the missing pi
#help-23
lean otter
ok
lemme see if i can use latex
$\ y = 5.465 cos(\frac{pi}{6.5}x - \frac{pi}{1.47}) + 7.485$
that did not work
lean thorn
there's a space at the end there, make sure there's not space around the $ 🙂
hmmm
ok seems like we're wrong in our calculation somewhere
lean otter
ok seems like we're wrong in our calculation somewhere
where
lean thorn
looking one moment
flat frigateBOT
Cy∀πid∑
lean thorn
oh ok so the book has that as the answer?
lean otter
oh ok so the book has that as the answer?
much appreciated m8
lean thorn
OH WHOOPS LOL
lean otter
OH WHOOPS LOL
what
lean otter
XDDD
lean thorn
yeah no problem! Glad you got it figured out
do the steps make sense on how to solve this?
lean otter
iT HaPpEnS tO ThE BeSt Of Us
lean otter
do the steps make sense on how to solve this?
ye i get it
step 1: LOOK AT THE FUCKING TBALE PROPERLY
no but fr i do
thx my guy
||also i sent you a friend req because you seem like a really chill person||
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topaz scarab
I've proven that the sequence in the denominator goes to infinity, but now I'm unsure how to proceed. I'm trying to use the Stolz-Cesaro theorem.
safe radishBOT
I've proven that the sequence in the denominator goes to infinity, but now I'm u...
quasi bison
hold on
ok so stolz cesaro is like the discrete analogue of l'hop i see
i think you'll have a bad time trying to apply this here. are you explicitly required to?
topaz scarab
No, it's just the only idea I had.
But I am indeed having a hard time trying to apply it. i believe I should turn it into something that'll give me e.
topaz scarab
We haven't seen this yet.
quasi bison
💀
topaz scarab
The problems we are dealing with seem to be related to Stolz, that's why I tried using that.
drowsy karma
The problems we are dealing with seem to be related to Stolz, that's why I tried...
This is a very nice limit, there are at least several ways to calculate it
topaz scarab
$\lim_{n \to +\infty} \frac{1}{\sqrt[n+1]{(n+1)!} - \sqrt[n]{n!}}$
flat frigateBOT
Oppenjaimer
topaz scarab
This is all I have so far.
drowsy karma
that is bad approach
if you want to use stolz
you need to work on modoficaion
of yoru sequence
otherwise you dig in troubles
topaz scarab
How would I go about modifying this? In a way that I get (1+1/n)^n or something similar, but where n is a sequence that goes to infinity?
drowsy karma
think ab your nominator n
and make it go udner radical
$\sqrt[n]{\frac{n^{n}}{n!}}$
flat frigateBOT
Joanna Angel
topaz scarab
Right.
topaz scarab
Wait, in this case I can use the root criterion too, right?
drowsy karma
yes too that is another approach and more elementary
there is a nice lemma:
and in soem way you told ab it now
that
let me praphrase it
If d'Alemebr then Cauchy
if you know what i mean
topaz scarab
d'Alembert is what I was thinking of using with that a^n/n! inside the n-th root, though I don't see what Cauchy has to do here.
drowsy karma
$\lim_{n \to \infty } \frac{a_{n+1}}{a_{n}}=q\Rightarrow \lim_{n \to \infty }\sqrt[n]{a_{n}}=q,\\text{ where}\text{ }\text{ }a_{n}>0$
flat frigateBOT
Joanna Angel
drowsy karma
this way is a way more elementary than stolz
topaz scarab
Yes, this is what I'd like to apply. But what does Cauchy have to do with it?
drowsy karma
implication works only in one direciton
topaz scarab
Ohhh I got it now.
drowsy karma
🙂
topaz scarab
$\frac{(n+1)^n}{n^n}$ tends to e, and so does the original sequence.
flat frigateBOT
Oppenjaimer
topaz scarab
Thank you so much.
drowsy karma
yvw 🙂
topaz scarab
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sour drift
I can’t figure out how the final answer was obtained. Does the bracket in the second last line not integrate to exp(x)cos(x) - exp(x)cos(x) = 0?
safe radishBOT
I can’t figure out how the final answer was obtained. Does the bracket in the se...
sour drift
Because from the first integration the answer is 0 so a second integration is done, why don’t I keep integrating infinitely since it always ends up = 0?
potent seal
I can’t figure out how the final answer was obtained. Does the bracket in the se...
After the first time, shouldn't it be
$e^x {(-cosx)} - \int e^x{(-cosx)} dx$?
flat frigateBOT
Lorentz
halcyon carbon
Another day another tabular method W
sour drift
Well I got sin(x)exp(x) - int exp(x)cos(x)
potent seal
$\int fg' = fg - \int f'g$
flat frigateBOT
Lorentz
sour drift
Ye
sour drift
Well in my calculation g’ was exp(x) as I was told to have that term be the simpler one to integrate since it stays the same
potent seal
Oh
halcyon carbon
just learn tabular method
sour drift
Does this equation work out using tabular method?
potent seal
Well in my calculation g’ was exp(x) as I was told to have that term be the simp...
Ok yeah it's correct then, I thought you took sin x
halcyon carbon
Its the same thing but a lot easier to 'visualise'
potent seal
I just don’t know how the did the last line
They got the same integral on left and right sides
sour drift
I keep getting 0 because the exp(x)cos(x) cancel each other out
So I let it all equal the original integral?
I get u
potent seal
I can’t figure out how the final answer was obtained. Does the bracket in the se...
$\int e^x sinx dx= e^x sinx - {(e^x cosx + \int e^x sinx dx)}$
flat frigateBOT
Lorentz
potent seal
Now let the part on the left be u(to make it simpler)
sour drift
So the point is to get the same integral again to bring it back over to the other side?
potent seal
So the point is to get the same integral again to bring it back over to the othe...
Yess
potent seal
F
sour drift
I’ll research the tabular method anyway
potent seal
Me too
sour drift
I know that’s what u were trying
halcyon carbon
damn i dont have the package installed
sour drift
Thanks guys
halcyon carbon
yeah just look it up
safe radishBOT
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lean otter
why a , c false ?
lean otter
any one got idea ?
granite idol
you could try a physics server if you get no help here: #old-network
lean otter
thank u
hard crest
think of two charges, positive and negative, both inside the surface
hard crest
net flux across the surface should be 0, right?
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@lean otter Has your question been resolved?
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marsh raptor
Hello, I am tryna prove the following -> If a circle A cuts two circles orthogonally then the radicle axis of the two circles passes through the centre of the circle A
safe radishBOT
Hello, I am tryna prove the following -> If a circle A cuts two circles orthogon...
marsh raptor
I have no clue how to proceed with this question
proud belfry
Neither do i good luck man
marsh raptor
Neither do i good luck man
thanks ig
nvm i figured it out, the tangent of the circles will be normal to circle A so both will be equal to radius hence will be equal and lie on radicle axis
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buoyant pond
@desert juniper sorry i was afk thanks but how do you know that x^4 is increasing slower than e^x?
safe radishBOT
<@189916659417481216> sorry i was afk thanks but how do you know that x^4 is inc...
fleet condor
<@189916659417481216> sorry i was afk thanks but how do you know that x^4 is inc...
$x^n$ always increases slower than $n^x$ for $n>1$
flat frigateBOT
desert juniper
that's only on limiting behaviour, bunny
fleet condor
oh so true
buoyant pond
so this explains how x^6 has 2 solutions and x^2 has 1 with e^x?
desert juniper
but yes. for any n>1, x>1, there's a point where at which n^x is higher AND grows faster than x^n
buoyant pond
but yes. for any n>1, x>1, there's a point where at which n^x is higher AND grow...
I c
desert juniper
so this explains how x^6 has 2 solutions and x^2 has 1 with e^x?
you mean the equalities x^6=e^x and x^2=e^x?
buoyant pond
Ye
desert juniper
that doesnt sound right, x^6=e^x should also have 3 solutions
buoyant pond
Because 2 is smaller than e
buoyant pond
that doesnt sound right, x^6=e^x should also have 3 solutions
Oh
but it shows 2
And for every even power greater than 6
Has 2
desert juniper
cuz the third is more to the right than for x^4
and then you should have 2 for every odd power n>3, since you dont have the one in the negatives
buoyant pond
Ye its like they increasing more steeply after 6
desert juniper
x^2 only has one because while it has the one in negative x, it doesnt grow fast enough early to catch up with e^x
desert juniper
anyways, most of this is not really formal enough for you to use in proofs and stuff. We simply know the functions because we've seen them a lot of times
buoyant pond
ohk
desert juniper
for example, if instead of x^2 you got 20x^2, it does have two
buoyant pond
Ah ye that too a thing
desert juniper
about the "differentiating way for knowing this"
You can differentiate both functions, and you will see for which values one grows faster than the other by graphing both derivatives
e^x has the exact same graph for every derivative, while the monomials have decreasing powers
usually that means that between 0 and 1 they grow faster the lower the exponent, but from x=1 onwards they grow faster the higher the exponent
buoyant pond
So x⁴ would have a lower slope than e^x after any x greater than 1?
desert juniper
no, x^4 has a lower slope after about x=7.38
buoyant pond
Ahh
desert juniper
if you graph x^4 and e^x, you will see that after x=7.38 roughly, e^x starts to catch up
buoyant pond
Still tho i cant graph e⁷ manually
desert juniper
and since that's the point where e^x starts growing faster than x^4, it catches up pretty damn quick
buoyant pond
Hmm yeah that explains
buoyant pond
Also i can use this right
desert juniper
you do get taught the kind of rules that bunny stated in limits
desert juniper
Also i can use this right
that's only when computing limits of functions when x tends to +-inf
but it's very useful for quick computing of them, or when you're analyzing computer algorithms for efficiency
desert juniper
algorithms are usually studied in their computation complexity, depending on size of input.
An algorithm that grows as fast as the factorial of the input usually means "get a better algorithm"
an algorithm that grows as a^n with n being size of input usually also means "get a better algorithm"
because even for very modest input sizes you already got absurdly high number of operations
safe radishBOT
@buoyant pond Has your question been resolved?
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unreal kindle
hello, how do i show that $int(\mathring{A}) = \mathring{A}$ ?
safe radishBOT
hello, how do i show that $int(\mathring{A}) = \mathring{A}$ ?
flat frigateBOT
lilisworld
unreal kindle
<@&286206848099549185>
remote hamlet
Try to show that the internal part of an open set is equal to the set itself
remote hamlet
Both stand for "interior of" I assume
unreal kindle
Both stand for "interior of" I assume
yes
halcyon carbon
interior of (interior of A) = interior of A?
unreal kindle
interior of (interior of A) = interior of A?
yes
halcyon carbon
boreal hornet
so basically a set is closed if its complement is open, by that definition set has border and int
lets say that that setis a, then $A=int(A)+(\mathring{A})$
flat frigateBOT
yariguh
unreal kindle
ok then
unreal kindle
lets say that that setis a, then $A=int(A)+(\mathring{A})$
what does this mean
boreal hornet
so by that definition A with math ring also contains border and its int
unreal kindle
int(A) is $\mathring{A}$
flat frigateBOT
lilisworld
unreal kindle
not the border
boreal hornet
**lilisworld**
thats not what it says here
boreal hornet
no its not the same
remote hamlet
- For any set A, int(A) is an open set
- If U is an open set, int(U)=U
Dividing the problem in these two steps is a nice way to address it
merry sleet
can the fact that $\mathring{A}$ is open be assumed for this proof?
flat frigateBOT
Benjamin
unreal kindle
1) For any set A, int(A) is an open set
2) If U is an open set, int(U)=U
Dividi...
okk
unreal kindle
can the fact that $\mathring{A}$ is open be assumed for this proof?
yes i think so
merry sleet
if you only think so
do the proof
to be sure of it
:p
whats the definition of the interior of a set?
boreal hornet
i was thinking you approach it by contradiction where if A mathring had int that would mean that that set contains elemtns of either complement or intA
unreal kindle
whats the definition of the interior of a set?
$\forall x \in \text{int}(A), \exists \varepsilon > 0 : B_\varepsilon(x) \subseteq A$
flat frigateBOT
lilisworld
boreal hornet
a set that has a supremum but not a maximum
unreal kindle
i'm going to do the proof and i come back
boreal hornet
so you mean to tell that set (0,1) has maximum and minimum on R?
merry sleet
a set that has a supremum but not a maximum
you didnt talk about mins
but what about in C?
whats an open set
merry sleet
i'm going to do the proof and i come back
you can stay here the proof shouldnt be long
unreal kindle
ok
merry sleet
so whats your definition of an open set?
unreal kindle
yes i'll imediately try to write the proof because i don't want to type the definitions separately
unreal kindle
wait, so $\mathring{A}$ is an open set if $\forall a \in \mathring{A}, \exists \varepsilon > 0, \forall x \in \mathbb{R}^n, | x -a| < \varepsilon \implies x \in \mathring{A}$ right?
flat frigateBOT
lilisworld
unreal kindle
@merry sleet
unreal kindle
it is the set $B(a, \varepsilon ) = { x \in \mathbb{R}^n, | x - a| < \varepsilon }$
if it's open
flat frigateBOT
lilisworld
unreal kindle
@merry sleet
merry sleet
yeah ok
merry sleet
**lilisworld**
can you translate this in terms of ball and then make the link with the defintion of the interior you gave?
unreal kindle
can you translate this in terms of ball and then make the link with the defintio...
the definition of the open set?
merry sleet
i mean you got to prove the interior is an open
you gave me a definition of the inerior involving a ball
but your definition of an open set didnt use it
so make the link using the definition
unreal kindle
so $\mathring{A}$ is an open set if $\forall a \in \mathring{A}, \exists \varepsilon > 0, B(a, \varepsilon) \subseteq \mathring{A}$?
flat frigateBOT
lilisworld
unreal kindle
@merry sleet
merry sleet
yes
unreal kindle
so the only difference is the $\mathring{A}$ in the second definition instead of A and since $\mathring{A} \subseteq A$, $\mathring{A}$ is an open set
flat frigateBOT
lilisworld
merry sleet
yeah so essentially the issue is that the ball could not be all inside the interior
so take a ball even smaller
like take the definition of the interior
get the epsilon
then talk about the epsilon/2
radius ball
and prove all elements in this ball are inside the interior
using the triangle inequality
unreal kindle
ok
safe radishBOT
@unreal kindle Has your question been resolved?
glossy fjord
can someone help evaluate the limit
safe radishBOT
can someone help evaluate the limit
drifting mountain
!show
safe radishBOT
Show your work, and if possible, explain where you are stuck.
thin bridge
show your attempt at that
drowsy karma
i tried factoring by grouping
it is enough to write your denomnator as (x - 7)(ax + b), and you just observe what a and be must be equal to, to make it equal to your trinomial
glossy fjord
it is enough to write your denomnator as (x - 7)(ax + b), and you just observe...
why did you choose (x-7) out of all numbers?
thin bridge
factor theorem
glossy fjord
are there any other ways to go about factoring without that theorm?
thin bridge
ac method/grouping, find the pair that satisfies the conditions
drowsy karma
why did you choose (x-7) out of all numbers?
becasue x = 7 is root ot trinomial
drowsy karma
why did you choose (x-7) out of all numbers?
so you found it ?
drowsy karma
(x-7)(3x+4)
glossy fjord
yeah
glossy fjord
so what next ?
drowsy karma
sometimes grouping is fast too, but the leading coefficient, if not 1, complicates
what next ?
then you can see
(x-7)
in both plaes
glossy fjord
they cancle
drowsy karma
in nominator
yesyes!
so now you have nice expresison
and you can plug in x = 7, since
glossy fjord
(x+3)/(3x+4)
drowsy karma
you compute the lkmit when x approaches 7
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steep summit
safe radishBOT
steep summit
i like made the triangle and then got sin u.. then idk how to get it to like sin u/2
steep summit
isnt that double angle formula
gentle herald
sin 2x =2 sin x cos x
U don't need this
gentle herald
isnt that double angle formula
Yes
steep summit
but it says to use half angle
gentle herald
Replace x by x/2
U have half angle formulas
steep summit
oh so like sin 2x/2
gentle herald
And cos 2x=2cos²x-1=1-2sin²x
cos x = 2cos²(x/2) -1 =1-2sin²(x/2)
steep summit
ic
gentle herald
Btw u have to know the sign of sin (x/2) and cos (x/2) from that diagram only
Formula won't give u that
steep summit
liek the +-?
gentle herald
Yes
steep summit
so i do it liek this? this feels wrong
earnest rapids
Yes
Don't you need sin u/2 ?
Replace u by u/2 and then it remains
Sin u/2 = √{(1-cos (2u/2))/2}
steep summit
yes
earnest rapids
And you have cosu
earnest rapids
xD I didn't equate it
Look at general form
Sin Θ = √{(1-cos 2Θ)/2}
Now here I kept Θ= u/2
earnest rapids
u is angle right? 💀
steep summit
yeah
earnest rapids
what is u though
Then it's angle
earnest rapids
xD I didn't equate it
Look at general form
Sin Θ = √{(1-cos 2Θ)/2}
Yes that is when Θ=u in this
Since you wanted half angle just Θ=u/2
steep summit
why isnt it 2u/2
earnest rapids
Who isn't 2u/2?
earnest rapids
How in the world you got 1/13 💀
steep summit
sqrt(1/169)
earnest rapids
169?
steep summit
yeah
earnest rapids
Dear 13*2 = 26 not 169
earnest rapids
Why -ve?
steep summit
what
earnest rapids
u ranges from 3pi/2 to 2pi
So u/2 ranges from 3pi/4 to pi
And sin is positive in that interval
steep summit
-ve?
earnest rapids
Negative
earnest rapids
Yes
steep summit
wtv its called
steep summit
3pi/4 to pi is so weird
earnest rapids
Its math
earnest rapids
Indeed
steep summit
yay
ruby edge
How do I do b?
safe radishBOT
How do I do b?
ruby edge
This was my working out, I used sum and product of roots but it got me the wrong answer
safe radishBOT
@ruby edge Has your question been resolved?
ruby edge
Oh wow of course, thanks a lot
solar bloom
safe radishBOT
solar bloom
i need help with finding the intervals for concave up and concave down
I already found the inflection point
but i must have gotten the wrong second derevative because when i plug it in i get the intervals of concave up as -infnity,0 and concave down as 0,infinity
any help is greatly appreciated
<@&286206848099549185>
safe radishBOT
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solar bloom
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frigid terrace
safe radishBOT
odd zenith
cos theta is equal to x, sin theta is equal to y
if the question is asking in which quadrant is cos negative and sin is positive etc.. just find which quadrant corresponds to a negative x and a positive y
frigid terrace
if the question is asking in which quadrant is cos negative and sin is positive ...
how abt tan??
frigid terrace
if the question is asking in which quadrant is cos negative and sin is positive ...
(btw thx is so super helpful)
frigid terrace
so tan positive, cos negative
sin / - cos = - tan??
quadrant 3 then right???
idk
odd zenith
hmm yeah the math is correct correct but
you wanna figure out basically
which one is negative
sin or cos
in the question you see cos is positive
so sin must be the negative one
so u can make -tan
frigid terrace
mb i misread that one
odd zenith
np!
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mortal thicket
how do i integrate $\frac {3(2\cos x - \sin x)}{2\sin x + \cos x}$
flat frigateBOT
Big Chicken
forest gust
how do i integrate $\frac {3(2\cos x - \sin x)}{2\sin x + \cos x}$
u sub
mortal thicket
for what
forest gust
?
forest gust
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quasi sentinel
g(x) isnt differentiable at x=a
because of an absolute value
lets say (f'(g(x))g'(x))' is defined in all x
that equals to f''(g(x))*(g'(x))² + f'(g(x))g''(x)
is f''(g(a)) = 0 because of the chain rule or does it not matter because its (g'(x))²
safe radishBOT
g(x) isnt differentiable at x=a
because of an absolute value
lets say (f'(g(x))g...
peak estuary
how can (f'(g(x)) g'(x))' be defined for all x, if g'(x) is not defined for x=a
safe radishBOT
@quasi sentinel Has your question been resolved?
light dagger
Find the volume of the given combined solid:
safe radishBOT
Find the volume of the given combined solid:
light dagger
volume is 1/3 π r² ( h₁ + h₂ )
light dagger
its a cone and a cone combined
merry sleet
it seems stuff has been erased
so its just two cones with the same base and different heights?
light dagger
yes
merry sleet
ok
light dagger
it seems stuff has been erased
just the ABCDs are eraased
light dagger
bro which is which i dont get'
light dagger
14 is the diameter ? so radius is 7?
merry sleet
yes
light dagger
22.5 is h1 and 22.5 is h2????
hearty egret
14 is the diameter ? so radius is 7?
if you have to ask that, maybe the problem is too advanced for you
merry sleet
so just replzce that
light dagger
if you have to ask that, maybe the problem is too advanced for you
i was just confirming
light dagger
im kid lol
merry sleet
but we know h1+h2 is 45
did you get this tseteg?
hearty egret
i was just confirming
have you wanted a confirm for diameter=2 times radius ? ...
lean otter
Diameter of a circle is 2 times of its radius
merry sleet
i think everyone got that
merry sleet
help me with the process
you havr h1+h2 in your formula , you know what h1+h2 is so just replace
light dagger
Diameter of a circle is 2 times of its radius
the problem is not the diameter
merry sleet
what dont you understand in what i said? or why are you not answering?
lean otter
Find the volume of the given combined solid:
There are 2 cones here right?
merry sleet
.close
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quasi sentinel
how can (f'(g(x)) g'(x))' be defined for all x, if g'(x) is not defined for x=a
does it not work when f'(g(a)) = 0
safe radishBOT
does it not work when f'(g(a)) = 0
@quasi sentinel Has your question been resolved?
quasi sentinel
.close
lean otter
hi
lean otter
sorry I’m looking for the pic thats why I start it with something
if the arabic is too distracting
wild copper
!status
safe radishBOT
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 I don’t know how to solve it all like the steps etc. It’s a life problem and I don’t know how to do those because I’m only good at outright problems I practiced on and this is new to me
plug it in how
I did the first question but I don’t know if it’s correct
but the second question I don’t even know
here let me show u my answer
second one just ignore the stuff I did bc I didn’t know what to do I just kept trying and nothing worked
if the first one is correct then I don’t need help on that but I do need help on the second one
wild copper
Don't post it in video, please?
lean otter
okay wait I’ll try
wild copper
For the second question, You know that L(I) = 60. Now, get the definition of L(I) and solve for I
lean otter
do you understand how I solved it
wild copper
,rccw
flat frigateBOT
lean otter
For the second question, You know that L(I) = 60. Now, get the definition of L(I...
oh okay lemme try
wild copper
Seems legit, but maybe it's better to put in more detail step-by-step
lean otter
I did it look
is this right so far
I don’t know what to do next
wait I got ot
right?
because its decimal so we make it subtraction instead
lean otter
Don't put log on the numerator of a fraction like that
what do you mean
wild copper
I mean write it like this:
$$\frac{60}{10} = \frac{10}{10}\log(\frac{I}{I_0})$$
flat frigateBOT
Xwtek
wild copper
Probably
lean otter
okay just did it
then I did this
the final step
is that correct
wait which exercise are you trying to solve, you sent us 2 pages
lean otter
wait which exercise are you trying to solve, you sent us 2 pages
theyre the same one just has arabic in it so it might distract non arabic speakers
wild copper
That's the same exercise, just that one doesn't have arabic text
lean otter
what do I do after I made the decimal into a subtraction
wild copper
$I_0$ Is defined right in the question, right?
flat frigateBOT
Xwtek
lean otter
$I_0$ Is defined right in the question, right?
I don’t know what that means sorry
just plug values and logarithmic algebra
I followed the natural logarithm thing where you make decimal into subtraction
I don’t know where to go from there
this is my last step
the one at the bottom
wild copper
Just plug the $I_0$
flat frigateBOT
Xwtek
lean otter
you mean replace the value of it?
to the one in the question?
I don’t know what plug is in arabic in terms of math
replace
so I make I_0 into 10^-12
wild copper
Yes
lean otter
and the Log I
what do I do with it
it is given in the problem question
so far is this good?
now I replace Log I with 60 right?
wild copper
,rcw
flat frigateBOT
wild copper
Yes
lean otter
then I move 60 to behind the equal right? so its 6-60 = Log 10^-12
lean otter
huh
wild copper
You are finding I
lean otter
yes
wild copper
Definition of log function.
wild copper
I mean, if you don't know how to apply a function on a constant, you need to review algebra.
lean otter
I don’t know what it means because I don’t speak English fluently all my work is in Arabic
I came here for help not so you can act conceited because I didn’t know how to solve something. this is a server to help people in math right? I just asked you to show me an example so I might remember it or know the meaning since like I said it’s not in my first language
but thats okay I’ll just ask somewhere else
wild copper
First of all, it's not acceptable to ask every step of the solution. It's basically a way to skirt around no solutions rule. You are expected to show your effort to solve the problem.
lean otter
First of all, it's not acceptable to ask every step of the solution. It's basica...
I want to learn how to solve it so in other equations I would remember the way to solve it and solve them
this isn’t a homework or a test my finals are tomorrow this was on one of the papers they gave us to get us ready for the finals
how would I know how to solve something if I never learned the steps?
wild copper
Math is like puzzle. The point of math is to discover the steps yourself, maybe with hints. If I give you the whole steps to solve the problem, you are not learning math.
lean otter
Math is like puzzle. The point of math is to discover the steps yourself, maybe ...
you’re right I should somehow find the way to solve something I have never seen before
<@&286206848099549185>
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grand schooner
im sutck on this question and would like some help with just working it out
unreal kindle
hello, how do show that $\bar{\bar{A}} = \bar{A}$
safe radishBOT
hello, how do show that $\bar{\bar{A}} = \bar{A}$
flat frigateBOT
lilisworld
unreal kindle
closure of A
peak estuary
which you defined how?
unreal kindle
x is in $\bar{A}$ if d(x, A) = 0
flat frigateBOT
lilisworld
peak estuary
have you already proved that the closure is closed?
unreal kindle
no, i need to prove that
i dont know how to do it
if i take y in $\mathbb{R}^n \textbackslash A$, i need to show that $\exists r>0, B(y, r) \subset \mathbb{R}^n \textbackslash A$ ?
flat frigateBOT
lilisworld
unreal kindle
@peak estuary
steep lily
that would show that A is closed, not the closure
unreal kindle
wouldn't that show that $\mathbb{R}^n \textbackslash A$ is open?
flat frigateBOT
lilisworld
steep lily
yes, which is equivalent to A being closed
unreal kindle
and the closure is closed if its complement is open
so it should be
if i take y in $\mathbb{R}^n \textbackslash \bar{A}$, i need to show that $\exists r>0, B(y, r) \subset \mathbb{R}^n \textbackslash \bar{A}$ ?
flat frigateBOT
lilisworld
unreal kindle
@steep lily
steep lily
yes
(hold on I'm trying to figure out what denascite was going to do with the fact closure is closed, or is that your actual question and not that closure of closure is just the closure?)
unreal kindle
he just meant that if i show that the closure is closed, then it will imply that the closure of the closure is the closure
so yeah i need to do that first i think
not sure if i also need to prove that if a set is closed then its closure is equal to the set itself
steep lily
he just meant that if i show that the closure is closed, then it will imply that...
you (probably) do, because I see no way of showing this implication without that
unreal kindle
ok
so if i want to prove that $\bar{A}$ is closed, i take y in its complement and then? how do i prove that the ball exists?
flat frigateBOT
lilisworld
steep lily
what does y being in the complement mean
unreal kindle
it means that y is not in bar(A)
steep lily
do you have another way of saying that?
steep lily
yes
unreal kindle
so we know that d(y, A) > 0, so forall a in A, || y - a || > 0?
steep lily
We in fact know forall a in A, || y - a || > d(y,A) > 0
unreal kindle
isnt d(y, a) the same as ||y - a||?
steep lily
d(y,a) yes, but not d(y,A)
unreal kindle
wait, is d(y, A) a set?
steep lily
no, d(y,A) is also a non-negative number
it's (idk your definition but it should be equivalent) the largest lower bound for d(y,a), as a ranges over A
steep lily
well we would like it to be
but all we know is that B(y,r) is disjoint from A, not necessarily from the closure of A
it does work, you just have to prove it
or you could make the r smaller, say d(y,A)/2, but you would still need to show it's disjoint from the closure (possibly easier idk)
unreal kindle
sorry im lost, B(y, r) is the set of all x in R^n that verify || x - y || < r, i dont see the correlation with what we just did
like where is the a?
steep lily
we're looking for $r>0$ such that $B(y,r)\subseteq\bR^n\setminus\overline{A}$. We've established that for $r=d(y,A)>0$, we have $B(y,r)\subseteq\bR^n\setminus A$, because for any $a\in A$ we have $a\notin B(y,r)$ as $d(y,a) \geq r$.
flat frigateBOT
Edward II
steep lily
(btw there is another I think easier proof of your original question I'm just running with this because a) it's an important fact and b) it's good practice)
unreal kindle
we're looking for $r>0$ such that $B(y,r)\subseteq\bR^n\setminus\overline{A}$. W...
wait im dumb, since || y - a || > r > 0 for all a in A, how do we get B(y, r) from that?
unreal kindle
ah ok yes, im so slow lol, so yeah B(y, r) in the complement of A, so now we need to show that it is in the compement of bar(A) right?
steep lily
yes
(not slow, you just haven't built up an intuition yet which can only imo be done through practice)
unreal kindle
since we know B(y, r) is disjoint from A, and closure(A) contains A .... we need to show its also disjoint from the interior?
does that make sense
or we dont need the interior
steep lily
not the interior, the bits of cl(A) that aren't A
btw
I can only see a proof by contradiction
unreal kindle
not the interior, the bits of cl(A) that aren't A
the border then?
steep lily
yes
unreal kindle
so by contradiction, we need to find a contradiction with the fact that B(y, r) is not disjoint from the border of A??
so we take k and k in B(y, r) and k in border of A?
steep lily
do you know that cl(A) is A along with its border?
unreal kindle
yes
steep lily
then yes that would work
it would be enough to say k in cl(A), but this should work too
unreal kindle
it would be enough to say k in cl(A), but this should work too
what is the contradiction if k is in cl(A)?
unreal kindle
if k in both B(y, r) and cl(A), we know k is not in A, but there's still the border so if k is in the border then it would not contradict B(y,r) disjoint from A unless A is closed... So i'm clearly missing something here
unreal kindle
you'll end up contradicting B(y,r) disjoint from A
why??
i can only see a contradiction if A is closed
steep lily
because k is in B(y,r), there's an even smaller s so that B(k,s) is in B(y,r). Separately, as k is in cl(A), we know that d(k,A) = 0, so there is some a in A s.t. d(k,a) < s (because if there wasn't d(k,A) > 0). But then this a lies in B(k,s) and hence in B(y,r)
steep lily
it goes in an entirely different direction from this
to show two sets are equal, we show the points are the same
now if $x\in\overline{\overline{A}}$, then by definition $d(x,\overline A)=0$
flat frigateBOT
Edward II
steep lily
this means that whatever positive r I choose, there is some y in cl(A) with ||x-y|| < r/2
and then you work with y being in cl(A), something similar to that ^ , and triangle inequality to show that in fact x is in cl(A)
steep lily
not sure if i also need to prove that if a set is closed then its closure is equ...
also this still needs to be done so the non-simple proof isn't actually finished 💀
If it's already been proven that the closure is closed, and the closure of a closed set is just the set, then yeah cl(A) is closed so it's closure is equal to it and we're done, which I think is why denascite asked if you knew (one of) those facts because then the proof would be nice and short
unreal kindle
ok im doing the simple proof first
steep lily
and then you work with y being in cl(A), something similar to that ^ , and trian...
and the other direction of x in cl(A) then x in the closure is clear
unreal kindle
@steep lily is this true first: x in cl(A) --> || x - a || = 0?
im lost with all of the definitions im not sure anymore
unreal kindle
d(x, A) = inf {d(a, x); x in R^n }
steep lily
cool
steep lily
<@365017527484416001> is this true first: x in cl(A) --> \|| x - a || = 0?
first of all no, because that means x = a
secondly infinum is indeed biggest lower bound
so x in cl(A) means the biggest lower bound for ||x-a|| is 0, as a ranges over A
so no matter what positive number ε I choose, there is some a in A such that ||x-a|| < ε as otherwise ε would be a bigger lower bound
unreal kindle
ok
so we know || x - y || < r/2, y is in cl(A) so there ε such that || y - a || < ε, is that true?
uhm no i think it's not clear enough
steep lily
y in cl(A) means for any ε there is some a, and you've said there is some ε and not explained a
unreal kindle
ok so for any ε, there is some a
so if ε=r/2, there is also some a
we then have || x - y || + || y - a || < 2(r/2)
and with the triangle inequality
we have || x - y || + || y - a || <= || x - y + y - a || < 2(r/2) ?
no its not that
it 's the opposite
steep lily
triangle ineqaulity goes the other way around, ||x-a|| <= ||x-y|| + ||y-a|| < r
steep lily
now r>0 was arbitrary, so this shows the biggest lower bound for ||x-a|| is 0, aka d(x,A) = 0 so x in cl(A)
unreal kindle
ok thank you very mucj
i still need to do all of this
lol
which one is easiest?
or are they all easy?
@steep lily
of course im not going to ask everything here but i just want to know which one is the easiest and which one is the hardest
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desert quarry
safe radishBOT
desert quarry
I don't know calculus 101 anymore
is there an elementary thing as to why this is true
solar hazel
looks like that integral is computable with FTC
if you evaluate it that way i presume you get log|x-y|
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lyric girder
Hey how would you guys solve this?
safe radishBOT
Hey how would you guys solve this?
lyric girder
My solution looks as follows but it really just relies on the n=4 case which I don’t think is correct
vagrant oracle
lyric girder
<@&286206848099549185>
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lyric girder
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plucky elk
You need to compute the lower sum using general n, not a chosen one
So your answer will depend on n
lyric girder
You need to compute the lower sum using general n, not a chosen one
Yes, so for example my intervals will be [-1,(-1+2/n)],[(-1+2/n),(-1+4/n)]…[(1-2/n),1] so how can I decide a lower value of the function at each interval when I don’t know what the values equal to?
Idk if that makes sense I don’t really understand how to tackle the problem
plucky elk
You're given f(x), so do cases on your x values
Sometimes f is 2, sometimes it's 1. It depends on x
And your x value depends on n
safe radishBOT
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fluid spoke
safe radishBOT
fluid spoke
where do i start
oblique goblet
So you have a function in terms of velocity, what do you know about velocity in terms of position?
Or better yet, how do you define velocity?
fluid spoke
s' = v
oblique goblet
Right, so we have a function for v, so we can sub that in and integrate both sides
You’ll get a +C on the right, which is where the second piece of information comes in (allows you to figure out what C is)
By second piece of info i mean the $S(\pi^2)=2$
flat frigateBOT
Sean
fluid spoke
Right, so we have a function for v, so we can sub that in and integrate both sid...
^^
oblique goblet
Then you know that integral of sin is -cos
And that when you make it sin in the derivation process 4/pi is multiplied bc chain rule
So you need a pi/4 term too
fluid spoke
oblique goblet
So overall its $-2\cos{\frac{4}{\pi}t}+c$
flat frigateBOT
Sean
oblique goblet
U substitution makes the process easier yeah but for trig if you just remember to divide the antiderivative by whatever the frequency is you’re good
safe radishBOT
oblique goblet
Looks right
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noble moth
I've done the other ones, pretty much this one is new to me I dont know where to begin, because right now from my understanding I have 4 different lines
safe radishBOT
I've done the other ones, pretty much this one is new to me I dont know where to...
noble moth
1/1 100/50 = 2/1 1000/250 = 4/1 10000/1000 = 10/1
I dont want the whole answer I was just hoping for someone to help me where to start
<@&286206848099549185>
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noble moth
All I know is that the cost increases proportionally with the shells produced
I have the slope as 0.01 dollars per shell
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fallen thunder
In sequence A, the first term is 1 and each term after the first term is 8 greater than the preceding term. In sequence B, each term is twice the corresponding term of A. In sequence C, the first term is 1 and each term after the first term is d greater than the preceding term, where d is an integer. If the sum of the first 100 terms of B is subtracted from the sum of the first 200 terms of A, the result is equal to 8 times the sum of the first 100 terms of C. What is the value of d?
safe radishBOT
In sequence A, the first term is 1 and each term after the first term is 8 great...
fallen thunder
😭 I'm sorry the question is kinda long
could someone help me find a non-algebra way to like get to the answer
my first deduction was if sum of the first 100 terms of B is subtracted from the sum of the first 200 terms of a then i'd be left with only the sum of A from 1 - 200
so i have:
$$\sum_{n=1}^{200} (1+8n) = 8\cdot \sum_{n=1}^{100} 1 + dn$$
flat frigateBOT
kanna
fallen thunder
would i be instantly able to somehow see that d = 8?
I realized i had a lot of typos hahah
solar hazel
your indexing is a little off
fallen thunder
I think it's okay because
i treated it as 1 + sum
and then got rid of it from both sides
wait
solar hazel
you are treating the first element of A as 9
is this right? $A_n = 1 + 8(n-1)$, $B_n = 2(1 + 8(n-1))$, $C_n = 1 + d(n-1)$, and $$\sum_{n=1}^{200}1 + 8(n-1) - \sum_{n=1}^{100}2(1 + 8(n-1)) = \sum_{n=1}^{100}8(1 + d(n-1))$$
fallen thunder
you are treating the first element of A as 9
no this is the result after my algebra
flat frigateBOT
chmonkey #1 simp
fallen thunder
i had 1 + that sum
fallen thunder
so i have:
$$1 + \sum_{n=1}^{200} (1+8n) = 1+ 8\cdot \sum_{n=1}^{100} 1 + dn$$
flat frigateBOT
kanna
solar hazel
but that's the first 201 terms
the sums should only go to 199 if you do it like that
solar hazel
which is probably nicer than my indexing
$$\sum_{n=0}^{199}1 + 8n - \sum_{n=0}^{99}2(1 + 8n) = \sum_{n=0}^{99}8(1 + d)$$
flat frigateBOT
chmonkey #1 simp
fallen thunder
**kanna**
but yeah i think it'll be an iteration of this but with correct indexing hmmm 
fallen thunder
**chmonkey \#1 simp**
mhm
solar hazel
if you just compute the sums i think you should get a quadratic equation in d
fallen thunder
**kanna**
wait wouldn't it still be this
but 199 would be the upper bound on the left hand side
solar hazel
**chmonkey \#1 simp**
also oops little typo there, forgot the n at the end
fallen thunder
and 99 on the right hand side probs
fallen thunder
also oops little typo there, forgot the n at the end
hmm okie np
solar hazel
maybe, i didn't do it haha
fallen thunder
$$\sum_{n=1}^{199} (1+8n) = 8\cdot \sum_{n=1}^{99} (1 + dn)$$
I think it's this yeah
flat frigateBOT
kanna
fallen thunder
there's no easy way to get d by observation right?
,w sum of (1+8n) from 1 to 199
flat frigateBOT
fallen thunder
,calc 159399/8
flat frigateBOT
Result:
19924.875
no idea but it's ok
fallen thunder
okay that's definitely 😭 not right
ughhh i'll just do this later
thxx for now
solar hazel
yea that can indeed not be right
plush grail
A company manufactures x units of one item and y unit of another item. The total cost in dolars, S, of producing these two items is $$ S = 5x^2 + 2xy + 3y^2 + 700 $$ If the production quota for the total number of items is 40, what is the minimum production cost ?
safe radishBOT
A company manufactures x units of one item and y unit of another item. The total...
flat frigateBOT
vuviCa
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@plush grail Has your question been resolved?
safe radishBOT
@plush grail Has your question been resolved?
safe radishBOT
@plush grail Has your question been resolved?
lean otter
@plush grail
Use x+y=40
Then replace y in terms of x in the equation
Then derivative to find minima
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plush grail
<@1157581073052602499>
I tried but since both x and y comes fractional I confused
lean otter
Oh can you send your solution
plush grail
Just a sec
I write the lagrange function:
L(x, y, z) = $$ 5x^2 + 2xy + 3y^2 + 700 $$ - landa(40-x-y)
flat frigateBOT
vuviCa
lean otter
Can we write y=40-x ?
crude star
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crude star
How would one do this
for 2, 1 I could add T(1, 0) and T(1, 1) and for T(0, 1) what would I do
oh nvm
.close
crystal belfry
Hi
icy lance
you forgot the +1
icy lance
shouldnt be a + at the end
icy lance
bingo
icy lance
no worries
crystal belfry
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