I am not good at explaining 
#help-43
blissful tangle
but you should be able to understand it once you read it from google
compact pewterBOT
@visual flint Has your question been resolved?
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distant arch
hi, can anyone explain how to get the transmission function?
compact pewterBOT
hi, can anyone explain how to get the transmission function?
silver canopy
hi, can anyone explain how to get the transmission function?
This question feels longer than it should be
Basically just convolve f and g
Utilise the sifting property of convolution
compact pewterBOT
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quartz yoke
Q9
unreal night
Q9 can be done using congruency
unreal night
I dont have my notebook with me right now
dense mason
Why don't you try to draw it?
unreal night
Its a parallelogram tho..
quartz yoke
Ik
unreal night
I mean trapezium
quartz yoke
Thats how bad my drawing is
unreal night
Ah. Try using a ruler
quartz yoke
NCERT class 10?
Yep
quartz yoke
Ah. Try using a ruler
Okay
But what sides will i take
I know there r 2 methods
One is similarity
And the other is BPT
quartz yoke
Im taking BPT cuz it's easier
quartz yoke
I am pretty sure you had to do some construction
You do?
bold latch
Like its top part is cut
quartz yoke
Can i see the figure plz
bold latch
quartz yoke
Awesome drawing skills prqt
Ooooooooookie I understand
Plz continue
quartz yoke
This is for BPT?
bold latch
This was just a basic properties which will be used, not the actual solving
quartz yoke
Alright
bold latch
Now we know that AB is parallel to DC
quartz yoke
Yeah we do
bold latch
That means that A and B points of that traingle must be the mid points
quartz yoke
But what about O?
bold latch
quartz yoke
Okieee
bold latch
Just because A and B were mid points of that traingle we can construct line MN
Which is mid points of AD and BC
quartz yoke
So Bo divided by Do would be equal to Bn by NC
quartz yoke
Uh
That wont work
Oh waitt
Ad divided by Md is 1
So Ao = OC right
Cuz the midpoints
unreal night
Charge your phone
quartz yoke
AM/MD = BN/NC
Trapszium?
bold latch
Charge your phone
Yes 😭
bold latch
Tysm prqt now charge ur phone
Yeah 😭
compact pewterBOT
quartz yoke
Whee
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inland kite
hello people, I absolutely have zero clue on how to start
compact pewterBOT
hello people, I absolutely have zero clue on how to start
inland kite
so, lemme pour my thoughts here indicating im trying lol
-
The function mentions R and Q, so density theorem might be useful
-
Since we need to prove f(x) -> 0 when x -> 0, epsilon-delta proofing style might be used
im talking abt (a) btw
any clue guys?
elfin finch
Well have you attempted using epsilon-delta proof yet
inland kite
no because idk how to write, lemme give u my draft
can I let delta be f(b)
I mean b
elfin finch
Sure
inland kite
ok, so its good so far?
elfin finch
What is this showing tho?
Also I don't think density theorem will be that important for finding the lim
wheat pasture
(also, as a tiny point, have you done epsilon delta proofs before for slightly more "basic" functions?)
inland kite
(also, as a tiny point, have you done epsilon delta proofs before for slightly m...
yes but im still a bit rusty, I know u must find some delta and then write the full proof
and that delta should have epsilon in it
elfin finch
full answer, pls fact check
Your error is going from x<b to |f(x)|<|f(b)|
wheat pasture
yes but im still a bit rusty, I know u must find some delta and then write the f...
Not to take over too much, but you may want to take a look at some of those examples if you have them - once you notice something, the proof here becomes relatively easy 
inland kite
Not to take over too much, but you may want to take a look at some of those exam...
examples such as lim x->2 2x+1 = 5?
smth like that?
alr I will look at some easier questions on yt brb...
dull moon
Yes, because rn you're not really writing all the required steps and its a bit difficult to follow your proofs.
wheat pasture
examples such as lim x->2 2x+1 = 5?
Yea, stuff like that 
linear functions are quite nice to work with
(what can you say about |f(x)|?)
inland kite
(what can you say about |f(x)|?)
wow u predicted what im gonna ask
I looked at bprp, he tried to replace |x-0| with delta
inland kite
(what can you say about |f(x)|?)
I mean it depends if x is rational or nah
wheat pasture
If it is? What about if it isn't?
wheat pasture
I'm happy 
wheat pasture
That's basically it, yea maybe put that you chose delta as epsilon at the start, and add a few more words, but basically you're there already
inland kite
I hope this makes u happy 🙏
wheat pasture
I would change it slightly - state that delta is > 0 (because we've already stated what it is, we don't really need to say that it "exists") but I am happy 
Of course, don't forget that there's one small extra part to the question - are we continuous at 0?
wheat pasture
There's one extra thing
You want the limit to exist (so both sided limits must approach the same thing), and it needs to match up to what the function value f(0) is
We just did the first part of showing the limit exists, but does that correspond to f(0)?
inland kite
I mean f(0) = 0 right? So we need to show RHS and LHS of x = 0 is equal to 0?
inland kite
We just did the first part of showing the limit exists, but does that correspond...
from the function itself, f(0) = 0
I think intuitively speaking, it is not continuous
wheat pasture
I think intuitively speaking, it is not continuous
Why not?
wheat pasture
What we want is that $\lim_{x\to 0} f(x) = f(0)$ here for $f$ to be continuous at $x = 0$, after all, and we just said that $\lim_{x\to 0} f(x) = 0$ \emph{and} that $f(0) = 0$
boreal girderBOT
@wheat pasture
inland kite
OHHH
wheat pasture
cuz it would jump abruptly
(and, well, more that the function basically "falls down" at the irrational points, so you're basically "f(x) = x but make it 0 at irrationals")
So, you can say that as you get closer to 0, f(x) is either "really small, or just 0"
inland kite
ohhh so for the continuity, can I say since lim x-> 0 f(x) = 0 = f(0), it is then continuous at 0
wheat pasture
Yep, that's it 
wheat pasture
The function will be discontinuous at any point apart from 0, yep - and that's gonna be our job to show 
Intuitively, you can probably see why it would be of course!
wheat pasture
We can do it by proof by contradiction, which might be the neatest way to lay it out, methinks!
inland kite
alr lemme make my draft
wait if it exists, what should it converge to
can I write it converges to c?
wheat pasture
Yea, you can just call it something like L or whatever you want
inland kite
Yea, you can just call it something like L or whatever you want <:catlove:920698...
im quite stuck on finding the upper bound
wheat pasture
What may help is to take a step back for a tiny second 
Saying that the limit exists is a more formal way of us saying that "as the input gets closer to the point we're approaching*, the output gets closer and closer to our limit"
(* we aren't too worried about the function value at the point in this case)
So what we take is that for any choice of $\epsilon > 0$ we pick, we can find some $\delta > 0$ such that whenever $0 < \abs{x - c} < \delta$, that we then have $\abs{f(x) - L} < \epsilon$
boreal girderBOT
@wheat pasture
wheat pasture
Not quite, we basically take the whole statement as true, and then try and get a contradiction
inland kite
of course
wheat pasture
I'm wondering whether negating the definition of convergence might work easier now a bit 
inland kite
there exists a positive epsilon, and for every positive delta, 0 < |x-c| < delta but |f(x) - L| >= epsilon
i think I have to find a bound for f(x) such that |f(x) - L| >= epsilon
is that a good motivation
wheat pasture
Basically, noting that we also want to start off with the fact that "for any L", because saying the limit exists says that some L exists that we converge to
inland kite
yes...
wheat pasture
<@788085606483361802>
Anyways, my thoughts for the original one - "if we take irrational points, we would be basically saying that L should be zero, but if we took rational points, we'd be saying that L is nonzero"
wheat pasture
That's basically the "informal" version of what we're going for
wheat pasture
but I still need to find some delta tho
We don't need to here - we are assuming, rather than trying to show, convergence
If we were showing convergence, we would need to find some delta of course (and the limit L too, for which epsilon-delta is not particulalry good at finding when it's for verifying)
wheat pasture
Also have you covered the sequential version of limits before?
inland kite
is this valid
inland kite
Also have you covered the sequential version of limits before?
yes, we were taught sequential limit theorem and stuff
wheat pasture
is this valid
It is worth being a bit careful - it is more that neither sided limits actually exist!
wheat pasture
yes, we were taught sequential limit theorem and stuff
That does make things a lot easier then - you can take a sequence of rational 
points approaching c, and then take a sequence of irrational 
points approaching c, which rules out quite a bit of the fiddly work
inland kite
here is my approach with SLT
inland kite
That does make things a lot easier then - you can take a sequence of rational <:...
oooo
inland kite
It is worth being a bit careful - it is more that neither sided limits actually ...
I don't get wym, we assumed lim f(x) exists when x -> c
wheat pasture
I don't get wym, we assumed lim f(x) exists when x -> c
True, but they actually don't exist (and we can get a contradiction from that front) - at the very least, it isn't "wrong" as such, but we can do better 
inland kite
a\
wheat pasture
here is my approach with SLT
And this is not bad, but not complete - we could have that L is actually 0 (see the point we made before, that L is both zero and nonzero)
inland kite
That does make things a lot easier then - you can take a sequence of rational <:...
ooo I get what u mean, I just form two subsequences of f(x) consisting of only rational numbers and one consisting only irrational ones, then those two subsequences will converge to c and zero, respectively, a contradiction
wheat pasture
Yep, that's basically it 
inland kite
holy cow
inland kite
And this is not bad, but not complete - we _could_ have that L is actually 0 (se...
how are u certain it is zero
wheat pasture
how are u certain it is zero
I'm saying that there is the possibility that it is zero, which we didn't rule out in that version, but that does get ruled out when we show it's actually [also] something nonzero
inland kite
I'm saying that there _is_ the possibility that it is zero, which we didn't rule...
I see
anyways, thanks for the tip, I didn't thought of SLT
is it valid 🙏
I think the first subsequence is not necessary, as long as there exists one subsequence that does not converge to L, its a contradiction already
wdyt
wheat pasture
One small change, to make the limits be of $f(x_{n_i})$, rather than just of $x_{n_i}$
boreal girderBOT
@wheat pasture
inland kite
One small change, to make the limits be of $f(x_{n_i})$, rather than just of $x_...
ah ye ur right
wheat pasture
I think the first subsequence is not necessary, as long as there exists one subs...
Well, more that the contradiction we have lies on that L actually doesn't exist
inland kite
ah right, this contradiction stuff twists my mind a lot
alright
I think that would be it
I thought I can continue for the next question, but its already late
thank you so much fellow moderator, it is actually my first time being answered by a mod
wheat pasture
we sometimes appear around and all, while we are mods, some of us do like answering questions too 
Have a good one, it was a pleasure working with you, hopefully you rest well 
compact pewterBOT
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umbral sand
Could anyone explain what a direct and inverse variation is
compact pewterBOT
Could anyone explain what a direct and inverse variation is
sonic wyvern
direct variation means that as a quantity increases or decreases, the linked quantity changes in the same direction.
inverse variation means that as a quantity increases or decreases, the linked quantity changes in the opposite direction.
paper rune
direct variation: can be modeled by y = kx. inverse variation: can be modeled by y = k/x
umbral sand
that makes a bit more sense
dull moon
For example, the number of workers and the amount of work they do in one day vary directly.
the number of workers and the time taken to do the entire work vary inversely
umbral sand
could you help me out with a question
sonic wyvern
please send it!
sonic wyvern
,rccw
boreal girderBOT
dull moon
umbral sand
alot of it becuase this is a new topic for me
sonic wyvern
alright, from the top then, what is the first line you don't get? also, do tell us what you understood as well.
sonic wyvern
well, x is directly proportional to F.
as in, if you apply more force to the spring, it will stretch longer. I believe you agree?
umbral sand
yea\
sonic wyvern
right. but we don't know that the amount of force in Newtons is directly equal to the extension length in cm.
we do know that if the force increases, the extension length does too, but it's not a straightforward relationship as x = F.
so we introduce an unknown scaling factor k into our equation to show that x could vary by a factor of F instead of directly by F itself.
sonic wyvern
correct. and that's all there is.
umbral sand
then could you help with this other question
sonic wyvern
you don't need to ask. just send it!
sonic wyvern
,rccw
boreal girderBOT
sonic wyvern
I'll start with the good old question again.
from top to bottom, what is the first line you don't get, and what do you already understand about the problem?
umbral sand
i know formula for the sphere and the V for it, then im stuck onwards
sonic wyvern
basically you want A in terms of V.
you don't want any other sneaky variables in the affair.
umbral sand
isnt 4 over 3 just the K
sonic wyvern
does that make the whole plan clear?
umbral sand
yea kinda
sonic wyvern
(4/3)pi is the k IF you were asked to find the relationship between the volume and the radius.
but you're now asked to find the relationship between the total surface area and the volume.
be very careful about the quantities you're relating.
umbral sand
yep okay
near bluff
umbral sand
i have to learn alot of this in next hour and later tonigh becuase i missed one lesson last week
tonight
sonic wyvern
basically you want A in terms of V.
so given this and this message up here, is the plan now clear?
umbral sand
yea
umbral sand
makes it easier
how
sonic wyvern
why do you want the r power 6 to appear
because if not, let's see what happens when you divide A by V.
you're gonna get r^2 in the top and r^3 in the bottom, so your resulting expression still contains r.
but we don't want r in our final expression.
umbral sand
i get that
barren gulch
you're gonna get r^2 in the top and r^3 in the bottom, so your resulting express...
nvm
umbral sand
so it is easier to cancel out
sonic wyvern
so there are two ways of getting those two r's to be equal so that they can cancel out.
the simpler and more straightforward one is to raise both A and V to some power such that the exponent of r in both expressions become equal.
sonic wyvern
for this problem, it's done by raising A to a power of 3, and V to a power of 2, so that you get r^6 on both sides that you can later cancel out.
of course, nothing is stopping you from raising A to a power of 3/2 while leaving V unchanged...
but that's very much asking for trouble, so.
(or you can even raise V to a power of 2/3 while keeping A unchanged.
but both of these methods require you to handle fractional exponents throughout the division step afterwards, so unless you love torturing yourself, it's generally not recommended.)
umbral sand
im doing a quiz thingy on direct variation
sonic wyvern
I'm not sure what I should do with that information, OP.
sonic wyvern
sure.
azure dust
Hy any needs help or you covered
sonic wyvern
well, how did you get 5?
umbral sand
looked at it to fast
sonic wyvern
then I suppose you've answered your own question.
umbral sand
i see where i went wrong
sonic wyvern
right, so I presume that's done.
umbral sand
yep until i get another question
sonic wyvern
okay.
tired bear
im not sure why but i keep getting the wrong answer for option A
compact pewterBOT
im not sure why but i keep getting the wrong answer for option A
carmine garden
tired bear
its not as hard as the clickbait videos say but annoying nevertheless
unreal night
Well I can't help you in this, but i think there might be a higher chance of getting help in a physics server
compact pewterBOT
@tired bear Has your question been resolved?
tired bear
well i thought angular velocity was mathy enough
molten badger
im not sure why but i keep getting the wrong answer for option A
that's $\sqrt{24}a$ right?
boreal girderBOT
Alexis_Fx
molten badger
or is it sqrt{24a}
molten badger
okay I did get the answer
spark folio
idk what all this calculations you did. My idea is simple. Angular velocity about z axis for the CM is same as that of the point of contact with ground right? So, lets consider the big disk of radius 2a. It moves omega times 2a per second on the ground. And to complete one circle on ground, it needs to move 10a times omega. so effectively, it takes 5 seconds to have same angular displacement wrt z axis as it does about the axis of omega
tired bear
oh i did the same but i took the motion of the center of the disc as reference instead of the bottom of the disc
i thought you just do omega_CM = v_CM/r_CM
wait hold on ill think on this and come back
.close
serene coral
Can anyone check question b to see why i'm wrong?
compact pewterBOT
Can anyone check question b to see why i'm wrong?
serene coral
Find EC so that area of ABFE is 2/3 of the area of ABC
boreal girderBOT
serene coral
The correct answer is $\frac{5}{\sqrt{3}}$ btw
boreal girderBOT
Thomas
molten badger
Find EC so that area of ABFE is 2/3 of the area of ABC
!xy please
compact pewterBOT
Find EC so that area of ABFE is 2/3 of the area of ABC
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.
molten badger
btw you can't just post non-english working lol
noble wing
can i ask question here?
noble wing
oh ok
serene coral
Given right triangle ABC with AB = 3, AC = 4. Pick a random point E on AC and draw EF perpendicular to BC. Find EC so that the area of ABFE is 2/3 of the area of ABC.
molten badger
Can anyone check question b to see why i'm wrong?
you did some calculation wrong that's all
I think...
compact pewterBOT
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heavy bolt
A drinking vessel is formed from a combination of a tube and a cone as shown in the picture. If the vessel is filled to the brim with syrup, the number of hemispherical glasses needed to transfer all the syrup is… glasses.
compact pewterBOT
A drinking vessel is formed from a combination of a tube and a cone as shown in ...
fair cradle
!status
compact pewterBOT
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
grizzled geyser
Did you just ask the same question already?
silver canopy
The amount of glasses needed to transfer the contents?
grizzled geyser
fair cradle
The amount of glasses needed to transfer the contents?
yerp
grizzled geyser
Also: #help-6 message
silver canopy
Which part
heavy bolt
like the part where we have to count the volume
grizzled geyser
calculate not count
heavy bolt
yeah that
grizzled geyser
How do you calculate the volume of combined shapes?
magic wren
if the volumes are disjoint you can add them together
heavy bolt
if the volumes are disjoint you can add them together
so like we calculate the volume first then we add them all up
?
grizzled geyser
correct
grizzled geyser
yes
compact pewterBOT
@heavy bolt Has your question been resolved?
heavy bolt
18 glass'correct?
grizzled geyser
First, react to the message above you
If you don't, this channel will auto-close
dry tide
12th one please i have no idea what to do
compact pewterBOT
12th one please i have no idea what to do
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
You know what reciprocal equation means right?
dry tide
yup
dry tide
You know what reciprocal equation means right?
i do
the reciprocal of a root is also a root of the equation
cursive harbor
,rccw
boreal girderBOT
cursive harbor
tired bear
12th one please i have no idea what to do
expand and take all the x^ terms common
tired bear
the coefficient of x^4 and the constant should be equal for one condition
tired bear
same for x^3 and x
tired bear
in other words the polynomial should have symmetric coefficients
dry tide
in other words the polynomial should have symmetric coefficients
yeah yeah thank you
tired bear
k
compact pewterBOT
@dry tide Has your question been resolved?
compact pewterBOT
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chrome frost
how do you know what root to take for b i
compact pewterBOT
how do you know what root to take for b i
fleet flare
f^-1 will be both the positive and negative answer (for the whole graph)
based on the restrictions of f that may change
ancient raft
how do you know what root to take for b i
Also what do you mean by root
fleet flare
f^-1 = a +- sqrt(c)
ancient raft
Ok
boreal girderBOT
KB
fleet flare
and we want to know whether its the +sqrt{...} or -sqrt{...} right?
f(x) in this case is not one-to-one so it doesn't have an inverse, so it cannot be both
instead, it has two one for the positive root and one for the negative root
in this case, our restriction is for $x \leq -2$
boreal girderBOT
KB
fleet flare
since we want f^-1(x) to be less than or equal to -2, we have to subtract values not add values
so we should get $f^{-1}(x) = -2-\sqrt{x+2}$
boreal girderBOT
KB
fleet flare
does that make sense?
compact pewterBOT
@chrome frost Has your question been resolved?
chrome frost
since we want f^-1(x) to be less than or equal to -2, we have to subtract values...
ohhh
okay
vapid yacht
Guys when finding the taylor expansion at infninty, I can introduce a variable t = 1/x as x --> infty , t goes to zero.
compact pewterBOT
Guys when finding the taylor expansion at infninty, I can introduce a variable t...
vapid yacht
I can then write the Maclaurin's series (Taylor series at 0) with respect to t
and then eventuallly I will find the taylor expansion of order 2 at infty for f
however does this trick always work? for all functions?
brisk axle
Just expand it and consider subbing in 0 and inf separately?
vapid yacht
Can i really sub in infty?
brisk axle
yh
just divide everything by greatest power of x
and then everything turns to 0
Except what u need
Cus u get 1/x^n
vapid yacht
I see what you mean
vapid yacht
I can then write the Maclaurin's series (Taylor series at 0) with respect to t
But this also works right?
In general, if $f$ admits a taylor expansion at $+\infty \iff \phi(t) := f \left( \frac{1}{t} \right)$ admits a taylor expansion at zero?
boreal girderBOT
Klein Bottle
vapid yacht
Is this statement correct or nah?
brisk axle
I think so yh
vapid yacht
Alright
brisk axle
Rather convoluted way of doing it tho
vapid yacht
yeah no that's fair
but my TA doesn't particularly like substituting infty directly
Anyways thanks for your assistance. Have a nice day!
.close
compact pewterBOT
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ornate sable
Any suggestions for 4b. I see that I can use the Taylor polynomial with x=r . But simplying it into something where it is clear to see that it is <1/200 is unclear
compact pewterBOT
Any suggestions for 4b. I see that I can use the Taylor polynomial with x=r . Bu...
quaint gulch
taylor's theorem give you a formula for the remainder too
ornate sable
Yes but what would the remainder do? Sure we can say sinx = Taylor polynomial+ remainder. But I don't see the benefit of this
quaint gulch
sin(r) = approx around r + remainder
sin(r) - approx around r = remainder
|sin(r) - approx around r| <= upper bound for remainder
and approx around r is r² since r is your approx from x² = sin(x)
ornate sable
sin(r) = approx around r + remainder
sin(r) - approx around r = remainder
|sin(r...
Why upper bound. For remainder? Doesn't it just have a value?
quaint gulch
yes but you don't know its value
ornate sable
But we can represent it
quaint gulch
the theorem gives a formula that says there exists epsilon such that the remainder...
blablabla
<@&268886789983436800>
ornate sable
Also your saying that at r our Taylor polynomial is approximately r^2 right?
ornate sable
But that is cubic not n
quaint gulch
I don't understand what you're trying to say
ornate sable
We used a cubic approximation. But don't we want to show b for a nth degree approximatation?
quaint gulch
(b) asks to show an inequality about part (a) approximation
you logically choose r = sqrt(15)-3 from r² = r - r^3 /6
because this r makes this equation true it's a root
so from sin(r) = r - r^3 /6 + remainder
you have sin(r)-r² = remainder
so |sin(r) - r²| <= upper bound of remainder
your problem just becomes showing the remainder is less than 1/200
ornate sable
you have sin(r)-r² = remainder
So we have |r-r^3/6+remainder-r-r^3/6|
So we have |r|^5/120
But how can we estimate r^5 without a calculator?
quaint gulch
from the formula of remainder you would get a cos that you can upperbound by 1 so the remainder is indeed smaller than r^5/120
and now you want to prove r^5/120 <= 1/200
r^5 <= 3/5
without a calculator I would calc an approximation of sqrt(15)
ornate sable
r^5 <= 3/5
How do you know this?
ornate sable
without a calculator I would calc an approximation of sqrt(15)
Well it's less than 4
quaint gulch
How do you know this?
it's just equivalences
I want to prove this
r^5/120 <= 1/200 is what we want to show
it's equivalent to r^5 <= 3/5
3.8² = 9+4.8+0.64
ornate sable
?
quaint gulch
= 14.44
quaint gulch
?
well you're asking me to show it without calculator
it's a pain to approximate square roots but I'll try
3.9² = 9+5.4+0.81
it's bigger than 15
so sqrt(15) < 3.9
r^5 <= (3.9-3)^5
81² = 6400 + 160 + 1 = 6561
* 9 = 65610-6561
/10^5
it's below 3/5
not sure why you would want to go through that without calculator but here it is
ornate sable
81² = 6400 + 160 + 1 = 6561
\* 9 = 65610-6561
Where's this from?
ornate sable
not sure why you would want to go through that without calculator but here it is
When is the cut off then between no calculator and calculator
quaint gulch
Where's this from?
me calculating
quaint gulch
When is the cut off then between no calculator and calculator
I didn't use any calculator because you asked me
But how can we estimate r^5 without a calculator?
well, we can like that
through calculations with your hands
like writing your results, doing the multiplications and divisions by hand, etc
ornate sable
me calculating
Where is 81 from?
quaint gulch
9², since I need to calculate 0.9^5, it's like calculating 9^5/10^5
and 9^5 = 81 * 81 * 9
so I need to calculate 81² and then 81² * 9
which is 65610-6561 (10*81²-81²), so 0.9^5 = 0.65610-0.06561
<= 0.6 = 3/5
ornate sable
Ok I see
Also our inequality then only holds for the third degree approximatation
Since we base on substitution on a which use the cubic Taylor polynomial
quaint gulch
(b) asks to show an inequality about part (a) approximation
^
also, with higher degree approximation you would have more accuracy, obviously
what would have given less accuracy is to approximate at degree 1
ornate sable
But then we would have to keep doing a with higher degree polynomials
quaint gulch
ie x² = x
if you want a better approximation yeah
fortunately the exercise doesn't ask for it since it wouldn't even be feasible to calculate roots of a polynomial after degree 4
<@&268886789983436800>
ornate sable
fortunately the exercise doesn't ask for it since it wouldn't even be feasible t...
Ok I understand
quaint gulch
fortunately the exercise doesn't ask for it since it wouldn't even be feasible t...
(without calculator*, you could always approximate)
ornate sable
Also for a when we are using the cubic aporoxation how do we know that we can have a=0. Since we don't know where the root is and approximating at a point farther from the root will lead to a worse approximatation
quaint gulch
that's why you take an upper bound and why there is a variable in the remainder formula
you know what the remainder looks like relatively to your point
but it isn't that precise
the formula does suffice to bound it
which is convenient if we want only some quite large approximation
ornate sable
We use an upper bound for a?. I thought it is just solving a cubic
quaint gulch
what are you calling a = 0
ornate sable
An approximatation of sinx when x=0
So I'm asking how do we know we use this approximation over any other arbitrary value of x
Is it just because it is simpler to work with?
quaint gulch
because the exercise is trying to make you do something cool like approximating solutions to non trivial equations by hand
so it shows you you can do it with some formula meant for approximation
ornate sable
But the formula requires us to choose where to approximate
quaint gulch
not really, the remainder accounts for how far you are
ornate sable
But we have infinity many amount of remainders. We could have choose to approximate for any value of x
quaint gulch
yeah but you wouldn't even get the same approx either
obviously the remainder changes when the approx does
if you're approximating at another point you get another approx
ornate sable
Ye so what I'm trying to figure out is how we know 0 is a good choice of point for part a
quaint gulch
I mean the max of sin on [0, pi/2] is at pi/2, where sin is 1
your solution can't be bigger than 1
if you didn't take 0, you would have taken 1?
yeah could work too, definitely
much more of a pain to do all the subsequent calculations tho
for a very minimal difference
0 seems to be a reasonable point of approximation of something around 0.9 to me
ornate sable
I mean the max of sin on [0, pi/2] is at pi/2, where sin is 1
your solution can'...
Why are you looking at maxes?
quaint gulch
your original eq is sin(x) = x²
the positive solution is definitely between 0 and 1
above 1, x² > 1
but sin(x) <= 1
ornate sable
Oh ok. That does cut the choices down
So any point between 0 and 1 the won't be too far for what x is
quaint gulch
well since you're between 0 and 1, 0 can't be that bad of a point fo approximation
ornate sable
Yes I see that now
quaint gulch
already a bit far for real use cases but for an exercise without calculator being on point to 10^-1 is already good
you could always get more accuracy tho
by taking things closer and closer
and being less lax with upper bound
ornate sable
How would we lower our upper bound of 1?
quaint gulch
I meant for the r^5 < 3/5 part
that's what the exercise asked for
but the more you approximated r precisely
the more it would be accurate for r^5
even on that part you could already refine your margin
r^5 is close to 1/2 which is already much less than 3/5
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mighty marten
I used to be seriously into math a while back, and made some stuff that I believe if properly formalized could be paper-worthy for my course completion work, or at least help with it. Issue is, idk if that's the case or not, and deciphering my scribbles on 5 year old notebooks is gonna take a while. So, may I ask here if whatever I do remember studying about is remotely worth exploring again?
compact pewterBOT
I used to be seriously into math a while back, and made some stuff that I believ...
mighty marten
For context, I'm in Computer Science now
The topics themselves are:
-
A function that iteratively finds the closed form of any x^n sum (so 1^n + 2^n + ... + x^n) using the result from the previous power
-
An interpolation function that creates a graph that passes through points in 2d space (don't recall if its a polynomial)
I do have the end product of these 2 in my desmos account, and likely some process behind them in an old notebook that is so unorganized it would be a waste of time to check it if not for good reason
pulsar wagon
these are both solved problems
they are still cool to explore. but just speaking on whether answering them will lead to anything paper worthy, probably not
short ferry
pulsar wagon
they are still cool to explore. but just speaking on whether answering them will...
because they are like, very well explored already
mighty marten
these are both solved problems
Gotcha. Just thought about these 2 randomly and figured they were likely just re-discoveries by an enthusiastic 17 yro
I'll look through some more stuff in my desmos and see if anything may warrant scrutiny tbh. Otherwise I'll leave it out for good
short ferry
unfortunately pretty much everything a normal person can discover was most likely already explored by some random oiler in the 17th century
pulsar wagon
mighty marten
Ok yeah not rly, lemme close this
mighty marten
unfortunately pretty much everything a normal person can discover was most likel...
True
Or is confidently incorrect, maybe
pulsar wagon
pretty much everything that would be significant enough for mathematicians to care about maybe
mighty marten
Yeah usually math really needs hilarious amounts of baggage to become relevant as a discovery
Unless its game theory but I suppose that field is really untouched for some reason
dawn oyster
The topics themselves are:
- A function that iteratively finds the closed form ...
I just wanted to say before you close
keep pursuing things independently, I think it's a really good way to learn!
just don't necessarily expect it to be original
pulsar wagon
yea, i hope nothing i said was discouraging
mighty marten
keep pursuing things independently, I think it's a really good way to learn!
It was yeah. I was just kinda dumb as a kid
dawn oyster
It was yeah. I was just kinda dumb as a kid
no worries I do the same and feel the same about math
pulsar wagon
tis good to explore already solved problems if you find them fun
dawn oyster
everything takes time
mighty marten
yea, i hope nothing i said was discouraging
Wasn't. Its just that as a kid I kept doing stuff I found cool, but when I eventually tried to show it to others, it'd be too rough/handwavy/wrong for it to be encouraged
short ferry
It was yeah. I was just kinda dumb as a kid
idt a dumb kid would be able to discover recursive formula for x^n sum, its definitely a cool discovery
mighty marten
idt a dumb kid would be able to discover recursive formula for x^n sum, its defi...
Yeah but whatever I did was, in fact, not formal in the slightest
So I'd always be stuck in the "boring part" of formalizing
pulsar wagon
there is always gang theory
mighty marten
Figured if that was what I was gonna do, I would not want to pursue math. I liked exploring puzzles, not creating intricate and excruciatingly formal theories
dawn oyster
The topics themselves are:
- A function that iteratively finds the closed form ...
more mathematically, I was also super interested in the first problem. I'd really recommend looking into "stirling numbers" and "discrete calculus"
really underrated pieces of math
mighty marten
tis good to explore already solved problems if you find them fun
I remember rebuilding calculus with some borrowed insight and then comparing it to the real thing. Ended up acing calc in uni
pulsar wagon
cool. i like to do the same with combinatorics
mighty marten
more mathematically, I was also super interested in the first problem. I'd reall...
I remember finding discrete calc I think? I believe this is similar
dawn oyster
there's some really interesting general theories in combinatorics yeha
mighty marten
Its a whole thing on "derivatives" being the inverse of summing the first n results of a function
dawn oyster
I think people outside the field just view it as a bunch of loosely stitched together problems but there's been a lot of recent work
mighty marten
I think people outside the field just view it as a bunch of loosely stitched tog...
Fair
I remember trying out game theory last time I was into math
I believe I also have a thing to publish about it when it comes to computer science so that's.. something?
At least enough for my future end of course dissertation
dawn oyster
I believe I also have a thing to publish about it when it comes to computer scie...
yeah dw about it not being "important"
mighty marten
If I may hijack my own post I kinda wanted some feedback for it but my uni has no game theory things besides ig economics
dawn oyster
I think being a young mathematician/computer scientist/whatnot is a lot about getting intuition in the field and grad schools probably like seeing people do that by doing this kind of work
dawn oyster
If I may hijack my own post I kinda wanted some feedback for it but my uni has n...
I can't personally help with game theory but I think that's allowed yeah
mighty marten
Sure. The gist of it is picking up games (be them algorithms, entertainment media, problems, etc) and doing a process to convert a game tree into 2 values that try to distill certain aspects of games. For recreational games for example, it could be the difficulty of the game (how hard it is to make the right choices to win) vs agency (average likelihood of shifting the game state into a less difficult situation). Then you can put those on a 2x2 grid and use that to classify new entries. So you can define certain entries as "exemples" in order to classify any given point in the graph into a category. My best guess on what that could be useful for is generating algorithms with a machine (say, an LLM), as they can grab their prototype, shove it into the category model and see how close/far it is to the intended result
strong veldt
there is always gang theory
sub gangs?
pulsar wagon
mighty marten
sub gangs?
Group theory, ring theory, gang theory
dawn oyster
Sure. The gist of it is picking up games (be them algorithms, entertainment medi...
I think these kinds of questions are allowed
but I'm not sure how much people can help since it's more of a soft question on whether an idea might work
rather than a cut and dry homework problem
mighty marten
Tbf I just wanted to know if that already was a thing more than anything
Likely isn't but what do I know yk
dawn oyster
okay fair enough
mighty marten
Cuz yeah I figured its kind of a weird question
dawn oyster
maybe the topic specific channels would be helpful?
mighty marten
tho only after u mentioned it was
mighty marten
maybe the topic specific channels would be helpful?
Fair, will go there. Might leave this channel open till it auto closes tho just in case
Thanks!
dawn oyster
there is a game theory channel: https://discord.com/channels/268882317391429632/1364706668960546906
but they're pretty advanced over there so fair warning
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ornate sable
Can someone tell me why $\frac{1}{4!}\int_0^r(r-t)^4 \arctan^{(5)} (t)dt$ is not $E_{5}(r)$
compact pewterBOT
Can someone tell me why $\frac{1}{4!}\int_0^r(r-t)^4 \arctan^{(5)} (t)dt$ is not...
boreal girderBOT
BigBen
ornate sable
Also $r=\frac{-3+\sqrt{21}}{2}$
boreal girderBOT
BigBen
kind crane
Can someone tell me why $\frac{1}{4!}\int_0^r(r-t)^4 \arctan^{(5)} (t)dt$ is not...
not enough context
ornate sable
Sorry. @kind crane @sacred lagoon . Question 5
Give me a moment to send what I have done
sacred lagoon
oh ok so E5 is the fifth remainder of Taylor's formula ok
makes sence now
and technically it is 5 IPP to arrive there
ornate sable
ornate sable
and technically it is 5 IPP to arrive there<:true:1467307461957648580>
What is IPP?
ornate sable
Well I just did a u sub
But I have checked on desmos on what the value should be and the integral I have set up does not match it
sacred lagoon
no just Taylor's formula with remainder can be easily proven by consecutives IBP
ornate sable
no just Taylor's formula with remainder can be easily proven by consecutives IBP
I see
sacred lagoon
it's kinda dumb but you just shifted the formula by one
ornate sable
it's kinda dumb but you just shifted the formula by one
? Isn't the degree for the Taylor polynomial for arc tan going to be 2n+1 so the error must be 2n+2
sacred lagoon
for the general formula the remainder is shifted by one
ornate sable
Well our n is 2n+1
sacred lagoon
means replace n+1 by n
ornate sable
Don't we just replace the n?
sacred lagoon
Well our n is 2n+1
yep but you went with 2n+2
ornate sable
Wait I mean 2n+2
sacred lagoon
under the integral should be f(2n+2)(t)
ornate sable
I'm confused. Why are you replacing the whole thing? We are just subbing in a certain n
ornate sable
Well isn't that why? Tn ends at n so r should be of degree n+1
sacred lagoon
Sorry. <@261933205387477002> <@723084918120513629> . Question 5
but here on the first document it's written En
ye no everything holds in all the documents except in this exercice
even right above it contradicts w the formula
im thinking it's just an error but that's kinda weird still
ornate sable
Here where he defines it as the integral
ornate sable
So what is going on? Shouldn't the way we talk about the error hold universaly?
sacred lagoon
yeah I think it's just an error in this exercice in particular
sacred lagoon
So what is going on? Shouldn't the way we talk about the error hold universaly?
no it's just in those 3 exemples
ornate sable
Wait I'm confused. Since the way I solved all the previous problems relied on the fact that we have a higher degree
It worked for exercise 4 though?
sacred lagoon
in the rest of all docs I didn't find any of this shift
well it may have worked by coincidence
technically it makes sense since higher degree = smaller impact
even if it's a little less precise
ornate sable
Wait can you see if there are any errors in the previous question? Since I went in with the same view regarding the error and concluded a right answer
ornate sable
even if it's a little less precise
So your saying that he was simply looking at the errors that were of higher degree but he could have also looked at the error of the same degree
ornate sable
Ok
sacred lagoon
I don't see any other errors
idk what else to say abt this soo hope you can clarify this out
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fiery raft
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fiery raft
Any hints? I have only written out the integral of how to find the length
@crimson notch
paper rune
,rccw
boreal girderBOT
fiery raft
@crimson notch bruva help
tranquil mesa
are u looking for the length? area under the curve?
fiery raft
are u looking for the length? area under the curve?
Length
tranquil mesa
integrate it with the formula
fiery raft
Yeah I know but I can’t seem to find the answer
The answer key says to use u-substitution or something…
cyan sluice
yea set u = the inside of the square root
then sub in the differential for dx
then integrate as normal
fiery raft
so dx = 9/4 du
Wait what
crimson notch
compact pewterBOT
@fiery raft Has your question been resolved?
rose fulcrum
I need help with a Dynamics question
compact pewterBOT
I need help with a Dynamics question
rose fulcrum
Here's what I've got so far:
I know my friction force is correct, our instructor walked us through the FBD in class, so I also know it's not going to slip
I'm just not sure about the acceleration.. clockwise rotation and left-pointing friction, so these values should be negative, and it's going to roll to the right with the direction of the 20 N force, so the linear acceleration should be positive (to the right)
But unless I make the mass moment of inertia negative, the linear acceleration comes out negative
Shouldn't the MMOI be positive? counterclockwise? Resisting clockwise rotation and movement to the right?
<@&286206848099549185> Any ideas?
rustic sigil
oh man maybe im in the wrost server dude 😭
I need help w sinusoidal functions this looks like the level 100 boss
unreal night
discord.gg/physics
Dude he gave you the server
rose fulcrum
All I saw was a server invite to some place I'm not interested in joining.
So.. yeah.. blocked.
.close
compact pewterBOT
rose fulcrum
Also, the server that was shared explicitly refuses to help with homework questions
So.. nice work, dip shit
elfin finch
Also, the server that was shared explicitly refuses to help with homework questi...
Huh did not know that
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strange pendant
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boreal girderBOT
Renato
dull moon
**Renato**
I'd say double-induction here. The inductive step includes the application of the properties of both $a_n$ and $a_{n-1}$ to $a_{n+2}$ but idk how this will actually help.
boreal girderBOT
Annie Maqionde
dull moon
specially proving 3^n | a_{n-1} => 3^{n+2} | [5a_{n-1}]^2
$3^n | a_{n-1} => 3^{n+2} | [5a_{n-1}]^2$
this?
boreal girderBOT
Annie Maqionde
dull moon
I mean if you assume $a_{n-1} = 3^n k$ for some $k\in \mathbb{N}$ then $(5a_{n-1})^2 = ?$
boreal girderBOT
Annie Maqionde
strange pendant
**Annie Maqionde**
this doesn't get to anything
I just tried it
you can say 3^{2n} | 5a_{n-1}^2
dull moon
specially proving 3^n | a_{n-1} => 3^{n+2} | [5a_{n-1}]^2
if this is your question, I do not know why it shouldn't work
dull moon
you can say 3^{2n} | 5a_{n-1}^2
correct.
your question is to prove $3^{n+2}$ is to be proved to divide $[5a_{n-1}]^2$
boreal girderBOT
Annie Maqionde
dull moon
if $3^{2n} \mid [5a_{n-1}]^2 \implies 3^{n+2} \mid [5a_{n-1}]^2$
boreal girderBOT
Annie Maqionde
dull moon
**Annie Maqionde**
now you'd have to explain why this is not possible according to you
strange pendant
**Annie Maqionde**
trying to prove a more strict condition from a less strict one is just not possible
dull moon
trying to prove a more strict condition from a less strict one is just not possi...
why, is it mathematically invalid?
dull moon
specially proving 3^n | a_{n-1} => 3^{n+2} | [5a_{n-1}]^2
by your inductive step, the stronger condition is justified
strange pendant
I never said it wasn't
dull moon
I never said it wasn't
Well yes, then, what is the issue?
strange pendant
I mean i can certainly know why it is true, is just that proving it is different
dull moon
**Annie Maqionde**
why would proving this be difficult? there's no proof necessary here, its quite evident from the fact $2n\geq n+2$
boreal girderBOT
Annie Maqionde
strange pendant
how do you show that 3^(n+2) | 3^(2n)
dull moon
how do you show that 3^(n+2) | 3^(2n)
$3^{2n} = 3^{n+2} \times 3^{n-2}$, and here $n>2$
boreal girderBOT
Annie Maqionde
strange pendant
if we can say 3^(n+2) | 3^(2n) then since 3^(2n) | 5.a_(n-1)^2 then 3^(n+2) | 5a_(n-1)^2
divisibility is a transitive relation
dull moon
if we can say 3^(n+2) | 3^(2n) then since 3^(2n) | 5.a_(n-1)^2 then 3^(n+2) | 5a...
correct.
strange pendant
alright I finished the proof
strange pendant
**Annie Maqionde**
this really helped thanks
strange pendant
I'd say double-induction here. The inductive step includes the application of th...
is possible yes, but i prefer using strong induction when the recurrence relation depends on the previous 2 terms
I will be closing now
.close
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strange pendant
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boreal girderBOT
Renato
strange pendant
brisk axle
not sure what exactly the question is asking but dont you just set the remainder 17 = a^2-2a and solve the quadratic for a?
dawn oyster
the remainder is for the difference of polynomials
so I'm pretty sure when you resubstitute you have 17 = 0 mod a^2 + 1
dawn oyster
yes those are the only solutions I believe
strange pendant
yes but how did you concluded that a^2 + 1 | 17
is not easy to arrive to that conclusion just from the long division
dawn oyster
so I'm pretty sure when you resubstitute you have 17 = 0 mod a^2 + 1
from here
if x = y mod m then m | x - y
dawn oyster
so the quadratic here has remainder 17 right
but a^2 + 1 = 0 mod a^2 + 1
so when you resubstitute that whole part is 0
strange pendant
what
dawn oyster
um
dawn oyster
you have a^4 - 11a + 5 = 0 mod a^2+1. then you have a^4 - 11a^2 + 5 = (a^2 - 12)(a^2+1) + 17
if you substitute the latter expression into the former you have (a^2-12)(a^2+1) + 17 = 0 mod a^2 + 1
dawn oyster
if you substitute the latter expression into the former you have (a^2-12)(a^2+1)...
but the whole polynomial part in a dies since it's a multiple of a^2+1
strange pendant
no
strange pendant
,w expand (x^2-12)(x^2+1) + 17
boreal girderBOT
strange pendant
x^4 - 11x^2 + 5 = 17 (mod x^2 + 1)
but x^4-11x^2 + 5 = 0 (mod x^2 + 1)
so 17 = 0 (mod x^2 +1)
<=> x^2 + 1 | 17
@dawn oyster
dawn oyster
I mean yeah that also works
dawn oyster
x^4 - 11x^2 + 5 = 17 (mod x^2 + 1)
wait where did you get this from
dawn oyster
oh wait yeah mb
dawn oyster
you have a^4 - 11a + 5 = 0 mod a^2+1. then you have a^4 - 11a^2 + 5 = (a^2 - 12)...
that's equivalent to this
strange pendant
if you substitute the latter expression into the former you have (a^2-12)(a^2+1)...
yeah
strange pendant
if you substitute the latter expression into the former you have (a^2-12)(a^2+1)...
then take mod a^2 + 1 you get 17 = 0 (mod a^2 + 1)
dawn oyster
so we agree?
strange pendant
yeah of course
dawn oyster
okay
strange pendant
as long as it's logical math always agrees
strange pendant
well yeah you kinda disagreed here though lol
I didnt understood the first time I read
dawn oyster
oh okay fair enough
strange pendant
I thought you were doing some monkey business
dawn oyster
you had me convinced I was doing monkey business lol
compact pewterBOT
visual flint
a group is cyclic and its order is 8
so i am looking for subgroups which are made of 'a^2'
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quick marlin
Let n belongs to N suff large prime p,there exist X,Y,Z belongs to {1,.....,p-1} then X^n+Y^n=Z^n (modp)
compact pewterBOT
Let n belongs to N suff large prime p,there exist X,Y,Z belongs to {1,.....,p-1}...
quick marlin
Is there any proof
quick marlin
And what about Ramsey theorem?
short ferry
hm, what about it?
quick marlin
There will be a monoX triangle in that theorem
short ferry
i dont get what youre asking
short ferry
https://amininima.wordpress.com/2013/05/20/fermats-last-theorem-mod-p/
this link explains it using ramseys thm
quick marlin
this link explains it using ramseys thm
Okie thank you bro
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✅ Original question: #help-43 message
quick marlin
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upper bane
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noble fossil
yo am i stupid or
compact pewterBOT
yo am i stupid or
tired willow
yo am i stupid or
independent rolls
noble fossil
id say its 10 but that foesnt make sense
nah
i don’t understand where the catch is
tired willow
so the "number of 3s in 2 rolls" follows binomial distribution
tired willow
ive never seen something like this
uh just let x be the number of 3s 2 rolls because thats what youre supposed to find here
and calculate P(3) and P(not 3)
keep in mind each roll is independent
tired willow
yes
noble fossil
yeah ill come back to this later this is stupidly hard
noble fossil
spark wigeon
(1/6)(5/6) + (5/6)(1/6) is probability of gettting one 3
tired willow
yeah ill come back to this later this is stupidly hard
can u try listing all possible outcomes for X?
the possible number of 3s in 2 rolls?
noble fossil
the possible number of 3s in 2 rolls?
2
spark wigeon
or 0 or 1, it's already in the table