how is adding these equations together gives 4.5?
#help-41
mighty harness
amber waspBOT
how is adding these equations together gives 4.5?
mighty harness
isnt it just
fo/(y-ai) + fo/(y+ai) = (f_o(y+ai) + f_o(y-ai)))/(y^2 + a^2) so i get
2f_o(y) / (y^2 + a^2)
substituting y and a values
.close
amber waspBOT
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karmic anchor
All I know is that this isn't an elliptic integral
amber waspBOT
All I know is that this isn't an elliptic integral
karmic anchor
any hint
amber waspBOT
@karmic anchor Has your question been resolved?
olive yacht
I'd try u = x^2 first and see where that gets you
karmic anchor
olive yacht
All I know is that this isn't an elliptic integral
ah I think I got it, use trig sub. let x^2 = tan(theta)
you should be able to use some trig identities to simplify after substituting
karmic anchor
Hm
amber waspBOT
karmic anchor
thanks for the help
karmic anchor
ah I think I got it, use trig sub. let x^2 = tan(theta)
half angle sub of tan or cot would have done the work
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mellow rapids
can a function be even at 1 point and odd at 1 point?
amber waspBOT
can a function be even at 1 point and odd at 1 point?
sterile nymph
can a function be even at 1 point and odd at 1 point?
The function f(x) = 0 is both even and odd.
But it is the only function that satisfies that condition
mellow rapids
The function f(x) = 0 is both even and odd.
cuz i have a question our teacher sent which defys that somewhat
or maybe thats what i think
sterile nymph
!xy
amber waspBOT
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.
fresh hatch
What is it
mellow rapids
at x = -3 f(-3) = 5 and at x = 3 f(3) = 5
but at x = -2 f(-2) = 3 and at x = 2 f(2) = -3
fresh hatch
No picture?
mellow rapids
f(-3) = f(3) which is even
but then it goes odd
tulip tapir
a photo or screenshot please
mellow rapids
tulip tapir
that's not necessarily continuous
and we can't tell unless it's only defined on integer values
mellow rapids
If it's not even and not odd, it's neither
but it oscillates between odd and even xD
so do i go neither?
mellow rapids
what about h(x) , its even at 0 , - 1 and -2 but at -3 it goes odd
sterile nymph
So if there is a point where the function is not even, then the entire function is not even
And so on
mellow rapids
so both functions are neither?
mellow rapids
So if there is a point where the function is not even, then the entire function ...
^?
sterile nymph
so both functions are neither?
This is correct
Both functions are neither even nor odd
mellow rapids
This is correct
ah ok , i was confused cuz it was going back and forth xD
ty
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amber waspBOT
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crisp totem
how would i use this identity to obtain a proof of the cauchy-schwartz inequality?
crisp totem
not really sure where to start
pallid canopy
use a^2 >= 0 for all real numbers a
crisp totem
use a^2 >= 0 for all real numbers a
what do you mean? im confused
pallid canopy
which part are you confused on
crisp totem
i understand that this is the proof, i just dont know how i'd use the identity to get that
crisp totem
how would i use this identity to obtain a proof of the cauchy-schwartz inequalit...
and i need to use the identity here to obtain a proof of cauchy schwartz proof ?
elder umbra
how would i use this identity to obtain a proof of the cauchy-schwartz inequalit...
the rightmost term is negative, therefore the right side of the equation is smaller than sum(x_i^2)*sum(y_i^2)
sullen tulip
Hi
sullen tulip
Can someone tell me why theyre not equalling each other if theyre both area formulas 😰😰
grizzled pagodaBOT
Result:
144
that doesn't equal 64
nimble parrot
yes the height should be 4 * sqrt(3)
amber waspBOT
@sullen tulip Has your question been resolved?
amber waspBOT
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tough mica
Let $\Pi : x + 2y - 2z = 6$ and $L : X = \lambda (2, -1, 1) + (0, 2, 1)$. Find all lines $L'$ such that $L' \perp L$, $L' \cap L \neq \emptyset$, and $d(P, \Pi) = 2$ for every $P \in L'$.
grizzled pagodaBOT
ඞඞඞ
tough mica
!status
amber waspBOT
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
1
tough mica
the direction vector of L is perpendicular to the direction vector of L'
let v1 be the direction vector of L and v2 of L'
v1.v2=0
they are orthogonal
also L' and L share at least a point
tough mica
How do you find L'?
maybe we can cross product the direction vector of L with the normal of the plane to get a vector that is orthogonal to both
that way we get a direction vector of L' but we still need a point that passes through L'
@patent raptor
@pallid canopy
patent raptor
what riemann do
tough mica
he is prolly online
and he prolly already take calc 3
no
2-11
no 021
@patent raptor
grizzled pagodaBOT
bacc the sigma😔🤞
tough mica
yes
grizzled pagodaBOT
bacc the sigma😔🤞
grizzled pagodaBOT
bacc the sigma😔🤞
patent raptor
Does this make sense?
tough mica
yes
patent raptor
What is the easiest choice
notice we cannot take x_2=x_3=0
that would be then the null vector
and the null vector, is perpendicular to every vector so that makes the whole thing pointless
So we take any number but 0 for x_2 and x_3 for example 1
tough mica
fair
patent raptor
(0,1,1) is a good direction vector
that is perpendicular
Now we have to make sure their itnersection is not empty
I think you pointed it out already
patent raptor
(0,2,1) is the point
this must also be a point on L'
other wise they would be perpendicular but they would have no solutions
in 3-dimensional space
pallid canopy
he is prolly online
Yea don't ping individual helpers
tough mica
sorry
tough mica
other wise they would be perpendicular but they would have no solutions
the plane is parallel to L' since all the points P in L' have same distance to the plane
patent raptor
,calc 12+2(-1)*(-2)*1
grizzled pagodaBOT
Result:
6
you were right with the cross product
patent raptor
the plane is parallel to L' since all the points P in L' have same distance to t...
very good observation
grizzled pagodaBOT
bacc the sigma😔🤞
tough mica
yeah
or we can use the normal of the plane
because they are parallel they have same direction
,w (1,2,-2)×(2,-1,1)
grizzled pagodaBOT
patent raptor
patent raptor
oh lol hmm
tough mica
ye
patent raptor
same as (0,1,1)
tough mica
ye
patent raptor
because you brought up a valid point
and i thought
what if L and L' are perpendicular
but L' intersects PI
tough mica
,w (0,1,1).(2,-1,1)
grizzled pagodaBOT
patent raptor
yea
So we have to find a point P where the distance to the plane is 2
and that point is also in L
tough mica
find lines L' such that . . . and d(P,Π)=2 for every P in L'
we need to find the lines
to get a line we need direction and a point
we got direction
but all the lines
could be below the plain
or above the plain
where the distance = 2 you know
patent raptor
yes
well basically no
if the line L' were say below the plane
but L is above?
no you are right
it depends if L intesects the plane or not
,w (1,2,-2).(2,-1,1)
grizzled pagodaBOT
patent raptor
L actually intersects the plane
cause the normal vector is not perpendicular to the direction vector
tough mica
patent raptor
So basically we just need to figure when does L have a distance 2 to PI
tough mica
,rotate
grizzled pagodaBOT
tough mica
idk it looks 2D
but basically there is prolly only 2 lines L'
the exercise says like find all L' such that
I just wanted to point that out
patent raptor
yes
ok I think i have an idea
i hope this kinda works out
In order to figure that vector x we would basically need a point from the plane and on L
tough mica
yes
patent raptor
thats the hard part
tough mica
agreed
patent raptor
actually we can take any point from PI I think
we just have to find t so that it matches
tough mica
okay
patent raptor
it can have from all direction a distance of 2
That's the vector
that we need to project on n
(1,2,-2)
grizzled pagodaBOT
bacc the sigma😔🤞
patent raptor
,, \operatorname{proj}_{\textbf{n}} \textbf{p} = \frac{\textbf{p} \bullet \textbf{n}}{\norm{\textbf{n}}^2} \textbf{n} = \frac{2\lambda+4}{9} \begin{pmatrix} 1 \ 2 \ -2 \end{pmatrix}
,w (1,2,-2).(6-2lambda,lambda-2,-lambda-1)
grizzled pagodaBOT
grizzled pagodaBOT
tough mica
patent raptor
,w simplify 6 - 2 (-1 - λ) + 2 (-2 + λ) - 2 λ
patent raptor
Now pythagoras
,, \sqrt{\norm{\textbf{p}}^2 - \norm{\operatorname{proj}_{\textbf{n}} \textbf{p}}^2} = 2^2
grizzled pagodaBOT
bacc the sigma😔🤞
patent raptor
,w simplify (6-2lambda,lambda-2,-lambda-1).(6-2lambda,lambda-2,-lambda-1)
grizzled pagodaBOT
patent raptor
,w simplify 1/9 * ((2lambda+4),2(2lambda+4),-2(2lambda+4)).((2lambda+4), 2(2lambda+4), -2(2lambda+4))
grizzled pagodaBOT
patent raptor
,w sqrt(73/81 (4 + 2 x)^2 - 292/729 * (x+2)^2) = 2^2
wtf
1.08
it worked
I assume it didnt round somewhere
Ok somewhere went wrong but we are very close
they should have cancelled out neatly
tough mica
wdym
patent raptor
the distance we calculated is 2.04
which is not 2
so somewhere the dot product messed up
patent raptor
,w simplify 1/9 * ((2lambda+4),2(2lambda+4),-2(2lambda+4)).((2lambda+4), 2(2lambda+4), -2(2lambda+4))
grizzled pagodaBOT
patent raptor
,w sqrt((6x^2-26x+41)-4(x+2)^2)=2
patent raptor
this looks good
patent raptor
this looks awful
grizzled pagodaBOT
patent raptor
this sucks fr cant tell my mistake
amber waspBOT
@tough mica Has your question been resolved?
patent raptor
and you posted it all along
patent raptor
It was so easy that's so embarassing ngl
amber waspBOT
@tough mica Has your question been resolved?
tough mica
**bacc the sigma😔🤞**
p ∈ L'
but you are using p ∈ L
we are being told distance between plane Π and p is 2 but p ∈ L'
tough mica
.solved
amber waspBOT
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dull nacelle
also did :(5/36)100 =13.88
dull nacelle
what did I do wrong
amber waspBOT
@dull nacelle Has your question been resolved?
dull nacelle
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elder fiber
amber waspBOT
split kraken
elder fiber
it doesnt say my units are incorrect
when i did mi/m it said my units were incorrect
split kraken
miles and minutes is what the units on the graph's axes are though 
your answer is numerically correct at least
idk what the website is on about then
amber waspBOT
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twilit cove
ok so um stats question
amber waspBOT
ok so um stats question
twilit cove
im reviewing from start
I disagree with this answer but I wanna hear your thoughts
this is the ans
but I dont think thats it..?
ivory pivot
but I dont think thats it..?
what does it mean for an event to be mutually exclusive
twilit cove
C as in Chen
but yea I think this also should work
Like I disagree that the sol is not only one that works
shadow stump
why would it be impossible for them to have been born in the same month?
twilit cove
no its not impossible
twilit cove
Im saying i dont think this is the only sol?
oh yea ofc this is not the only sol
does mine work then?
I think so
ok
valid remnant
all 3 students could have been born in the same month
twilit cove
yup
that event and the event below it are disjoint tho?
cause an outcome would be A1B1C1 for the first but A1B1C2 for the second
you're never gonna have an intersection
shadow stump
i think we are considering just the events A_i, B_i, C_i, not combinations of them
amber waspBOT
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amber waspBOT
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loud crag
hello I have no idea how to start this
amber waspBOT
hello I have no idea how to start this
neat wind
hello I have no idea how to start this
as hinted in the diagram, it may be useful to consider square cross sections
amber waspBOT
@loud crag Has your question been resolved?
loud crag
as hinted in the diagram, it may be useful to consider square cross sections
get a area of the cross scetion using squares?
stoic ivy
They’re trying to hint at the little black square in the picture. If you were to somehow determine the area of the squares as a function of the z value then you could integrate the squares on the z axis
Assuming z is the vertical axis
I guess I’m being confusing cause they defined y as the vertical axis 💀
fringe cloak
try to think about one of the cylinders as a square based prism, with the side of this square being 2r
neat wind
wouldn't use 2r here (would rather use 2x or smth instead) since the problem already defines r as the radius of the cylinders
fringe cloak
wouldn't use 2r here (would rather use 2x or smth instead) since the problem alr...
i agree, but then x would equal r anyway in this analogy. What i mean is that then, in this case, S would be much easier to find as it'd be 2R^3π, so the diference between this and real S is just 2x 8R^3 - 2R^3π. Idk if I explained it well
I actually don't think I explained it well
amber waspBOT
@loud crag Has your question been resolved?
wheat oasis
Two identical rectangular prisms each have a height of 90 centimeters (cm). The base of each prism is a square, and the surface area of each prism is Kcm. If the prisms are glued together along a square base, the resulting prism has a surface area of 92/47 K cm2. What is the side length, in cm, of each square base?
amber waspBOT
Two identical rectangular prisms each have a height of 90 centimeters (cm). The ...
@wheat oasis Has your question been resolved?
split sail
Consider this to be the rectangular prism.
Let the side of the below square (base here) be s.
Area of a square = s^2
So the length, width and height are s, s and 90 cm respectively.
ATQ, 2s^2 + s(90)*4 = K
2s^2 + 360s = K
Now after the gluation the total surface area will be:
s^2 + s^2 + 180(s) * 4 = 92K/47
This is a 2 variable equation
if you solve it you will get s = 4
wheat oasis
Consider this to be the rectangular prism.
Let the side of the below square (bas...
Cant' it be 2s^2 + s(90)4 = K ?
because
there is 4 lateral faces
split sail
oh
wheat oasis
and 2 base areas
wheat oasis
Thank you so much 🩷
split sail
you're welcome
wheat oasis
This is a 2 variable equation
I can put s^2 + s^2 + 360(s) for 'k'
split sail
mhm
split sail
same for the second, it should be 180(s) 4 = 720s
split sail
And find 's'
yes by using 2 equations formed
wheat oasis
same for the second, it should be 180(s) 4 = 720s
the glued prism u mean?
split sail
yup
split sail
there will be no K
wheat oasis
Why
split sail
since you replaced it with 2s^2 + 360s
split sail
no problem
wheat oasis
.close
amber waspBOT
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fossil shell
I'm not so sure how to do part a
amber waspBOT
I'm not so sure how to do part a
fossil shell
Ah alright that's btr now
spiral zealot
have you done anything for it so far?
fossil shell
I do know angle mpq and angle xqn are the same
But I'm not sure how to set up both of them in an eqn that can help me find the angle
spiral zealot
those angles aren't the same
spiral zealot
yes
fossil shell
Not sure if I rmb the trapezium angles correctly but pmn+mpq=180
spiral zealot
yes
spiral zealot
yes
fossil shell
So I don't understand how the other would not be 90 deg
spiral zealot
you'd need to be aware of where the parallel lines are
angle mpq and angle xqn
would sum to 180°
(also capitals)
anyway to start this, apply the properties of circles/radii
fossil shell
> angle mpq and angle xqn
would sum to 180°
So they both add up to 180 deg?
fossil shell
anyway to start this, apply the properties of circles/radii
Hmm how so they don't have the same area of circumference for their arc length right
spiral zealot
can you rephrase that question
fossil shell
Like how do I start this by applying the properties of circles
spiral zealot
what do you know about the radius of a circle
arc MX is drawn with P as the centre
imagine that as a cropped out portion of a full circle
and use that to detremine the length of PX
fossil shell
Px=8 too right
But I'm not sure how I'm still supposed to find the angle with both the lengths
Or do I imagine it as a right angled Triangle
spiral zealot
draw a line from Q thats pependicular to PM
fossil shell
Hmm ok
So now ik the hypotenuse of the triangle which is 13 right
Nvm got it tks
.close
amber waspBOT
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mellow rapids
@coral wigeon got a question to find domain of (x/1+x^2)
mellow rapids
now this goes into imaginary numbers
coral wigeon
it does not
mellow rapids
do i still write it as -inf , sqrt-1
coral wigeon
also, x/(1+x^2)?
strange field
x/(1+x^2)?
coral wigeon
x/(1+x^2) parses for any real number
mellow rapids
i thought it should be 1-x^2
coral wigeon
1+x^2 is never 0
mellow rapids
1+x^2 = 0
x^2 = -1
x = +- i
do we not talk about complex numbers then?
it wont go 0 for any real number tho , i see that
shadow stump
if you get an imaginary solution when the domain we're looking for is a subset of the real numbers, we conclude that the solution is irrelevant
mellow rapids
if you get an imaginary solution when the domain we're looking for is a subset o...
ok so i should just look for real numbers?
shadow stump
yes, these types of function questions are all about functions which take real numbers as inputs and outputs
coral wigeon
you can make functions with whatever domain and codomain you want
coral wigeon
yes
mellow rapids
even complex domains?
coral wigeon
yes
mellow rapids
damn
split sail
layla
mellow rapids
k ty
coral wigeon
yajat
mellow rapids
why are u here
split sail
why i cant be
mellow rapids
stop ruining my learning experience
coral wigeon
why is your name sweatyballs_123
shadow stump
absolutely we can consider complex domains in general, that's just not the intent for these types of questions
split sail
why is your name sweatyballs_123
can i not have that name?
mellow rapids
absolutely we can consider complex domains in general, that's just not the inten...
ah ok
strange field
why is your name sweatyballs_123
because their balls are sweaty obviously
mellow rapids
ty for the help ❤️
coral wigeon
i'll ask the questions here
mellow rapids
can i close?
split sail
yea if ur done 😭
mellow rapids
.close
amber waspBOT
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chilly jackal
amber waspBOT
chilly jackal
dont 1 and 2 contradict each other 
coral wigeon
no
chilly jackal
why?
indigo cloud
its bad wording
chilly jackal
if 1 is true 2 cannot happen can it?
indigo cloud
they mean that q is not +-1
chilly jackal
oh so the root cant be a fraction if the polynomial is monic with integer coefficient right?
hllo???
amber waspBOT
chilly jackal
.reopen
amber waspBOT
✅
chilly jackal
hold up
isnt the proof for part 1 just
RRT says its gon be x=+/- p so it cant be a fraction as p is an integer?
coral wigeon
if you can use that sure
chilly jackal
💀 stupid question this is
coral wigeon
1 and 2 seem to be pretty much the same thing
chilly jackal
if you can use that sure
yeah i can
amber waspBOT
indigo cloud
yeah i can
are you sure? the point of the problem seems to be proving rrt
chilly jackal
are you sure? the point of the problem seems to be proving rrt
i just did that in the previous one
.reopen
amber waspBOT
✅
chilly jackal
i can see why 4 is true; but i cant write out the proof for it
could someone help?
like what do i use? strong or weak induction?
indigo cloud
i just did that in the previous one<:devastation:898725325335298088>
well 1 and 2 are literally just quoting rrt and you are done. whats the point of that
chilly jackal
i have no idea personally
indigo cloud
like what do i use? strong or weak induction?
well why do you think its true
chilly jackal
well why do you think its true
if u shift k terms to the RHS, then u can take alpha^k common and we are done for a specific k
indigo cloud
well then write that down
chilly jackal
thats it?
that works?
sorry I am self-studying and the book has no solutions
and I am new to proof-writing
amber waspBOT
@chilly jackal Has your question been resolved?
amber waspBOT
tulip tapir
...
a, b, and c can be negative...
it's not like they must be positive?
e.g. -x^2 - 3x + 2 has sols
no
you might get roots of negatives, but those are allowed
amber waspBOT
red knot
It generally refers to functions that are measurable but do not necessarily satisfy any integrability conditions.
It just represents the space of all measurable functions on
[a,b].
Yes
indigo cloud
the book should say somewhere what order they put on the space
its not like you found that definition in a vacuum
so this is the first definition in the entire course?
damn some courses suck
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Available help channel!
ancient raft
Could I write $$cotx=tan^{-1}x$$
amber waspBOT
Could I write $$cotx=tan^{-1}x$$
grizzled pagodaBOT
Good
patent raptor
yes but notice that people might confuse it with arctan(x), unfortunately
I would rather write it as (tan(x))^(-1) or 1/tan(x)
amber waspBOT
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empty wyvern
How do you find the next 3 terms if the sequence isn't geometric, or arithmetic?
hot aspen
In a bag, there are 3 red marbles and "B" blue marbles. Two marbles are randomly selected from the bag without replacement. The probability that the two marbles are the same color is 0.5. Calculate the sum of all possible values of B
amber waspBOT
@empty wyvern Has your question been resolved?
graceful badger
In a bag, there are 3 red marbles and "B" blue marbles. Two marbles are randomly...
Open a new help channel, this is occupied
graceful badger
How do you find the next 3 terms if the sequence isn't geometric, or arithmetic?
Well you can still just recognize the pattern
late rock
solve for all real values of $x$: $\frac{x}{1-x}+x=\qty(\frac{x}{1-x})^x$
grizzled pagodaBOT
niom
late rock
i tried combining the left side into one fraction and then ln both sides but it doesnt seem to get anywhere
and i also cant use series expansions to do stuff, if that helps in any way
amber waspBOT
@late rock Has your question been resolved?
late rock
<@&286206848099549185>
pseudo flint
I am looking at it right now
late rock
late rock
solving
split sail
which part are u having a hard time
late rock
i have no idea how i can approach this
late rock
i have no idea how i can approach this
just this i guess?
late rock
wdym by expanding
split sail
making it longer
late rock
yea but how
late rock
late rock
from my brain
patent raptor
This has one ugly solution
patent raptor
You can show that it has at least one solution but getting it with bare hands seems very difficult unless you use numerical approximations
late rock
im looking for an exact form
formal dome
oof
late rock
closed form i guess
patent raptor
of what
late rock
yea cuz or else desmos wouldve done it for me
distant quartz
This has one ugly solution
Zero works
patent raptor
Zero works
no
distant quartz
Why
formal dome
Zero works
at 0, lhs is 0, and rhs is either 1 or undefined
patent raptor
left side is 0 right side is undefined
limber star
Zero works
NO it does not
limber star
0^0 is an indeterminate form mate
distant quartz
0^0 is an indeterminate form mate
Forgot the power
patent raptor
ja man
formal dome
0^0 is an indeterminate form when we're talking about limits.
Do you see a limit there?
formal dome
that was for TeufeL, not rish
coral wigeon
late rock
are we not interested in the question
patent raptor
not really
late rock
formal dome
honestly, getting an analytical solution on that one is way beyond my ability
patent raptor
Maybe the Lambert function is somehow applicable
formal dome
You can show that it has at least one solution but getting it with bare hands se...
can you show (reasonably easily) that it has only one tho?
late rock
desmos shows a this, by the looks of it the upper part is never going to intersect
yea
formal dome
eeeeh, i would not say that's never gonna intersect without proof tbh
patent raptor
I did f-g and seems to have one root
late rock
eeeeh, i would not say that's never gonna intersect without proof tbh
just speculation
patent raptor
also the derivative seems to be always negative implying that it's stricly decreasing
but yea we would need a proof technically
late rock
also the derivative seems to be always negative implying that it's stricly decre...
derivative of which function?
patent raptor
I am currently trying something
patent raptor
derivative of which function?
Assume f(x) = g(x) then define h(x) = f(x)-g(x)
late rock
oh
patent raptor
of h the derivative
late rock
dunno what that does
patent raptor
dw
late rock
okey
would some algebraic geometry help
just curious
or something crazy woo
i heard algebraic geometry can smash some 4th degree complex polynomial so it sounds crazy
patent raptor
What could help is an approximation of a taylor series around x = 1
late rock
approximation
patent raptor
ok that looks actually stupid
late rock
patent raptor
around x = 0.5 might work
late rock
can you show (reasonably easily) that it has only one tho?
is there something we can do if we know theres only one
late rock
around x = 0.5 might work
why do you think changing the value works?
patent raptor
because of the domain
late rock
ah
patent raptor
0 and 1 are bad as well negative numbers
late rock
but doing a taylor series on that thing is crazy 
patent raptor
yea we would need at least degree 6 i think to get a good approximation
late rock
but that means no exact form
patent raptor
but that means no exact form
around this neighboorhood if we use 1 or 2 more degree's it will be the same basically
the quadratic is already pretty decent
late rock
okey
patent raptor
the error seems to be pretty close to 0
late rock
how do i apply this tho
i dont understand geogebra
its showing a numerical value?
in the last line
patent raptor
it's very long
late rock
uh ok
limber star
no the last line is the taylor expansion of (x/(1-x))^x
at x = a
late rock
and a=0.2
patent raptor
late rock
wth
patent raptor
0.3 works better
late rock
uh ok
patent raptor
the derivatives are not nice
late rock
indeed
limber star
Is this homework?
limber star
Oh then its fine
late rock
why do you ask?
patent raptor
yea
pseudo flint
exact solution may not be expressible by elementary functions
late rock
most likely yea
pseudo flint
for $y = x^x$ we use Lambert W function
grizzled pagodaBOT
Kayra Bolat
pseudo flint
to solve for x
late rock
how does that help
patent raptor
It does but I couldnt get it in such form
late rock
so its not clear
pseudo flint
if you manipulate the expression to that form you can use Lambert function
limber star
<:catthin4K:811354320460578896> why do you ask?
Juz curious.
late rock
thats a weird thing to ask
pseudo flint
But lambert function itself is essentially an infinite series so you can only approximate its value
limber star
thats a weird thing to ask
I suppose.
limber star
for $y = x^x$ we use Lambert W function
I don't think that would work here mate
pseudo flint
But using calculus to show that it has 1 real solution would be a good exercise I suppose
patent raptor
Yes
pseudo flint
I don't think that would work here mate
why?
patent raptor
just interested in some algebra thing
Do you have the impression that in math everything is solvable algebraically?
limber star
But using calculus to show that it has 1 real solution would be a good exercise ...
But would it help in 'solving for x'?
late rock
Do you have the impression that in math everything is solvable algebraically?
idk how i can answer that, but i know not everything in math is solvable
patent raptor
ok good
late rock
actually this problem stems from x+y=x^y and x-y=xy
pseudo flint
W(xe^x) = x
yeah I know
patent raptor
hoomans are thinking
your name backwards literally means hello in German but yea I agree
patent raptor
moin 👋🏻
pseudo flint
$x^x = y \implies ln(x)e^{ln(x)} = ln(y) \implies ln(x) = W(ln(x)e^{ln(x)}) = W(ln(y)) \implies x = e^{W(ln(y)}$
grizzled pagodaBOT
Kayra Bolat
pseudo flint
if we can manipulate the right side to take the form x^x we can use W maybe?
late rock
do you have a more concrete path?
pseudo flint
nope
late rock
:(
patent raptor
do you have a more concrete path?
concrete sometimes breaks too
late rock
nyeh
pseudo flint
I still believe that the exact solution is not expressible in simple terms
late rock
i agree
late rock
i think math gave me a really hard problem
pseudo flint
Just like some polynomials, no matter what manipulation you do you cannot find roots concretely
late rock
we can solve this instead $\frac{x}{x+1}+x=x^{\frac{x}{x+1}}$
grizzled pagodaBOT
niom
late rock
patent raptor
<:catgiggle:912459497954500659>
I think you should do linear algebra instead of this "algebra"
patent raptor
yes
late rock
im interested but not interested enough
limber star
patent raptor
i see
late rock
im not interested in algebra specifically also
limber star
late rock
and its easy for me to come up with this type of algebra question
late rock
im looking for an exact form
pseudo flint
I think I solved it but the expression is ugly
late rock
show it?
pseudo flint
1 min cleaning up
late rock
late rock
oh..
late rock
do we agree to close
pseudo flint
waitt
late rock
okey
late rock
looks cool
patent raptor
looks cool
what math level are you
late rock
erm
i know quite a bit of integration techniques, and touched a big of multi variable calc
and a bit of group thoery
and a bit about doing basic operations on matrices
i mean all basic operations on matrices
craggy sundial
got to this for the first eq
I don't think it's any better
late rock
if its fun then sure
pseudo flint
oh okay then
late rock
out of the owijenocwmec-phism stuff i only know the basic definition fo isomorphism
late rock
got to this for the first eq
I don't think it's any better
what is u here?
craggy sundial
u=1-x
limber star
I see
limber star
only touched a bit
That's some really good stuff you have been dabbling in
late rock
limber star
Do you like Real Analysis?
late rock
not in particular
patent raptor
do you like fake analysis?
late rock
aka complex analysis
limber star
aka complex analysis
Complex is not fake tho
late rock
just kidding
limber star
just kidding
Yes I understand.
amber waspBOT
@late rock Has your question been resolved?
late rock
Do we agree to close
patent raptor
Do we agree to close
if you want
amber waspBOT
amber waspBOT
Available help channel!
split sail
How to prove this is a non-regular language?
amber waspBOT
How to prove this is a non-regular language?
lethal steppe
How to prove this is a non-regular language?
I suppose you could show that no DFA exists for this
That would be my intuition at least, I don’t know if that’s the classical approach
amber waspBOT
@split sail Has your question been resolved?
split sail
I suppose you could show that no DFA exists for this
Yes but how would I show that
lethal steppe
Yes but how would I show that
Depends on how rigorous you’re supposed to be
When I had to do it I didn’t have to do it rigorously, so I basically just used intuitive reasoning
Along the lines of assuming m states and showing that a certain term can’t be detected correctly
amber waspBOT
@split sail Has your question been resolved?
half narwhal
how do you solve this
amber waspBOT
how do you solve this
half narwhal
im lost
foggy kayak
put the values of x and see where the discontinuity exists
drifting fiber
is this precalc?
drifting fiber
oh
half narwhal
oh alr
drifting fiber
wait i think i got this
half narwhal
put the values of x and see where the discontinuity exists
which graph should i put it in thgouth
wait yo this the graph
miss input
drifting fiber
gimme a second to process
half narwhal
alr
drifting fiber
the way collegeboard words things is horrible
drifting fiber
it’s asking that in both of these graphs, at what x value is there a jump discontinuity
drifting fiber
one more
drifting fiber
no it’s at x=2
-1 and 2
because at both of those x values on bothe graphs there’s a jump
half narwhal
but it asked only to check the 3 values
half narwhal
it gave 3 x values
half narwhal
wdym?
drifting fiber
think of it as an mcq. it’s asking “which of these values” (x = -1, x = 2, and x = 4) have jump discontinuities
at x = 4 for both of the graphs, g(x) is continuous
for the other 2 there’s a jump for both graphs
spiral zealot
incorrect
drifting fiber
i don’t have my ipad on me but i have my phone
spiral zealot
the question isn't asking about where f,g individually has jump discontinuitites
but rather where **f(x)+g(x) ** has jumps
half narwhal
how would one do that
spiral zealot
use one sided limits as the question mentioned
consider
$$\lim_{x\to -1^-} (f(x)+g(x))$$
$$\lim_{x\to -1^+} (f(x)+g(x))$$
$$f(-1)+g(-1)$$
repeat for the other locations
drifting fiber
the wording is bad lowk it got me struggling 😭
grizzled pagodaBOT
ℝαμOmeganato5
spiral zealot
did you do all 3 i've listed up there?
half narwhal
how do i find what i get when i add from each individual side
spiral zealot
from the graph
half narwhal
limx--1+ (f(x) + g(x))
spiral zealot
look at what's happening from each side
drifting fiber
how do i find what i get when i add from each individual side
f(-1) = 0 and g(-1) = -1
so if u added those it’s -1
spiral zealot
what?
drifting fiber
if there’s a jump discontinuity, then yeah the limit doesn’t exist. if there’s a hole, the limit would still exist.
spiral zealot
can you show your work
spiral zealot
can you show your work for
$$\lim_{x\to -1^+} (f(x)+g(x))$$
grizzled pagodaBOT
ℝαμOmeganato5
half narwhal
for that side
as x approaches -1
from the positive side
f(x) approaches 1 and 1
oh
drifting fiber
WAIT I SEE WHAT WENT WRONG
half narwhal
its 2
drifting fiber
i think i have the answer
half narwhal
@spiral zealot
spiral zealot
yes, f(x) and g(x) both approach 1 from the right side,
the overall right side limit will be 2
half narwhal
so tis a jump
spiral zealot
yes
spiral zealot
approach the other locations in the same way
drifting fiber
wait @spiral zealot i still don’t understand what the question is asking
collegeboard words things badly
spiral zealot
i've seen bad wording/marking systems from them
but this seems worded fine to me
drifting fiber
what’s the question asking though 😭
spiral zealot
to investigate continutity for f(x)+g(x)
drifting fiber
so confirm if they’re jump discontinuity?
spiral zealot
yes
drifting fiber
oh so i was originally on the right track, just not with the math part though 😭😭
drifting fiber
are both -1 and 2 jump discontiniouties
yes
spiral zealot
can you show your work
drifting fiber
HOW 😭
spiral zealot
oh so i was originally on the right track, just not with the math part though 😭...
you're only considering the jumps for each function individually rather than for f(x)+g(x) as a whole
drifting fiber
you're only considering the jumps for each function individually rather than for...
so how do i find k(x)i’m so lost
spiral zealot
how are you getting
2 + 2
for the left side limit at 2
half narwhal
how are you getting
2 + 2
for the left side limit at 2
that is a marvelous question
i have now rectified that
and they are both 1
so its continious
spiral zealot
misspelled jump for the second one
drifting fiber
misspelled jump for the second one
can u help me with calc ab later? 🙏🙏🙏 i’m doing implicit differentiation
spiral zealot
follow the guidelines for asking for help
and someone will assist
drifting fiber
follow the guidelines for asking for help
and someone will assist
k i’ll do it later
ima eat
half narwhal
.close
amber waspBOT
amber waspBOT
Available help channel!
chilly jackal
Find the minimum x such that 65x+528 is a perfect square
amber waspBOT
Find the minimum x such that 65x+528 is a perfect square
finite agate
One message removed from a suspended account.
chilly jackal
More generally, given the expression ax+b find the minimum x such that ax+b is a perfect square
Where a and b and x are all pos integers
neat wind
without programming?
chilly jackal
Yes
quick spoke
Find the minimum x such that 65x+528 is a perfect square
mod 5 means there exists no such x
chilly jackal
mod 5 means there exists no such x
Wait what why?
quick spoke
mod 5:
0^2 = 0, 1^2=4^2=1, 2^2=3^2 = 4
then if u mod 5 ur expression
65x + 528 mod 5 = 3
hence there exists no such x since 3 is not a q.r. mod 5
tulip tapir
factorize?
chilly jackal
Won't work I thonk
neat wind
k = 4
tulip tapir
mhm
tulip tapir
Won't work I thonk
it does lmao
chilly jackal
it does lmao
4k + 33 must be a perfect square how does that help
tulip tapir
...?
split sail
?
tulip tapir
you know that k > 0
chilly jackal
I took 16 common
chilly jackal
Hmm what about 2k+33 being a perfect square?
chilly jackal
Oh shit am stupid
split sail
Yes just imagine an odd perfect square
amber waspBOT
tulip tapir
Hmm so reducing the coefficient like that hmmm
what?
are you just relating branches of ax + b
regal glacier
$$ax + b = n^2 \iff \exists \sqrt{b} \pmod a$$
grizzled pagodaBOT
Pluton
chilly jackal
are you just relating branches of ax + b
Yes
chilly jackal
**Pluton**
Isn't that just saying b is a QR mod a and x congruent to b mod a?
quick spoke
ur effectively finding rational points on y^2 = ax + b
regal glacier
The first part
quick spoke
so maybe u can borrow some techniques from elliptic curves?
regal glacier
b is QR mod a
quick spoke
like i'm pretty sure it'll still be true that the the set of rational points + infinity should form an abelian group
amber waspBOT
Available help channel!
fallen goblet
am i allowed to do 7days/1animal or am i supposed to do it some other way
cold sparrow
am i allowed to do 7days/1animal or am i supposed to do it some other way
That is what you're actually doing but sometimes they explain it differently
But that chart there
Imagine each column is a fraction
That's the "real" calculation
fallen goblet
if i did this how would i cancel out the animal part
cold sparrow
Well if i was doing it by hand
I would use the
2.3kg of corn / 1 animal
That information lets you cancel the animal unit
fallen goblet
.close