (or can't) use them, and how you should use them given what you know the formula represents.
so, do you have any other questions?
flat tartan
if possible, rederive certain key formulas, like the quadratic formula, yourself...
JK
thanks that helped alot
midnight plankBOT
Available help channel!
inland patio
$X_n \to X$ in distribution iff $F_{X_n}(x)\to F_X(x)$ at each point of continuity of $x$. I have a question about this claim in the forward direction. We know from the Portmanteau theorem, that if $X_n\to X$ in distribution, then $P_{X_n}(B)\to P_X(B)$ for every Borel set $B$ whose boundary has zero $P_X$ measure. I'm a bit uncertain how to proceed further here. Any help is appreciated.
midnight plankBOT
$X_n \to X$ in distribution iff $F_{X_n}(x)\to F_X(x)$ at each point of continui...
grand pondBOT
psie
inland patio
.close
shut canyon
midnight plankBOT
shut canyon
monic f ∧ monic g → monic g ∘ f
runic hamlet
write down the definitions of things
midnight plankBOT
@shut canyon Has your question been resolved?
shut canyon
runic hamlet
thats not the def
shut canyon
dont talk about individual elements
composing two arrows with a monic if they merge, then they were the same to start with, which is strange, becaue first they are two arrows but then they merge and they were the same?
runic hamlet
just because we give things different names doesnt mean they are different
we just didnt know yet that they were the same
shut canyon
just because we give things different names doesnt mean they are different
you start with two strings with two beads, and then you find they merge into a single bead and singe string, and deduce the two strings were the same? it's like maybe you change context when you say fx=fy
this seems similar to the pullback idea, you pull back truth or something
and the constructivist logic idea, where you construct logic as you go?
runic hamlet
how are you doing cat theory but the idea of
(something involving x)=(something involving y) =>... => x=y
is strange to you
twilit jetty
I get the impression youre trying to learn all the big words before learning all the common meanings
there are several wordless ideas that Ive consistently see you not having that you just have to build up as you go through a standard math course
shut canyon
Maybe it's like we start with one bead, and one string fx=fy, fgx=fhy
and we care to know whether there were two strings to start with
but I still sense there is something to the idea of going from the 0 set, two strings, two beads, and change context, which constructing the proof path
lyric charm
kytsu you cannot be doing these things with no grasp of basic discrete math / basic proofs
shut canyon
like this seems like a pullback
twilit jetty
Im curious actually
,,|x|=\begin{cases}x&\text{if }x\ge0\-x&\text{if }x<0\end{cases}
grand pondBOT
mtt
twilit jetty
@shut canyon can you justify this definition of |x| for R -> R?
frank wolf
like this seems like a pullback
Woah pullbacks
Really this is just a commuting square
runic hamlet
but thats not such a fancy word
frank wolf
Wdym
shut canyon
Really this is just a commuting square
f ∘ g = f ∘ h is not necessarily pullback?
frank wolf
f ∘ g = f ∘ h is not necessarily pullback?
No, since a pullback is a kind of “universal” commuting square
shut canyon
No, since a pullback is a kind of “universal” commuting square
if f ∘ g = f ∘ h , then in that context it is universal isn't it?
shut canyon
there is a universal arrow? here also C⟶B is universal maybe
frank wolf
Well what you’d need is a universal pair of arrows
But there’s no reason to expect g and h form that universal pair
shut canyon
Well what you’d need is a universal pair of arrows
Alright, thank you for your help @frank wolf , I appreciate it. I don't yet understand why this square is not a pullback, I'll look into it
midnight plankBOT
@shut canyon Has your question been resolved?
midnight plankBOT
@shut canyon Has your question been resolved?
midnight plankBOT
midnight plankBOT
Available help channel!
midnight plankBOT
high wedge
I did a conversion to chomsky normal form but I don't know if my step where I replaced AC with Y is correct without making a new rule for Y -> AC
but maybe it's fine idk
midnight plankBOT
@high wedge Has your question been resolved?
high wedge
<@&286206848099549185>
runic pebble
Long question
high wedge
Long question
Yeah but I got all the steps listed out so shouldn't be that bad
Mainly I am concerned I did some wrong thing going to chomsky normal form
runic pebble
Alright
high wedge
.close
midnight plankBOT
midnight plankBOT
Available help channel!
edgy crater
would part (i) be right? i feel like it doesnt feel right since i have only considered 1 type of possibility, as in ive considered u roll 6 unique in the first 6 rolls, then 6 unique in the second 6 rolls, so is this wrong or right, ty
midnight plankBOT
would part (i) be right? i feel like it doesnt feel right since i have only cons...
wise flicker
would part (i) be right? i feel like it doesnt feel right since i have only cons...
Why do you think you should write 6/6 for 1
edgy crater
oops 1 doesn't rly correspond to rolling a 1, its like the first roll has to be unique so 100% chance
so theres 6/6 chance the first is unique, then 5/6 for the second roll, so on
lethal path
oops 1 doesn't rly correspond to rolling a 1, its like the first roll has to be ...
you're forgetting the fact that there are a lot of ways the rolls could have a different order
do you know what a factorial is?
lethal path
e.g if you had 6 rolls and 2 of each number (1, 2, 3)
let's say you have some ordering: you could swap 2 of the 1s and that would still be the same ordering
you could also swap 2 of the 2s, or 2 of the 3s
and that's all the ways you've overcounted
wise flicker
let's say you have some ordering: you could swap 2 of the 1s and that would stil...
Doesn't matter it corresponds to nth roll for 1st 6 and next 6 so you would have unique rolls
they missed cases where you would get 2 of a number eg1 in first 6 and 2 of another in next
edgy crater
ive kinda thought abt it differently, not sure if right, ive just considered that there are 6^12 possiblities and we want an arrangement consisitng of 1 1 2 2 3 3 4 4 5 5 6 6 in any order, i know that there are 12! ways of doing this if all were unique, but they aren't so would i have to divide by 2? i feel like its more to it than dividing by 2
wise flicker
divide by 2^6 since there are 6 identical cases
lethal path
also the name of this concept is multinomial probability if you want to learn more
lethal path
also the name of this concept is multinomial probability if you want to learn mo...
you're going to get a lot of uni-level stuff on distributions; ignore that
you just need the multinomial coeffficient
hoary drift
midnight plankBOT
hoary drift
I tried expanding it using sin(A + B)
wise flicker
I tried expanding it using sin(A + B)
brodie ts blur
dreamy lichen
I tried expanding it using sin(A + B)
and what did you get after doing that?
t should be separable
hoary drift
lethal path
I think it's also homogeneous?
hoary drift
t should be separable
How is it separable
dreamy lichen
I think it's also homogeneous?
I dont see it smh
lethal path
nvm
dreamy lichen
How is it separable
either substitution, or applying sin(x+y) formula
hoary drift
either substitution, or applying sin(x+y) formula
My brain is not braining rn because I can't figure out how to separate it
dreamy lichen
have u applied the formula? What did you get?
hoary drift
dy/dx = sinx cosy + cosx siny
dreamy lichen
wait im stupid, i mixed up the formula, sorry
lets go sub then
keep it in the dy/dx = sin(x+y)
there is one sub that can be done and will do quite a lot of simplification
not too hard to guess
do u know what sub could it be?
hoary drift
No
hoary drift
Ive never used substitution in differential equations before
Ill look into this method after this question
dreamy lichen
its not that common ig, but it can help sometimes
lethal path
omg I feel so dumb after realising
wise flicker
omg I feel so dumb after realising
?
hoary drift
try u = x + y
Do I differentiate both sides wrt x?
dreamy lichen
Do I differentiate both sides wrt x?
nope, the trick is to notice that du = dx + dy
so dy/dx = ?
hoary drift
du/dx - 1
dreamy lichen
yeah
thorn sorrel
nope, the trick is to notice that du = dx + dy
i mean yes you do differentiate both sides do you not
dreamy lichen
well
yeah
i thought we meant the og equation, sorry
differentiating u = x + y works just as fine
dreamy lichen
omg I feel so dumb after realising
I feel dumb after realizing that sin(x+y) isnt 2sin(x)cos(y)
dreamy lichen
np
hoary drift
.close
rancid vigil
midnight plankBOT
fallow scarab
,tex .demoivre
grand pondBOT
riemann
rancid vigil
So z+1 = e^{ix} ?
fallow scarab
midnight plankBOT
toxic veldt
So z+1 = e^{ix} ?
C'est pas exactement vrai. L'equation (z+1)^n=e^{ix} admet n solutions donc il faudrait ajouter les racines n-iemes en facteur
On aurai plutot $z+1=e^{ix}e^{\frac{2ik\pi}{n}}, 0\le k<n$
grand pondBOT
Discusser
midnight plankBOT
Available help channel!
edgy crater
hi would this be right or wrong, i tried using a tree diagram to visualise it initially, then realised that in any scenario, bob has to lose all his money, so theres always going to be (0.5)^b involved, and alice can lose up to a-1 times. so i thought initially it would be just (0.5)^b + (0.5)^b(0.5)^1 + ... + (0.5)^b(0.5)^a-1, but i suppose that for example with (0.5)^b(0.5^1, there's a-1 ways of placing alices' loss, and similar for let's say where alice loses twice, we want to choose 2 places for her to lose in a-1 ways
midnight plankBOT
hi would this be right or wrong, i tried using a tree diagram to visualise it in...
wise flicker
U going right direction
edgy crater
did i not reach the destination / went off course
wise flicker
You include cases where you place alice loses once in a row :
eg alice loses = A
Bob loses = B
BB.....BAB...
ABAB...
Alternate AB
AAABB...
BB...BAAAB... is also possible
Do you know stars and bars method?
edgy crater
yes ive heard of it
edgy crater
You include cases where you place alice loses once in a row :
eg alice loses ...
wouldnt these cases be accounted for by the n choose r
wise flicker
Elaborate
edgy crater
for instance, say alice loses twice, by including the coefficient n-1 choose 2, then ive accounted for any way that alice loses twice, e.g. aabbb ababbb bbbaba, and ive further accounted for all the possible amount of ways she can lose, by the (0.5)^1 up to the (0.5)^a-1
edgy crater
Do you know stars and bars method?
id be interested by this way
ive heard of stars and bars method kinda being used for n choose r but probably not enough depth
wise flicker
for instance, say alice loses twice, by including the coefficient n-1 choose 2, ...
What are you choosing when a-1 choose n
Arrangements of alice losses?
wise flicker
yeah think so theres a-1 possible slots but now i dont think there is
Hmm.. slots are b coz after that you stop
edgy crater
wouldn't it be b + (r), where r is the amount of times alice loses perhaps
wise flicker
wise flicker
wouldn't it be b + (r), where r is the amount of times alice loses perhaps
Mm yes but with conditions
edgy crater
yh crucially last penny is with bob
wise flicker
yh crucially last penny is with bob
After bob's last penny we can't put alice
edgy crater
is it like for example in the case of 1, it would be ((b choose 1, 1) - 1) (0.5)^b(0.5)^1
wise flicker
is it like for example in the case of 1, it would be ((b choose 1, 1) - 1) (0.5)...
Why - 1
edgy crater
because we can't have the last penny being alices'
so im discounting for that case but second thoughts something feels off
wise flicker
so im discounting for that case but second thoughts something feels off
Go on
edgy crater
i think i kinda see the stars and bars aspect as the n in (n choose r) is increasing like alice in the star and bob is the bar
ohhh
maybe
wise flicker
Yes, but if we work with not two losses of alice one after another your first method is correct
edgy crater
we account for one less bob at the end
wise flicker
we account for one less bob at the end
Right
Lmao ts getting sus, nvm
Now apply stars and bars
There is one condition number of stars changes
midnight plankBOT
wise flicker
Np
midnight plankBOT
Available help channel!
verbal pumice
Prove that $\displaystyle \sum_{k = 1}^{\infty} \frac{1}{\lambda_k^2} = \frac{1}{4(p+1)}$ where $\lambda_n$ is the $n$th root of the Bessel function of order $p$.
midnight plankBOT
Prove that $\displaystyle \sum_{k = 1}^{\infty} \frac{1}{\lambda_k^2} = \frac{1}...
grand pondBOT
house
midnight plankBOT
midnight plankBOT
Available help channel!
undone cipher
I'm trying to understand the combinatorial argument for Binomial theorem
midnight plankBOT
I'm trying to understand the combinatorial argument for Binomial theorem
undone cipher
So the idea is I suppose
if you have (a+b)^n
and you want a^k
then since multiplication is commutative you can effectively re-arrange n terms in nchoosek ways such that the top term is a^k with ccoefficient 1
but how do you avoid double counting?
suppose k is 2
and n is 3
you have (a+b)_1(a+b)_2
(a+b)_2(a+b)_3
(a+b)_1(a+b)_3 i suppose
where in each case top term is a^2
candid gull
what do $a, b, n, k$ represent in this context
grand pondBOT
Mirror
undone cipher
So I am looking at expansion of:
(a+b)^n
and then looking for the coefficient of a^k where k <= n
eternal pawn
then since multiplication is commutative you can effectively re-arrange n terms ...
you mean a^k b^(n-k) right?
undone cipher
you mean a^k b^(n-k) right?
Sure I mean it sort of follows in a sense
lament knoll
a^k=(a+b-b)^k idk if it would help?
eternal pawn
so you're trying to get the number of subsets of [[1, n]] where a appears/is chosen k times
undone cipher
a^k=(a+b-b)^k idk if it would help?
mr avignon
undone cipher
so you're trying to get the number of subsets of [[1, n]] where a appears/is cho...
sure NcK ?
undone cipher
You know binomial theorem right ?
eternal pawn
which one
undone cipher
Have you heard or seen of the combinatorial argument/proof for it ?
undone cipher
there is a combinatorial argument for it being true
eternal pawn
I mean sure
so you want the number of products of the form aabbababb... where a appears exactly k times
undone cipher
well (a+b) appears k time ye
eternal pawn
you might need to illuminate me on your combinatorial argument
undone cipher
So I suppose the argument is (I'm not entirely sure) that if you look at (a+b)^n
you have n factors
if you pick k factors
then you have say (a+b)^k where you are guaranteed 1a^k
and I suppose the argument is that there are n choose k ways of finding these factors so your coefficient of the term containing a^k will be (n k)
because you used k factors, there are n-k factors left for b
so your final term is
(n k)a^kb^n-k
this is what i believe is the argument
sleek zinc
but how do you avoid double counting?
why do you feel that double counting is occurring in doing $\binom{n}{r}$
undone cipher
why do you feel that double counting is occurring in doing $\binom{n}{r}$
Well take the case where n = 3 k = 2
grand pondBOT
improbable
undone cipher
You have:
(a+b)_1(a+b)_2(a+b)_3
you can rearrange them in different ways sure, but I feel like once you multiply the first factor with second one, you can't just consider the case where you multiply the first one with the third one in the same 'count'
sleek zinc
think like this
there are n brackets of (a+b)
we need to choose k brackets from which we'll take a and from the rest n-k brackets we'll take b
to give us the product of a^k
undone cipher
yes sure
undone cipher
to give us the product of a^k
bro just left
sleek zinc
yeah i mean there is nothing more
we wanted the coeff of a^k
and we chose k brackets
that will give us a
could you clarify what is not clear to you ?
midnight plankBOT
@undone cipher Has your question been resolved?
undone cipher
could you clarify what is not clear to you ?
there are n choose k ways of chosing k bracquets
but like how can you do all of em simultaneously
sleek zinc
okay
we want the term a^k
for one term let's suppose we choose the first k brackets
this gives us one term
then we choose another k brackets
that gives us another term
we need to know how many such terms are there
that is equivalent to the no. of ways of choosing brackets
small warren
Can I please get help with knowing what I did wrong and what I should've done?
small warren
My logic behind the first question is that y=45, x=2 and so the interval should be between 0 and 2.
This result for the 2nd question seemed a bit absurd to me, so I just looked at each interval and saw a pattern that every next y value decrease by 50%, I didn't use the formula for computing percent change for the initial and final values given.
small warren
This result for the 2nd question seemed a bit absurd to me, so I just looked at ...
Should I have used this formula?
civic lynx
My logic behind the first question is that y=45, x=2 and so the interval should ...
take a closer look at the units - how many minutes is 2 ticks on the x-axis?
civic lynx
This result for the 2nd question seemed a bit absurd to me, so I just looked at ...
i think, because they asked for a "percentage decrease," they want a positive value. try 50% instead?
small warren
Every 3 minutes is 1 x value
small warren
i think, because they asked for a "percentage decrease," they want a positive va...
ooo ty, i can't retry it but that's probably why
civic lynx
the hardest part of online assignments is the computer software 
small warren
Ah and you have to convert the x values to minutes?
civic lynx
if it said percent increase then maybe i would put -50%