vernal ruin
yeesssss love you guys ❤️
#help-28
balmy peak
did u account for the 2 singles having to be different from the triple and the other single?
sand stone
did u account for the 2 singles having to be different from the triple and the o...
yes
its here
balmy peak
o
pulsar sand
Simulation error eh?
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agile vigil
For arbitrary alphabet $\Sigma$ and language $L \subseteq \Sigma^$, let $\sqrt{L} = { w \in \Sigma^ \vert ww \in L}$.
Let $\Sigma $, $\Gamma$ are alphabets, $L \subseteq \Sigma^$ language and $h: \Sigma^ \mapsto \Gamma^*$ homomorphism. Compare languages $h(\sqrt{L})$ and $\sqrt{h(L)}$.
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Michal
agile vigil
any idea how to do this?
im stuck on both inclusions
$\subseteq:$ $w \in h(\sqrt{L}) \implies \exists x \in \sqrt{L}: h(x) = w \implies x = yy$, where $y \in L \implies w = h(x) = h(yy) = h(y)h(y)$
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Michal
agile vigil
dont know how to move futher
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agile vigil
<@&286206848099549185>
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knotty plank
knotty plank
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humble steppe
Find $f’(x)$, then check at which point(s) is the derivative not defined.
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jimmy1234
knotty plank
Would f’(x) be y^2 2
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simple totem
torn jolt
how would i go about working out (c)?
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how would i go about working out (c)?
torn jolt
this is what i can think out so far
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rapid rain
this is what i can think out so far
What is F(n+2) ?
torn jolt
What is F(n+2) ?
fn+1 + fn but i wrote the bottom part to show what i want to get to
rapid rain
Let's try to calculate (fn+1)^2 - fnfn+2
I didn't tell you but the trick is to use induction I believe
torn jolt
ye i figured
rapid rain
Let's try to calculate (fn+1)^2 - fnfn+2
because you get that this is equal to -fn^2 + (fn-1)(fn+1)
let's try to prove this last statement
rapid rain
Let's try to calculate (fn+1)^2 - fnfn+2
starting from this, (fn+1)^2 - fnfn+2 = (fn+1)^2 - fn^2 - fnfn+1
this is equal to -fn^2 + fn+1(fn+1 - fn)
and fn+1 - fn = fn-1 for n > 1.
so finally, (fn+1)^2 - fnfn+2 = -fn^2 + fn-1 x fn+1
for the starting step, f2^2 - f1f3 = 1 - 2 = -1 = (-1)^1
and you're done
torn jolt
for the starting step, f2^2 - f1f3 = 1 - 2 = -1 = (-1)^1
im supposed to get to -f(n+2)^2 + f(n+1)f(n+3) though right
which is the index shifted up 3
rapid rain
not always
you can also prove induction like this :
"let n such that fn^2 - (fn-1)(fn+1) = (-1)^(n-1) (so property true for n-1)
Then (fn+1)^2 - fnfn+2 = ..." and you get the calculations we did a bit before
And you prove the property true for n
@torn jolt this way works equally
torn jolt
hmm
alternatively could i just say that if n then n-1 which implies that if n then n+1
wait actualy hold on
rapid rain
no, there are only two ways
"if n-1 then n" OR "if n then n+1"
it just depends on which one is easier to write in terms of indexes
rapid rain
In the bit we dit above we only showed that (fn+1)^2 - fnfn+2 = -(fn^2 - fn-1 x fn+1)
So, if fn^2 - fn-1 x fn+1 = (-1)^(n-1), then :
(fn+1)^2 - fnfn+2 = -(fn^2 - fn-1 x fn+1) = -(-1)^(n-1) = (-1)^n
and we have the induction step with no problem :)
torn jolt
like this?
@rapid rain
so i bascailyl show that the n statement is equivalent to the n-1 statement and the n-1 statement is equivalent to the n+1 statement?
thats strange
rapid rain
no, you either show n-1 equivalent to n or n equivalent to n+1
more simply, either show "n => n+1" or "n-1 => n"
you only need one of them
rapid rain
You're not trying to go backwards, you're trying to go forwards
so "if n, then n-1" doesn't help us at all
All we did was to show (fn+1)^2 - fnfn+2 = -(fn^2 - fn-1 x fn+1)
If it amuses you, you can replace "n" with "n+1" :
(fn+2)^2 - fn+1fn+3 = -(fn+1^2 - fn x fn+2)
rapid rain
If it amuses you, you can replace "n" with "n+1" :
(fn+2)^2 - fn+1fn+3 = -(fn+1^...
and now this formula applies for all n positive integer
so you can use this formula to prove "if n, then n+1"
Was I clear enough @torn jolt ?
torn jolt
All we did was to show (fn+1)^2 - fnfn+2 = -(fn^2 - fn-1 x fn+1)
but doesn't this technically just translate to "if n then n-1"?
rapid rain
no, this only gives the formula in n in terms of the formula in n-1
torn jolt
sorry this is confusing to me
rapid rain
so if we have a value for the formula in n-1, we instantly get the formula in n
torn jolt
no, this only gives the formula in n in terms of the formula in n-1
oh so i basically start with if n then (-1)^n and then turn n into n-1
rapid rain
no, you start with (-1)^n-1 and THEN turn n-1 into n
you want to go forwards
so you increase the value of n
torn jolt
All we did was to show (fn+1)^2 - fnfn+2 = -(fn^2 - fn-1 x fn+1)
ok so i start with this to get n-1
and then work from there to get to n?
rapid rain
wait a second
torn jolt
but if i use n to get to n-1 isnt getting back to n just the same thing in reverse
rapid rain
We use n-1 to get n
not the opposite
Just like we use n to get n+1
not the opposite
rapid rain
sure
As I said, there are two ways
"Assume n-1 is true. Then n is true"
OR
"Assume n is true. Then n+1 is true"
It is the exact same
just add "+1" to both sides
torn jolt
All we did was to show (fn+1)^2 - fnfn+2 = -(fn^2 - fn-1 x fn+1)
so if we start with assuming n-1 is true then wouldn't this be fn^2-fn-1fn+1
as opposed to the negative and the other way around
rapid rain
so if we start with assuming n-1 is true then wouldn't this be fn^2-fn-1fn+1
if we start by assuming n-1 is true then yes we do get fn^2-fn-1fn+1 = (-1)^n-1
rapid rain
So then (fn+1)^2 - fnfn+2 = -(fn^2 - fn-1 x fn+1) = -(-1)^n-1 = (-1)^n
we did "Assume n-1 is true. Then n is true"
torn jolt
So then (fn+1)^2 - fnfn+2 = -(fn^2 - fn-1 x fn+1) = -(-1)^n-1 = (-1)^n
but isnt this just starting with n again
sorry im really confused
i thought we start with fn^2-fn-1fn+1 = (-1)^n-1 and work our way towards (fn+1)^2 - fnfn+2
rapid rain
no, we didn't "start" with n, we're trying to get to it
torn jolt
So then (fn+1)^2 - fnfn+2 = -(fn^2 - fn-1 x fn+1) = -(-1)^n-1 = (-1)^n
this is the other way around
rapid rain
sure, LHS = RHS is the same as RHS = LHS
torn jolt
ok so u just wrote it the other way around
rapid rain
In order to show that A = B, you can start from A and get to B OR start from B and get to A
If you want to start from B and get to A, it's your choice
I prefer to start from A and get to B
We wanna prove (fn+1)^2 - fnfn+2 = (-1)^n, so I start with "(fn+1)^2 - fnfn+2 = ..." and I try to find the correct path
rapid rain
All we did was to show (fn+1)^2 - fnfn+2 = -(fn^2 - fn-1 x fn+1)
As it turns out, the path is already traced for us because we showed this previously
torn jolt
im still not quite understanding things
is it possible to do this with traditional induction in which you show if n then n+1 or is this the only way
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regal tinsel
One bag and one book together cost $140. Two bags and three books together cost $380. What is the cost of one bag?
plush egret
what have you tried?
regal tinsel
i havent
buoyant tinsel
try first
buoyant tinsel
set up a systems of equations
balmy peak
One bag and one book together cost $140. Two bags and three books together cost ...
one bag + one book = 140
two bags + three books = 380
a+b = 140
2a+3b = 380
and then just solve
regal tinsel
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rugged wharf
im trying to simplify this boolean expression:
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im trying to simplify this boolean expression:
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woeful eagle
find the minimum value of f(x) = eˣ + e⁻ˣ
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find the minimum value of f(x) = eˣ + e⁻ˣ
gray hamlet
1
gray hamlet
oh wait, ignore me Im dumb
frozen spade
you also can use the inequality $a+b \geq 2\sqrt{ab}$, $a,b>0$
glossy valveBOT
秋水
woeful eagle
how do i prove that though
frozen spade
all i can do is monotonicity, 1-1, basic stuff
you can consider the function $f(x) = x+\frac{1}{x}$
glossy valveBOT
秋水
frozen spade
how do i prove that though
expand $(\sqrt{a}-\sqrt{b})^2 \geq 0$
glossy valveBOT
秋水
frozen spade
yeah, do you know the intervals of increasing and decreasing of this function?
woeful eagle
nope, all i know is the expression
frozen spade
you can do some calculation to find that.
let $x_2>x_1>0$, simplify
$f(x_2)-f(x_1)$
glossy valveBOT
秋水
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torn jolt
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torn jolt
how can I show that red and blue area are the same
woven snow
topaz valley
theif 
woven snow
also iirc they aren’t op
#help-41 message this was the first when i discovered
i just stole their diagram
#help-21｜아리스킨충1 message next was this
topaz valley
wtf 
and then the next was this https://discord.com/channels/268882317391429632/1028593858109505546
woven snow
torn jolt
ahh okay thank guys
topaz valley
by none other than me 
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@wary merlin Has your question been resolved?
hot bone
f(x)=2x-1, and you need to prove that
the inverse of f^2 is equal to the inverse of f squared ?
it seems to me that your process of getting f^2 is wrong
f^2=(f(x))^2 = (2x-1)^2
im not familiar with f^2 as a symbol for f(f(x)), it is assumed though when dealing with linear transformations
Unless of course something else is implied by your book
void arch
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void arch
I don’t see how this proposition is true
fast peak
any statement of the form "forall x in the emptyset" is always true
void arch
It means for all x in the empty set, 2+2=5 is true
Logically I struggle to understand why?
What is x in the empty set? It wouldn’t even be zero so what would it represent since there are no elements in the empty set
torn jolt
there is no x, thats why its true trivially
it might help to look at the negation
"there exists an x in empty set, so that 2+2≠5"
gray hamlet
lol this is a math joke
torn jolt
but that statement is false obviously
there are no x in empty set
so the original statement it true
void arch
I understand the negation part
gray hamlet
and 2+2 only = 5 in george orwells 1984
void arch
But thinking of just the initial statement logically I can’t seem to get my head around how it’s true
gray hamlet
@void arch is this for a test or something? Whats the question?
torn jolt
there are exactly 0 elements in the empty sets, so the statement P(x) = (2+2=5) needs to be checked exactly 0 times, so we never actually have to check this false statement
fast peak
$\forall x\in A: P(x)$ is shorthand for $\forall x: x\in A\implies P(x)$
glossy valveBOT
Denascite
fast peak
if A is the emptyset then x in A is false and the implication (false)=>(anything) is true
torn jolt
well that begs the question why implication works like that, but in the end thats just how its defined
fast peak
well it makes sense
I never really got why people struggle with this part of implications
torn jolt
idk, if i ask someone
"does 6 is a prime number imply 1+1=3" they would probably say no
void arch
So all propositions are implications?
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void arch
if A is the emptyset then x in A is false and the implication (false)=>(anything...
I’ve become more confused from this, if you also consider the logical negation as an implication statement
void arch
"there exists an x in empty set, so that 2+2≠5"
^
There exists an x in empty set is a false proposition, 2+2=5 is a true proposition
So false implies true
Making the overall statement true meaning that by contradiction the original statement is false
torn jolt
wait, the forall quantifier indeed gives an implication, but i think for the exists quantifier its different
fast peak
what do you mean, considering logical negation as an implication statement
glossy valveBOT
rbit ✨
void arch
what do you mean, considering logical negation as an implication statement
Is that not correct to do? You treated the initial statement as an implication though?
fast peak
yes but you can't do that with any statement. only with these specific "forall" type statements
fast peak
well thats just what these forall statements are
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spare nacelle
hello my result for this is ${2 \over {r^2}} \vec{e_r}$ can this be correct?
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b3s4d
sacred fog
context?
spare nacelle
i have to calculate this in spherical coordinates
gritty rose
Is $\Delta$ the Laplacian?
glossy valveBOT
riemann
gritty rose
Did you use this formula (1) ?
https://mathworld.wolfram.com/LaplacesEquationSphericalCoordinates.html
spare nacelle
oh wait yes i did
thats what i did: $\laplacian \vec{e_r} = {\laplacian\vec{r} \over r } + \vec{r}~\laplacian\cdot{1\over r} = {2 \over r^2 } \vec{e_r}$
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b3s4d
spare nacelle
or does this only work with divergence?
native briar
should this be c > 0 ?
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should this be c > 0 ?
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dull spade
i need help with this exercise
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i need help with this exercise
dull spade
from the answer in my book it says 193km
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dull spade
ive tried it and 3 times ive obtained answers around 132 km. i tried using cosine rule, component vectors, simple bearing solving techniques.. I Still couldn't figure out
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unreal bloom
Not sure if I used the wrong method :/. I have to find the solution of the system
shrewd hamlet
Negative on the right side
-18y=-64
Because u have to subtract 75 from both sides
11-75 is -64
That’s the right method
@unreal bloom
cold tangle
Substituting x into function where you derived it from will not work.
Wait im dumb
Ok my eyes might be broken
shrewd hamlet
Lol I was bouta say
cold tangle
Lol I was bouta say
I have no idea how i saw that he substitited x in the second equation but i did
I was even questioning why isnt he getting 0 = 0?
safe trench
Don't assume gender
unreal bloom
Oh no wonder, I feel dumb when it’s a sign issue 
wait but I get a decimal when I divide
@shrewd hamlet
unreal bloom
Oh, so I’ll have to use the decimal to plug in x for the second equation?
unreal bloom
A fraction? So -64/-18
cold tangle
32/9
unreal bloom
How did you get that
cold tangle
-64/-18 = 64/18 = 32/9
unreal bloom
I don’t understand
safe trench
32/9 is the simplest fraction form
If you were asked to simplify -64/-18 how would you do?
cold tangle
$$\frac{-64}{-18} = \frac{\frac{-64}{-2}}{\frac{-18}{-2}} = \frac{32}{9}$$
glossy valveBOT
Pluton
unreal bloom
Oh I see, I was tryna do it on the calculator. Idk how to do it manually
unreal bloom
I’m in college
safe trench
💀
cold tangle
This is elementary school math
cold tangle
Its not judgement its reality
unreal bloom
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safe acorn
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safe acorn
how to get the nth term for this
i have no idea
It's a multiple choice problem tho
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lilac valve
i have no idea
Well, remember that in a telescoping series, there are consecutive terms that cancel each other out. For example, in the first and second terms of this series, the sum would be 8arctan(k+1)-8arctan(k)+8arctan(k+2)-8arctan(k+1). Both 8arctan(k+1) terms cancel
safe acorn
Oh yes
gaunt helm
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gaunt helm
how am i supposed to prove this?
lilac valve
there are two things you should think about when you try to prove this:
- that △AWX and △AYZ share a common angle on the vertice A
- that the sum of all angles in any triangle is equal to 180 degrees
eager anchor
this isnt really a math question but I thought I'd ask to see if anyone here knew anyways
eager anchor
The language is Java
Is this a mistake or am I missing sometihng
bb is a clone of a so changes to it should not affect aa right?
neat bay
@eager anchor yeah def a Java question
Seems like clone is doing some sort of alias thing
eager anchor
oh.
so the outer list has it's own ID but the inner list namely the second one still has the same reference from aa?
I guess the first list had it's alias overwritten with a new integer array
I get it, thanks.
neat bay
Cool
torn jolt
i dont understand why are there two possible tangent lines
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i dont understand why are there two possible tangent lines
buoyant tinsel
and 5x-13 isn't one of them
it's the tangent line going through (2, -3), it never said f(x) had to be (2, -3)
torn jolt
it's the tangent line going through (2, -3), it never said f(x) had to be (2, -3...
i thought it was point
u derive the parabola and i substituted in the x coordinate
light sonnet
This is calculus, right?
torn jolt
yes
torn jolt
and 5x-13 isn't one of them
how
buoyant tinsel
5x-13 is not tangent to x^2 + x
do the steps of how you would normally find a tangent at a point, except the point this time is your unknown
and you will instead be able to substitute x and y with the given point (2, -3)
torn jolt
5x-13 is not tangent to x^2 + x
u derive it, so its 2x + 1, substitute 2 for x then u get your slope
right?
need help on 11 and 15
torn jolt
my fault
buoyant tinsel
np
buoyant tinsel
u derive it, so its 2x + 1, substitute 2 for x then u get your slope
you can check it yourself, where does 5x-13 and x^2 + x intercept and does the derivative there match
torn jolt
you can check it yourself, where does 5x-13 and x^2 + x intercept and does the d...
i mean is my process correct??
u get 2x + 1
its just graphically it looks wrong lmao
buoyant tinsel
2x+1 is the first step yes
torn jolt
2x+1 is the first step yes
thats equation for finding any slope based of x right
so isnt it right to substitute 2
buoyant tinsel
no, because again, (2, -3) does not necessarily lie on f(x)
it only has to be on the tangent line
torn jolt
no, because again, (2, -3) does not necessarily lie on f(x)
omg wtf
okk
sooo what would be the next step if u need to substitute
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winter verge
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rocky vale
,rcw
glossy valveBOT
deft zodiac
which
winter verge
7
rocky vale
I'm sure you know how to compute the area of a rectangle
Have you tried finding an expression for the area of that rectangle in particular?
winter verge
No?
rocky vale
Well, that would be the first step. That's essentially what part a is asking you to do
Do you understand how you did number 6? Your answer for that one is right but your work is missing the most important step
deft zodiac
hm but we dk the width right
winter verge
Do you understand how you did number 6? Your answer for that one is right but yo...
That's fine
winter verge
hm but we dk the width right
Don't know width, no y axis
rocky vale
hm but we dk the width right
Oh, I guess we're meant to assume that the bottom left corner of the rectangle is the origin
Otherwise it's not possible
I didn't even notice it wasn't there
winter verge
I knew it, lol
rocky vale
Yeah without knowing where the rectangle is with relation to the origin it's not possible
But the most likely thing is that the y-axis is along the left side of the rectangle, and someone just forgot to print it
winter verge
That's all I needed
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grand whale
I need Help with this problem in calc 1. I dont know what im doing.
grand whale
clarification:
I know how to do differentiate, I just cant seem to get the chain rule down with y as a prime. is there a tutor/helper that can show me or explain to me what I did wrong? also I understand that msubtan=yprime and that im creating a equation with y-y1=m(x-x1) equation*
nevermind
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dense edge
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dense edge
Is it possible to shift a vertical asymptote right?
In order for a vertical asymptote to appear, a factor must be irreducible. That leads me to believe any (x-3) can be factored by the difference of two squares
In other words (x+3) is irreducible (shift left 3), but (x-3) is reducible
light sonnet
(x-3) can't be factored as a difference of two squares
dense edge
it can't?
dense edge
(x-3) can't be factored as a difference of two squares
(sqrt(x) + sqrt(3))(sqrt(x) - sqrt(3))
eager violet
(x-3) can't be factored as a difference of two squares
(√x + √3)(√x - √3) 
light sonnet
Why the hostility
light sonnet
(sqrt(x) + sqrt(3))(sqrt(x) - sqrt(3))
Yes that is possible but normally difference of squares has integers for exponents
dense edge
so it is possible. which is what i was getting at. (x+3) is not possible
but (x-3) is possible
light sonnet
It's really rare to do difference of squares with fractional exponents
dense edge
so how do we shift right on this if the definition for having an asymptote is "cannot be factored"
eager violet
so it is possible. which is what i was getting at. (x+3) is not possible
(√x + √3i)(√x - √3i)
light sonnet
Let's rewind, what exactly are you trying to ask?
dense edge
Is it possible to shift a vertical asymptote right?
this
eager violet
(x-3) is reducible
It's not reducible because the factors aren't polynomials
light sonnet
Are you asking to shift it to the right?
dense edge
(√x + √3i)(√x - √3i)
wait, this will make (x+3)?
glossy valveBOT
light sonnet
What?
eager violet
,w expand (√x + (√3)i)(√x - (√3)i)
glossy valveBOT
dense edge
wow
light sonnet
Ah brackets
dense edge
but not using the real number system
eager violet
The factors have to be polynomials for it to be reducible
light sonnet
You can factor anything using the complex number system
dense edge
(x-3) is reducible using the real number system
eager violet
It's not because it's factors aren't polynomials
dense edge
(√x + √3)(√x - √3) both of these factors are polynomials, are they not?
light sonnet
Is it possible to shift a vertical asymptote right?
Anyways you can shift the vertical asymptote but you'll end up with a different function
dense edge
why?
eager violet
(√x + √3)(√x - √3) both of these factors are polynomials, are they not?
Polynomials have nonnegative integer powers of x
dense edge
leading term cannot be sqrt?
eager violet
√x is fractional
eager violet
leading term cannot be sqrt?
No term can be
dense edge
woah, i didn't know that
light sonnet
Which is why I stated that
Yes that is possible but normally difference of squares has integers for exponents
eager violet
Because the definition of polynomial requires nonnegative integer powers
dense edge
What about 5^-3?
eager violet
even written as 5^1/3 ?
The restriction is only on powers of x, not other numbers
eager violet
It wasn't a polynomial then
dense edge
What would it be called?
eager violet
An expression? Idk if it has a special name
dense edge
but it would still count as a function if f(x) = blah blah blah, you name it
it's just polynomial.. interesting. they need to stay pretty simple
linear polynomial being the simplest form of a polynomial
or maybe constant would be
f(x) = 1
wait maybe that's linear too
1 to the power 1 constant function
light sonnet
I am pretty sure I have seen polynomials with negative powers
If it was this, like $5x^{-2} - x^{-1}$ that's just a rational expression because $5x^{-2} = \frac{5}{x^2}$
glossy valveBOT
dldh06
dense edge
polynomials can get crazy but not too crazy.. can't have sqrts or inverse
afaik
i have seen inverse functions f^-1(x) = polynomial, but maybe that's different
eager violet
Polynomial is only referring to the powers of x. There's no reason the inverse of some function can't have natural number powers of x
gritty rose
Inverse of square root for the first nontrivial example
shrewd hamlet
Why does it say O for the series expansion
tulip marlin
by definition, polynomials only have non-negative powers
also, not everything has a name
polynomials are just so widely used that they do
vernal lagoon
Anyone good with epsilon delta proofs? Finding the delta to prove other limits is pretty confusing for me
vernal lagoon
Example:
Kinda stuck here
Should I be getting |x| alone on the left side of the epsilon inequality to get delta (which is whatever would end up on the right?)
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vernal lagoon
Reopening the above question
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Reopening the above question
vernal lagoon
Same thing 😅
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vernal lagoon
then factorize by 1/|b|
Like take 1/|b| out from both sides?
then you could get |f(x)-L| < epsilon if I'm reading that right?
glass crystal
like | something * (1/b) + somethingElse * (1/b)| = |1/b| * |something + somethingElse|
vernal lagoon
Yeah gotcha
vernal lagoon
then you could get |f(x)-L| < epsilon if I'm reading that right?
Would this be the next step then
Or like Im confused how to actually PROVE it
or get a delta value
vernal lagoon
can you get |bf(x)- bL| <something?
I think you could get that statement < (epsilon)(b)
vernal lagoon
so that would be delta
glass crystal
you almost got what you wanted
vernal lagoon
p sure
glass crystal
but its b*epsilon
and not epsilon
so just go at the start of your definition
and instead of using epsilon as your positive number
use epsilon/b
for any epsilon >0
epsilon/b >0
vernal lagoon
ohhh
vernal lagoon
yeah
glass crystal
and you use epsilon/b
and you started with
for any epsilon >0
you found your delta
vernal lagoon
and then prove it with epsilon/b with the standard kinda pattern right
or you would have to prove it for both maybe
glass crystal
for both what?
vernal lagoon
for both what?
both of the initial equations
like the if ... then ...
or would the way to do it just to start with the if and transform it into the then
glass crystal
do it like this
the trick of finding |f(x)-l|<a*epsilon
and then changing the original epsilon is very useful
vernal lagoon
the trick of finding |f(x)-l|<a*epsilon
oh so you get delta which is epsilon / |b| and then you replace that with the right side of the equation which was originally just < epsilon?
I think I see
glass crystal
delta is delta
glass crystal
its not epsilon/b
do you understand the roles of delta and epsilon in this kind of proof?
vernal lagoon
I thought u choose something to use as delta which typically has epsilon in it
vernal lagoon
do you understand the roles of delta and epsilon in this kind of proof?
epsilon is the margin of error and delta is the max distance between x and whatever it's approaching right
glass crystal
it can be like that in practical cases
but in general
epsilon delta proofs are essentially
"for epsilon as small as you want if i can get close enough (meaning distance <delta) then something that depends on epsilon holds
glass crystal
well delta is defined
because
f(x) --> L
you use the same delta
for bf(x)-->bL
elfin sinew
when writing a tangent and cotangent function from a tangent line, what would the phase shift be?
elfin sinew
how do i find the phase shift
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gilded urchin
i need help finishing this argument, its the last part
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i need help finishing this argument, its the last part
gilded urchin
i don't quite get what im supposed to do to end the argument
<@&286206848099549185>
<@&286206848099549185>
<@&286206848099549185>
brave blaze
what is the main question ?
gilded urchin
here
i already did this
@brave blaze
i just dont know how to end the argument
brave blaze
$\lvert (-1)^n n! x^{-(n+1)} \rvert \leq n!$
glossy valveBOT
Ｈｅｒｅｌｓ
brave blaze
$\lvert (-1)^n n! x^{-(n+1)} \rvert = \frac{n!}{x^{n+1}}$
glossy valveBOT
Ｈｅｒｅｌｓ
gilded urchin
why can i rewrite it like that
brave blaze
$\lvert (-1)^n \vert = 1$
glossy valveBOT
Ｈｅｒｅｌｓ
brave blaze
thats what I just did
brave blaze
idk fr
gilded urchin
oh okay
can you help me with somehting else then
something a bit easier
what is the conclusion to this
do i write that the limit is 0? or do i just write that the expression is positive for all r>=1/2
brave blaze
im not quite good with multivar calculus so i cant help you
gilded urchin
do you know somebody else in the server how might be able to help me then 🙂
@hot herald can you maybe help here if you have time?
<@&286206848099549185>
ocean horizon
my knees are absolutely weak and shaking right now
gilded urchin
hahaha
ocean horizon
my heart is palpitating
gilded urchin
they give point
and i gotta get them all
so i gotta make sure i have a proper conclusion
dont wanna get kicked out maynn XD
gilded urchin
lmao
ocean horizon
XD
first one
the limit is sqrt(6r-3)
that is the limit
that is all you need to do
gilded urchin
okay
down here
how do i go from n! to n!/x^n+1
and then i gives n!<=n! in the end right?
ocean horizon
Yes
Hold 9n
Im outside eating
Ill be back in like 45 mins and ill give u a solid answer
gilded urchin
okay okay
ocean horizon
but u should really try to understand it
marks for assignments is great but try to understand
gilded urchin
yeah, i know
i will read them through in the vacation next week
revise the stuff i already had, so im strong
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austere rampart
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onyx glen
is RQ = (3/5)b known?
austere rampart
Yes i think so
onyx glen
that sounds like you are not 100% certain of it
austere rampart
onyx glen
ok, so it is information from a previous subproblem of the same problem.
in that case,
triangles WRQ and WOP are similar with scale factor 5/3
you can use this to find the ratio of OW to RW and hence that of OW to OR
austere rampart
I did that so RW is 3/5a right?
onyx glen
how did you arrive at that?
austere rampart
If it's scale factor of 5/3, doesn't that mean i have to divide that by a? Or did i get that wrong? TT
onyx glen
you got it wrong by attempting to jump the gun
austere rampart
Then how should i do it?
onyx glen
find the ratio of OW to RW
torn jolt
Let's say our vector space is $V$ (with field of scalar $F$) and we have vectors $v_1, v_2, \dots, v_n$.
A linear combination of the vectors $v_1, v_2, \dots, v_n$ is a vector of the form
[ a_1 v_1 + a_2 v_2 + \dots + a_n v_n, ]
where $a_1, a_2, \dots, a_n$ are scalar from $F$.
The span of the vectors $v_1, v_2, \dots, v_n$ is the set of all linear combinations of the vectors:
[ \text{span}(v_1, v_2, \dots, v_n) = { a_1 v_1 + a_2 v_2 + \dots + a_n v_n: a_i \in F }. ]
The set ${v_1, v_2, \dots, v_n }$ is linearly independent if
[ a_1 v_1 + a_2 v_2 + \dots + a_n v_n = 0 \implies a_1 = a_2 = \dots = a_n = 0. ]
Vectors are linearly independent if every vector in their span has a unique representation as a linear combination, by which I mean there is only one way you can choose the scalars to get that specific vector.
A basis for your vector space $V$ is a linearly independent set $B = { b_1, b_2, \dots, b_n }$ with $\text{span}(B) = V$.
So, the fact that $\text{span}(B) = V$ means that every vector can be written as a linear combination of your basis vectors, and that the basis is linearly independent means that there is a unique way to write each vector as a linear combination of the basis vectors. So, for every $v \in V$, there is a unique choice of scalars $a_1, a_2, \dots, a_n$ such that
[ v = a_1 b_1 + \dots + a_n b_n. ]
So, because of there being a unique choice of scalars, we call those scalar "coordinates" and then coordinate vector with respect to B of v is just the column vector containing those coordinates, so
[ [v]_B = \begin{bmatrix} a_1 \ a_2 \ \dots \ a_n \end{bmatrix}. ]
oop wrong channel
glossy valveBOT
Frandom
Compile Error! Click the 
reaction for more information.
(You may edit your message to recompile.)
torn jolt
mb guys
ebon path
Thales theorem
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torn jolt
Hi how did
⁵√7,776 = ⁵√r⁵
6 = r
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Hi how did
⁵√7,776 = ⁵√r⁵
6 = r
magic nymph
what's 6^5 babe
elfin vector
Is anyone do chemistry
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Is anyone do chemistry
elfin vector
Emperical formula
What is the empirical formula if a compound is consists of 21.2% N, 6.1.% H, 24.2% S and 48.5% of O?
full marsh
I don't think this is the place to ask that
In #old-network you'll find a chemistry server. Might want to ask there.
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merry atlas
Can I ask quick questions here without a specific problem? Sorry if no but here goes:
Does a tangent to a 3rd degree function intersect the function's graph a specific number of times? Or is it just "at least once" ?
merry atlas
And is there a rule about this for higher functions too if so
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@merry atlas Has your question been resolved?
pale birch
Can I ask quick questions here without a specific problem? Sorry if no but here ...
I think you can check this by making the graph of x^3 and then making tangents at multiple points and seeing if they meet at any point on the curve other than the point where they are tangent to
but thats the lazy way
merry atlas
See, when I used the tangent's equation $y=p(x)=f(a)+f'(a)(x-a)$ I get one with two intersections. But I can easily imagine a tangent (graphically) that only has one. But is that not a tangent then because of the degree and only one point?
glossy valveBOT
Kienai
merry atlas
Specifically, I am wondering if I can claim the tangent to this function (the one I found is blue) has at least one point where it intersects, or if I should claim it has two (before finding it)
the coordinate (0,1) is part of the requirement
pale birch
Interesting
aren't tangents found by taking derivative of the function of x with respect to y
so shouldnt the tangent at a point be found by the equation 6x^2 - 3
@merry atlas
merry atlas
yeah, like the equation says the derivative is part of it
$y=p(x)\iff p(x)=f(1)+f'(1)(x-1)\iff p(x)=(2-3+5)+(6-3)(x-1)\iff p(x)=3x+1$
glossy valveBOT
Kienai
merry atlas
Sorry wrong order, this part is first $1=(2a^3-3a+5)+(6a^2-3)(-a)\iff 4a^3-4=0 \iff a^3=1 \iff a=1$
glossy valveBOT
Kienai
merry atlas
But I don't know if that means I am right about claiming it has at least one point where it intersects, or should say two
merry atlas
Sorry, maybe the whole thing provides some more context... hang on
pale birch
I only know the basics of tangents to curves
merry atlas
It is in Swedish but it's what I've said before, going into excruiciating detail in parts. Between (1) and (2) I claim the tangent p(x) intersects f(x) at least once and that's the part I'm not sure if I should claim at least twice or what
pale birch
The tangent to a curve can intersect the curve at infinitely many points.04-Sept-2021
got this from google
merry atlas
Tried to google and got that but nothing about "at least" many times :/
pale birch
it says "can" so it depends upon the curve i think
and the tangent taken
at a point
depending on the point
So i think you should write Twice
since thats what youre getting
merry atlas
yeah, my feeling is "at least once" is right but I dunno
Since the one time is the part where it is a tangent, and the other point is just crossing the graph because this third degree function extends toward infinity
merry atlas
Yeah, but I claim this before I know what the tangent is
In order to figure out how to find it
It is the assumption behind p(x)=f(x)
pale birch
is there anyway to translate all that into english
that'd help me in understanding better lol
merry atlas
Haha, okay okay hang on
pale birch
or you could wait for an helper
merry atlas
I can write it out
pale birch
just write out the parts needed
merry atlas
Determine tangent $p(x)$ till $f(x)=2x^3-3x+5$ going through coordinate $P=(0,1)$. Observe the tangent's coordinates can be written as $p(0)=1$
\bigskip
Assumption about one intersection here, p(x)=f(x)
\bigskip
We write f(x) as f(a) until we find x=a to let us solve p(x)=f(x) and use the tangents equation to do it:
\newline $y=p(x)=f(a)+f'(a)(x-a)$ (3)
\bigskip
We find $f'(a)=6x^2-3$ and rewrite (3) as:
\newline $1=f(a)+(f'(a))(0-a) \iff 1=(2a^3-3a+5)+(6a^2-3)(-a)$
\bigskip
Simplify:
\newline $1=(2a^3-3a+5)+(6a^2-3)(-a)\iff 4a^3-4=0 \iff a^3=1 \iff a=1$
\bigskip
a=1 gives us
\newline $y=p(x)\iff p(x)=f(1)+f'(1)(x-1)\iff p(x)=(2-3+5)+(6-3)(x-1)\iff p(x)=3x+1$
That didn't like my spacing at all
pale birch
reading
merry atlas
Now the bot cleanup is readable
glossy valveBOT
Kienai
merry atlas
Sorry, it kind of says multiply by (3) but the (3) is the equation's number, sorry
pale birch
I figured, its okay
I am just now finding out the tangents equation
aren't tangents found using dy/dx for a curve
merry atlas
You can yeah, that is baked in here
So at the point where f(x)=p(x) their derivatives are equal
pale birch
Can I ask quick questions here without a specific problem? Sorry if no but here ...
well then the answer to your intial question is, yes it intersects the function more than 1 times
also sorry for the late replies ,I was studying for my chemistry test tmrw
pale birch
Thank you
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pulsar cradle
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pulsar cradle
Is this correct so far
vast fossil
Derivative of x^2y isn't 2x dy/dx
vast fossil
Yes
pulsar cradle
So the answer would be (-6x+2xy)/(x^2+4y^3)
Also i can rewrite that as
(2xy-6x)/(4y^3-x^2)
and does the 2: * dy/dx just turn into dy/dx
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simple latch
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@simple latch Has your question been resolved?
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@simple latch Has your question been resolved?
willow badger
Close your ticket 😠 @simple latch
spare nacelle
@simple latch if you know a) and b) why is g) so hard?
simple latch
Close your ticket 😠 <@1024469482606972938>
how
willow badger
“.close”
simple latch
.close
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spare nacelle
wait did u solve it?
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maiden panther
quick question: i get -2/-3 as the slope but that isnt listed
onyx glen
how are you getting -2/-3?
maiden panther
plugging in the y1 y2 x1 and x2
burnt breach
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burnt breach
how?
dull void
I assume you would go about this by
writing sin(3theta) as sin(theta + 2theta)
and grind out a bunch of identities
if you know sin(a + b) identity and the double angle identities it should be doable
dull void
uh
easiest is probably writing them as sin(3theta)/cos(3theta)
and you calculated both
but honestly that looks gruesome
right
there is a tan(a + b) identity too
it's probably in ur textbook somewhere
do tan(theta + 2theta) like the others
I think all 4 of these you use theta + 2theta trick
burnt breach
dull void
and use double angle and addition identities
right
you can also use these to recover double angle identities, by writing sin(2theta) as sin(theta + theta) and similar for the others
so technically, just by these addition formulas all those should be doable
just very algebra heavy
dull void
np
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torn jolt
i need help, i have tried taking 2019^x as x but didnt get Any answer ......
tiny merlin
I need help. Can someone help me pls🙏
tardy quiver
Cot c = b^2 + a^2 - c^2 / abc
Cot a = b^2 - a^2 + c^2 / abc
Cot b = a^2 + c^2 - b^2/ abc
tardy quiver
cot c/cot a + cot b
tiny merlin
i get ans, thx
tardy quiver
is
tardy quiver
i get ans, thx
ou ok
tardy quiver
is
a^2 + b^2 - c^2/a^2 + b^2 + 2c^2 - a^2 - b^2
this gives
2c^2 / (17/9c^2 - c^2)
which gives 4/9
tiny merlin
i think (19/9c^2-c^2)/2c^2=5/9
tardy quiver
i got 17/9 instead of 19/9
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tiny merlin
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torn jolt
Could anyone give a small hint?
fast peak
so given a sequence $\mu(A_n)$, can you get some coefficients so that $\sum c_n \mu(A_n)$ converges?
glossy valveBOT
Denascite
torn jolt
Weird... I had actually went down this line of reasoning and for some reason concluded it not to work.
But ye, that works. Thanks.
.close
willow fern
Just wanna know if this is right (the whole thing could be wrong I'm just guessing)
tardy quiver
how come 300+1 = 901
gaunt lynx
wow
quick maths
let’s go
not teasing u btw bro
I did this mistake many times lmao
willow fern
Lol 301*
gaunt lynx
yeye
let’s go
second part
also nah
u know intially
at the beginning
u had
how much bacteria
301 right
what’s 3x that
willow fern
That's it
willow fern
903
willow fern
Ohhh
gaunt lynx
$903= 300 + 5^t$
willow fern
Thanksss
glossy valveBOT
Springsskateboard
willow fern
.close
pallid sentinel
I have a question on related rates, but I take maths in Arabic so I’m going to try to translate it to English with my best knowledge on maths terms.
Basically the questions says that there is a candle infront of a round disc which has a radius of 10cm, the disc is 60cm away from a wall. The candle was moved towards the disc with a rate of 5cm/s. the question asks me to calculate the rate of change in the radius of the circle (the shadow) on the wall when the candle is 20cm away from the disc.
pallid sentinel
abit stuck on where to start but i think it should be done with using similar triangles?
calm osprey
sorry but the question still feels abit ambiguous. would you mind clarifying more? (drawing a picture might help)
calm osprey
This is how I understood the question
okay so something like this then (this is a 2d projection from the side cuz i'm bad at drawing)
calm osprey
yeah okay then we just need to in cooperated the fact the candle move
pallid sentinel
I took y/10 = (60+x)/x
calm osprey
yeah exactly problem is that does not describe the "candle moves a 5m/s towards the disc" criteria so we need to work that in. note sure how to do this fully either gotta figure this out
calm osprey
ohh wait is this a differential? ohh god i'm not awake enough for this
calm osprey
well rate of change and differentials are different things, but it might just be rate of change things.