Im right here you know
#help-49
ebon atlas
and try to solve this :p
twilit jetty
tell you what it is
twilit jetty
its right there in the picture
thats called "chain rule"
say youre trying to take the derivative of one function inside another
usually youre taking derivatives of functions like f(x) * g(x), or f(x) + g(x)
ebon atlas
yea thing with me tho ill prolly ask u why that works now 😔
twilit jetty
those are functions that are multiplied or added
now consider composite functions, f(g(x))
do you know what composite functions are
ebon atlas
yes
twilit jetty
now composite functions have their own derivative
if you want to get a sense of this, consider differentiating f(2x)
and think of this conceptually
ebon atlas
If it's not f(x) it's composite even f(-x) or f(x²)
how does that work
twilit jetty
I want you to focus on specifically finding the derivative of f(2x)
you should be able to see a visual pattern here
so in red here is f(x)
and in green is f(2x)
youll notice f(2x) is just f(x) but horizontally squished, right?
ebon atlas
youll notice f(2x) is just f(x) but horizontally squished, right?
yeah
twilit jetty
so lets say we were looking at f(x) at x = 0.9
the slope of f(x) at x = 0.9 would be f'(0.9), right?
ebon atlas
mhm
twilit jetty
now take a look at the green function
its marking a particular point
to have a particular derivative
for starters, where is this point at
ebon atlas
are u referring to 0.9/2 btw
ebon atlas
https://cdn.discordapp.com/attachments/1399517652317114393/1405237030233047271/8...
OH WAIT
IK THAT
twilit jetty
thats not going to explain what the question really wants though
which is why the chain rule has to work
ebon atlas
https://cdn.discordapp.com/attachments/1399517652317114393/1405237030233047271/8...
but i never learned it properly i legit js thought of du as variables that cancel out 💀
twilit jetty
right we'll need to get to that later
ebon atlas
thats not going to explain what the question really wants though
yeah pls keep going tho
ebon atlas
f(0.9/2)
twilit jetty
are you sure?
twilit jetty
you could just say f(0.9) you know
ebon atlas
yea ok
twilit jetty
f(2*0.9/2)
that you said this though is interesting, its also correct
ebon atlas
that you said this though is interesting, its also correct
yea it kinda clarifies what function ur using :p
so the two coordinates fall on a horizontal straight line?
cuz the output is the same
ebon atlas
alr
twilit jetty
at 0.9, the gradient is f'(0.9)
now compare that to the green point
what do you think the gradient would comparatively be?
we know its steeper for one
ebon atlas
yea
itd be larger but
if ur looking for me to say the exact value i will probably disappoint u
twilit jetty
you already know what this value is
consider this: if it takes 30mph to get from A to B
and you see someone else travel from A to B in half the time you did,
they went 60mph didnt they
twilit jetty
no you didnt
you said cos(x)
you forgot a key factor here
you didnt say cos(something else)
you forgot the "something else" part of it
ebon atlas
hmm
twilit jetty
they went 60mph didnt they
youre saying they went 60mph but took as long as you did
thats not correct
twilit jetty
now lets focus on the derivatives
this does mean though you have a good sense of what this slope would be
given the red point's slope is f'(0.9),
what would the green point's slope be
twilit jetty
no it isnt, you need to stick with me before you talk
ebon atlas
yeah sorry
twilit jetty
notice here youre taking derivatives at two different points
you found the slope of a different point at x = 0.9/2 instead of x = 0.9
we know the slope must be double that of the slope at x = 0.9
so if you know the slope f'(a),
and h(x) = f(2x),
then we're saying h'(a/2) = 2 f'(a)
twilit jetty
so since h'(x/2) = 2 f'(x),
ebon atlas
but algebraically im lost
ebon atlas
we js rewrote the same thing in a diff way?
twilit jetty
you say that like this is the same thing, I havent shown you that thing yet
let me continue
twilit jetty
then we're saying h'(a/2) = 2 f'(a)
you can see this fact shouldnt really depend on a
it should be true for any number in general
so h'(x/2) = 2 f'(x) for any x
right?
ebon atlas
nah nowhere ignore that part i was trying to link them wrong...
i thought you introduced a new function
but you were
describing
an existing one
wed been talking abt
anyways
twilit jetty
and people wonder why wearing glasses counts as a disguise
twilit jetty
so h'(x/2) = 2 f'(x) for any x
anyways since this is true for any x,
doubling x wouldnt change how true it is
so we also know h'(2x/2) = 2 f'(2x) for any x
does that make sense
ebon atlas
yes it does
twilit jetty
and so h'(x) = 2 f'(2x)
previoulsy you were just doing h'(x) = 2 f'(x)
you didnt write down the 2x part of it
here,
h'(0.45) = 2 f'(0.9)
if you had h'(0.45) = 2 f'(0.45) instead thatd be this slope instead, which is not the correct one to double
ebon atlas
if you had h'(0.45) = 2 f'(0.45) instead thatd be this slope instead, which is n...
ohh i just understood it rn
when you put it like that
ok yes i got it (ill still have to teach myself it again 🥀 )
twilit jetty
in general,
if you have h(x) = f(g(x)),
and you want to find its derivative,
its f'(g(x)) g'(x)
ebon atlas
ur a patient person
twilit jetty
sure
twilit jetty
in general,
if you have h(x) = f(g(x)),
and you want to find its derivative,
its...
if you think about it, g(x) should be differentiable if you zoom in on any section and it looks like a line
so that g'(x) would then be the slope of that line
so if you want the derivative of f(g(x)), and g(x) has a slope of g'(x) at x,
we use the same pattern as with 2x
f(2x) has derivative 2 f'(2x)
f(g(x)) has derivative g'(x) f'(g(x))
this is where the f'(g(x)) g'(x) formula comes from
ebon atlas
in general,
if you have h(x) = f(g(x)),
and you want to find its derivative,
its...
okay that js made a lot more sense.. in the case u were talking about btw like f(x) and f(2x) is g(x) essentially =2x and so the derivative would be f'(2x)*2?
checks out :o
twilit jetty
yes
now for the formula you did learn, which is dy/du du/dx
this is the same idea as before, but you missed a key area
ebon atlas
now for the formula you did learn, which is dy/du du/dx
seems drastically different than this one idk why..
twilit jetty
this formula hides how you plug in each variable
its $\frac{dy}{dx}(x)=\frac{dy}{du}(u)\cdot\frac{du}{dx}(x)$
grand pondBOT
mtt
twilit jetty
you plugged in x instead of u into the function dy/du
now how does this link with the original definition we wrote down? we have to carefully change each part so that they line up as expected
ebon atlas
i also get how it was wrong to say the derivative of sin(pi*x/180) would be (pi/180)(cos(x)) btw cuz that makes no sense rn and i see how itd mess up the whole graph
twilit jetty
**mtt**
right, but you still need to use this formula correctly so that you dont make this mistake again
we can do this by matching it up with the other formula d/dx f(g(x)) = f'(g(x)) g'(x)
first, Ill change the letters to match closer with the definition:
d/dx y(u(x)) = y'(u(x)) u'(x)
wait this isnt correct
this isnt going to be easy
twilit jetty
you sort of cant
ebon atlas
lets see
twilit jetty
this is a very precise kind of difference here
first, do you know how to do $\frac{d}{d(2x)}f(2x)$?
grand pondBOT
mtt
grand pondBOT
mtt
twilit jetty
what would the derivative be
ebon atlas
that is correct
ok in that case i have a sense of it
ebon atlas
what would the derivative be
wait ur not supposed to ask me that 😰
this brings my middle school math anxiety back
twilit jetty
I just replaced f with sin...
its the exact same question but you know what f is now
and you already know d/dx sin(x) = cos(x)
do the exact same thing again
twilit jetty
yep its cos(2x)
d/d(2x) sin(2x) = cos(2x)
now lets replace 2x with u
d/du sin(u) = cos(u), it still lines up
twilit jetty
thats certainly one of the things to say of all time
we're not going to mention that until later
ebon atlas
d/du sin(u) = cos(u), it still lines up
thats how i thought of it too
ebon atlas
thats certainly one of the things to say of all time
😭 that bad?
twilit jetty
yes, but when you did d/dx sin(pi x/180), you put in x instead of u into cos(x)
tribal temple
this brings my middle school math anxiety back
(for what it's worth, this should be a safe space where we won't be too hard on you or judge you if you get the wrong answer, we want you to learn as much as possible 
)
twilit jetty
it wouldve been pi/180 cos(u), or pi/180 cos(pi x/180)
so here its no more than a typo, but its a very easy to forget typo
so you can see here we have a reason for going about this chain rule idea more literally so that we can avoid this mistake with more intuition
ebon atlas
yes, but when you did d/dx sin(pi x/180), you put in x instead of u into cos(x)
yea i understood why that was wrong like 20 messages back but it makes more sense now that you put it this way ^o^ (i was thinking in terms of graphs but i need the algebra to check out too)
twilit jetty
😭 that bad?
its characteristically good but a major "abuse of notation"
ebon atlas
(for what it's worth, this should be a safe space where we won't be too hard on ...
thats so sweet ☹️ tysm 💛
twilit jetty
that means youre using the words in unintentional ways which wont work in general
ebon atlas
so here its no more than a typo, but its a very easy to forget typo
when i first said it, it was like a deep error in my foundations i think
i get it now to some extent
ebon atlas
its characteristically good but a major "abuse of notation"
yea i js hear of it when it comes to integration
WHICH I HAVENT DONE YET
i think my original question is answered too in terms of why the red graph on the right looks scaled down (while still holding the features of the red graph on the left)
tysm @twilit jetty
twilit jetty
back
@ebon atlas yea by this point the original question has long been answered, but we still gotta make sure the mistake wont come up again
so thatll come in terms of making extra sure that we get dy/du du/dx
knowing that d/d(2x) sin(2x) = cos(2x) is what we expect
ebon atlas
<@930150118131376179> yea by this point the original question has long been answ...
i probably wont be making the same mistake anytime soon :p
twilit jetty
alr
sin(2x)
= dsin(2x)/d(2x) * d(2x)/dx
= cos(2x) * 2
very similar to the
sin(2x) = cos(2x) 2 we'd get anyway for f'(g(x)) g'(x)
twilit jetty
junior year
twilit jetty
3rd out of the 4 years we stay in uni
twilit jetty
I mean you pieced together what you got wrong and got correct almost every big leap we did
ebon atlas
hmm idk i could be a faster learner
ebon atlas
I mean you pieced together what you got wrong and got correct almost every big l...
OK WAIT if im entering 11th grade in liek a month do you think i could keep up with uni math by the time i finish hs
ikk u probably couldnt give a judgement of that but meh
twilit jetty
if you know calculus when you come out of hs youll already be better than most people
so youll be good for uni
ebon atlas
ok tyy ^^
twilit jetty
np
ebon atlas
.close
grave kernel
midnight plankBOT
grave kernel
any ideas with this, i was trying to think with examples but theres too many of them,
maybe something with gcd or taking 7 common from the expression but i cant seem to figure that out
eternal pawn
grave kernel
no its the product
of abc
i mean
yes
its over all the values
mb, i read your question wrong
like 1,1,1
thatll be 1
then 1,1,2
thatll be 2
so then 1 + 2
like that
twilit jetty
yes its a sum over all $(a,b,c)\in T$, or $S=\sum_{(a,b,c)\in T}abc$
grand pondBOT
mtt
fathom onyx
its over all the values
i.e. your condition for a, b, c being natural numbers should be inside the set defined as T
grave kernel
yes
feral sedge
consider the action of (7 - a, 7 - b, 7 -c) on T
grave kernel
wont i get the same set T then?
feral sedge
assuming you sort the triples in non-decreasing order yes it is a simply map that sends T to T
the point is the properties of this map will allow you to show 7 | S
grave kernel
so then i replace abc with (7-a)(7-b)(7-c), and then that product will supposedly give me a 7 to factor out right?
from which i can say its true?
feral sedge
can you spell out more explicitly what you think the argument is
grave kernel
like, wdym
feral sedge
write it out very clearly
grave kernel
since (7-a),(7-b),(7-c) gives me the same set(kinda) if i show that the sum of (7-a)(7-b)(7-c) gives me a result which is divisible by 7, it should prove the original one too, but trying it out now, i think im going in the wrong direction
feral sedge
i see
grave kernel
i cant figure out more
feral sedge
so you agree
$S = \sum_{(a, b, c) \in T} abc = \sum_{(a, b, c) \in T} (7 - a)(7 - b)(7 - c)$
grand pondBOT
Mqnic_
feral sedge
this follows from the fact: the action from T to T is a bijection
grave kernel
right
feral sedge
now if you take mod 7...
grave kernel
okay, really sorry to interrupt, i still havent worked with mods, its right after this tho, so, i do have a hard time understanding them
feral sedge
hm
okay well that is fine
for simple purposes you do not need to know much aside from
(a + b) mod n = a mod n + b mod n
ab mod n = (a mod n)(b mod n)
feral sedge
**Mqnic\_**
so taking this mod 7 something important happens
grave kernel
-abc?
feral sedge
yes
feral sedge
yes, so 7 | S because 7 is odd, to clarify
chrome swift
help
chrome swift
midnight plankBOT
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
small jasper
Your inequality is the wrong way around
Also put parentheses around your numerator for clarity
But you can't solve this inequality algebraically (at least with "standard" tools)
your best bet is to use trial and error
the solution is quite small
chrome swift
But you can't solve this inequality algebraically (at least with "standard" tool...
how'd you determine this because i was trying to solve it algebraically
i thought there was a way with logs or something
small jasper
how'd you determine this because i was trying to solve it algebraically
This is a standard case of using the lambert W function
which I doubt is assumed knowledge (also you'd still have to estimate the solution after)
small jasper
why are we bringing colors here bro 😭
complex functions normally require 4 dimensions to graph
we don't draw in 4 dimensions
so that's the work around
chrome swift
you smart as hell
u studying string theory or something
the answer is first term btw thanks
small jasper
u studying string theory or something
eh it's one of the weird internet math things that people like
idk why they picked this specifically
also had this show up in my exams lol so
small jasper
the answer is first term btw thanks
yup it's just the first term
hence why I said the solution is small (and thus guessable by trial and error)
midnight plankBOT
If you are done with this channel, please mark your problem as solved by typing .close
chrome swift
wait wait
im getting tripped up
ugh
why does the question say how many terms to sum together
the error bound is just the next term right
no addition needed right
or am i over thinking it
small jasper
I assume you're referring to this
This says that the error of approximating a sum by adding the first n terms is the (n+1)st term
chrome swift
yeah
small jasper
the error bound is just the next term right
which is what I assume this is about
small jasper
why does the question say how many terms to sum together
now back to this
This says that the error of approximating a sum by adding the first n terms is the (n+1)st term
so let's say we got something like n > 2.85 and we answered n=3
this means you would have to add the first 3 terms to get within your desired error
here the answer was n=1
chrome swift
i thought n+1 meant you just add 1 to the term
small jasper
so you're just "adding" the first term to get a sufficient estimate
aka just using the first term and nothing else
which is why the idea of "adding/how many terms" is a bit odd here
small jasper
i thought n+1 meant you just add 1 to the term
no - it means that you calculate the (n+1)st term
Ex. If a(n) = 1/n and we had n = 3, then a_(n+1) would be a(4) = 1/4
chrome swift
ok i kinda get it
chrome swift
which is why the idea of "adding/how many terms" is a bit odd here
yeah tripped me up
small jasper
yeah tripped me up
chrome swift
no - it means that you calculate the (n+1)st term
Ex. If a(n) = 1/n and we had ...
ahhh
hold on lmk if i get this wrong
let's say u solve for n given a remainder of .01
and n = like 20.54
u round up to 21
chrome swift
well it would be an inequality
oh yeaj
small jasper
,w (n+1)/6^(n+1) \leq 0.01
grand pondBOT
small jasper
so you can just trust the inequality sign
it says n is at least 2.22343
the smallest integer n satisfying that is 3
done
chrome swift
ok
and wait
u don't need to add one to that cuz u already did the n+1 when setting up the intequality right
and this means that when you add n amount of terms in the series you get a remainder of what ever you were given
small jasper
u don't need to add one to that cuz u already did the n+1 when setting up the in...
yes as per the statement of the test
chrome swift
alright thanks
small jasper
and this means that when you add n amount of terms in the series you get a remai...
you get a bound on the remainder
chrome swift
and this means that when you add n amount of terms in the series you get a remai...
and this right?
small jasper
you get a *bound* on the remainder
not the exact remainder
because if you could get the exact remainder
chrome swift
oh yeah i keep forgetting mb
small jasper
then calculating any infinite series would be very easy
chrome swift
then calculating any infinite series would be very easy
u right it would be too OP
small jasper
oh yeah i keep forgetting mb
chrome swift
@small jasper i need a sanity check
the question said no more than .1
so it's .1 >= 1/6
right
small jasper
so it's .1 >= 1/6
yes
,w |-\frac{(-1)^1 * 1}{6^1}+\sum^{\infty}_{n=1} \frac{(-1)^n * n}{6^n}|
chrome swift
what
grand pondBOT
chrome swift
oh that's the partial sum
dude idk if my brain is really fatigued but
how is .1 greater than 1/6
small jasper
how is .1 greater than 1/6
,calc 1/6
grand pondBOT
Result:
0.16666666666667
u change the signs when inverse?
i thought only when u multiply or divide by negative
small jasper
i thought only when u multiply or divide by negative
You say "only" probably cause you weren't taught what happens if you take the reciprocal of both sides
only what happens when you add/subtract/multiply/subtracft
I assume you're American so this sounds reasonable
idk
chrome swift
so reciprocal changes signs
small jasper
whatever
anyway a < b implies 1/a > 1/b if a and b have the same sign
and if a > b, then 1/a < 1/b
this comes from dividing both sides of a < b by ab (and similar for a > b)
so in essence it is dividing both sides by a (positive) number ab
but having the shortcut offhand is nice
chrome swift
naw i must be brain fatigued if i can't even understand this
how long can u do math for without having a headache
small jasper
how long can u do math for without having a headache
depends
I mean I've had six hour exams before lol
which would take even longer if you tried to finish
chrome swift
maybe the power of negative 1 flips the sign and the negative there makes it flip idk
small jasper
but when actually learning math
eh depends
is it a reading comp exercise or does it actually require thought
chrome swift
it's all practice problems then i dug into rabit holes
small jasper
maybe the power of negative 1 flips the sign and the negative there makes it fli...
this comes from dividing both sides of a < b by ab (and similar for a > b)
chrome swift
rabbit holes for me may not seem like rabbit holes for u btw
small jasper
it's all practice problems then i dug into rabit holes
eh imma count this under "learning" since it's new, so again it comes back to "is this reading comprehension/direct application or does it actually require thought"
which ofc changes overtime
chrome swift
why did this guy not flip it
grand pondBOT
small jasper
,w 1000 \leq (n+1)^2
grand pondBOT
small jasper
the solutions aren't the same
small jasper
wait he's wrong?
yeah that's what being a muppet means lol
small jasper
I assume you're referring to this
chrome swift
why it say this
small jasper
it's remainder >= a(n+1) right
Also if it was this
this would be useless since the remainder could go off to infinity
chrome swift
okayy
why in the problem it's .1 >= a(n+1)
is it cuz u want an error of no more than
small jasper
why in the problem it's .1 >= a(n+1)
$|s-s_{n}|<|s_{n+1}| \leq 0.1$ is the usual way you write the string of inequalities
grand pondBOT
Civil Service Pigeon
small jasper
and then just take the last bit
chrome swift
yeah i don't think im truly understanding the theorem like you are im just memorizing it which is a mistake
small jasper
is it cuz u want an error of no more than
uh this is what the alternating series error bound is for
so
sure ig
small jasper
yeah i don't think im truly understanding the theorem like you are im just memor...
tldr the absolute difference between a sum and the nth partial sum is at most the absolute value of the (n+1)st term
chrome swift
ur hurting my brain
chrome swift
tldr the absolute difference between a sum and the nth partial sum is at most th...
yeah i get that
wtf
chill yo i get it
like
u have the whole series with all it's terms
if u subtract Sn from it ur gonna have a(n+1)
that's the bound right
small jasper
tldr the absolute difference between a sum and the nth partial sum is at most th...
small jasper
if u subtract Sn from it ur gonna have a(n+1)
$|s-s_n|=|a_{n+1}+a_{n+2}+\cdots| \leq |a_{n+1}|$
grand pondBOT
Civil Service Pigeon
small jasper
$|s-s_{n}|=R_n \leq |s_{n+1}| \leq 0.1$
grand pondBOT
Civil Service Pigeon
small jasper
if you want to throw that in go ahead
but usually we define R_n to be the absolute difference
aka $|s-s_n|$, not necessarily $s-s_n$
grand pondBOT
Civil Service Pigeon
small jasper
ofc this applies for when said difference is negative
chrome swift
if the problem says u want an error of let's say .0001
would u write it as |an+1| <= .0001
sry if im being a dumbass
small jasper
now it's good
chrome swift
i was doing math the whole day
what if the question was an error no more than .0001
still the same?
small jasper
what if the question was an error no more than .0001
this is what I assumed you meant in the first place
because again, finding the exact error would mean you found the exact value of the series itself
chrome swift
yeah im being dumb my fauly
small jasper
which defeats the entire purpose of bounding the error
chrome swift
ok i'll take ur advice of
Rn <= |an+1| <= (given remainder)
like i'll remember that for ever
cuz that helps a lot
small jasper
Rn <= |an+1| <= (given remainder)
fix the first less than sign to a $\leq$
grand pondBOT
Civil Service Pigeon
chrome swift
**Civil Service Pigeon**
wait why'd u write s
small jasper
typo
chrome swift
oh
alrighty dude thanks
i'll see if i can do another problem that's similar real quick
small jasper
alrighty dude thanks
small jasper
i'll see if i can do another problem that's similar real quick
k
chrome swift
alright GN dude thanks
chrome swift
i'll see if i can do another problem that's similar real quick
naw man i've been doing math for so long
my brain hurts
wish me luck on my exam
thanks a lot
.clos
.solved
midnight plankBOT
midnight plankBOT
Available help channel!
fringe pine
what happened to the first t inside the integral? 
(not cheating btw i’m just trying to learn how to integrate)
midnight plankBOT
what happened to the first t inside the integral? <a:spongebob_sadge:10212296595...
sharp coral
when you perform the u-substitution the dt has to be replaced with something involving du
fringe pine
when you perform the u-substitution the dt has to be replaced with something inv...
so how did they eliminate the t to make it du instead of dt?
sharp coral
what do you get when you take the derivative of u = t^2 - 1 with respect to t?
fringe pine
what do you get when you take the derivative of u = t^2 - 1 with respect to t?
2(du/dt)?
sharp coral
no, it should be du/dt = ...
fringe pine
i kinda see it
midnight plankBOT
@fringe pine Has your question been resolved?
viral dagger
find all irrational $x$ such that $x^3-15x$ and $x^4-56x$ are both rational
midnight plankBOT
find all irrational $x$ such that $x^3-15x$ and $x^4-56x$ are both rational
grand pondBOT
ihave<skissue>
viral dagger
err whar?
pearl hull
Damn
viral dagger
some sort of factoring maybe?
lapis lynx
the irrationals must be of the form $a+b \sqrt{c}$ and the square root terms must cancel
grand pondBOT
cytos
viral dagger
the irrationals must be of the form $a+b \sqrt{c}$ and the square root terms mus...
proof?
lapis lynx
no need. it’s all in my head
viral dagger
no need. it’s all in my head
?? 😭
feral sedge
find all irrational $x$ such that $x^3-15x$ and $x^4-56x$ are both rational
a = x^3 - 15x, b = x^4 - 56x
continue
viral dagger
lapis lynx
no need. it’s all in my head
might’ve been wrong icl tbh ong
viral dagger
a = x^3 - 15x, b = x^4 - 56x
what do you want me to do?
feral sedge
reduction
viral dagger
yeah thats kinda what i was guessing, but how?
i mean im guessing you want rational=(something in x)
viral dagger
i mean im guessing you want rational=(something in x)
something in x thats easier to solve for rationals (like linear?)
feral sedge
yeah thats kinda what i was guessing, but how?
x^4 = x * x^3
viral dagger
the proof is cbrt2 is not of that form and therefore its wrong
i think he was going for irrationals that might solve the problem are in that form?
hard shard
oh fair
viral dagger
x^4 = x * x^3
you want to multiply a by x?
oh wait
b-ax=15x^2-56x
then solve for x using quad eqn?
$x=\frac{56+a\pm\sqrt{a^2-112a+3136-b^2}}{30}$
grand pondBOT
ihave<skissue>
lapis lynx
$x^3-15x=a,a \in \mathbb{Q} \ x^3-15x-a=0 \ (x-\frac{a}{5})(x+\frac{a}{10})^2=0$
hard shard
**ihave<skissue>**
gross
lapis lynx
would this then just not have any irrationals that produce a rational
viral dagger
there will be an a^3 term in the expansion no?
hard shard
,w expand (x-a/5)(x-a/10)^2
grand pondBOT
hard shard
eh?
lapis lynx
did wikipedia lie to me
hard shard
did you try applying cardanos
feral sedge
you want to multiply a by x?
you have proven
lapis lynx
oh no I’m stupid
lapis lynx
wait wait
viral dagger
,w expand (x+a/5)(x-a/10)^2
hard shard
sus
viral dagger
aight
grand pondBOT
cytos
lapis lynx
surely I got it this time
hard shard
you can tell just by the coefficients this is gonna be wrong
feral sedge
x^2 = px + q
using this the idea is to get equations in p and q
then solve those
again use the given conditions x^3 - 15x = a rational, etc
viral dagger
using this the idea is to get equations in p and q
p and q are rationals? right
viral dagger
again use the given conditions x^3 - 15x = a rational, etc
a=x^3-15x=px^2+qx-15x=p^2x+qx-15x+pq
0=p^2x+qx-15x+pq-15p-56
pine thicket
i was too obsessed with doing some ratio thing but it turns out it was adding
deep vine
Am I the only one who thinks you guys are massively overcomplicating things? Unless you guys are done and I should shut up
hard shard
is there a simpler solution?
feral sedge
a=x^3-15x=px^2+qx-15x=p^2x+qx-15x+pq
0=p^2x+qx-15x+pq-15p-56
don't think it's p^2q at the end but sure this fine
viral dagger
a=x^3-15x=px^2+qx-15x=p^2x+qx-15x+pq
0=p^2x+qx-15x+pq-15p-56
15p+56=a?
feral sedge
don't go back to a and b
there is something fairly important that equation you have tells you
viral dagger
wait wrong variable
deep vine
is there a simpler solution?
- Find all irrational x such that x³-15x is rational
- Find all irrational x such that x⁴-56x is rational
- Take the intersection
viral dagger
don't think it's p^2q at the end but sure this fine
oh its pq right
feral sedge
no the point of this is to get rid of a and b
a and b are very inexplicit things that don't help you when you're trying to find x explicitly
and to do this you want to find p and q explicitly
viral dagger
a=x^3-15x=px^2+qx-15x=p^2x+qx-15x+pq
0=p^2x+qx-15x+pq-15p-56
like this?
feral sedge
is that true
well wtv the point is
an irrational + a rational can never be rational
feral sedge
which are
hard shard
1. Find all irrational x such that x³-15x is rational
2. Find all irrational x s...
how do you sppose we do that?
viral dagger
-p^2=q
viral dagger
oh wait
exception at p^2+q-15=0, which we can basically isolate q=15-p^2 then sub back?
feral sedge
i wouldn't do that
do the same process for the other initial condition you are given
viral dagger
mk
b=x^4-56x=x^2(px+q)-56x=px(px+q)+q(px+q)-56x=p^2(px+q)+pqx+pqx+q^2-56x
coefficient of x is p^3+2pq-56=0
this ones ugly :(
viral dagger
oh yeha
viral dagger
uhh how do you solve it exactly?
feral sedge
exactly how you'd expect (substitution to remove one of the variables)
midnight plankBOT
@viral dagger Has your question been resolved?
viral dagger
exactly how you'd expect (substitution to remove one of the variables)
p=-cbrt(56), which gets a=-15cbrt(56)+56
viral dagger
wait -cbrt(56)
feral sedge
that doesn't make sense either
try the substitution again and recheck very carefully
remember your equations are
p^2 + q - 15 = 0
and
p^3 + 2pq - 56 = 0
feral sedge
hm
viral dagger
=0
feral sedge
looks good
viral dagger
,w solve x^3-30x+56=0
grand pondBOT
viral dagger
so p=4
feral sedge
correct
viral dagger
sorry smth came up, back
feral sedge
nw
viral dagger
so, we got 15p+56=a <=> a=116 <=> x^3-15x-116=0
viral dagger
uhh did i plug it wrong
feral sedge
tbh i'm not sure where 15p + 56 = a came from to begin with
anyway you should find q not a
viral dagger
b-ax=15x^2-56x
oh wait and the constant term (aka q) here is b, so -1=q=b
feral sedge
remember what the definition of p and q are
feral sedge
but does finding a and b really help you
this entire time the point was finding p and q
feral sedge
x^2 = px + q
.
feral sedge
yes so you are done
feral sedge
🤔
viral dagger
b-ax=15x^2-56x
i underatand why would somebody try this up to here, whats the (incentive? encouragement? what makes you think to do that?) to make it do the x^2=px+q and get equations interms of p and q
and after getting the equations i think its reasonable on how one would think to complete it
feral sedge
tbh the stuff with a and b is just too messy
you want to make use of 15x^2 - 56x + ax - b = 0 somehow but this form takes too long to do calculations with
so supposing x^2 = px + q is just as well
and solving for x before knowing p and q (or a and b) is no good either, since then you can only proceed by plugging x into x^3 - 15x, which is completely terrible
even if you guessed x = (a + bsqrt c) you would still not have such an easy way to proceed
so after having x^2 = px + q you should think to make further reductions
hard shard
real quick skish can you do me afavor and just make sure you transcribed the problem properly
viral dagger
real quick skish can you do me afavor and just make sure you transcribed the pro...
yeah gimme a sec
viral dagger
real quick skish can you do me afavor and just make sure you transcribed the pro...
as far as i understand the problem, yes
hard shard
ok thanks
viral dagger
why btw? im curious
viral dagger
so after having x^2 = px + q you should think to make further reductions
ohh ok, ill try to keep that in mind
feral sedge
so after having x^2 = px + q you should think to make further reductions
and in this case you did that via 0 = sx + t where s and t are rational so s = 0
hard shard
why btw? im curious
seems like a lot of work
feral sedge
really? there is not much
hard shard
youve been here for an hour already
feral sedge
so? i could probably write out this solution in full in 5 minutes
hard shard
ok fair
viral dagger
so? i could probably write out this solution in full in 5 minutes
yeah mb
.solved ty guys!
midnight plankBOT
viral dagger
i mean like 2/3rds of it was me not responding
feral sedge
oh ok
midnight plankBOT
Available help channel!
last slate
midnight plankBOT
dusty arrow
im guesing [.] denotes gif
last slate
im guesing [.] denotes gif
Yess
twilit jetty
not a good idea
last slate
Sorry forgot to mention
last slate
not a good idea
All of the subsequent questions were done in the same manner
twilit jetty
split the fraction into $\frac{\sin x}x\cdot\frac{\tan x}x$
grand pondBOT
mtt
twilit jetty
what do you think
do sin x / x and tan x / x remind you of anything
remember this is a limit as x approaches 0
twilit jetty
yes
dusty arrow
All of the subsequent questions were done in the same manner
never just adopt to a pattern , i see what you are trying to do with that but thats not needed
last slate
Wrong
dusty arrow
sigh
twilit jetty
so this limit is?
last slate
Actually the second limit is not 1
last slate
so this limit is?
1
dusty arrow
Actually the second limit is not 1
yeah we know
last slate
How tho
twilit jetty
,,\frac{x^2}{\sin x\tan x}=\frac1{\frac{\sin x\tan x}{x^2}}
grand pondBOT
mtt
last slate
This is 0
twilit jetty
no
twilit jetty
what was the limit of this, again?
dusty arrow
it was not mentioned in the question
last slate
1
last slate
What about the gif of this expression
What would be the limit of that is the question
twilit jetty
answer the question
last slate
1
dusty arrow
ah ok mb
twilit jetty
and since these two are equal,
last slate
Yes
twilit jetty
what would this limit be
last slate
Idk
twilit jetty
you dont know?
twilit jetty
well "Dude,"
last slate
Stop and observe for a sec
twilit jetty
and since these two are equal,
these expressions are exactly the same
dusty arrow
these expressions are exactly the same
floor function-
twilit jetty
first off
last slate
@twilit jetty
twilit jetty
whats the floor of 1?
twilit jetty
so its either 1 or its DNE
last slate
Limit of sinx/x means that it approaches to 1
last slate
Stop and listen for a sec
twilit jetty
zyro you
listen to me
we do this one step at a time
and the first step is assuming that the limit exists
then the second step is verifying that the limit exists
the first step, you havent done yet
last slate
U are not understanding the question
twilit jetty
zyro you seem to be seeing some sort of idiot in me that isnt real
just say the number 1 ok?
last slate
Ok
twilit jetty
what point do you think I am making?
last slate
Go ahead
twilit jetty
you keep typing over me like you know
take the reins: what was I going to do with this?
last slate
I think that u are saying that because the limit of sinx/x is 1 and tax/x is 1 the value of the question would be Q
twilit jetty
wrong
last slate
1
twilit jetty
but we've already verified that the number inside the limit would be 1 regardless
if its 0 and its 1,
thatd be DNE
twilit jetty
no I see what youre getting at
dusty arrow
dusty arrow
yes
last slate
Yes
twilit jetty
I was thinking itd either be 1 or DNE
last slate
Thats what I am trying to say this whole time
twilit jetty
but sin(x)/x approaches from below
dusty arrow
so the limit would be 1 but a lil shorter than 1
zyro was trying to compare functions with sinxtanx - x^2
last slate
And since tanx>x the limit of its floor would be 1
dusty arrow
so he can conclude if sinxtanx > x^2 or the other way arround
last slate
Yes
dusty arrow
yeah?
last slate
Thats where I am having problem
dusty arrow
you can just check whoese rate is more , tan or sin
twilit jetty
lets not use derivatives on a question that doesnt require them unless absolutely necessary
especially with floor
dusty arrow
but to compare it with x^2 woud be differnet
not really you would be comparing
Tanx /x and sinx/ x just check who rate more , so you know which one dominates in the gif function
last slate
i think taking f(x) = sinxtanx-x^2 and checking the signs of the function would be better
dusty arrow
i think taking f(x) = sinxtanx-x^2 and checking the signs of the function would ...
thats works too but you are effectively doing the same , if its subjective paper and step marking is there go for it
but its not required to do all this when you can judge the whoese rate would be more
twilit jetty
the only answers possible are 0 or 1 for each limit
twilit jetty
so theres in total 4 possible answers
so it being a multiple a choice wont belp us
if one of the answers is a DNE then we can use that to eliminate up to 2 of the answers
dusty arrow
also i dont get graphing calculators in my exams
you know the range na , thats all thats required to determine the rate tbh
twilit jetty
since the limits are 0 and 1 or 1 and 0
last slate
how do i compare the rates
dusty arrow
how do i compare the rates
you can do that roughly too ig , we alr know tanx is increasing in x belonging to 0 to pi/2 , same for sinx , so just assume value at end points and yeah rates
if you wanna be precise and sure , derivative is the way
dusty arrow
you can do that roughly too ig , we alr know tanx is increasing in x belonging t...
average would work too tho tbh
twilit jetty
average wont tell you the behavior near 0
dusty arrow
yeah i gotchu jee
last slate
so i want to try to find the most precise way
twilit jetty
theres a precise way I want to avoid as a last resort
last slate
taylor can be used
twilit jetty
you get to do that?
last slate
but the thing is that is not in our syllabus
twilit jetty
oh that wouldve been powerful
dusty arrow
but the thing is that is not in our syllabus
it is? tf
last slate
so we cant relly use them
twilit jetty
you can consider taking the double derivative of (sin x)(tan x) - x^2
last slate
i did that
dusty arrow
did i get which exam you prep for wrong?
twilit jetty
was it positive?
twilit jetty
oh right itd be 0
last slate
but the thing is taylor hsnt been taught
dusty arrow
yeah taylor is very much relevant
last slate
rn
last slate
so i cant use it
dusty arrow
tanx > x > sinx , its a very comon inequality
twilit jetty
youd have to take 4 derivatives then
last slate
i wanna do it by analyzing the function
twilit jetty
thats not enough
dusty arrow
i can just take 2 to check the behaviour
show your work
twilit jetty
oh heres something easier
you can take two derivatives of (sin x)/x and of (tan x)/x
(tan x)/x has a larger second derivative than (sin x)/x
problem is thatll tell you what happens for (tan x)/x - (sin x)/x, not (tan x)/x * (sin x)/x
@last slate I found a function you can d/dx twice for
we need to find out whether this is true or not in the neighborhood of x=0
now if you write it like this,
that just means we check the same thing for (sin x)/x - sqrt(cos x)
due to the square root and dividing by x, this is a function with a nonzero second derivative (as you can check if you d/dx it twice)
midnight plankBOT
@last slate Has your question been resolved?
twilit jetty
however it takes quite a while before you get somewhere useful
each of these are either special limits or you can plug in x=0 into it
this though, you need l'hopital for this
jade magnet
midnight plankBOT
lyric charm
the proof? or the choice of weights?
jade magnet
the proof? or the choice of weights?
mostly the proof
so to be able to get every distinct weight you need to have the weights for (2^n)-1 of 2^0, 2^1, 2^2 etc up to 2^(n-1)
lyric charm
first you say mostly the proof then you talk about the other part of the question...
jade magnet
first you say mostly the proof then you talk about the other part of the questio...
sorry idk
i feel like its both
lyric charm
anyway, with n weights you have n binary choices to make for weighing: either you put the weight on the scales or you dont
which means 2^n different configurations
you cannot have more than this many different achievable weights
(but you can certainly have less)
jade magnet
anyway, with n weights you have n binary choices to make for weighing: either yo...
oh yeah thats true
jade magnet
which means 2^n different configurations
yeah ok
i forgot you can also include 0 as a configuration
so ig i felt confused because you could only go up to (2^n)-1 but i forgot 0 is included
jade magnet
which means 2^n different configurations
also this is something but then how do we actually prove this
because writing a few sentences isnt a proof but i honestly dk how to do a proof for this either
lyric charm
writing a few sentences isnt a proof
i'd beg to differ depending on what's contained in these sentences
you can write a formal proof by induction if you wanna spend time on that
that n (distinguishable) weights yield 2^n configurations
jade magnet
> writing a few sentences isnt a proof
i'd beg to differ depending on what's con...
oh so you can just write a proof like that?
I always preferred like formal mathematical proofs rather than just writing/explaining one
jade magnet
you can write a formal proof _by induction_ if you wanna spend time on that
I mean if I don't have to for the question then it's fine
lyric charm
I always preferred like formal mathematical proofs rather than just writing/expl...
what do you think of as a "formal mathematical proof"
do you think it's something completely devoid of words and is instead just symbol soup
jade magnet
do you think it's something completely devoid of words and is instead just symbo...
im gonna be honest
kind of yes
or at least very algebraically
lyric charm
well then you have the wrong idea about proofs
real mathematicians write "prose proofs" all the time
jade magnet
well then you have the wrong idea about proofs
Yeah probably
jade magnet
real mathematicians write "prose proofs" all the time
I guess it's something I need to get used to
because idk for some reason I wasn't a huge fan of prose proofs
whenever i write one it's very hit or miss
lyric charm
heres what a proof from my own master's thesis looks like. it's not important what it's talking about but i just wanted to show you what a theorem & proof in real, serious research looks like.
don't mind the "through joining" 
jade magnet
heres what a proof from my own master's thesis looks like. it's not important wh...
huh interesting...
where did you do your masters btw
just curious
jade magnet
in bulgaria, where i live now
yeah fe
jade magnet
I'm going to probably try the last part now
jade magnet
this time there can only be one weight on one side, one weight off or one weight on the other side unlike last time where it was a binary true or false statement so that's why you can't weigh more than 3^n different weights
midnight plankBOT
midnight plankBOT
Available help channel!
hoary hamlet
I dont understand this explanation. I understand everything up to where alpha is solved
midnight plankBOT
I dont understand this explanation. I understand everything up to where alpha is...
midnight plankBOT
@hoary hamlet Has your question been resolved?
worldly pine
what am i doing wrong here
midnight plankBOT
what am i doing wrong here
slender walrus
using percentages in the equations by themselves
not using variables
its unclear what they represent and also makes the equations invalid
start with
clearly introducing variables and what they represent
here to start a good idea would be to have two variables
to represent:
amount invested at 15%
amount invested at 12%
lyric charm
i think i might know what's happening
wizard might be attempting to reproduce an algebra-free solution to this exact question that i gave some time ago
only he's decided to completely strip away every single word of it
.
this is what i wrote 5 days earlier
you can follow this method, it's fine -- but YOU HAVE TO SAY ALL THE WORDS AND TALK THROUGH ALL THE REASONING. absolute, non-negotiable, hard requirement.
otherwise you suffer consequences.
the consequences are:
- exactly zero people understand your work -- this means even you do not understand it.
- you get things wrong.
slender walrus
3 years,