$\frac{\cos\left(x\right)-1+x^{2}}{\left(x+\sin\left(x\right)\right)^{2}}=\frac{-\frac{x^{2}}{2}+x^{2}}{\left(x+x\right)^{2}}=\frac{\frac{x^{2}}{2}}{\left(2x\right)^{2}}=\frac{x^{2}}{8x^{2}}=\frac{1}{8}$
#help-13
dusk finch
wraith daggerBOT
MæthIsAlwaysRight
dusk finch
yep
knowing the approximations is extremely useful
you can just eyball the limits then
inner solar
yes and its 2in1 cuz if you can you them for trig identiites
cedar kilnBOT
dusk finch
anyway thanks for help have a nice christmas and a happy new year
You too :)
cedar kilnBOT
Available help channel!
dreamy void
Let $H$ be a semi-group with neutral element $e$. Then there is
[ G := G(H) := {g \in H \mid \exists h \in H s.t. gh = hg = e }. ]
Show that $G$ is a group.$\$
So far I managed to show that the composition of two elements of $G$ is in $G$ again (closure).
Now I would like to show associativity, but I am stuck, and in the solution it just says that associativy follows immediately from $H$ which I don't see how. Any help would be cool!
wraith daggerBOT
𝔸dωn𝓲²s
proper mortar
copper cove
Think about how the operation in G relates to the operation in H. Since H is a semigroup and associative
oh, he answered first
cedar kilnBOT
rigid dome
can you help me with this
trim tendon
can you help me with this
!msgdel
cedar kilnBOT
can you help me with this
The original post of this help channel has been deleted, and it will abruptly close and possibly lock. (This is irreversible.) Please claim a new channel, and don't delete the first message of any future channel you claim.
trim tendon
I think it'll do it automatically
cedar kilnBOT
Available help channel!
rigid dome
acctually i didn't find the solution i typed solved just cus it seems that there was a problem with this channel
timid flower
grim yacht
Are you able to find the cdf of max_n (U_n) with n just being some arbitrary integer less or equal to k?
How can you use this cdf to potentially solve 1?
cedar kilnBOT
@rigid dome Has your question been resolved?
rigid dome
Are you able to find the cdf of max_n (U_n) with n just being some arbitrary int...
i didn't get you
could you elaborate ?
grim yacht
Let K be a random variable defined as K = max_n (U_n). If K is less than k then this is equivalent to U_1, U_2 , ..., U_n all being less than k
If you assume each U is independent of each other how could you derive the distribution for K?
i.e. calculating P(K<k)
If you're completely clueless to this then you could also look up ordered variables on the web
*order statistics
rigid dome
If you assume each U is independent of each other how could you derive the distr...
U_n are not independent of each other
grim yacht
U_1 , U_2, ..., U_n aren't independent of each other?
rigid dome
U_1 , U_2, ..., U_n aren't independent of each other?
yep
grim yacht
What's their relation?
rigid dome
What's their relation?
the only thing that is given is that they have the same distribution
grim yacht
If they're dependent but we aren't given a joint distribution then we can't solve the problem
cedar kilnBOT
rigid dome
If they're dependent but we aren't given a joint distribution then we can't solv...
.reopen
cedar kilnBOT
✅
rigid dome
If they're dependent but we aren't given a joint distribution then we can't solv...
i see got you, then there is a mistake in the exercise
grim yacht
Often those who make these problems are just a little lazy and leave that detail out, because it's usually always taken for granted
rigid dome
Let K be a random variable defined as K = max_n (U_n). If K is less than k then ...
but i still didn't get how it will lead to the solution 😅
grim yacht
Are you able to find the cdf of max_n (U_n) with n just being some arbitrary int...
Try applying the rule P(A n B) = P(A|B)P(B) and formulate it in a possible way that could make use of this
rigid dome
Try applying the rule P(A n B) = P(A|B)P(B) and formulate it in a possible way t...
ok i'll try with it, thank you so muchhh
grim yacht
Try applying the rule P(A n B) = P(A|B)P(B) and formulate it in a possible way t...
(Using this method to solve the problem you'd have to do this recursively, about k times or something, for it to work)
rigid dome
(Using this method to solve the problem you'd have to do this recursively, about...
you're referring to this formula right?
grim yacht
Yup that's right
I will also recommend looking at the Wikipedia page on order statistics since the first problem is basically just a case of this
Could give you some hints, even tho there aren't any super good explanations of the formulas there
rigid dome
Could give you some hints, even tho there aren't any super good explanations of ...
thank you so much i just checked it yeah it already gave me an idea on how it could be solved. thank you so much 🙏
grim yacht
np
rigid dome
.close
cedar kilnBOT
cedar kilnBOT
Available help channel!
cosmic umbra
help me to prove .
cedar kilnBOT
help me to prove .
dawn junco
have you seen eigenstuff ?
grim yacht
What property does the eigenvalues of T have?
grim yacht
have you seen eigenstuff ?
What's eigenstuff?
dawn junco
eigenvalues eigenvectors, was writing lazily
grim yacht
Oh should have been able to guess that. Thought it was some ytber for a sec
dawn junco
Well there’s eigenchris 
grim yacht
Learned topology in 5 min from that guy
cosmic umbra
im sorry i dont understand
why eigen here
what do you mean
to prove invertible we have rank n =n or ker={0}or dim !=0 right
dawn junco
There's a ton of other criteria you know
cosmic umbra
There's a ton of other criteria you know
what can you help me
dawn junco
What does it mean that det(T - sqrt(2)I) = 0 ?
cosmic umbra
no invertible
dawn junco
Regarding eigenvalues/vectors
cosmic umbra
why eigen here
dawn junco
What does it mean that det(T - sqrt(2)I) = 0 ?
Well that looks a lot like the characteristic poly of A yk
Except instead of x we plugged in sqrt(2)
grim yacht
why eigen here
To find the elements of kerT you're basically solving for x in Tx = 0 right
grim yacht
To find the elements of kerT you're basically solving for x in Tx = 0 right
You can rewrite this as Tx = lambda x with lambda equal to zero
Thus if ker T contains a nonzero element it must have an eigenvalue equal to zero
What can you then say about the eigenvalues if T was invertible?
cedar kilnBOT
@cosmic umbra Has your question been resolved?
dawn junco
Yeah i guess you don’t need to mention eigenvalues directly if you don’t want to @cosmic umbra
trim wagon
cedar kilnBOT
worldly chasm
You multiplied by 2
undone star
how the hell did -2 appear
2(sqrt(z-3)-1)=2
expanding the bracket
the 2 gets distributed
cedar kilnBOT
cedar kilnBOT
Available help channel!
pine brook
[
\lim_{x \to +\infty} \left( \sqrt{x^2 + 3x} - \sqrt{x^2 + x} \right)^x
]
wraith daggerBOT
maxy
pine brook
How can i solve this with taylor?
cedar kilnBOT
@pine brook Has your question been resolved?
grim yacht
Ok so I should know this but ngl I'm kinda struggling too. Have you tried Taylor expanding the first term with anchor point at sqrt (x^2 + x) @pine brook ?
You will then get a power series with z = 2x
heady salmon
Why does I(this is an i not an L) disappear?
cedar kilnBOT
Why does I(this is an i not an L) disappear?
heady salmon
So its cancels out… like 1-1?
vestal bluff
it cancels out like $2 * 2^{-1} = 1$
wraith daggerBOT
ashy!
vestal bluff
and $1 * x = x$
wraith daggerBOT
ashy!
rocky harness
type .close
heady salmon
.close
modest hamlet
Hello, I need help with understanding a question in a math textbook regarding the set of all elements which aren’t an element of themselves:
modest hamlet
It’s question 5
I don’t understand how one can define x as E since that breaks the rule that builds the set
Since E contains itself
So I don’t understand how that invalidates the existence of the set containing all elements which aren’t an element of themselves
dire geode
,rccw
wraith daggerBOT
solemn trail
Sa l'aire compliqué
modest hamlet
Is it because since a set is defined only by its elements, the property that all elements of a set have in common doesn’t apply to the set itself, which is the cause of the paradox in the case of “the set which contains all elements which don’t include themselves”?
Ah tu parles français?
modest hamlet
L1
solemn trail
ah sa doit etre francais moi chui au quebec
modest hamlet
Oui, mais l’anglais me va aussi
solemn trail
non jveut dire L1 sa doit etre un systeme francais et non quebecois parce que jen ai jamais entendu parler
modest hamlet
De la théorie des ensembles?
modest hamlet
Ah ok
solemn trail
en se moment je suis au cegep, on a fait les dérivées, quand même assez interessant
modest hamlet
Nous on a fait ça en terminale
solemn trail
terminale?
modest hamlet
Oui
solemn trail
sa veut dire quoi
modest hamlet
Last year of high school
solemn trail
ah wow
modest hamlet
🙂
solemn trail
bon le system francais est plus en avance que celui quebecois
solemn trail
j'aimerais visiter la france une journée, pas pour la tour eiffel mais vraiment pour voir les différences culturelles entre les deux pays
solemn trail
ah bon moi chui pas rendu la dans mon acheminement mathématique, du coup je te souhaite bonne chance et salut
cedar kilnBOT
cedar kilnBOT
Available help channel!
silent island
Hello, I can't seem to figure out where I'm going wrong with this question (I'm trying to get from RHS to LHS)
silent island
2 * 4 * ... * 2n
isn't
2 * ( 1 * 2 * ... n)
wait why not?
muted timber
because you're taking out a 2 per factor
livid hound
2 * 2 isn't 2 * (1 * 1)
silent island
because you're taking out a 2 per factor
ohhh
i was treating it as if it was an addition
but with multiplcation i would have to take out 2^n not just 2
yes that makes it work, thanks!
.close
odd flower
Nine sixteenths two thirds is
What is that?
void sand
*Nine sixteenths two thirds* is
What is that?
I also don't know what the question here is 
is this nine sixteenths times two thirds?
nine sixteenths plus two thirds?
what are you asking?? 
void sand
I can't read... Polish I think this is?
void sand
well uh, I don't exactly feel compelled to 
may I ask if you could state your original question more precisely?
odd flower
🤣
brave copper
hollow trail
looks like some sort of game show
odd flower
hiii lilith
hollow trail
without further context i would assume it means something like "nine sixteenths of two thirds" which would correspond to multiplying them
dusk goblet
english
opal pebble
If it is then you multiply 9/16 by 3/2
dusk goblet
is this like a game show?
odd flower
No
opal pebble
Rioke
If it is (9/16)/(2/3)
Then you multiply by the inverse of the bottom fraction
so now it's (9/16) X (3/2)
cedar kilnBOT
opal pebble
Dude I am trying to help you
odd flower
👺
odd flower
Dude I am trying to help you
I just don’t understand
opal pebble
When you are dividing by a number less than 1 youre making the number bigger
Like 1 divided by 0.5
1/0.5
1/(1/2)
so instead you multiply by the inverse of the fraction
odd flower
But both lower than 1?
odd flower
,calc 0.6/0.6
wraith daggerBOT
Result:
1
Bo…
opal pebble
,calc (9/16)/(2/3)
wraith daggerBOT
Result:
0.84375
,calc (9/16)*(3/2)
wraith daggerBOT
Result:
0.84375
See
odd flower
,calc 9/16
wraith daggerBOT
Result:
0.5625
,calc 3/2
wraith daggerBOT
Result:
1.5
Do you see how it flips when you multiply by the inverse of the fraction
odd flower
okey
opal pebble
does it make sense now
cedar kilnBOT
Available help channel!
ancient ore
I've done the first two parts. I don't know how to approach the third part.
dire thorn
plug in t=1 for the 2 above integrals, also notice 1! = 1, 2! = 2, 3! = 6
ancient ore
ohhh because LHS of eq2 is the derivative of LHS of eq1
and further, LHS of eq3 is the derivative of LHS of eq2
ok then it can be solved from here, if t=1
ancient ore
plug in t=1 for the 2 above integrals, also notice 1! = 1, 2! = 2, 3! = 6
this might be a dumb question, but how do you know to plug in t=1? i'm really bad at maths
dire geode
1+tx = 1+x when t=?
cedar kilnBOT
@ancient ore Has your question been resolved?
dire thorn
this might be a dumb question, but how do you know to plug in t=1? i'm really b...
You want the bottom integral to match the above integrals so you do what riemann said
tidal vapor
"an" is a unit vector. does last sentence mean we scaled "an" vector with |b|*cos(theta) to get the desired vector?
quasi merlin
"an" is a unit vector. does last sentence mean we scaled "an" vector with |b|*co...
Yes it does
(This is known as vector projection)
tidal vapor
if it wasn't projected onto a unit vector,
would that change anything in the equation?
quasi merlin
The projection of some vector b unto another vector a would still involve the unit vector for a.
If it wasn't a unit vector (call it a), you would end with the new vector scaled by |a|
tidal vapor
a.b=|a||b|*cos(theta)
quasi merlin
Yh
tidal vapor
The projection of some vector b unto another vector a would still involve the un...
to cancel out we would have to divide right?
quasi merlin
Yep
tidal vapor
|b|*cos(theta) = a.b/|a| where left side is length of projected matrix?
aah my brain fried
nvm i give up
.close
cedar kilnBOT
cedar kilnBOT
Available help channel!
void sand
how does this proof of mine look?
cedar kilnBOT
how does this proof of mine look?
void sand
lmk if you spot any glaring mistakes
prisma laurel
looks fine
prisma laurel
I guess there's some nitpick to be done if you take boundary points, in the case of manifolds with boundary
limber marsh
void sand
void sand
I guess there's some nitpick to be done if you take boundary points, in the case...
bleh 
prisma laurel
but I don't think the proof really changes at all in that case
limber marsh
how does this proof of mine look?
isn’t this just immediate from the definition? i didn’t read the proof and assumed it was right
prisma laurel
I think ISM just does those by taking charts in the half-plane right
void sand
isn’t this just immediate from the definition? i didn’t read the proof and assum...
idk, nothing is immediate for me 
void sand
ic
limber marsh
neat
limber marsh
the exercise tells you that it doesn’t matter what chart you pick
smooth functions are smooth in every atlas chart
void sand
the exercise tells you that it doesn’t matter what chart you pick
I noticed that it was like, very close to the def of smoothness
but the main thing I see missing is the F(U') \subseteq V'
I didn't conclude that during my proof
limber marsh
well you can just restrict U’ if F(u) is outside V’ for u in U’
everything is so nice in smooth manifolds
limber marsh
because you can just restrict to a smaller subset
void sand
so smoothness doesn't depend on what charts we use
limber marsh
yeah and the proof will pretty much use the composition of smooth functions being smooth
like not even reading your proof too hard i bet that’s one of the only things needed
void sand
everything is so nice in smooth manifolds
I feel like the definitions are so... bulky?
like there's just so many moving parts
void sand
no like, all the smooth manifolds defs I've ran into so far take like
3 sentences to state
so when I need to prove smth, there's a lot to do
prisma laurel
I feel like the definitions are so... bulky?
geometric ideas can be hard to make precise and sometimes result in very verbose mathematical statements
limber marsh
because the moral is much shorter
void sand
because the moral is much shorter
right, I get the feeling that a large part of the definitions are just there for technical reasons
limber marsh
right, I get the feeling that a large part of the definitions are just there for...
do you know vakil
void sand
the author, or his AG book?
limber marsh
the author
void sand
sure I do
limber marsh
vakil came to give talk at my university and one of the main points that he really wanted to hammer home is that people prove things before they come up with precise definitions sometimes
which was a cool concept
he gave this idea when he was talking about “bott periodicity in AG” which they just released a paper about
but the whole point was about talking about what it should “look like” first and only after doing that coming up with “good” definitions
void sand
vakil came to give talk at my university and one of the main points that he real...
does he mean like, people will try to prove smth based on a moral defn, and then slowly add conditions to the defns of the objects in order to make the proof work?
limber marsh
yeah
void sand
I see 
limber marsh
but this is true of many definitions in math and you should be looking for morals in all of them
and especially something so closely related to physics like manifold theory
this is part of why history of math is such a good subject to pair with the math itself
like it can really put your definitions into context
not that you would have to do that
just knowing some sort of morals is good enough
void sand
gotcha 
hold on, I gotta go shovel snow
if you have more to say, say it here and I'll be back in 15min to read it
limber marsh
if you have more to say, say it here and I'll be back in 15min to read it
hats it
that’s
void sand
okie dokie
I'll close the channel then
thank you derivada and smay for the help and the insights!
.solved
cedar kilnBOT
cedar kilnBOT
Available help channel!
frozen spindle
http://www.cs.rpi.edu/~magdon/courses/cf/notes/american.pdf
I need help understanding the proof of Exercise 4.1 here. There is some subtlety once it introduces the expectation under a measure that loses me.
frozen spindle
in particular, it is not clear to me what we are integrating over in there
like I can see that our probability space is the space of paths, but are the variables functions from paths to reals?
something about it isn't clicking
cedar kilnBOT
http://www.cs.rpi.edu/~magdon/courses/cf/notes/american.pdf
I need help understa...
Please post images (such as PNGs or JPGs) of the question rather than other filetypes such as PDFs which have to be downloaded. Non-image downloads can potentially contain viruses or other security risks.
cedar kilnBOT
@frozen spindle Has your question been resolved?
cedar kilnBOT
@frozen spindle Has your question been resolved?
dire geode
Try uhhhh #advanced-probability
fair wing
is it possible to find a way to go from two points by only moving in certain increments? ex: moving 1 away from where you started, but you can only iterate by 0.3 units
mystic egret
is it possible to find a way to go from two points by only moving in certain inc...
in a 2-dimensional space?
fair wing
yes
mystic egret
well then intuitively yes
twilit bison
if you're closer than .3 you can get there in 2 steps
fair wing
yeah ik cuz you'd just make an isosceles triangle
twilit bison
yeah
fair wing
i should ask is there a formula or something for it
well thats what i meant to ask rather
twilit bison
for what, the least number of steps?
i think you can most likely achieve a minimum if you just go directly towards your destination
fair wing
i guess so, i've just realized if it's less than the distance you can just b-line it and then make an isosceles triangle
let me visualize it rq
fair geyser
it doesn't feel optimal to do 3 steps
fair wing
if you have 0.3 unit-long steps to go 1 unit total then if you do the simple method you have 5 steps total
but 4 is possible
fair geyser
exactly
fair wing
so is there a way to find the angles needed to go 4 units
wait nvm i see it all
im overcomplicating it
.solved
dry pawn
The 1st image is the question, 2nd image is the answer from this book to the question. I am confused at the answer, why the n(-4/n) becomes 4, why no negative?
upper laurel
The 1st image is the question, 2nd image is the answer from this book to the que...
looks like a mistake
cedar kilnBOT
@dry pawn Has your question been resolved?
dry pawn
looks like a mistake
I see, thanks
untold lichen
https://en.m.wikipedia.org/wiki/Heptadecagon
do anyone know how they "solved for X"?
I thought that was the part the derivation should be explaining
teal wraith
💀
cedar kilnBOT
@untold lichen Has your question been resolved?
fresh patio
<https://en.m.wikipedia.org/wiki/Heptadecagon>
do anyone know how they "solved f...
You will probably want to consult one of the references
light knoll
chatgpt is giving me this:
cedar kilnBOT
chatgpt is giving me this:
light knoll
but on wikipedia I'm seeing this:
so it would seem that chatgpt is giving me the negation of the actual formula on wikipedia?
nimble veldt
dont trust chatgpt.
light knoll
dont trust chatgpt.
I'm just trying to confirm it's formula is incorrect, and there isn't something I'm not seeing
prisma laurel
I believe here ${\bf r_{21}}={\bf r}_2-{\bf r}_1$
wraith daggerBOT
derivada.schwarziana
prisma laurel
while $\widehat {\bf r_{21}} = \dfrac{{\bf r_{21}}}{|{\bf r_{21}}|}$
wraith daggerBOT
derivada.schwarziana
prisma laurel
dont trust chatgpt.
also 
, can be an useful tool at times but only if you know enough to be able to fact-check its answers. LLMs aren't well suited for mathematical reasoning since they're basically text prediction with a lot of resources lol
light knoll
also <:this:489575168314966026>, can be an useful tool at times but only if you ...
I'm just using it to guide myself, and fact checking everything it says, which is why I'm here, because it looks to be exactly the same as the second form given on wikipedia, given r21 = r2 - r1, but the answer is it's negation, so the F12 from chatgpt is incorrect?
dire geode
Reading books is a much better guide because they don't waste your time being wrong
prisma laurel
I'm just using it to guide myself, and fact checking everything it says, which i...
that's okay
I think it depends on the order convention of the indices
nimble veldt
I'm just using it to guide myself, and fact checking everything it says, which i...
if you wanna check chatgpts calculation post the whole calculation not only the result.
prisma laurel
**derivada.schwarziana**
this would be a force vector pointing from r_1 to r_2, so if F_{21} is the gravitation force that body #2 exerts on body #1 then that's fine
if not then it's the other way around. boils down to convention
light knoll
if you wanna check chatgpts calculation post the whole calculation not only the ...
I was asking about it's recreation of the formula, but I'll definitely keep that in mind in future
.resolve
thank you!
prisma laurel
(and admittedly I'm not a physicist so I'm rusty on this lol)
light knoll
.close
cedar kilnBOT
cedar kilnBOT
Available help channel!
light knoll
for
dire geode
(Delta t)^2
cedar kilnBOT
cedar kilnBOT
Available help channel!
lilac flame
a child has 6 pockets and 5 coins and in how many ways can he put the coins into the pockets
lilac flame
i wanna understand the logic
fair oracle
the coins are all the same i assumev
steel heart
There are 5 coins and 6 pockets
hexed vortex
The key to understanding the logic behind combinations is to first understand the logic behind permutations
dusk finch
this doesnt have much to do with combinations i believe
coral jewel
can 1 pocket contain multiple coins
hexed vortex
if the coins are identical, where there is no order then wouldnt that be a combination?
coral jewel
if yes, then ⭐ 📊
steel heart
this doesnt have much to do with combinations i believe
Stars and Bar theorem?
dusk finch
if yes, then ⭐ 📊
Yay someone understood it
steel heart
That is combinatorics
dusk finch
combinations are the binomial coefficients
hexed vortex
Combination is like C(2,3) right
dusk finch
ye
hexed vortex
Is this question dealing with that?
like wouldnt it be similar to saying if you have 6 positions and 5 people, how many different ways can you position those people
dusk finch
can 1 pocket contain multiple coins
we need answer to this
coral jewel
@lilac flame
fair oracle
like wouldnt it be similar to saying if you have 6 positions and 5 people, how m...
ye
steel heart
There are 5 coins and 6 pockets
*****|||||
dusk finch
you only need 5 seperators
the pockets will be between them
e.g.
* | * * | * | | * |
hexed vortex
this is similar to a problem I've encountered before dealing with functional antennas lol
coral jewel
op asks question and then just dips 
dusk finch
e.g.
\* | * * | * | | * |
this would mean
1st pocket has 1 coin
2nd pocket has 2 coins
3rd pocket has 1 coin
4th pocket has 0 coins
5th pocket has 1 coin
6th pocket has 0 coins
steel heart
op asks question and then just dips <:P_SobbingPien:1283294945779912798>
He got scared by 4 helper tryna help him
dusk finch
Let's just finish it anyway lol
soo we are essentially just counting the number of strings with 5 coins (*) and 5 seperators (|)
there are 10 total characters in such string
and we need to choose 5 characters to be the seperators
hence 10 choose 5
this is where the choice function comes in
and that should be it
coral jewel
this is not the way
wraith daggerBOT
The following error occured while calculating:
Error: Undefined function C
dusk finch
,w 10 choose 5
wraith daggerBOT
dusk finch
252
coral jewel
this is the way
dusk finch
⭐ & 📊
hollow trail
,calc combinations(10,5)
wraith daggerBOT
Result:
252
question its the same as having aaaaabbbbb and finding how many ways you can like scramble them right
dusk finch
yep
fair oracle
so yea 10!/((5!)(5!)) would also work
cedar kilnBOT
252
The purpose of this server is to help you learn, not to hand out answers. Do not ask someone to give you the answer directly.
fair oracle
which is the same as 10 choose 5
coral jewel
!noans <:KEK:586240877358350341>
!nosols
cedar kilnBOT
!noans <:KEK:586240877358350341>
As a helper, please do not give out answers that could be copied as a homework solution. Have the student work through the problem themselves and guide them along the way.
steel heart
Wait those two are different
hexed vortex
reminds me of this question I had did a while back that I was struggling with lol
still dont quite understand the solution
dusk finch
reminds me of this question I had did a while back that I was struggling with lo...
of this or of mistav's problem?
hexed vortex
oh wait no nvm it make sense looking at it again
hexed vortex
reminds me of this question I had did a while back that I was struggling with lo...
this one
mistav still hasnt come back lol
fair oracle
i skipped combinatorics and probability so im not good at it
hexed vortex
reminds me of this question I had did a while back that I was struggling with lo...
I havent taken a class in it either so I've been trying to read this book on my own
I think its a really intersting book
has some pretty intriguing questions and examples
fair oracle
how do you translate no two defectives are consecutive
dusk finch
I think that you could start by arranging the functional ones
and then there will be gaps
in each gap, you can place at most 1 defective
and this way, you will ensure that no defectives will be consecutive
hexed vortex
how do you translate no two defectives are consecutive
It means this is not possible | * | ** | * | * where * is defective and | is functional
dusk finch
is it n - m + 1 choose m btw?
dusk finch
yep
cedar kilnBOT
@lilac flame Has your question been resolved?
fair oracle
It means this is not possible | * | ** | * | * where * is defective and | is fun...
what if it said no three consecutive
hexed vortex
what if it said no three consecutive
Then | * | ** | * | * would be possible and that means instead of allowing only one non-functional position between each bar there can be two, which in turns double the number of positions you can have for non-functional antennas, so it'd be
C(2(n-m+1), m)
hexed vortex
Then | * | ** | * | * would be possible and that means instead of allowing only...
@dusk finch does this sound right
dusk finch
Then | * | ** | * | * would be possible and that means instead of allowing only...
sounds right
hexed vortex
thanks
fair oracle
oh awesome ty
cosmic umbra
can someone help me 108
cedar kilnBOT
can someone help me 108
hexed vortex
can someone help me 108
You want to minimize the distance from some general point (x,y,z) to (4,2,0), thus you'd be minimizing the distance formula sqrt((x-4)^2 + (y-2)^2 + z^2) given the constraint z^2 = x^2 + y^2
cosmic umbra
what do you mean sqrt((x-4)^2 + (y-2)^2 + z^2) you mean the answer should be small as much as posible ??
upper ruin
Minimize means that, indeed
hexed vortex
what do you mean sqrt((x-4)^2 + (y-2)^2 + z^2) you mean the answer should be sma...
This is an optimization problem and with these problems the goal is to identify your contraint function and your objective function. Here the constraint function is stated in the question but the objective function, the function we wish to minimize or maximize is given through words. It says we want to find the closest point to (4,2,0) given that we are constrained to our constraint function $z^2 = x^2 + y^2$. So out of all points that lie on this cone we want to find the point closest to (4,2,0), meaning we need to find the point that has the minimum distance to (4,2,0). We find the distance between points using the distance formula $\sqrt{(x-{x_o})^2 + (y-{y_o})^2 + (z-{z_o})^2}$ where $(x_0, y_0, z_0) = (4,2,0)$. If we want to find the closest point then we must find (x,y,z) that minimizes this distance formula. Think back to finding the minimum of a function in early calc 1 by seting the first derivative to 0 and applying the first derivative test
wraith daggerBOT
Olivia
hexed vortex
Here with multivariable functions you might use something like lagrange multipliers
cedar kilnBOT
@cosmic umbra Has your question been resolved?
cosmic umbra
.close
lilac flame
a boy has 6 pockets and 5 coins in how many ways can he put the coins in his pockets, answer is 6^5 i wanna understand the logic
hollow kiln
first coin can go to 6 different pockets
lilac flame
i assume they did first coin can go into 6 pockets and so on, but whats wrong with the approach of first pocket can get 5 coins
hexed vortex
i wanna understand the logic
up here
lilac flame
we answered this one
oh yeah sorry i went to eat lunch mb
hollow kiln
and so on
fair oracle
There are 5 coins and 6 pockets
```*****|||||```
think of it like this
lilac flame
yeah i understood that
lilac flame
" but whats wrong with the approach of first pocket can get 5 coins" whats wrong with this
fair oracle
" but whats wrong with the approach of first pocket can get 5 coins" whats wrong...
its not wrong it just takes too long to go case by case
the second pocket also can get 5 coins
and the other ones too
then umd have to do case 4 coins 1 coin
then 3 2
hollow kiln
first coin can go to 6 different pockets
you read that?
lilac flame
Yep
cedar kilnBOT
cedar kilnBOT
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hollow kiln
ABC is an equilateral triangle
how to show BD+CD=AD
cedar kilnBOT
ABC is an equilateral triangle
how to show BD+CD=AD
cedar kilnBOT
@hollow kiln Has your question been resolved?
hollow kiln
<@&286206848099549185>
uneven quartz
ABC is an equilateral triangle
how to show BD+CD=AD
Use the law of cosines.
Firstly, you can infer that angle ADB = angle ACB = 60 deg and that angle ADC = angle ABC = 60 deg (why?)
AB = AC
Use law of cosines on triangle ADB and triangle ADC with the angles ADB and ADC which are 60 deg each as deduced earlier.
You can easily eliminate other variables so that only AD, BD and CD are left in the system of equations.
You'll be left with $a^2 - c^2 = ab - bc$ which implies $(a+c)(a-c) = b(a-c)$ and so $(a+c) = b$ where a, b and c are BD, AD and CD respectively.
wraith daggerBOT
anshul6368
cedar kilnBOT
uneven quartz
No problem.
cedar kilnBOT
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worldly socket
a^(2n+1) = a - ab^(2n)
a - ab^(2n) + b^(2n+1) = 1
a = (1 - b^(2n+1))/(1-b^(2n))
I found a in terms of b and n. Does this mean that ive found the solutions?
worldly socket
mighty shuttle
You could instead say $a^{2n}=x; b^{2n}=y$. Then $x+y=1, ax+by=1$
solve for x and y now
and then n
crimson delta
You could instead say $a^{2n}=x; b^{2n}=y$. Then $x+y=1, ax+by=1$
think about that second equation again
wraith daggerBOT
ƒ( wai ina teacup)= I don't know
worldly socket
ok thanks
why does my solution not work?
it says there are infinite solutions and i have given the expression for a in terms of any b and n
crimson delta
but are there infinitely many solutions?
worldly socket
yeah
crimson delta
presumably you are supposed to solve for a,b for a fixed n
worldly socket
That's when a,b,x,y are all unrelated
oh
earnest geyser
Wdym
dusk finch
Are you supposed to solve them simultaneously?
worldly socket
yeah
dusk finch
Is that x and y thing even necessary then?
earnest geyser
Not rly
worldly socket
ok
earnest geyser
I think there are no answers apart from a=1 and b=0 or vice versa
Because suppose either of them are not 0
worldly socket
yeah
earnest geyser
a^2n is a positive value as well as b^2n
worldly socket
yeah
earnest geyser
a^(2n+1)<a^2n since a<1
worldly socket
yea
earnest geyser
Same with b
earnest geyser
U got it?
worldly socket
yep thanks
earnest geyser
K
worldly socket
.solved
cedar kilnBOT
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crimson sedge
i need to study the variations of f
but first i wanted to prove that whats inside the ln is positive(or at least what the domain is so that it stays positive)
aaand i cant quite get my hand on it
onyx dove
u=e^x substitution
cedar kilnBOT
@crimson sedge Has your question been resolved?
crimson sedge
u=e^x substitution
god im so stupid
cedar kilnBOT
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crimson sedge
hey
crimson sedge
sorry about last time my internet was awful
i wanted to ask about injective and surjective functions
can someone please explain them to me? all i get on google are those stupid circles with arrows...
like... the two seem VERY similar to me
from the explanations that i read
crimson sedge
i am here @mighty shuttle
crimson sedge
okayy
mighty shuttle
Essentially in a one one function, each output hasa. a unique input
crimson sedge
okay so no more and no less than one input
crimson delta
wai you need to be more precise
crimson sedge
cuz it can be 1 with different inputs
crimson delta
injective and surjective depend hugely on domain and codomain
crimson sedge
how so
mighty shuttle
Technically speaking a functions is said to be one -one , if an element in the codomain has at most one pre-image in the domain
crimson sedge
okay
and surjective is really similar... or is it not?
cuz the way we prove it
what
crimson sedge
what's your defn
each y in the codomain has an x in the domain
mighty shuttle
each y in the co domain has ateast one x in the domain
crimson sedge
and with injective, it has just one?
one x in the domain?
is that the difference?
mighty shuttle
at most one x
mighty shuttle
cool, then can I quiz you to ensure you've understood?
crimson sedge
sure
crimson sedge
define $f(x) to be -x^2 , if x<0, and x^2 if x \geq 0
define $f(x) to be -x^2 , if x<0, and x^2 if x \geq 0$
wraith daggerBOT
CUPS
crimson sedge
okay hmmm
mighty shuttle
This is a bit hard, so don't be discouraged
mighty shuttle
awesome!
cedar kilnBOT
agile swallow
injective and surjective depend hugely on domain and codomain
Best way to go about this
Imo
cedar kilnBOT
Available help channel!
weary thistle
how do I differentiate arctanx with respect to tanx? (for xε(-π/2,π/2))
lyric narwhal
do you know what a derivative of a function with respect to another function means in general
weary thistle
yes, but im not familiar on how to approach problems that require to take the derivatewith respect to a whole function
lyric narwhal
sure, but we'll write it a bit differently
Let $f(x)=\arctan(x)$ and $y(x)=\tan(x)$. Then $$\dv{f}{y}=\dv{f}{x}\dv{x}{y}$$
wraith daggerBOT
kheerii
lyric narwhal
does this make sense
weary thistle
if we treat it like a fraction too, yes
lyric narwhal
this is precisely what the chain rule says
lyric narwhal
$\dv{x}{y}=\frac1{\dv{y}{x}}$
wraith daggerBOT
kheerii
lyric narwhal
yes
lyric narwhal
in general $\dv{y}{x}=\frac{\dv{y}{t}}{\dv{x}{t}}$ where $x$ and $y$ are functions of another variable $t$
wraith daggerBOT
kheerii
weary thistle
in the task, it also says: consider known the derivative of tanx. (doesnt say anything about the derivative of arctanx with respect to x) With the help of the definition of the derivative find d/dy(arctanx)
does it want me to calculate in some other way?
@lyric narwhal
cedar kilnBOT
@weary thistle Has your question been resolved?
cedar kilnBOT
@weary thistle Has your question been resolved?
cedar kilnBOT
@weary thistle Has your question been resolved?
weary thistle
<@&286206848099549185>
weary thistle
it wants me to find
in the task, it also says: consider known the derivative of tanx. (doesnt say anything about the derivative of arctanx with respect to x) With the help of the definition of the derivative find d/dy(arctanx)
austere ember
okay, so you know the derivative of tanx is (secx)^2
if y = tanx, then arctan(y) = arctan(tan(x)) = x
(assuming that x has an appropriate domain, which it does here)
weary thistle
oh yes
austere ember
actually wait sorry i did that backwards, let y = arctanx, then tan(y) = tan(arctanx) = x
you ultimately want to find y'
weary thistle
no y=tanx
austere ember
implicitly differentiate and use your known tan derivative and chain rule
weary thistle
it says here
austere ember
okay this problem is written very poorly, d/dy (arctanx) is clearly 0
are you meant to find the derivative of arctanx?
like d/dx
austere coyote
No
austere ember
ohh
weary thistle
how am i supposed to do that
austere coyote
Look
austere coyote
Neil if I may
austere ember
**kheerii**
then you're looking for this
austere coyote
Xtra write x in respect of y
Saying that x equals arctan y
Plug that into arctan x
Then you have the arctan of arctan of y
Then differentiate with respect to y
It should be 1/1+(arctany)^2 times 1/1+y^2
If not check wolframalpha
weary thistle
in the task, it also says: consider known the derivative of tanx. (doesnt say an...
this is what the task is saying
austere coyote
It might just be another way to get the same answer
weary thistle
what does it mean "definition of derivative?"
also it doesnt say consider known the derivative of arctanx like it does for tanx
so how will i say that d/dx(arctanx)=1/1+x^2
maybe it wants me to do it some other way around?
austere coyote
It looks like I’m right I googled it
weary thistle
It looks like I’m right I googled it
i know, but is there any othe rway?
austere coyote
Honestly I don’t know I’d have to think about it
Would it matter if you did it differently?
weary thistle
i dont know, thats the thing haha
weary thistle
thanks. You can DM me if you want. If you dont think anything else, thats ok. im tired of this problem, been looking it for 2 days now. im closing this channel, and plz let me know if anything
.close
cedar kilnBOT
ocean zenith
Is the ans to 6÷2 (1+2)... equal to 9 or 1 ?
lyric kite
9
lyric kite
6/2*3
lyric kite
the parentheses say that you do 1+2 first, and then you do left to right
steel crest
Yes
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fallen trail
Why isn't the first one correct but the second one is?
cedar kilnBOT
Why isn't the first one correct but the second one is?
fallen trail
Btw how do we graph implicit differentiation in desmos?
worldly chasm
I do not think there is a way to do it. But I could be wrong.
dire geode
,w diff cos(xy(x))
fallen trail
I do not think there is a way to do it. But I could be wrong.
ooh
wraith daggerBOT
fallen trail
dire geode
Use y' instead of dy/dx
fallen trail
Why isn't the first one correct but the second one is?
we want to find dy/dx of xy=sin^-1(2x)
fallen trail
Use y' instead of dy/dx
ooh
fallen trail
Why isn't the first one correct but the second one is?
appears I'm correct
are these results equal?
For some reason the book does it this way
odd verge
,w diff arcsin2x/x
fallen trail
are these results equal?
def not 
Unless if I'm graphing these implicit equations wrong
I'm so conflicted
Do I believe myself and wolframe or the book 💀
odd verge
,w diff arcsin2x/x
wraith daggerBOT
fallen trail
fallen trail
For some reason the book does it this way <:thinkies:827642567775748163>
This is the book's answer btw
odd verge
Hmm okay
odd verge
I think this is correct
cedar kilnBOT
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fallen trail
You should probably get other opinions too though.
Honestly it's pretty weird I think what would've been most helpful would've been to be able to graph the equations
fallen trail
.close
cedar kilnBOT
odd verge
<:thinkies:827642567775748163>
odd verge
Same as this
odd verge
All three answers are literally the same.
They just have different styles.
Cosxy = √{1-sin²(xy)}
@fallen trail
fallen trail
Cosxy = √{1-sin²(xy)}
OOOOOOOOOOOHH
cedar kilnBOT
OOOOOOOOOOOHH
If you are done with this channel, please mark your problem as solved by typing .close
fallen trail
If you are done with this channel, please mark your problem as solved by typing ...
.close
.close
.close
😭
odd verge
odd verge
Oh
cedar kilnBOT
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twilit badger
Question: is there an intuitive way to understand why the chain rule works for find the derivative and integral of a function: like x² for example. Why is it that the derivative is 2x and the integral is 1/3x² ?
dreamy void
Like many things in mathematics, these are just results of applying the definition of something; here the definition of a derivative which is essentially a limit.
Applying this definition leads to results such as power, product, chain rule etc. or specifically that derivative.
[ f'(x) := \lim_{h \to 0} \frac{f(x+h)-f(x)}{h} ]
So with $f(x) = x^2$ you'd have
[ f'(x) = \lim_{h \to 0} \frac{(x+h)^2-x^2}{h} ]
and if you calculate that limit you get $f'(x) = 2x.$
wraith daggerBOT
𝔸dωn𝓲²s
dire geode
Question: is there an intuitive way to understand why the chain rule works for f...
The answer is gonna depend on what's intuitive for you
twilit badger
So would it be difficult to make a logical conclusion that doesn't rely on proofs?
twilit badger
The answer is gonna depend on what's intuitive for you
Hmm I suppose so
I have researched many ways to interpret integrals and derivatives
derivates fundamentally I roughly understand
dire geode
If you don't understand the algebraic steps then you're not gonna have a good time in calculus and beyond
dreamy void
If you yearn for greater understanding, I would try to analyze the definition rather? Why would someone define exactly like that a derivative and what is the intution behind that, instead of a specific example that is really just computation.
twilit badger
The algebra isn't a problem
twilit badger
If you yearn for greater understanding, I would try to analyze the definition ra...
So you mean that the chain rule is something too specific to try to intuitively understand?
dreamy void
No I meant your example with x²
twilit badger
Ah yes well that is just and example for what I meant
dreamy void
There are a lot of intuitive videos that explain the idea and definition of a derivative
twilit badger
But now that I think about it it does seem a bit arbitrary
twilit badger
There are a lot of intuitive videos that explain the idea and definition of a de...
Derivative is less so a problem for me
I more struggle with integrals
and also the relation between the two
dreamy void
@dire geode sums haha
Integrals are also based on limits, the idea behind integrals is to find the area under a curve vaguely speaking
But before this you should be familiar with limits
twilit badger
Limits are also fine for me
I'm going to take my example again but tell me if I am thinking to deep about it
But why is it that 1/3x³ describes the area under x² ?
dreamy void
I hate to say it but that is the result from applying the definition
twilit badger
But I suppose like you said that that is just the result of doing the math after taking the conclusion
Hmm yes
dreamy void
There is not really something special about why it's x^3 now or 1/3 instead of any arbitrary other term
twilit badger
Okay
I guess I'll just think about it a little more
I appreciate your and the other persons help immensely!
dreamy void
It's like say you have or define X and from X there results something new called Y.
Now you are asking me how and why Y exactly? But see this is just a consequence of something deeper, here X, or in your example the definition of an integral.
dim tiger
btw f(x)=x^2 isnt a very good example to see the chain rule
twilit badger
It's like say you have or define X and from X there results something new called...
Yes I suppose so
twilit badger
btw f(x)=x^2 isnt a very good example to see the chain rule
How so?
dreamy void
btw f(x)=x^2 isnt a very good example to see the chain rule
2x * x' = 2x * 1 = 2x
dreamy void
it's like a trivial chain rule
opaque root
twilit badger
What would be a better example?
dreamy void
f(x) = x² and g(x) = sin(x)
Consider g(f(x)) = sin(x²)
dim tiger
f(x) = x² and g(x) = sin(x)
Consider g(f(x)) = sin(x²)
thats exactly the same example i was thinking of 
twilit badger
f(x) = x² and g(x) = sin(x)
Consider g(f(x)) = sin(x²)
Okay, thanks!
dreamy void
Also here you can see the "chain rule" is based again on that very definition of derivative
https://en.wikipedia.org/wiki/Chain_rule#First_proof
You don't have to understand it but be grateful that such results make life more abelian (simple)
twilit badger
Standing on the shoulders of giants I suppose
Alright, I appreciate the help from everyone
.close
cedar kilnBOT
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neat fern
anyone can help me with that?
1.- DERIVAR:
f(x)= 〖(2X-3)〗^5 〖(X^2+4X)〗^6
2.- DERIVAR:
f(x)= 〖(4X-3)〗^3/〖(2X-3)〗^4
dull oxide
anyone can help me with that?
1.- DERIVAR:
f(x)= 〖(2X-3)〗^5 〖(X^2+4X...
!sttaus
!status
cedar kilnBOT
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
dull oxide
You will need chain rule for both problems
you'll nee product rule for the first problem, and quotient rule for the second
dull oxide
espanol?
neat fern
yes
cedar kilnBOT
@neat fern Has your question been resolved?
cedar kilnBOT
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upbeat lotus
cedar kilnBOT
cedar kilnBOT
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noble trench
I need to prove that the angle phi between the medians of the triangle is equal to the term in the image. The sides of the equilateral triangle are b and the base is a. I have a feeling I need to use the law of cosines but I can't seem to materialize it. Sorry for the low quality
worldly chasm
Isosceles?
noble trench
Yes sorry I'm not familiar with the terms in English