and \bZ for integers
#help-0
vale wigeon
stable night
OH
ah okay yes
so from here im not exactly sure how to show thats its not a perfect square
vale wigeon
you showed that a+b is 2 times an odd number
show that no perfect square can be 2 times an odd number
perhaps by showing that all even perfect squares are necessarily divisible by 4, and two times an odd number never is
ocean sealBOT
HellO
stable night
show that no perfect square can be 2 times an odd number
hmm ill try this
wait so im proving something inside my proof?
like proving a general statement inside my proof
vale wigeon
i think the proof of the general statement would best go BEFORE your proof proper
so like
Lemma: Let n be an odd number. Then 2n is not a perfect square.
Proof.
[proof of lemma goes here]
-----------
Claim: The sum of two odd perfect squares is never a perfect square.
Proof.
[your work thus far]
By the Lemma proved above, [...], being 2 times an odd number, is not a perfect square.
QED.
alright
one sec
Lemma: Let n be an odd number. Then 2n is not a perfect square.\
n = 2k + 1, k $\in \bZ$ (by definition of odd numbers)\
2n = 2(2k + 1) = 4k + 2 (by basic algebra)\
Let a, b be odd perfect squares
Then, a = $(2m + 1)^2$ and b = $(2k + 1)^2$, m, k $\in \bN$ (by T1)
a + b = $4m^2 + 4m + 1+ 4k^2 + 4k + 1 = 2(2m^2 + 2k^2 +2m +2k +1)$ (by basic algebra)
a + b = $2(2(m^2 + k^2 +m +k) +1)$
ocean sealBOT
HellO
stable night
okay Im actually not sure how to prove that something is/is not a perfect square.. my idea is that if the value of 2n was odd or something I could show that 2n cant be a perfecr square but now its even so im kinda lost
marsh rapids
Can 4k+2 be a perfect square ? That's all you need to answer
The rest is already done
marsh rapids
Suppose it is
Then we can write 4k+2 = p^2, and we know p is even
Can you finish ?
p^2 = (2q)^2 = 4q^2
stable night
wait idk if im missing the picture but i still dont see why that proves a contradiction
stable night
OH
wait so to prove contradiction
we first assume something wrong
then try to show that the result doesnt make sens
marsh rapids
If P if the property we want to prove, we can show that (not P) => false, then by contraposition, true => P, therefore P
stable night
ahhh i see
marsh rapids
Wait
Did you just not understand how to prove by contradiction but understood a formal logic proof ?
That's kind of funny
stable night
oh i knew how to prove by contradiction but i generally did it by proving something thats "related"
like um
marsh rapids
If P if the property we want to prove, we can show that (not P) => false, then b...
If it can't be wrong it has to be true. That's basically the idea
stable night
the 4k+2 is a multiple of 4 doesnt really link to the perfect square stuff and i didnt know we could do taht
stable night
If it can't be wrong it has to be true. That's basically the idea
ahh yes sir
me understand
lemme try to write out the full proof
marsh rapids
But the contradiction can be anything
stable night
Then we can write 4k+2 = p^2, and we know p is even
but what gave u the idea of going towards this
marsh rapids
like i cant think of it
When you have no idea you give yourself a starting point. (not P) is a starting point
marsh rapids
That way you see if it leads to some nonsense
What proof works best depends on the property. Look at what you need to prove with that method and ask "does that look easy" and pick the best option
And if it proves hard, try something else
marsh rapids
the 4k+2 is a multiple of 4 doesnt really link to the perfect square stuff and i...
Any contradiction works btw, that's what makes it "strong". A lot of proofs for the irrationallity of π construct a sequence of nonzero integers that goes to zero, which is impossible. That is very distant from what it proves
stable night
Lemma: Let n be an odd number. Then 2n is not a perfect square.\
n = 2k + 1, k $\in \bZ$ (by definition of odd numbers)\
2n = 2(2k + 1) = 4k + 2 (by basic algebra)\
Assume 4k + 2 is a perfect square.\
By definition, 4k + 2 = $(2q)^2$ (by T1)\
$(2q)^2 = 4q^2$\
$4q^2$ is divisible by 4, but 4k + 2 isnt. Contradiction.\
Hence, 2n must not be a perfect square.
Let a, b be odd perfect squares\
Then, a = $(2m + 1)^2$ and b = $(2k + 1)^2$, m, k $\in \bN$ (by T1)\
a + b = $4m^2 + 4m + 1+ 4k^2 + 4k + 1 = 2(2m^2 + 2k^2 +2m +2k +1)$ (by basic algebra)\
a + b = $2(2(m^2 + k^2 +m +k) +1)$
stable night
Any contradiction works btw, that's what makes it "strong". A lot of proofs for ...
ah okay i see
so i should always try to use contradiction
instead of direct proofs
marsh rapids
No
ocean sealBOT
HellO
marsh rapids
Direct works well sometimes too
For proving equalities for instance
Direct is still the usual method
But if the result is obscure, contradiction can be nice because it gives you a starting point
stable night
ahh i see
okie dokie
Lemma: Let n be an odd number. Then 2n is not a perfect square.\
n = 2k + 1, k $\in \bZ$ (by definition of odd numbers)\
2n = 2(2k + 1) = 4k + 2 (by basic algebra)\
Assume 4k + 2 is a perfect square.\
By definition, 4k + 2 = $(2q)^2$ (by T1)\
$(2q)^2 = 4q^2$\
$4q^2$ is divisible by 4, but 4k + 2 isnt. Contradiction.\
Hence, 2n must not be a perfect square.
Let a, b be odd perfect squares\
Then, a = $(2m + 1)^2$ and b = $(2k + 1)^2$, m, k $\in \bN$ (by T1)\
a + b = $4m^2 + 4m + 1+ 4k^2 + 4k + 1 = 2(2m^2 + 2k^2 +2m +2k +1)$ (by basic algebra)\
a + b = $2(2(m^2 + k^2 +m +k) +1)$ \
By the lemma proven above, a + b is not a perfect square.
ocean sealBOT
HellO
stable night
is this okay?
marsh rapids
It works
marsh rapids
It's far more detailed than I'd make it, but it depends on the expectations
marsh rapids
What level are you at ?
stable night
im turning 15
marsh rapids
That's quite young to know that much logic
stable night
idk my mom sends me for extra math classes lol
marsh rapids
Cause for HS level stuff, there's Khan academy, but it won't help you for proof based math
stable night
oh i wanna self learn like discrete maths and linear algebra kind
some people said its good to learn these early on
halcyon niche
definitely
marsh rapids
Proofs are done by practicing, finding new techniques, getting experience and intuition for things. That kinda of stuff
Are you French?
marsh rapids
That's a weird "about me" lol
marsh rapids
Don't really have any non-french resources for proofs unfortunately
stable night
aww
how about youtube
ive seen some nice videos on graph theory but not much on proofs
marsh rapids
Try to find an online pdf on your topic, maybe some teacher's website or some academic resource
That explores a topic and includes proofs for everything
Don't have any of that in English though
stable night
hmm ill try my best to find some
marsh rapids
And exercises to practice it yourself
stable night
yessir
marsh rapids
See #resources maybe
stable night
should i also try studying calculus if i have time
marsh rapids
Or #books-old
stable night
marsh rapids
Calc is very important
It's usually studied partially in high school, while a lot of the proof based stuff only comes after it
marsh rapids
Not very surprising
stable night
im rly bad at it 
stable night
woooot
marsh rapids
Calc 2 is where thinking is required and it can get tough if you don't have the right ideas
marsh rapids
woooot
It's just knowing elementary derivatives, differentiation rules, basic properties and theorems. Limits can be more involved, but basic calc 1 limits are hardly a problem, though they are more like calc 2 for the approach
stable night
i see
so ill try finding resources online
and read thru and practice some qns
lemme try finishing discrete maths first xD
okay tysm sir
ive learnt alot
lone heartBOT
@stable night Has your question been resolved?
alpine sable
lone heartBOT
marsh rapids
The line above is what we want, but we don't do any computations with it here. We start from P(k) and add k+1 to get the LHS, and then we transform the RHS into what we want
alpine sable
when/where do they add the k term, which we say to be equal to k(k+1)/2
oh i see, its added to the right below my red line
so then we add our k+1 term..
so where does the (k+1)(k+2)/2 = k+1 come back in
marsh rapids
That last message makes no sense. This equality doesn't even hold
marsh rapids
The sum is equal to the product. Not k+1 alone
alpine sable
ok the k+1 version of the P(n) expression
P(k)
can you just help me i don't understand
every time i come to this server people talk down to me without helping at all
.close
lone heartBOT
lone heartBOT
Available help channel!
misty notch
i need to calculate the area of that triangle
lone heartBOT
i need to calculate the area of that triangle
misty notch
abit confused but i think angle BCA is 90
vale wigeon
that is indeed true given the equations you've written
misty notch
yeh i figured cuz of the slopes
vale wigeon
it also appears that your triangle has a height of 4 and a base of 21 - 1/5
misty notch
so just 1/2 x 4 x 20.8
if i considered AC to be the base
i would get the same answer right?
lone heartBOT
@misty notch Has your question been resolved?
distant cove
lone heartBOT
keen socket
why cant u just take natural logs on both sides
distant cove
we can't use logs yet unfortunately
keen socket
oh
distant cove
we aren't allowed to, to be exact
alpine sable
You want to solve for x?
distant cove
You want to solve for x?
Yes
keen socket
You wrote T for some reason
he want a T substitution
to reduce to a linear expression
distant cove
to reduce to a linear expression
yea
alpine sable
maybe (3/2)^x
keen socket
wat
alpine sable
or (3/2)^(x/2), shouldn't matter much.
distant cove
this is the original
alpine sable
Just look at this, (3/2)^(x/2) should work nicely.
You get something like 4t^2 - 9 = 5t.
distant cove
let me try
t=(3/2)^(x/2)?
yep
turns out good and correct
thank you!
@alpine sable
.close
.close
tough tangle
i need help understanding this
lone heartBOT
i need help understanding this
tough tangle
if probability of smallR shooting the target is 1/2, and proability of shooting the target for Zanoes is 1/2, then how is the probability of SmallR winning 2/3
alpine sable
SmallR goes first, so in order for Zanoes to be able to have n atempts, SmallR must get N+1 atempts
(SmallR has to miss for Zanoes to get a chance, Zanoes has to miss for SmallR to get a chance)
That diagram was wrong
tough tangle
P(SmallR doesnt hit, Zanoes hits) == 1/4
P(SmallR hits on first try) == 1/2
P(SmallR hits, Zanoes doesnt hit) == 1/4
right?
alpine sable
Not quite
SmallR has a 1/2 to hit on first try
SmallR miss, Zanoes hits 1/4
SmallR miss, Zanoes miss, SmallR, hits, 1/8
And so on
tough tangle
those get added?
and honestly there is no bound to this problem so isnt this an infinite sum
alpine sable
It could be thought of like that
But the numbers very small very fast (1/(2^n)), so anything over 10 throws would never happen in reality
lone heartBOT
@tough tangle Has your question been resolved?
alpine sable
Throw 1, 3, 5... is made by SmallR
So id assume the sum of the probability of all odd throws are 2/3
alpine sable
those get added?
Here is a sketchy desmos sum
-1+2n makes it evaluated for throw 1,3,5,7...
tough tangle
so is that just the sum of P(SmallR hits)^(odd numbers)?
i am doing a problem where this probability changes so
alpine sable
Yes, asuming a different probability change it acordingly
(1000 should be infinity, but desmos isnt good at that)
The total sum of all the probabilites is 1, someone will hit some time
tough tangle
i see
alpine sable
If someone has 0.3 odds one time and 0.2 odds another time, their total odds of winning is 0.3+0.2 = 0.5
tough tangle
ohh
if its just P(smallr hits)^odd
then why was zanoe's probability given @alpine sable
alpine sable
Problems are not meant for you to walk right into the solution, in that case you would have had an easier time and learned less
Also this assumes SmallE and Zanoes has the same probability of winning, which may be a simplification
lone lotus
Question: "Write down the steps and process you follow to determine the total number of manhole covers in the county." This county would be San Joaquin County in California. Do I start by getting the population of the county? I'm not even sure how to start going about this. Thank you all in advance!
alpine sable
are you trying to do a fermi estimate?
or are you actually trying to get an accurate number
lone lotus
The questions says “total” so I’m guessing they do want an accurate number but it may be hard to do as a lot of estimation is gonna need to be done @alpine sable
alpine sable
google maps seems to have your county fully mapped, shouldnt take more than a few days ¯_(ツ)_/¯
perhaps something more reasonable is to get an average manhole count per street/intersection and then count the number of streets
tacit arch
sounds like a fermi problem for a job interview
lone lotus
That's exactly what it is lol
lone lotus
I wasnt told it was a fermi question but by the looks if it I think it is, I appreciate your help!
tacit arch
I wasnt told it was a fermi question but by the looks if it I think it is, I app...
It's deliberately open ended and the answer that will get you the offer will be dependent on your ability to impress the team which isn't really a math problem. Soooo good luck
lone lotus
@tacit arch Yea im reading through this rn and it seems like its just testing my ability to guesstimate and the method im using to do it. So I could safely get the average amount of manholes off lets say 10 streets and then times that by number of streets in county and Ill have a very rough answer right? Its kind of insane how interview processes have changed for me going from blue collar jobs to higher level corporate ones. Like wtf is this question lol.
worn fox
they're not really bothered how close you get to the answer unless you're doing stupid estimates of things
they're just trying to see how you're thinking about the problem
lone heartBOT
@lone lotus Has your question been resolved?
lone lotus
So i'm sure if I just type out my reasoning as to how id go about and give my answer I should be fine.
Thank you everybody i know this turned out to be pretty unrelated to actual math but I appreciate it all the same.
Good day all.
lapis ingot
are parametric equations and vector valued functions the same
mortal trellis
no. equations and functions are different things
lapis ingot
no. equations and functions are different things
aren't parametric equations technically functions of a variable though
lapis ingot
like you define them in terms of t
but then what's the difference between parametric equations and vector valued functions
mortal trellis
well on one hand you have eg two equations $x=\cos t, y=\sin t$
ocean sealBOT
Denascite
mortal trellis
and on the other hand you have a function $f(t)=(\cos t, \sin t)$
ocean sealBOT
Denascite
mortal trellis
ok maybe the distinction I want to make is too nitpicky
lapis ingot
so basically
they are pretty much the same
except
that the parametric equation
isn't really a function?
because it's given as equations
?
mortal trellis
yeah you can say it like that
calm swift
hi is anyone free to help?
lone heartBOT
hi is anyone free to help?
fierce prairie
calm swift
The diagram shows a triangle with coordinates A(–2, –1), B(6, –1) and C(3, 4)
im stuck on part c
im not sure how to find CX
this is where im stuck
alpine sable
It just so happens that the x and y coordinates of A and C both have a difference of 5, so you can skip the distance formula and avoid messy numbers
otherwise, you need to find the point on AC that is 4.24264 units away from A by considering the formula of the line AC (or any parametrisation of it)
calm swift
It just so happens that the x and y coordinates of A and C both have a differenc...
how would i continue with this?
alpine sable
since AC is a straight line, the ratio 3:2 can be taken componentwise
ie find the x coordinate of X first by looking at the x coordinates of A and C. Then do it for y
calm swift
so ill basically finding the coordinates for point X?
alpine sable
yes
alpine sable
yw :)
alpine sable
I get (-2+3,-1+3)=(1,2)
tacit arch
,w integral e^(x/2) / (1+e^x) dx
ocean sealBOT
tacit arch
might be?
alpine sable
yup i tried wolram alpha but its different from mine
tacit arch
try differentiating your answer to get the integrand
some inverse trig and inverse hyperbolic trig functions relate to each other
alpine sable
try differentiating your answer to get the integrand
gotcha, but can you please check if my substitution are correct
tacit arch
u sub looks good
alpine sable
some inverse trig and inverse hyperbolic trig functions relate to each other
we dont learn hyperbolic functions, can i self learn it?
tacit arch
,w integral 1/(x * sqrt(x-1)) dx
ocean sealBOT
alpine sable
the form i know is arcsec
tacit arch
,w derivative arcsec(x)
alpine sable
i memorized that
ocean sealBOT
tacit arch
yea looks the same
alpine sable
so its correct?
ocean sealBOT
riemann
tacit arch
not quite your integrand
alpine sable
side thought, integration by parts is easier than substitution
tacit arch
tough find
alpine sable
yup
hey someone can you please help me with this integral
$\frac{1}{x\sqrt{x - 1}}$
ocean sealBOT
plumbe_r
alpine sable
I know wolfram alpha can give me the answer but i just want to know how i can do it on my own
using u-sub i simplified it to $\frac{1}{\sqrt{x^3} -\sqrt{x}}$
help with integral of the above
$\int{\frac{1}{\sqrt{x^3} -\sqrt{x}}}dx$
ocean ravine
.
alpine sable
<@&286206848099549185> i need help please
ocean ravine
<@&286206848099549185> i need help please
i wouldnt count on it
ocean ravine
i got help earlier
usually people get mad when you ping them
ocean sealBOT
plumbe_r
alpine sable
usually people get mad when you ping them
oh
alpine sable
why they have the helper ping then?
lone heartBOT
tired echo
using u-sub i simplified it to $\frac{1}{\sqrt{x^3} -\sqrt{x}}$
which u sub did you use ?
lone heartBOT
Available help channel!
alpine sable
which u sub did you use ?
i didnt i just want to know which technique i should use
alpine sable
oh sorry i thought it was closed
lilac nest
.close
lone heartBOT
lone heartBOT
Available help channel!
nimble bane
How would I use my calculator to get this answer
lone heartBOT
How would I use my calculator to get this answer
tacit arch
newton rhapson method?
nimble bane
whats that
tacit arch
it finds roots of functions. in your case, $f(x) = 6(x^{15} - 1)/(x-1) - 29$.
ocean sealBOT
riemann
tacit arch
How would I use my calculator to get this answer
i guess it depends on your calculator. newton rhapson is just the method. how it gets implemented on calculators is up to the manufacturers
nimble bane
oh
nimble bane
<:thinkies:827642567775748163> newton rhapson method?
is there any other method?
I dont think our teacher is gonna teach us this method in class and
For me its a bit too complicated lol
tacit arch
then your calculator probably has a solver that you'll have to google how to use
lone heartBOT
@nimble bane Has your question been resolved?
nimble bane
Thanks ig
empty plover
show than un and vn are adjacent and that they both tend to the thing on the right
ive tried to show something but i cant. Cant find out which sequence is increasing and which is decreasing, and cant think of anyway they might relate to trigonometry
ripe jasper
prepa? bac?
empty plover
prepa
empty plover
thanks xD
ripe jasper
so you want to prove they are adjacent?
empty plover
yeah
ripe jasper
aight so did you that u_n is increasing and v_n decreasing?
Aight first of all show that u_n and v_n are positive -i think you can just say it since its kinda obvious-
empty plover
its kinda obvious
ripe jasper
its kinda obvious
what is obvious?
empty plover
they're positive
ripe jasper
they're positive
Ok great
empty plover
first terms are positive and recurrence directe
ripe jasper
then show this u_n < v_n
ripe jasper
but i am not getting anything i can know the sign of
use induction
ok check my "proof"
base case n=0 $u_0 < v_0$ "trivial"
ocean sealBOT
Hellboy
ripe jasper
assume $u_n < v_n$
ocean sealBOT
Hellboy
empty plover
Ah fuck didnt think of induction this always slips me :p
ripe jasper
$v_{n+1} - u_{n+1} = \sqrt{\frac{u_n v_n + v_n ^2}{2}} - \frac{u_n + v_n }{2}$
ocean sealBOT
Hellboy
ripe jasper
ill try to see where this lead
you can start from your induction hypothesis tho
empty plover
ok easy
yeah it work
s
simply replace vn by un and we'll have that its bigger than 0
marsh rapids
btw on appelle leur limite commune la moyenne arithmético-géometrique de a et b
empty plover
btw on appelle leur limite commune la moyenne arithmético-géometrique de a et b
non c'est une suite differente
c'est racine(un+1*vn)
pas racine (un*vn)
i think its different
marsh rapids
Oh exact
empty plover
t'as des idess comment proceder?
marsh rapids
**Hellboy**
Pour montrer que ceci est positif ?
empty plover
Non ca va ca marche j ai juste remplace vn par un
Apres ca on montre que un et vn sont respectivement croissant et decroissante resp. n est ce pas?
marsh rapids
Donc un_n croit c'est clair aussi ?
ripe jasper
t'as des idess comment proceder?
for proving adjencency?
marsh rapids
C'est un peu la 2e partie de la question quoi
ripe jasper
this is after showing u_n and v_n are resp. increasing/decreasing
empty plover
Un is obvious
marsh rapids
v_n décrois t'a la démo ou pas encore
empty plover
Okay nevermind lol
XD
ok j ia eu que c'est plus petit que 0 alors egal a 0 ca va?
I found its smaller than 0 so its 0 right?
un-vn
marsh rapids
vn > un !=> vn = un
marsh rapids
Sois plus précis alors
empty plover
vn-un dsl
vn-un is positive and smaller or equal than 0
(when i say smaller it includes equality, when i dont want to include i say strictly smaller)
ripe jasper
meh look gros pq tu veux v_n - u_n?
marsh rapids
We already know that u_n <= v_n. You said it earlier
marsh rapids
meh look gros pq tu veux v_n - u_n?
Pour mq que ça tend vers 0 non ?
empty plover
vn-un is positive and smaller or equal than 0
because if i replace un by vn it's 0 and its smaller than that
empty plover
because if i replace un by vn it's 0 and its smaller than that
is this correct reasoning it feels false
marsh rapids
jsp détaille rigoureusement et regarde
ocean sealBOT
Hellboy
empty plover
right
ripe jasper
oh also
you can do something more direct
you have u_n < v_n
thus
v_n - u_n > 0
ripe jasper
**Hellboy**
or to make a clear picture
ocean sealBOT
Hellboy
ripe jasper
since u_0 < u_n then -u_n < -u_0
ocean sealBOT
Hellboy
marsh rapids
On sais mais faut mq que la lim vaut 0
ripe jasper
On sais mais faut mq que la lim vaut 0
oui oui tkt on arrive
empty plover
everything is clear until now and i am looking forward to how this will lead to it going to 0
ripe jasper
everything is clear until now and i am looking forward to how this will lead to ...
so if i show |v_n - u_n| < e , e > 0
then this shows lim v_n - u_n = 0
empty plover
right
ripe jasper
**Hellboy**
can you figure it out from this ? or should i spoil?
empty plover
no ive got nothing go ahead
ripe jasper
no ive got nothing go ahead
no think more
empty plover
okay
ripe jasper
meh
empty plover
Hahaha
empty plover
yeah how?
ripe jasper
you have $|v_n - u_n| < e$ what you waiting for?
ocean sealBOT
Hellboy
empty plover
it should be for all e>0 :p
marsh rapids
That's for one epsilon and only 1
empty plover
lol
ripe jasper
That's for one epsilon and only 1
i mean you can pick any a,b , you find the epsilon
lol
empty plover
we have no control over a or b
marsh rapids
So if I fix epsilon, for what n do we have v_n - u_n < epsilon
ripe jasper
idc im just talkin
ripe jasper
So if I fix epsilon, for what n do we have v_n - u_n < epsilon
tbh for any n ig
pick e = (b-a)/n maybe?
lone heartBOT
@empty plover Has your question been resolved?
ripe jasper
meh
just look if u dont want to do it this way
then maybe use logic , assume its not converging to 0 , find contradiction
ocean sealBOT
Hellboy
empty plover
yes
ocean sealBOT
Hellboy
ripe jasper
@empty plover you see this?
empty plover
Yeah i see it
but i dont think thats the correct contraposition to the limit of a sequence
ripe jasper
Why
marsh rapids
Very weirdly formulated. More like wrong
empty plover
there exists an epsilon there exists an n for which n>N and un-l>e
ripe jasper
but i dont think thats the correct contraposition to the limit of a sequence
negation
marsh rapids
there exists an epsilon there exists an n for which n>N and un-l>e
No
ripe jasper
Very weirdly formulated. More like wrong
why wrong
marsh rapids
There exists epsilon > 0 st for all N, there exists n >= N st |v_n - u_n| > epsilon
Know your fundamentals guys
ripe jasper
There exists epsilon > 0 st for all N, there exists n >= N st |v_n - u_n| > epsi...
literally what i wrote
ripe jasper
There exists epsilon > 0 st for all N, there exists n >= N st |v_n - u_n| > epsi...
also its >=
marsh rapids
**Hellboy**
The existence of n is pretty much nonsense
marsh rapids
also its >=
Actually irrelevant
ripe jasper
The existence of n is pretty much nonsense
?
ripe jasper
Actually irrelevant
wut?
empty plover
Its fine i read the correction to this exerciseee during this time it's all good, tho thanks a lot for the help so far @ripe jasper
empty plover
and mateo ofc
ripe jasper
You can't define N if n doesn't already exist.
i mean we assumed n is there alrdy so idk what u get to say
marsh rapids
wut?
For all these limit things, pretty much every inequality can be taken strict or large without consequences
marsh rapids
i mean we assumed n is there alrdy so idk what u get to say
Where did you define n in that statement ? Nowhere so it doesn't exist to me
ripe jasper
Where did you define n in that statement ? Nowhere so it doesn't exist to me
i mean it was directed to him, not you "cuz i wasnt tryin to be any formal tbh"
marsh rapids
By forgetting to be formal you forgot to be correct. That's bad
lone heartBOT
ripe jasper
Its fine i read the correction to this exerciseee during this time it's all good...
send me correction tho
ripe jasper
By forgetting to be formal you forgot to be correct. That's bad
why do u think that was directed to u?
also what i was doin wasn't tryin to do a proof so its just an idea, idk why ur skeptic asf ptn
real gazelle
okay the channel is closed let's move on
marsh rapids
Who it's directed to doesn't change what it says
real gazelle
y'all can settle this in DMs
lone heartBOT
Available help channel!
ripe jasper
bruh this guy got skeptical rq but meh
lone heartBOT
Available help channel!
versed widget
lone heartBOT
versed widget
I need help finding the equation of this graph
I know it’s a cosine graph but I don’t know how to find the rest of the numbers
lone heartBOT
@versed widget Has your question been resolved?
lone heartBOT
@versed widget Has your question been resolved?
versed widget
<@&286206848099549185>
jade bloom
jade bloom
Yes but make sure to watch your signs so a = 4 and also c is the x coordinate of the max
versed widget
How is a=4?
jade bloom
[5-(-3)]/2
versed widget
Oh okay
jade bloom
a is the amplitude of the graph
versed widget
For the period how is it found.
jade bloom
the period is the length of one cyvle of the graph so for you graph its 3 and you can subract the two max point to find your period
versed widget
Oh okay, I’ll do that thank you
jade bloom
Yeah youre welcome
jade bloom
your k and c are wrong and you need to included the paranthesis
versed widget
I forgot to divide the 2pi/3
jade bloom
your max coordinate is at 0 and 3 so you could choose one of those for your c
versed widget
Alright I got y = 4cos(2pi/3(x+3))+1
versed widget
Oh okay, i saw the 3 which was positive. If it was negative 3 it would have x +3?
jade bloom
Yes
cobalt ridge
What is y = mx + b google is too complex for my brain to comprehend
abstract fractal
It's the equation of a line
cobalt ridge
Wat
What is slope again, Ik rise over run, but I coding and the problem is every x must have a y and somehow that is related to y = mx + b
abstract fractal
The m represents the slope and the b represents the y intercept
cobalt ridge
Ok but what is the meaning of every x must have a y
elfin snow
yeah
abstract fractal
Ok but what is the meaning of every x must have a y
If you plug in something for x, you'll never get "undefined" or sumn
cobalt ridge
Ok so I am plotting a MatPlot
Assume we have a List called Weeks: [1,2,3,4,5,6,7,8] (X axis)
And a List called Profit/Dollars: [1000, 2567, 5678, 8907, 13476, 11993] (Y axis)
We plot that in the MatPlot and we should have no problem or i am wrong? In normal math grid or whatever
wary stream
I don't think you have enough values
cobalt ridge
Ohhhhhhhhh
abstract fractal
How else would you define f(8). If the y values line up with the x values from left to right, then the x values 7 and 8 don't have a y value, because there's not enough y values
cobalt ridge
Yea but in my mind u could still technically plot it no?
cobalt ridge
Waht
wary stream
You need an equal amount of x values to y values
cobalt ridge
Why
wary stream
To plot it
abstract fractal
What's the corresponding y value to x = 7, if you can plot it?
wary stream
You can either get rid of the extra x values or add more y values, but every x should have a y value
abstract fractal
The matplot code (maybe idk I haven't looked at it) assumes the x and y values are the same length
abstract fractal
".close"
minor osprey
lone heartBOT
minor osprey
i dont get how to do this
alpine sable
Bro is this physics lol? @minor osprey
alpine sable
Oh I don’t know this one can’t help sorry
minor osprey
ok
alpine sable
Never did this one before
Sorry just wait for someone up
I can defenitrly try helping if you’d like
worn fox
I'm assuming you're gonna want to show that dE/dt=0
minor osprey
tbh im a little confused
because i think e is a constant
at least thats what they say in the notes
worn fox
Right that's what this question is making you show
If its derivative is zero, it must be constant
minor osprey
so how would i do that
worn fox
What's the equation for energy
minor osprey
e= mv^2/2 + v(x)
the farthest i got is to get rid of m as it is a unit mass particle
minor osprey
im confused as to why i would need to differentiate f(x)
and what the derivative of the kinetic energy is
worn fox
You just gave me the equation for energy
We want to show that energy is constant during motion
worn fox
Why do you think you need to?
minor osprey
as the derivative reprents the change
and if the change is 0 then the energy is constant
worn fox
Yes but where have we said anything about differentiating force
worn fox
That's just a general fact you should know if you're doing physics
minor osprey
im not
worn fox
"Force is the negative derivative of the potential"
minor osprey
okay so what do i after this part
worn fox
Well what do you have so far
minor osprey
v^2/2 + -dv/dx=E
worn fox
Eh huh
worn fox
e= mv^2/2 + v(x)
Start with this
minor osprey
okay
minor osprey
v for potential or veloctiy?
worn fox
e= mv^2/2 + v(x)
Okay start with this, and differentiate both sides, carefully
minor osprey
the problem is i dont know how to differentiate the right side
is it just (dv/dx)/2 + dV/dx
worn fox
We can set m to 1 to make our life easy
youre gonna need to use the chain rule on the first term
Youre close
minor osprey
so v*dv/dx?
minor osprey
im still a bit stuck on what to do after
worn fox
Yikes sorry we should be differentiating with respect to time!
Everything is basically the same
minor osprey
sorry so is x just t
minor osprey
does d/dt just mean the derivative of what is after
minor osprey
okay
worn fox
d/dt(stuff)
minor osprey
im with you so far
minor osprey
dV/dx(t)?
worn fox
But position is a function of time x(t)
minor osprey
ah okay
worn fox
So V(x(t))
worn fox
Chain rule!
minor osprey
so dV/dx*dx/dt?
minor osprey
is that dV/dt?
worn fox
Yes, by the chain rule
worn fox
Oh hang on
worn fox
is that dV/dt?
We just showed that this
worn fox
so dV/dx*dx/dt?
Is this by the chain rule
minor osprey
i thought that you could just get rid of the dx
as there was one on the bottom and one on the top
worn fox
You can, but we don't want to
worn fox
0?
worn fox
Nice
minor osprey
thanks dude this made a lot of sense
worn fox
No worries!
minor osprey
btw is it okay if i privately dm you? dont know if its allowed in this server
worn fox
I have dms turned off for this server sorry
minor osprey
ah
so theres no chance of asking for you to look over answers ive written down for questions?
i would pay
worn fox
You'd get quicker and more consistent replies posting them here in this discord with thousands of people my friend
minor osprey
i tried earlier but i couldnt get anyone to answer an applied maths quesiton
worn fox
Alright friend me and you can DM me stuff but can't guarantee replies, not right now anyway its 3AM I'm about to sleep
worn fox
Sweet, have a nice sleep when you do!
minor osprey
.close
inland vessel
I was wondering, should you index using the set of real numbers or should you avoid that on all costs. Are there limitations of what set of numbers to use while indexing things?
I'm not an expert on indexing, my creative mind randomly decides using the set of reals for indexing a collection of sets. Here is an image (v) The big question is, is this valid to do?
rose sigil
you can index a collection of sets with the real numbers but I'm not sure what your example is
inland vessel
you can index a collection of sets with the real numbers but I'm not sure what y...
I was trying to make a class A_i, whose sets share a common arbitrary property. I think this is called a class. My question was, is it valid to use any "set of numbers" for indexing.
or does it depend on the situation.
rose sigil
umm your notation, at the very least, could have some problems
and maybe other things
rose sigil
but yes, you can use any set of numbers for indexing (if you express it clearly)
inland vessel
Are the brackets and the random "|" the main issue, (pointing out my notation?)?
rose sigil
I don't understand either of those
inland vessel
uh...
rose sigil
or the i in R with what comes after it
inland vessel
i should space it
actually
$ \begin{array}{l}
\ i\ \in \ R\ \ \ \
\
A_{i} \ =\ {A_{1} ,A_{2} ,A_{3} ,A_{3} ,A_{3} ...........A_{i}}
\end{array}$
ocean sealBOT
nesymerp1
inland vessel
Oh i get it now, yeah mb i thought you knew set theory. But, it doesn't matter I think? Since you knew indexing...
inland vessel
or the i in R with what comes after it
ye ye, ts fine
rose sigil
**nesymerp1**
what is this supposed to mean?
inland vessel
So, basically.
i in R means that i is any real number (Or: i is a member of the set of reals)
And what I'm doing is indexing using "i". To classify that A_i is a set containing other sets indexed by i.
rose sigil
why does A_i reference itself in its definition?
inland vessel
I feel like it is used to explain what the "class" is made of in terms of indexing
but maybe its the wrong notation? im new to all of that still
rose sigil
ignoring that for now, can you tell me what the set A_{1/2} is?
or A_{pi}
or A_{-10}
inland vessel
Well I said the entire class contains sets that share a 'arbritrary property', like all the sets in the class contain elements that for example: Reals,Rational.
rose sigil
or class as you want to call it
inland vessel
i also could refer it as a family of sets.
rose sigil
the notation is still meaningless to me. You are trying to, for each real number i, define a collection of sets A_i, right?
inland vessel
the notation is still meaningless to me. You are trying to, for each real number...
yeah
rose sigil
ignoring that for now, can you tell me what the set A_{1/2} is?
so..?
inland vessel
this is why im asking if it was recommended to index like this
rose sigil
then no, I'm not sure what you are trying to define, but the notation is meaningless right now
inland vessel
so..?
I don't know specifically, it could be any collection of numbers that share a common property with let's say A_{pi}
but it may be because it's very ill defined and such, which causes you to not understand it well
rose sigil
but what is A_{pi}
inland vessel
same answer, any collection of numbers that share a common property/properties with any set A_[i]
rose sigil
are you familiar with any examples of classes?
inland vessel
no sadly not
im new to all of this and im trying to build my set builder comprehension by examples and such
rose sigil
then you might want to try to understand some. This doesn't look like any way to construct a class to me
inland vessel
Yeah I came here to ask if it was valid
to build a class like this, honestly.
but maybe like someone who knows set theory well could answer it to me, which would be an additional great help.
inland vessel
then you might want to try to understand some. This doesn't look like any way to...
Do you have any background in set theory by any chance?
rose sigil
yes
rose sigil
I've taken one grad level class in axiomatic set theory and used it in some other places
inland vessel
I've taken one grad level class in axiomatic set theory and used it in some othe...
Oh I see, well did I say anything incorrect about uh, classes? As in like, the explenation of them.
just, making sure im no dumbo
rose sigil
not sure what to say to that, some things you said were intuitively fine but you seem to have some misconceptions on what a class is
inland vessel
as all beginners do sometimes yeah...
inland vessel
not sure what to say to that, some things you said were intuitively fine but you...
Rightly... Well thanks for giving me a reality check about it. Now I'll be heading off to sleep, since its 4:00AM
rose sigil
no problem, good night 😁
frank nacelle
lone heartBOT
naive valley
does 115 refer to the previous problem? presumably 1.1.5?
otherwise i have no idea, 115 is obviously not a solution
tight locust
elfin snow
there are 115 hints
lone heartBOT
@frank nacelle Has your question been resolved?
lone heartBOT
last ether
115 v bucks
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Available help channel!
safe mural
hi can someone please help me with part 3 of this question
lone heartBOT
hi can someone please help me with part 3 of this question
safe mural
so i've got this for k hat
and i think that i can follow something similar to this from stackexchange, but i'm confused on how to make it work
lone heartBOT
@safe mural Has your question been resolved?
safe mural
<@&286206848099549185>
lone heartBOT
@safe mural Has your question been resolved?
lone heartBOT
@safe mural Has your question been resolved?
lone heartBOT
@safe mural Has your question been resolved?
lone heartBOT
@safe mural Has your question been resolved?
safe mural
.close
astral portal
lone heartBOT
marsh rapids
Try to find information on the coefficients of P, then draw some conclusions
astral portal
Try to find information on the coefficients of P, then draw some conclusions
well discriminant of p(x) is less than 0
marsh rapids
You can tell a lot about P just from the fact its roots are imaginary
A lot more than delta <= 0
astral portal
the graph does not touch the axis
astral portal
A lot more than delta <= 0
not equal* coz if equal it will be touching x axis
marsh rapids
Is 0 purely imaginary ?
astral portal
Is 0 purely imaginary ?
no
marsh rapids
Ok
astral portal
all i am gonna say the answer will be either c or d
according to my calculation
as p(x) has to be imaginary
marsh rapids
Why not a ?
astral portal
to make p(p(x))=0
astral portal
Why not a ?
the answer is d
marsh rapids
Yes but why ?
astral portal
idk
marsh rapids
Try to find information on the coefficients of P, then draw some conclusions
Really investigate this. You can tell more than you think
Especially since the roots are complex conjugates of each other
mellow tusk
pp(x)=0 would have solutions [ if possible ] at x=p(x) right?
marsh rapids
ai is weird because then you can't write ax^2 + bx + c though
astral portal
pp(x)=0 would have solutions [ if possible ] at x=p(x) right?
yea so p(x) has to be imaginary to make it possible
marsh rapids
pp(x)=0 would have solutions [ if possible ] at x=p(x) right?
It's a solution iff p(x) = +- ki, where ki is the root
astral portal
yea
mellow tusk
yea so p(x) has to be imaginary to make it possible
and x=p(x) can have either 2,1 or 0 real solutions
marsh rapids
So you know it's complex and need to disprove it's purely imaginary
astral portal
-ak^2+bki +c = 0
-ak^2 - bki + c = 0
bki = 0 => b = 0
marsh rapids
But b=0 is the useful one
marsh rapids
It's a solution iff p(x) = +- ki, where ki is the root
Think about that and the fact b = 0
astral portal
p(p(x)) = a(ax^2+c)^2 + c = 0
ax^2 + c = +-sqrt(-c/a)
hence answer d as c and a are real
while sqrt c/a is imaginary
hoping c not equal to 0
am i missing smth
or this is it
marsh rapids
Think about that and the fact b = 0
We know x can't be real. If x = hi for some h in R and p(x) = +- ki, then -ah^2 + c = +- ki, impossible
frail lantern
@marsh rapids Can I learn along and ask questions?
marsh rapids
Not starting from 0 on the complex numbers but if there's a step you didn't understand feel free to ask
But I just finished the solution
frail lantern
c would never be equal to 0 because product of 2 purely imaginary numbers would never be 0, am I correct in assuming this?
marsh rapids
!= means not equal
frail lantern
okay thanks a ton
marsh rapids
!= means not equal
You have that in a lot of programming languages
astral portal
i amma close it now
astral portal
.close
lone heartBOT
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Available help channel!
empty plover
show that this is bijective and calculate for any given n (x,y)
lone heartBOT
@empty plover Has your question been resolved?
mortal trellis
note that the second part is a triangular number
show that you can write each natural number uniquely as a triangular number plus a certain rest
mortal trellis
-n is not a natural number
alpine sable
ah damn
lone heartBOT
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Available help channel!
keen furnace
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keen furnace
qu 11
This is what I came to conclusion so far
I have no idea what’s my next step
vestal locust
Rather than horrible calculations, maybe show that both sides have the same derivate
which proves that left_side = right_side + constant
and then show that constant=0
That may be better
maybe not
You also have $\sin^2 \cos^2 = (\sin (2x))^2$
ocean sealBOT
Guilhem
vestal locust
...
keen furnace
**Guilhem**
oh is that another trigo identity?
I haven’t seen that one
yea I am kinda ignoring the other side too much
.close
shrewd whale
lone heartBOT
languid bolt
is that the whole question?
shrewd whale
r = -2pih +- (sqrt(4(pi^2)(h^2)+8piA))/4pi
yes
thats it
the whole thing
solve for r
shrewd whale
yeah sry
gray isle
still missing ()
gray isle
$$r = \frac{-2\pi h \pm \sqrt{4\pi^2h^2 + 8\pi A}}{4\pi}$$
ocean sealBOT
ℝamonov
shrewd whale
yes
gray isle
you can factor sqrt(4)=2 out of the sqrt
shrewd whale
ok
gray isle
cancel the common factor in both the numerator and denominator
and then there isn't really much to do after that
shrewd whale
what would it be?