yes, cos(x) and sin(x) have the same maximum y=1
#help-36
soft zealotBOT
drowsy epoch
idk what makes you that think
sin(x) and cos(x) have both the same image [-1,1]
you only need to know that there is this maximum, you dont need a calculator and specific value
if you know 2cos(x) is at most 2, then 2cos(x)=3 should immediately ring bells to you
yeah...
final saddleBOT
final saddleBOT
Available help channel!
little hill
$(f'x) \rightarrow \frac{x^-6}{-e^x} \rightarrow e^-x(6x^-7+x^-6)$
final saddleBOT
$(f'x) \rightarrow \frac{x^-6}{-e^x} \rightarrow e^-x(6x^-7+x^-6)$
soft zealotBOT
Σ'(∟μτ∫i)
final saddleBOT
Please show the original problem, exactly as it was stated to you, with the entire original context. A picture or screenshot is best. If the original problem is not in English, then post it anyway! The additional context might still be helpful. Do your best to provide a translation.
little hill
I just wanted to know how to solve it
leaden moon
What is this mess
final saddleBOT
Please show the original problem, exactly as it was stated to you, with the entire original context. A picture or screenshot is best. If the original problem is not in English, then post it anyway! The additional context might still be helpful. Do your best to provide a translation.
@little hill Has your question been resolved?
final saddleBOT
Available help channel!
ornate void
yo
leaden moon
!da2a
final saddleBOT
Asking the actual question right away is more likely to get responses.
Asking "Can I ask...?" or "Does anyone know about...?" doesn't give people enough information to decide whether they can help, and answering can feel like a promise to help with the actual question, which they might find themselves unable to.
ornate void
ya got any question to ask?
yea how do u solve implicit differentiation if there's a dy/dx on both sides of the equation
x³ + y3 = 6xy
that for example
grizzled tusk
differentiate both sides and put everything on the same side (or put everything on the same side and differentiate)
ornate void
x³ + y³ = 6xy
3x² + 3y²(dy/dx) = 6(1)(y) + (x)(dy/dx)
3x² + 3y²(dy/dx) = 6y + 6x(dy/dx)
3y²(dy/dx) -6x(dy/dx) = 6y -3x²
dy/dx(3y² - 6x) = 6y -3x²
dy/dx = (6y - 3x²) / (3y² -6x)
is this correct
grizzled tusk
seems correct to me
grizzled tusk
yes
ornate void
alr thanks goat
radiant igloo
x³ + y³ = 6xy
3x² + 3y²(dy/dx) = 6(1)(y) + (x)(dy/dx)
3x² + 3y²(dy/dx) = 6y + 6x...
2nd line is missing a 6 but otherwise perfect
ornate void
2nd line is missing a 6 but otherwise perfect
6[(1)(y) + (x)(dy/dx)]
does this work too
thx
final saddleBOT
@ornate void Has your question been resolved?
jagged flare
$\mathbb{F}_p$ means a polynomial modulo a prime $p$ right?
final saddleBOT
$\mathbb{F}_p$ means a polynomial modulo a prime $p$ right?
soft zealotBOT
ihave<skissue>
jagged flare
what does $\mathbb{F}_p/\langle x^d-1\rangle$ mean then? isnt it like both modulo $p$ and $x^d-1$?
soft zealotBOT
ihave<skissue>
jagged flare
yeah i dont get any of this set theory (?) notation 
azure hawk
**ihave<skissue>**
No it’s \mathbb{F}_{p^n} if I’m not wrong
Oh it’s not working like that
drowsy epoch
no spaces
azure hawk
$\mathbb{F}_{p^n}$
soft zealotBOT
ℳ
drowsy epoch
what does $\mathbb{F}_p/\langle x^d-1\rangle$ mean then? isnt it like both modul...
yes that's it
jagged flare
does it even make sense to be reduced by 2 modulos at once
drowsy epoch
well you have coefficients mod p and polynomials that satisfy x^d=1
jagged flare
er
what does the second part mean
so like, under (is that the term?) $\mathbb{F}_p/\langle x^d-1\rangle$, the coefficients of the polynomial is reduced mod $p$, and the degrees is reduced mod $d$?
soft zealotBOT
ihave<skissue>
jagged flare
and the end result would be some polynomial which is equivalent to p(x) mod x^d-1? but what about the mod p
drowsy epoch
**ihave<skissue>**
you could think of it like that i guess
drowsy epoch
you apply that on the coefficients
jagged flare
the coefficients only?
soft zealotBOT
jagged flare
if it was like 4x^3 then would it become like x or something
soft zealotBOT
jagged flare
mmmmmm i see i see
drowsy epoch
here we used also x^2-1=0
jagged flare
ok wait maybe im silly but what does F represent
drowsy epoch
some field
drowsy epoch
**ihave<skissue>**
also you should use F[x] to represent the field on polynomials
jagged flare
oh yeah i forgot about that, mb
drowsy epoch
and whats a field <:blobcry:971797574916907028>
it's a ring where every non-zero element has a multiplicative inverse
so on R we can divide for example as we usually do
you can also work with rings btw
jagged flare
and... whats a ring 
drowsy epoch
well you basically have some set where you can multiply and add stuff, but you dont have necessarily multiplicative inverses, unlike in fields
jagged flare
hmmm ok i see
kindred mortar
For exemple $\mathbb{Z}$ is a ring
drowsy epoch
you should review groups, rings and fields, they come like in this: groups > rings > fields, like each structure builds up on the previous
drowsy epoch
For exemple $\mathbb{Z}$ is a ring
\
soft zealotBOT
Lin Xia
kindred mortar
yes sorry
drowsy epoch
all g
jagged flare
for say like F[x], what would be the multiplicative inverse of x? would it be like 1/x? or does the notion of multiplicative inverse need some modulo?
jagged flare
you should review groups, rings and fields, they come like in this: groups > rin...
honestly i was just trying to solve a problem, and the only solutions i found involved the notations of fields. i havent studied group theory yet, so thats probably why im so clueless rn
drowsy epoch
for say like F[x], what would be the multiplicative inverse of x? would it be li...
uhm F[x] is not necessarily a field, it means you have polynomials with coefficients from the field F
it's a polynomial ring
jagged flare
oh?
jagged flare
ok ok i think i get it
kindred mortar
actually Z isnt a field right
yes for the same reason F[X] is not a field
jagged flare
one last thing, do you even need to use F, or is it just an arbitrary name to call a field
drowsy epoch
it's a convention
kindred mortar
one last thing, do you even need to use F, or is it just an arbitrary name to ca...
In your document it is $\mathbb{F}_{p^n}$ that is a specific field
jagged flare
ok i see
soft zealotBOT
Lin Xia
jagged flare
huh why p^n
or is that just another convention, where were only specifically dealing with n=1
drowsy epoch
no it's based on a theorem, that a finite field has order of some prime power
kindred mortar
It is a construction : for any $n$ you can build a field whose cardinality is $p^n$
soft zealotBOT
Lin Xia
kindred mortar
And it is named $\mathbb{F}_{p^n}$
drowsy epoch
{}
soft zealotBOT
Lin Xia
kindred mortar
ahah yes
inland kettle
generally if i have an n-degree polynomial irreducible mod p, then i can quotient Z/pZ[x] wrt said polynomial to get a finite field of order p^n
you can try it yourself, too; for example, get a quadratic irreducible mod 2 and quotient Z/2Z[x] wrt said quadratic. you'll get a field with order 4
and so on
jagged flare
what the whar
what does Z/2Z mean
ohh
nvm
ok interesting
thank you so much you guys!
.solved
final saddleBOT
grim nebula
both the solutions on aops are like written in such a convoluted way 
drowsy epoch
both the solutions on aops are like written in such a convoluted way <:bending_s...
maybe you need to apply a convolution
final saddleBOT
Available help channel!
fringe bane
final saddleBOT
supple jolt
,status
soft zealotBOT
supple jolt
Boooooooo 👎👎👎👎
fringe bane
Boooooooo 👎👎👎👎
?
final saddleBOT
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
Oh
tiny kraken
you can just plug alpha=0 and alpha=1 and find the area separately
fringe bane
you can just plug alpha=0 and alpha=1 and find the area separately
im cofused
cuz when u plug alpha =0,1
u dont get f(0) + f(1)
tiny kraken
you don't have to find closed form of f(x)
if they say find f(0), then replace alpha with 0
fringe bane
if they say find f(0), then replace alpha with 0
and then wht to do
@tiny kraken can u give me 10 min, gotta eat
brb
sryy
tiny kraken
what would the last equation be if alpha=0
tiny kraken
and then wht to do
okay
fringe bane
thansk
final saddleBOT
@fringe bane Has your question been resolved?
fathom jackal
bhai fir integrate kar from the corner points
tiny kraken
<@669797638526468106> im back
okay just replace alpha with 0 then find the area, really
I don't know how to explain it better
south dirge
bhai fir integrate kar from the corner points
finn ka method samjh aaya ya mein jee point of view se explain karun ??
fathom jackal
finn ka method samjh aaya ya mein jee point of view se explain karun ??
bhai mera doubt nhi he 🙏
south dirge
Acha sorry wrong tag
south dirge
<@669797638526468106> im back
finn ka method samjh aaya ya mein jee point of view se explain karun ??
final saddleBOT
@fringe bane Has your question been resolved?
final saddleBOT
hollow dune
I am stuck with comparing linear rates and probably worded alegbra questions in general im struggling to visualise and form equations and suitable variables.
final saddleBOT
I am stuck with comparing linear rates and probably worded alegbra questions in ...
vital crag
I am stuck with comparing linear rates and probably worded alegbra questions in ...
you should just show the original question and your attempted work
final saddleBOT
@hollow dune Has your question been resolved?
quartz vigil
final saddleBOT
soft zealotBOT
quartz vigil
soft zealotBOT
blissful meadow
This is 50
onyx peak
but the answer on key is different
what does the answer key say?
blissful meadow
Ok well is 50 ok or it's also different from the key? And how is it different?
soft zealotBOT
Azyrashacorki
blissful meadow
It's useful. In any case what you wrote wasn't wrong per se they just had a different way of writing it.
tranquil pine
,rccw
soft zealotBOT
tranquil pine
where are you stuck
soft zealotBOT
tranquil pine
hm.. you can write x and y in terms of t and set them equal
or just find the slope of the line by taking any two coordinates
and form the equation
final saddleBOT
@quartz vigil Has your question been resolved?
quartz vigil
Got it
weary meteor
What is -3i times 3i
final saddleBOT
What is -3i times 3i
south dirge
well what is "i"
weary meteor
Imaginary
south dirge
sqrt of -1 right?
weary meteor
Yeah
south dirge
Okay so we can multiply the real parts seperately here right?
south dirge
What is -3i times 3i
so what would you be left with after simplifying this?
weary meteor
-3i?
south dirge
uhm sorry?
south dirge
okay lets say you have to find the product of -3x and 3x what would it be?
weary meteor
Uhh lemme think
weary meteor
not quite right
That’s weird cs in the video I’m watching
The equation goes like this
(2-3i)(2+3i)
And they get
Using foil
They get
4+6i-6i-3i^2
weary meteor
south dirge
wouldnt it be -9i^2?
well?
weary meteor
That’s what I initially thought as well
But no according to her it’s -3^2
Should we get a third opinion
?
south dirge
Okay whats 3^2?
south dirge
Should we get a third opinion
not required for now
vital crag
,calc 3i * -3i
soft zealotBOT
Result:
9
9
vital crag
did you just interpret your teacher wrong
weary meteor
-3i times 3i is -9i
south dirge
-3i times 3i is -9i
what no?
weary meteor
This is her work
vital crag
this is wrong yes
south dirge
Its hould have been 9
weary meteor
this is wrong yes
How do I tell my teacher she is wrong?
vital crag
not that hard. be a good person about it
south dirge
How do I tell my teacher she is wrong?
Ah well point out politely that it should have been 9
vital crag
ask your parents how to be nice
weary meteor
No the thing is I’m missing 13 assignments in her class so why would she listen to me 😭
south dirge
If shes a good teacher she will own up to her mistake dw you are not the first person to be in such a situation
south dirge
No the thing is I’m missing 13 assignments in her class so why would she listen ...
Thats fine!
Atleast doing this would show her that you actually care about the assignments!
But on the othwr hand do please complete your assignments!!
weary meteor
Yeah I’m trying to do that I have an f atm tryna get to a D before grade are put in which is Friday
Alright thanks guys I felt like something was off, appreciate you guys for discussing with me!
.close
final saddleBOT
south dirge
sure anytime
final saddleBOT
Available help channel!
keen ocean
.close
jagged flare
given $$x^2=-1 \mod p$$
$$y^2=-1 \mod p$$
$$xy=-1 \mod p$$
then
$$xy=-1 \mod p$$
$$y=-x^{-1} \mod p$$
and
$$x^2=-1 \mod p$$
$$x=-x^{-1} \mod p$$
$$x=y \mod p$$
is this true?
final saddleBOT
given $$x^2=-1 \mod p$$
$$y^2=-1 \mod p$$
$$xy=-1 \mod p$$
then
$$xy=-1 \mod p$$...
soft zealotBOT
ihave<skissue>
loud sundial
given $$x^2=-1 \mod p$$
$$y^2=-1 \mod p$$
$$xy=-1 \mod p$$
then
$$xy=-1 \mod p$$...
Seems good to me 
trail mango
for invertible x sure
jagged flare
yay
deep condor
$\pmod p$ for better tex'ing btw
soft zealotBOT
حسیب ♥
jagged flare
for invertible x sure
er invertible x just implies x isnt 0 mod p right
trail mango
yea
jagged flare
$$x \mod p$$
$$x \pmod p$$
soft zealotBOT
ihave<skissue>
final saddleBOT
deep condor
oh i forgor about \mod, dont mind me
obtuse hedge
\pmod better for equations
\mod is technically the binary op, in accordance with the word of ☁️
trail mango
snow i've had it with you
obtuse hedge
\mod was the computer science operation? 
I don't remember anymore
grim nebula
\mod and \pmod are the same
formal trail
\mod is technically the binary op, in accordance with the word of ☁️
\mod and \pmod and \pod are all for the equivalence relation, \bmod for binary operation, \operatorname or \DeclareMathOperator for 2-argument function
deep condor
\mod and \pmod are the same
what about the humble 
final saddleBOT
Available help channel!
unkempt fern
Hello
unkempt fern
Can someone explain me how to use laplace transform
gleaming anchor
(in the future, please lead with the question so that the bot pins it!)
gleaming anchor
Can someone explain me how to use laplace transform
unkempt fern
Ok
formal trail
it mostly boils down to reading the relevant table (see the table on the right of the image)
formal trail
if you look at the first two entries of the left table, you ssee that you can distribute the laplace transform across constant multiples and addition. after distributing out, you should be able to get to one of the entries on the right
you should treat all constant functions as multiplied by u(t) for this exercise
unkempt fern
For F1(t) and F2(t) u mean ?
But what is the "a" in the first thing on the right table
vital crag
But what is the "a" in the first thing on the right table
A number like 3
vital crag
I understand nothing how i should do
read the sample exercises
https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx
unkempt fern
Ok thanks
final saddleBOT
@unkempt fern Has your question been resolved?
urban niche
Im working on my homework for AP Calculus BC, and I am struggling with some basic questions that are kind of awkward to Google. Right now I am doing some very basic Antiderivates and Indefinite integral homework. The first problem is to find the indefinte integral of x^3 -2. I know to separate the two into two separate integrals and then solve. If I make Constants for both of those does my answer become 2C...? I have a very lackluster foundational understanding of calculus because I didn't take precalc and I am basically self teachign the course... sos pls send help :(
final saddleBOT
Im working on my homework for AP Calculus BC, and I am struggling with some basi...
formal trail
C stands for anything, so you should really have a different constant for every integral
but then if you add two constants, you get another constant
so really you can just use one constant for the whole thing
urban niche
so I should be getting [x^4/4 + C] + [-2x + K]. Which i could rewrite to be x^4/4 -2x + C + K?
candid pulsar
yeah
formal trail
sure
urban niche
would K be negative? according to the integral rule my notes have, the negative signs gets pulled in front of the integral
which would make it distribute to the whole chunk after I integrate
formal trail
it doesn't really matter because both K and -K are arbitrary constants
urban niche
oh true
formal trail
similarly, C + K is another arbitrary constant
urban niche
On an AP test would there be a huge chance I get marked points for not writing dx at the end of all my integrals if they are "basic" ones
final saddleBOT
@urban niche Has your question been resolved?
final saddleBOT
Available help channel!
dark osprey
Given integers a, b, and c greater than 1 satisfying a(abc+2c-b^2)=a+2b-3c, prove that abc is the square of an integer.
final saddleBOT
Given integers a, b, and c greater than 1 satisfying a(abc+2c-b^2)=a+2b-3c, prov...
dark osprey
help me plsss
fallen aurora
Well maybe you could start by distributing the a within the first equasion and moving everything to one side?
You could also note how abc = d^2 where d is an integer greater than 1
(Personally not sure how to cintinue after that but it's a start)
dark osprey
Well maybe you could start by distributing the a within the first equasion and m...
i try it but it dont success
can u help a new way
cosmic warren
can i pinging ADMIN
No
fallen aurora
I recommend against it
final saddleBOT
dark osprey
.reopen
final saddleBOT
✅ Original question: #help-36 message
sturdy flax
why
because the job of moderators is to ensure no one is being rude or misbehaving, not to offer help.
at least, that is what that ping is for.
dark osprey
at least, that is what that ping is for.
so can u help me
trail mango
sturdy flax
unfortunately, I am unable to for this question. sorry.
final saddleBOT
sturdy flax
<@&268886789983436800> user insinuated helpers are stupid simply for being unable to help. check delete logs if you have them - the user typed and deleted the word 'stupid' in six individual messages.
radiant igloo
<@&268886789983436800> user insinuated helpers are stupid simply for being unabl...
breuh
trail mango
you can’t talk about yukari this way blud
inner kettle
Best Book for Probability
radiant igloo
Best Book for Probability
high jolt
it's true, I am stupid.
trail mango
it’s ok altanis you’re a comedian
tulip coyote
or mod
If you really wanted our attention, spelling out "stupid" letter by letter isn't the way to do it 
If you're gonna be like that, you can come back in a few hours 
high jolt
so good at my job i got banned from meta-discussion
trail mango
i should do that
sturdy flax
I would gladly take that label if not for the fact that I was not the only helper here and given the context...
tulip coyote
I would gladly take that label if not for the fact that I was not the only helpe...
Oi 
trail mango
average hanako disciple
fathom jackal
i can help clean the trash 👍
trail mango
fathom jackal
cant do math help tho...
timber leaf
thank you for your input
inland kettle
i can help clean the trash 👍
lmao what 
trail mango
slow help channel night
fathom jackal
lmao what <:opencry:586078614865444865>
small jobs have great impact 🔥
timber leaf
wise
trail mango
take it to the philosophy server
fathom jackal
take it to the philosophy server
socrates guided me here to spread wisdom 🙏
trail mango
a dead tree can't learn new tricks. but neither can a living one
final saddleBOT
Available help channel!
pine sand
final saddleBOT
south dirge
!status
final saddleBOT
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
a little bit yeah
kindred mortar
a little bit yeah
can you explain ?
pine sand
W is subspace of V if
a,b belongs to W then a-b belongs to w
a belongs to filed and alpha belongs to w then aalpha belongs to w
kindred mortar
ok
pine sand
So here
x,y belongs to R
alpha,beta belongs to W
xalpha+ybeta belongs to W then it is subspace of W
kindred mortar
btw you can just verify that $x\alpha + y\beta$ belongs to $W$ for every $x,y\in\mathbb{R}, \alpha,\beta\in W$ it will prove both of your conditions
soft zealotBOT
Lin Xia
kindred mortar
xalpha+ybeta belongs to W then it is subspace of W
How do you continue ?
pine sand
I am not getting w in the question properly
kindred mortar
$W$ is a subset of $\mathbb{R}^3$ so you can describe its elements with triplets of elements of $\mathbb{R}$ right ?
kindred mortar
How will i add triplets?
$(a,b,c) + (a',b',c') = (a+a',b+b',c+c')$
final saddleBOT
@pine sand Has your question been resolved?
final saddleBOT
pine sand
.reopen
final saddleBOT
✅ Original question: #help-36 message
kindred mortar
How will i add triplets?
.
pine sand
I am still confused with variables
kindred mortar
ok so you want to show that $x\alpha + y\beta$ belongs to $W$ for every $x,y\in\mathbb{R}, \alpha,\beta\in W$
soft zealotBOT
Lin Xia
kindred mortar
Every element of $\mathbb{R}^3$ can be written with triplet of $\mathbb{R}$
soft zealotBOT
Lin Xia
kindred mortar
So you take $\alpha,\beta\in W$ and you can say there exist $(a,b,c)\in\mathbb{R}^3$ and $(a',b',c')\in\mathbb{R}^3$ such that $\alpha = (a,b,c)$ and $\beta = (a',b',c')$
soft zealotBOT
Lin Xia
kindred mortar
Now you just have to verify that $x\alpha + y\beta$ belongs to $W$
soft zealotBOT
Lin Xia
kindred mortar
Is that clear ?
pine sand
x(3a-4b+c)+y(a+2b-c) belongs to W
We have to check this am i right?
@kindred mortar
a(3x+y)+b(-4x+2y)+c(x-y)
kindred mortar
x(3a-4b+c)+y(a+2b-c) belongs to W
no
this are the relations that the coordinates verify to be in W
You have to compute the coordinates of $x\alpha + y\beta$
soft zealotBOT
Lin Xia
pine sand
That is why i am confused with triplets addition properly
kindred mortar
Then you will check if this element verify the conditions to be in W
kindred mortar
That is why i am confused with triplets addition properly
Yes i understand
pine sand
Can you write xalpha for me?
kindred mortar
Sure
pine sand
Yes please
kindred mortar
If $\alpha = (a,b,c)$ then $x\alpha = (xa,xb,xc)$
soft zealotBOT
Lin Xia
pine sand
I see
kindred mortar
basically it is operations coordinate by coordinate
pine sand
x(3a-4b+c),x(a+2b-c)
kindred mortar
Same principle for addition of elements of R^3
kindred mortar
x(3a-4b+c),x(a+2b-c)
nonono
First of all you have to compute coordinates of $x\alpha + y\beta$
soft zealotBOT
Lin Xia
kindred mortar
Lets say $\alpha$ is (a,b,c) and $\beta = (a',b',c')$
soft zealotBOT
Lin Xia
kindred mortar
What are the coordinates of $x\alpha + y\beta$ ?
soft zealotBOT
Lin Xia
kindred mortar
Do you think you can find $x\alpha + y\beta$ coordinates ?
soft zealotBOT
Lin Xia
pine sand
alpha=(a,b,c)
xalpha+ybeta=x(3a-4b+c,a+2b-c)+y(3a'-4b'+c,a'+2b'-c')
kindred mortar
alpha=(a,b,c)
xalpha+ybeta=x(3a-4b+c,a+2b-c)+y(3a'-4b'+c,a'+2b'-c')
Forget about the definition of W
Lets say W is any subset of R^3
You want to verify if it is a subspace
The method is the same for every subset W
You have to take two elements (\alpha,\beta) of W and verify that for any (x,y)\in R, x\alpha + y\beta is in W
So the first step is always to compute $x\alpha + y\beta$
soft zealotBOT
Lin Xia
kindred mortar
and then we verify if it is in W
SO
$x\alpha + y\beta = x(a,b,c) + y(a',b',c') = ...$
Can you continue this ?
soft zealotBOT
Lin Xia
kindred mortar
a,b,c are elements of $\mathbb{R}$
soft zealotBOT
Lin Xia
kindred mortar
c is missing
wdym ?
pine sand
I am lost
pine sand
I can't explain
kindred mortar
Do yo agree with the method i exposed ?
pine sand
Yeah
pine sand
alpha=(a,b,c)
xalpha+ybeta=x(3a-4b+c,a+2b-c)+y(3a'-4b'+c,a'+2b'-c')
Please make it correct
kindred mortar
The answer is $x\alpha + y\beta = (xa+ya',xb+yb',xc+yc')$
soft zealotBOT
Lin Xia
kindred mortar
Please make it correct
What you wrote here cant be correct and it's easy to detect because alpha and beta are triplet while your LHS is composed of doublet.
kindred mortar
The answer is $x\alpha + y\beta = (xa+ya',xb+yb',xc+yc')$
@pine sand does it make sense to you ?
final saddleBOT
@pine sand Has your question been resolved?
halcyon badge
Hello everyone, in my study I have to do a project in the theme of optimisation, efficiency and soberity.
And I thought of optimize the Traveling salesman problem to reduce the travel so I would like to have your help if it is a good subject and if it is relazable. ( excuse me for my English if I do mistake).
halcyon badge
.close
kindred acorn
.open
radiant igloo
just post your question
final saddleBOT
just post your question
final saddleBOT
timber leaf
Whats your question blud
radiant igloo
Whats your question blud
i was tryna help him open a channel
final saddleBOT
Available help channel!
misty vale
hello; is this, like, something I should be able to figure on my own? It's from a textbook:
"0.5 If C is a set with c elements, how many elements are in the power set of C? Explain
your answer."
Below I will use n instead of c to denote the cardinality of C.
for {} → { {} } it's just 1
for {x} → { {}, {x} } it's 1 + 1
Clearly the power set will always contain an empty set, and the set C itself.
for {x, y} → { {}, {x}, {y}, {x,y} }, so 1+n+1
We get an additional term, n, because there will be n sets containing 1 element
for {x, y, z} → { {}, {x}, {y}, {z}, {x,y}, {x,z}, {z,y}, {x,y,z} } so 1 + n + n(n-1)/2 + 1
We get an additional term, n(n-1)/2, because the amount of subsets containing 2 elements is the same as the amount of edges in an undirected graph excluding self-loops (n^2 total edges minus n self-loops).
for {a, b, c, d} →
{ {},
{a}, {b}, {c}, {d},
{a,b}, {b,c}, ...,
{a,b,c}, ...,
{a,b,c,d} }
Again, I have 1 + n + n(n-1)/2 + 1 as before except I add the amount of subsets with 3 elements. In this case (n=4) there is 6n/6 = n of them (i figured it out graphically but I've no idea if it holds for n>4).
i don't see a pattern nor can I think of a different approach. I have already checked the answer: the cardinality the power set of A is 2^n where n is the cardinality of A. But I don't see it at all
final saddleBOT
hello; is this, like, something I should be able to figure on my own? It's from ...
barren hound
instead of all this n stuff.... how about you make a table of number of elts in C vs number of elts in powerset of C
0 -> 1
1 -> 2
2 -> 4
continue?
misty vale
well, I don't really see a reason for the pattern to hold indefinitely
barren hound
what pattern?
misty vale
the cardinality of the power set going: 1 → 2 → 4 → 8 → 16 → 32 → ... → n^2
barren hound
doubling each time?
misty vale
yeah
barren hound
each elt can either be in the set or not
stable helm
This is an elementary result about powersets
plucky rover
the cardinality of the power set going: 1 → 2 → 4 → 8 → 16 → 32 → ... → n^2
Minor correction: 2^n, not n^2
misty vale
Minor correction: 2^n, not n^2
yeah I mistyped ;-;
plucky rover
This is an elementary result about powersets
Unfortunately elementary results aren't axioms. They can and should be proven and have intuition behind them
plucky rover
each elt can either be in the set or not
Do you see why Hayley said this
-# (also hey Hayley, if you want to continue here say the word and I'll leave)
misty vale
i'm thinking about it but I'm not sure
plucky rover
Have you done any combinatorics
(since you have the undergrad math role, I'm assuming you're in college. Please correct me if I'm wrong)
misty vale
i'm studying but biology; The only math education I received is what I believe is equivalent to pre-calc. I kinda understand what a limit/differential/integral is (though that's probably irrelevant here). I'm aware the amount of permutations of n elements is n! & I understand why. I've seen the formula for combinations but I'm not sure if I could explain why does it look like that
plucky rover
I see I see
No worries! We can go through this slowly
Okay so a power set is the set of subsets of a given set right
misty vale
I do kind-of see the relation to permutations
misty vale
by taking 0 or more elements from the set and putting them in sets of their own
well, between 0 and n, inclusive, elements
plucky rover
I mean that's an easy way to think of it but isn't very helpful
So let's do it more systematically
We have n elements
We can arrange them somehow
Now let's pick the first element
We can either put this in the subset, or we can choose to not put this in the subset
With me so far?
misty vale
so far, yes
plucky rover
So if you choose to not pick the first element, that is one branch of choices
If you choose to pick it, that is another branch
Now let's check the second element
We have the same choice here right?
misty vale
sure
plucky rover
So for every element we have two choices
And the choices for each element do not depend on the choices for the other elements
misty vale
oh... so it branches like this?
misty vale
alright, I get it now; thank you so much
plucky rover
Np!
If you have any more questions, feel free to tag me
And if not, .close to close the channel
final saddleBOT
final saddleBOT
Available help channel!
left trail
Is this a valid proof for part b. I am only concerned that I did not include the full representation of the nth Bernoulli polynomial. But I don't know how to represent that right now and I think that since its highest term is the nth one that is the only one we should focus on for trying to prove the n+1 polynomial is of degree n+1
final saddleBOT
Is this a valid proof for part b. I am only concerned that I did not include the...
desert mantle
just write $P_n(x)=x^n+\sum_{i=0}^{n-1}a_i x^i$ or something along those lines
soft zealotBOT
Denascite
final saddleBOT
@left trail Has your question been resolved?
left trail
Ok so then it is fine to leave it as $x^{n+1}+\int \sum_{i=0}^{n-1}a_ix^i$
soft zealotBOT
BigBen
vital crag
Ok so then it is fine to leave it as $x^{n+1}+\int \sum_{i=0}^{n-1}a_ix^i$
What is "it"
left trail
What is "it"
The second to last line where I have the text in brackets
somber fog
**BigBen**
Better apply the integral directly to your sum and thus show that the principal coefficient of P_(n+1) is exactly 1.
left trail
Better apply the integral directly to your sum and thus show that the principal ...
By principal coefficient you mean the coefficient for x^n+1?
somber fog
By principal coefficient you mean the coefficient for x^n+1?
Yeah!
kindred mortar
$P_{n+1}(x) = \int P_{n+1}'(x)dx$ looks a bit evasive. You should be more precise I think.
soft zealotBOT
Lin Xia
kindred mortar
Even if it doesn't change the global reasoning
left trail
Yeah!
But we have $(n+1)\int x^ dx which is x^{n+1}$ and we can see that it will have coefficient 1
soft zealotBOT
BigBen
left trail
Even if it doesn't change the global reasoning
Are you sure because all I'm saying is that the P is equal to the integral of the deirvative of P
kindred mortar
Are you sure because all I'm saying is that the P is equal to the integral of th...
Yes but this is true up to a constant
somber fog
But we have $(n+1)\int x^ dx which is x^{n+1}$ and we can see that it will have ...
The problem is that the integral of a derivative is P_{n+1}(x) + C. You have to prove that constant C exists and is unique using the problem condition otherwise your definition of P is incomplete.
left trail
The problem is that the integral of a derivative is P_{n+1}(x) + C. You have to ...
What do you mean a unique constant. Unique in comparison to who? Also can't we just leave it as an arbitrary constant c since it has degree 0
final saddleBOT
vital crag
The second to last line where I have the text in brackets
Do your formulas for P_n match the ones in part a) for n=1,2... 5
final saddleBOT
Available help channel!
hexed aspen
I've learned that sine is the y axis of the rotating point that goes counter clockwise around a circle (of radius 1 unit)
And cosine is the x axis.
and that these are helpful in the forming of waves (sine waves and cosine waves).
I need some help in clearly understanding:
How these form the waves I mean, do u take the crest and trough as 1 unit or something and as sine reaches 1 u go to crest and - 1 u go to trough or something? not really sure how that works. If there's a significance of taking the radius of the circle of more than one unit as well.
And, what exactly is cot, tan, cosec, sec (I might know them in terms of fractions 1/sine, sine/cos etc.. But what are they)
What significance are they of
final saddleBOT
I've learned that sine is the y axis of the rotating point that goes counter clo...
final saddleBOT
grim nebula
.close
this is not an appropriate use of the .close command...
users are already automatically limited to 2 help channels at a time by the bot
loud sundial
this is not an appropriate use of the `.close` command...
mb
I've seen others do it before as well
granted I have bad memory so specific names don't come to mind
grim nebula
the guidelines are in #helpers-info
grim egret
They also asked different questions in both channels.
final saddleBOT
Available help channel!
final saddleBOT
loud sundial
<@&268886789983436800>
fathom owl
HOW SOLVE
fathom owl
thorny otter
Send working^
abstract bramble
what does the first few terms look like?
fathom owl
abstract bramble
hmm yes but i'm talking about the summation
fathom owl
Ouhh uhhh for sn it's using the formula for summation of AP and for Sn, ihni
thorny otter
What are the first few partial sums?
fathom owl
I don't know 😭
fathom owl
1
fathom owl
3 i think? Cause S2 would be 9 and s2 would be 3
winter dagger
ok great so it can't be c
That's not how you should be solving it tho
thorny otter
3 i think? Cause S2 would be 9 and s2 would be 3
2nd partial sum is S1/s1 + S2/s2
thorny otter
That's not how you should be solving it tho
but also this
winter dagger
Wrong channel ig
fathom owl
Girl how is this chem
abstract bramble
That's not how you should be solving it tho
of course not
muted prairie
!occupied
final saddleBOT
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).
thorny otter
What is Sn
What is sn
Divide Sn/sn, this is your summand
Use linearity of finite sums
Use known results for summing 1, r, r^2 etc.
fathom owl
What is Sn
What is sn
Divide Sn/sn, this is your summand
Use linearity of fini...
Dumb it down like I'm new to this world because I might as well be
muted prairie
basically, do you know a formula for the sum of the first n cubes
fathom owl
basically, do you know a formula for the sum of the first n cubes
No, there's a formula?
fathom owl
thorny otter
No, there's a formula?
Do you know the formula for the first n numbers?
fathom owl
Do you know the formula for the first n numbers?
Yes
heady portal
This is the formula for sum of cubes of n natural numbers
thorny otter
This is the formula for sum of cubes of n natural numbers
Let them work it out
heady portal
Or maybe derive it urself
fathom owl
YOU'D THINK I'D KNOWN THERE WAS A FORMULA BEFORE BUT NO THEY DON'T TEACH US NUTHIN IN SCHOOL
Anyways, thank you all for your patience and cooperation 
.close
final saddleBOT
final saddleBOT
Available help channel!
brazen talon
final saddleBOT
brazen talon
guys i understand the question, got to somewhere and got stuck ill explain what i did
first took a general point on parabola, drew a foot of perpendicular to y=x, intersected, found points, added and divided by two to get mid point and then tried to eliminate the variable
did all that, reached nothing
final saddleBOT
@brazen talon Has your question been resolved?
loud sundial
first took a general point on parabola, drew a foot of perpendicular to y=x, int...
This should work.
In the future, please show what you actually did rather than just sketching it. Because if you did something like a small algebra error, we’d have no idea where.
crystal lion
for part b, im not too sure why i went wrong because the answer is positive 1/48 pi
final saddleBOT
for part b, im not too sure why i went wrong because the answer is positive 1/48...
versed crater
,calc atan(100)
soft zealotBOT
Result:
1.5607966601082
,calc atan(101)
soft zealotBOT
Result:
1.5608956602069
,calc atan(1)
soft zealotBOT
Result:
0.78539816339745
,calc pi/4
soft zealotBOT
Result:
0.78539816339745
What happened to the first arctan
timber leaf
From step 3 takes 3/4 out first to make it less complex
crystal lion
What happened to the first arctan
i just assumed it goes to infinity so i just ignored it
versed crater
i just assumed it goes to infinity so i just ignored it
What makes you able to ignore it
timber leaf
And tan^-1(3t/4) is pi/2 btw
whole halo
cmon man
timber leaf
A oke
whole halo
also
timber leaf
Didnt see
quartz flax
💔
whole halo
thats not even correct
so technically its not spoiled
also wdym ^1
its tan^-1 or arctan
versed crater
@crystal lion “ignoring” a term is not a trivial thing, yo need a justification for it
timber leaf
Cuz im editing
timber leaf
you still left the ^1 in there too
whole halo
drowsy epoch
https://youtu.be/FtVllgTPyfk?t=23 moment
💀
crystal lion
arctan tends towards pi/2?
timber leaf
I dont understand english much so the video is for?
whole halo
youre typing in english bro
whole halo
"blud" isnt in google translate
and neither is ^1 lol
just let frosst cook with the remaining ingredients
timber leaf
I know, some slangs in English i know is skibidi and kirk and blud
quartz flax
just let frosst cook with the remaining ingredients
Frosst is gone tbf
timber leaf
and neither is ^1 lol
Didnt bro see the word typo
drowsy epoch
Frosst is gone tbf
on ice
bold turtle
arctan tends towards pi/2?
Yes, as t tends towards infinity, that arctan tends to pi/2
But that hasn't been implemented in the working out (it looks like you've sent it to 0 instead)
quartz flax
but 3/4 arctan on the other hand...
crystal lion
But that hasn't been implemented in the working out (it looks like you've sent i...
ohh okk
quartz flax
😁☝🏾☝🏾3/4☝🏾☝🏾⚠️
whole halo
what do you think 3/4 arctan would go to if arctan goes to pi/2
crystal lion
3/8pi?
bold turtle
(yeah, to clarify, I'm referring to the arctan() itself, not the term that contains the arctan() 
)
whole halo
arctan(t), arctan(3t/4) both go to pi/2 btw
why do you think that is
(for reference, if you had arctan(1/t) instead, that would instead go to 0 as t -> infinity)
crystal lion
why do you think that is
bc theyre still the same graph whereas arctan(1/t) is a different graph?
whole halo
wdym by same graph
quartz flax
looks same enough 😊
crystal lion
same shape of graph
soft zealotBOT
mtt
whole halo
arctan(t) and arctan(3t/4) would not go to the same value
however, arctan(t) and arctan(8 - t) would
quartz flax
youll need to be more specific
transformations!
whole halo
again no spoilers please
I gotta see what yuchanie is thinking, not what you are thinking
crystal lion
so for arctan(1/t), just 1/t is a reciprocal graph
but in arctan(t) and arctan(3t/4), both the inner functions for both behave like a straight line?
whole halo
heres a hint
$\lim_{t\to4}\arctan(t)$ and $\lim_{t\to4}\arctan(8-t)$ are both the same number
soft zealotBOT
mtt
whole halo
now theres no special tricks happening right now, can you tell me what number they both go to?
crystal lion
4
whole halo
I didnt say t and 8 - t
I said arctan(t) and arctan(8 - t)
what number would both of these limits be?
(you can just leave the word "arctan" in your answer)
crystal lion
arctan(4)?
soft zealotBOT
mtt
crystal lion
arctan(3)?
whole halo
yep
now you can see here t, 8-t, 3t/4 are all lines
but somehow the answer became the same for t and 8-t and different for 3t/4
now try finding this limit: $\lim_{t\to4}\arctan(t^2/4)$
soft zealotBOT
mtt
crystal lion
arctan(4)
whole halo
and if I change the /4 to a /3?
what would the answer become instead
t^2/4 and t^2/3 are both parabolas btw
crystal lion
arctan(16/3)
whole halo
now see here there should be a different reason than "shape" for telling apart what stays at arctan(4) and what doesnt
try saying what that reason is
crystal lion
now see here there should be a different reason than "shape" for telling apart w...
is that assuming that t always tends to 4 in this case?
whole halo
yep
you would imagine that would always work, right
if the inside tends to 4, it should always be arctan(4)
or in general, its arctan(what the inside tends to)
crystal lion
yeahh
whole halo
you learned this as the limit chain rule or the limit composition rule
it works since arctan is continuous
,,\lim_{t\to4}\arctan(f(t))=\arctan(\lim_{t\to4}f(t))
soft zealotBOT
mtt
whole halo
(oh and f is also continuous)
there might be more specific details Im leaving out here by accident, but this is the general idea
now given we've already got a rule like this,
the same also works for infinity in a way
,,\lim_{t\to\infty}\arctan(f(t))=\arctan(\lim_{t\to\infty}f(t))
soft zealotBOT
mtt
whole halo
so think of it this way
we know arctan(t) -> pi/2
now arctan(3t/4) -> pi/2 too
however, arctan(-t) -> -pi/2 even though -t is a line just like t is
arctan(1/t) -> 0 since 1/t -> 0 as t -> infinity
also, arctan(ln(t)) -> pi/2 too
seeing a pattern here?
crystal lion
pi/2?
crystal lion
so if the power of the inner function of t within arctan is a positive power then it will always go to pi/2 and if the power is negative then it goes to 0?
whole halo
thats correct
since as t -> infinity, t^(positive) -> infinity and t^(negative) -> 0
so you get pi/2 and 0 when through arctan
and remember, this isnt special to arctan, this would work for other functions too
crystal lion
ahhh okk
whole halo
requirement is that the functions are continuous enough
I believe its "continuous at the limit youre taking"
for example, lets say f(t) -> C
then if youre doing f(t)^2 for example, f(t) needs to be continuous at C for you to say that f(t)^2 -> C^2
(for that one, you could also just multiply f(t) by itself for product rule)
you need "f(t) -> C" for you to take the limit t -> C for arctan(f(t)), or ln(f(t)), or etc.
then itd be arctan(C), ln(C), so on
whole halo
np
crystal lion
.close
final saddleBOT
final saddleBOT
Available help channel!
final saddleBOT
soft zealotBOT
pine whale
!noai
frail parcel
final saddleBOT
frail parcel
how would i know something is growth or decay
fallow valley
sturdy flax
if you need concrete confirmation, test with increasing values of x. if you get larger values of f(x), it's growth. otherwise, if you get smaller values of f(x), it's decay.
frail parcel
so what value would i look at
sturdy flax
if you mean what values of x, any two values would work.
frail parcel
if you mean what values of x, any two values would work.
i’m lost..
my teacher sent this as the key
and she’s just looking at one valid
value
sturdy flax
then let's make it simple.
substitute x = 1 into the function, and then compute the value.
then substitute x = 2 into the function, and compute the value again.
if the value of the function at x = 2 > the value of the function at x = 1, then it's growth.
if it's the other way around, it's decay.
frail parcel
but my teacher is doing it by just looking at one value and that’s it
sturdy flax
of course, this is if you want concrete confirmation.
if you are confident, you can just stare at the exponent and the base and figure out whether it will be a growth or decay that way.
sturdy flax
what's the value of e?
frail parcel
what's the value of e?
2.something
sturdy flax
and is that greater than or less than 1?
sturdy flax
anything else?
frail parcel
sturdy flax
,rcw
frail parcel
how does LN turn into E?
soft zealotBOT
sturdy flax
for which?
frail parcel
21
sturdy flax
21?
frail parcel
i’m so cooked for this test
sturdy flax
do you agree that if $a = b$ then $c^a = c^b$?
soft zealotBOT
Yukari
sturdy flax
(for some number c)
frail parcel
i don’t agree
sturdy flax
cool. now, in your attempts at rearranging for x, you should get $\ln(x) = y + 3$.
soft zealotBOT
Yukari
frail parcel
i need to get rid of ln
old gale
Hi
sturdy flax
now, a core idea behind logarithms is that $b^{\log_b(a)} = a$.
soft zealotBOT
Yukari
old gale
Do you need help with 21 question?
sturdy flax
do you see where we're going?
frail parcel
**Yukari**
is this written right
sturdy flax
yes.
frail parcel
i do not see whgere ur going
sturdy flax
note that if a log and only a log is present in an exponent, and the base of the log is the same as the base of the exponent (the b here), the answer will always be the argument of the log (the a here).
frail parcel
i kinda get it
sturdy flax
now, you have $\ln(x) = y + 3$. you know that raising any nonzero real c to the power of both sides will still be equal.
and you also know that if you set this c to be equal to the base of the log on the left, then you can get x by itself.
soft zealotBOT
Yukari
sturdy flax
so the idea is now to make both sides exponents. the question you should now answer is, what is the base c? that is, $$(?)^{\ln(x)} = ?^{y+3}$$
soft zealotBOT
Yukari
frail parcel
i am so beyond lost
sturdy flax
**Yukari**
right, so you understand this at least?
frail parcel
yes i do
sturdy flax
**Yukari**
and this?
sturdy flax
so you swapped x and y. sure, if that's the method you're used to, we can work with that.
<@&268886789983436800>
sturdy flax
I do not like that way of solving for an inverse, but sure, I'll use what you've been taught.
you get then, after swapping, x = ln(y) - 3.
in your process of solving for y, you will get ln(y) = x + 3.
do you agree thus far?
sturdy flax
why not?
frail parcel
beacuse dont we have to switch it? why would we switch it back to Y
sturdy flax
**Yukari**
good. now, did you get this?
sturdy flax
sure. you know that $\log_3(81) = 4$. then, $3^{\log_3(81)} = 3^4 = 81$.
soft zealotBOT
Yukari
frail parcel
why is ur log on the top
sturdy flax
because we are using it as an exponent.
sturdy flax
yes.
frail parcel
okay yes i have never seen that before but yes i understand
frail parcel
yes i just want Y
sturdy flax
**Yukari**
you know also that if $\ln(y) = x + 3$, then for some base c, $c^{\ln(y)} = c^{x + 3}$, as per this rule here.
soft zealotBOT
Yukari
sturdy flax
**Yukari**
you might be wondering, why bother to do this, right?
this is where this rule comes into play.
if we pick c such that it matches the base of the log we have, then we can cancel the base with the log, leaving us with y = something.
which is what we want.
so the final step here is: what c do we pick?
frail parcel
so its e ^ ln(y) = e^x+3
sturdy flax
**Yukari**
absolutely. then on the left, you'll see this pattern.
you are now justified to cancel the base with the log in the exponent, leaving you with the y.
frail parcel
im not sure why we cancel but sure
sturdy flax
**Yukari**
because of this rule.
sturdy flax
**Yukari**
an example of which is seen here.
frail parcel
yea but how does that show canceling out e for the natrual log?
sturdy flax
$\ln(x) = \log_e(x)$, I believe you agree?
soft zealotBOT
Yukari
frail parcel
yes beacuse the base of ln is e
sturdy flax
then, compare these two forms:
$$b^{\log_b(a)} = a$$
$$e^{\log_e(a)} = a$$
frail parcel
wouldnt it be ln (base e) = x
soft zealotBOT
Yukari
frail parcel
or does ln turn into log when you add the base
frail parcel
**Yukari**
its the same thing but with e
sturdy flax
exactly. then, replace $log_e$ with ln to get $e^{\ln(a)} = a$.
soft zealotBOT
Yukari
sturdy flax
and that's how that rule came to be.
frail parcel
i see i see
sturdy flax
ln is really just a shorthand for log base e. they mean exactly the same thing.
frail parcel
okay i see so if you start with e for example right and the base is log (base e)
it cancels out
cuz e = log(base e)
right
sturdy flax
specifically, if you start with ln(x) and then raise e to that ln(x), it cancels out.
sturdy flax
sure.
frail parcel
quick question since its ln(x+5) i cant take out 5 since x+5 is togethet right
so i take the whooe thing out
frail parcel
oh i removed the y+5 first
sturdy flax
I'll need to see what you meant.
frail parcel
sturdy flax
,rccw
soft zealotBOT
frail parcel
but u just said to remove ln
sturdy flax
remove it by following the rules above, not just kick it out!