1 divided by 0 equals Infinity
#help-27
woven radishBOT
rain summit
(degrees)
rain summit
**1 divided by 0 equals Infinity**
that's for your situation
tall trellis
yea i know sin+cos=90 but how does thta factor into these indeinites
rain summit
yea i know sin+cos=90 but how does thta factor into these indeinites
hm
lemme think rq
fervent ravine
yea i know sin+cos=90 but how does thta factor into these indeinites
u can prove these on unit circle
even chatgpt can prove it for u try texting it tbh
tall trellis
u can prove these on unit circle
bro would you shut up
rain summit
AI's not recommended tho
tall trellis
u can prove these on unit circle
how i not stupid i already tried for 2 hours i dont get it
crystal dawn
even chatgpt can prove it for u try texting it tbh
so I usually don't ping mods over this kind of stuff, but <@&268886789983436800> rather poor behaviour in help channels, including encouraging OP to turn to AI for help.
past poor behaviour: #help-12 message and #help-12 message.
mortal tendon
how i not stupid i already tried for 2 hours i dont get it
sinx is the y coordinate,
trace an angle A, and then angle B after A
the net y coordinate would be sin(A+B)
unique canyon
so I usually don't ping mods over this kind of stuff, but <@&268886789983436800>...
I dont support use of AI, but in this case I guess, use of AI is fair , cause it actually helps you understand proofs
unique canyon
Do yk e^ix=cosx+isinx?
it isn't neccesary here tho
rain summit
so I usually don't ping mods over this kind of stuff, but <@&268886789983436800>...
thank you!
chrome sorrel
Do yk e^ix=cosx+isinx?
damn das crazy
soft umbra
This is getting messy
chrome sorrel
i mean it can be a way to prove the uhh thing
tall trellis
Do yk e^ix=cosx+isinx?
no didnt leanr complex trig
crystal dawn
I dont support use of AI, but in this case I guess, use of AI is fair , cause it...
if the OP decides to use AI himself, that's his choice to do so, but as helpers directing them to AI is never a good idea.
I believe OP came to this server and asked questions specifically in this server for humans to answer.
rain summit
if the OP decides to use AI himself, that's his choice to do so, but as helpers ...
strong point that i agree
unique canyon
if the OP decides to use AI himself, that's his choice to do so, but as helpers ...
yeah, I also agree tbh
fathom verge
if the OP decides to use AI himself, that's his choice to do so, but as helpers ...
This is the rule of thumb indeed
rain summit
This is the rule of thumb indeed
is the user handled may i ask?
fathom verge
Now please, go back to either helping or not participating to noise in this channel
unique canyon
is OP's doubt cleared btw?
tall trellis
sinx is the y coordinate,
trace an angle A, and then angle B after A
the net y ...
for y but how does x work
i get y but not x part here
fathom verge
bro would you shut up
Also, how silly that suggestion may have been, there's no need to resort to this
mortal tendon
x coordinate is for the cosine
same process trace an angle A, then B, and the net x coordinate is cos(A+B)
tall trellis
x coordinate is for the cosine
yea so why do we combine cosb with sina instead of writing sina+ sinb +- cos a cosb
becuasse in cosine we dont split it but in sin we do
?
chrome sorrel
so ur problem is understanding that trig identity?
tall trellis
yea
chrome sorrel
i think theres a geometric way of looking at this
However, its like the middle of the night for me rn
tall trellis
becuase to form the right angle we have to take the radius of 1 here whcih i get but i dont get the resk of it becuasse the deigtram is like sin45+cos45 = the full 90 but i odnt get the split
tall trellis
becuase the sin of 1 angle is the cosine of the other they add to 90 becuase of right triangles so sin(45)+cos(45)=90 but then if we take hl to prove 1 line is relfective how does sin and cosine land on the same line
<@&286206848099549185>
rain summit
are we doing the geometry or unit circle here?
tall trellis
are we doing the geometry or unit circle here?
trig
tall trellis
but geomerty og course
rain summit
here $a = \angle DBA$ and $b = \angle CBE$
woven radishBOT
1 divided by 0 equals Infinity
tall trellis
yea
rain summit
new diagram
from this diagram, can you find $\sin A$, $\sin B$, $\cos A$, $\cos B$ for me?
woven radishBOT
1 divided by 0 equals Infinity
rain summit
im trying to build up my proof now so don't judge me if im wrong
i haven't looked into the proof yet
chrome sorrel
interesting, this the first time ive seen this proof for it
tall trellis
the sin of a would be the lenght of segment be/be
since a is laregst angle here
in <abe
oh i mean <bae
rain summit
hm
chrome sorrel
i think it be better to think of it as in unit circle
but maybe im just biased tho
rain summit
but maybe im just biased tho
i don't do much trig so i'd do geo more to prove this
rain summit
oh okay
tall trellis
well be is reflective here no
rain summit
lemme see, $\sin A = \frac{AD}{DB} = \frac{DE}{CD}$
woven radishBOT
1 divided by 0 equals Infinity
rain summit
$\sin B = \frac{CE}{BC}$
woven radishBOT
1 divided by 0 equals Infinity
rain summit
$\cos A = \frac{AB}{BD} = \frac{CE}{CD}$
woven radishBOT
1 divided by 0 equals Infinity
rain summit
$\cos B = \frac{BE}{BC}$
woven radishBOT
1 divided by 0 equals Infinity
devout snowBOT
Available help channel!
magic echo
what do I do if im trying to use the quotient rule but the numerator equation is raised to a power like (# +- #x)^#/x^#+-#
devout snowBOT
what do I do if im trying to use the quotient rule but the numerator equation is...
magic echo
do I just move the raised power to each variable inside the parenthesis ?
green crypt
using hashtags is crazy
magic echo
;-;
woven radishBOT
Azyrashacorki
trail eagle
It becomes $n (a+bx)^{n-1} \cdot (a+bx)' = n(a+bx)^{n-1} \cdot b$
woven radishBOT
Azyrashacorki
trail eagle
You should practice the chain rule. $(f(g(x)))' = f'(g(x)) \cdot g'(x)$.
woven radishBOT
Azyrashacorki
trail eagle
Here $f(x) = x^n$ and $g(x) = a+bx$
woven radishBOT
Azyrashacorki
magic echo
f'(x) = nx^n-1 g'(x) = b
a+bx^n-1 * b
I feel like I know how to use the chain rule it's just a bit hard when trying to represent it only using letters
trail eagle
a+bx^n-1 * b
n(a+bx)^(n-1) * b
magic echo
ok so If i took (9x+10)^6 and applied this I would get 6(9x+10)^5 * 6(9)^5
trail eagle
No. (x^6)' = 6x^5 and (9x+10)' = 9, so the derivative is
(derivative of outside with inside plugged in) * derivative of inside = 6(9x+10)^5 * 9
gritty moat
i need to prove these by induction i had help doing the first one but im still a little lost on the rest
devout snowBOT
i need to prove these by induction i had help doing the first one but im still a...
polar chasm
i need to prove these by induction i had help doing the first one but im still a...
Hi, next time please send it as a picture
polar chasm
i need to prove these by induction i had help doing the first one but im still a...
Okay, so for the 2nd one, do you know what the big O notation means?
definitorically
gritty moat
definitorically
f(n) = O(g(n)) where f(n) is always less than g(n)
polar chasm
um not quite
f(n) = O(g(n)) if there are constants c and k, such that
f(n) <= c*g(n) for all n > k
in words, if f(n) is eventually smaller than constnat multiple of g(n)
gritty moat
okay
polar chasm
So for what constant c do you think that n^2 < c*n! will eventually hold?
gritty moat
well it depends on what value n is
polar chasm
well it depends on what value n is
all n
we want it to work for all sufficiently high n
now you can just take a guess, we are only trying to find the constant c. We dont need to prove it yet
gritty moat
im not sure but ill guess 5
polar chasm
uh, that works but is somewhat arbitrary
do you think that n^2 <= n! will work? n^2 <= 3n! will work? n^2 <= 5n! will work?
by "will work" i mean will be true for large enough n
gritty moat
okay i think every value but 0 will work for large enough n
polar chasm
Great, so lets just pick the simplest one, 1
gritty moat
okay
polar chasm
5 works too, but is kinda arbitrary. You could use it though, its just quite random
so we will try to prove that n^2 <= n!
how large does n have to be for it to work?
Just try few small values of n
until you think it will work
gritty moat
i think 5 will work
polar chasm
yeah, sure, so let's try proving that it works for n>=5
i think it actually works from n = 4, but we can do 5 as well
so let's start by the base case. Can you state it and verify that it works?
gritty moat
5^2 = 25 and 5! = 120 and 25 <= 120 is true so the base case where n = 5 is true
gritty moat
im not sure how
polar chasm
im not sure how
Inductive step works by assuming that it works for n, and then proving that it works for n+1
If we can manage to prove that, then we are done
because it working for 5 implies that it also works for 6
and then it working for 6 implies that it works for 7
...
polar chasm
Inductive step works by assuming that it works for n, and then proving that it w...
So we need to prove this
So suppose that it works for n
that means n^2 <= n!
and our goal is proving that it works for n+1, that is
(n+1)^2 <= (n+1)!
is this part clear? If not, what specifically is unclear? If yes, try doing some algebra and try to prove it, or at least simplify it a bit
gritty moat
i understand this part its the next part im unsure about
polar chasm
okay so you dont know how to prove that (n+1)^2 <= (n+1)! ?
gritty moat
yea i dont know
polar chasm
In induction, we always try to use the inductive hypothesis somehow, in our case the hypothesis is n^2 <= n!
so we will probably need n^2 and n! to appear somewhere in the inequality
(n+1)^2 <= (n+1)!
Can you try simplifying this so that it has n^2 somewhere and n! somewhere?
gritty moat
what do you mean by n^2 somewhere and n! somewhere?
polar chasm
what do you mean by n^2 somewhere and n! somewhere?
Just simplify it until you see n^2 and n! somewhere in the inequality
like you can expand (n+1)^2
and you can also rewrite (n+1)! using the recursive defn of factorials
gritty moat
idk how to do that sorry
polar chasm
do you know how to expand (n+1)^2
there is an (a+b)^2 identity ||(a+b)^2 = a^2 + 2ab + b^2||
try using that
or just write it as (n+1) * (n+1) and carry out the multiplication
gritty moat
sorry i got no idea whats going on why does (n + 1) * (n + 1) equal n^2
polar chasm
sorry i got no idea whats going on why does (n + 1) * (n + 1) equal n^2
it doesnt equal n^2
what is it equal to?
Can you expand it
polar chasm
there is an (a+b)^2 identity ||(a+b)^2 = a^2 + 2ab + b^2||
using this identity
gritty moat
n^2 + 2n1 + 1^2
polar chasm
perfect
2n1 is just 2n
because 2n * 1 = 2n
so its n^2 + 2n + 1
what about (n+1)!, can you simplify that somehow?
k! = k * (k-1)!
this is the relevant identity here
gritty moat
n * (n -1)!
gritty moat
sorry i dont understand what any of this means
polar chasm
hm, you should probably practice algebra a bit then
(n+1)! would be (n+1) * n!
that's what we get when we replace k by n+1 in here
k! = k * (k-1)!
(n+1)! = (n+1) * (n+1-1)! = (n+1) * n!
does it make more sense now? It's just algebra with squares and factorials
gritty moat
how did you get (n + 1) * (n + 1-1) wouldnt it be n * (n + 1-1)
polar chasm
not really, because k was replaced by n+1
so k * (k-1)! turns into (n+1) * (n+1 - 1)!
we replace all the k's by n+1
that's how substitution works
gritty moat
oh okay
polar chasm
so we get this
notice that now we have n^2 and n! in the inequality, so we have a little more hope on applying the n^2 <= n! inductive hypothesis
gritty moat
yea
polar chasm
but we'd still need the n! to stand separately, and not in a product with n+1
so we can distribute it
do you know how to expand / multiply out (n+1) * n!
gritty moat
(n+1)/(n+1)?
polar chasm
not quite
do you know how to expand
3 * (x+2)
or rather (x + 2) * 3 (which is the same thing)
gritty moat
3x + 6
polar chasm
so you probably did sth like this in ur head
multiplied x by 3 and then 2 by 3
and then added it up
can you do something similar with
(n+1) * n!
gritty moat
2n! + n!?
polar chasm
nearly
it would be
n * n! + n!
you cant really simplify n * n!, so you gotta keep it as it is
gritty moat
okay
polar chasm
$n^{2}+2n+1\le n\cdot n!+n!$
woven radishBOT
MathIsAlwaysRight
polar chasm
**MathIsAlwaysRight**
hence it suffices to prove
2n + 1 <= n * n!
because then you can add it with n^2 <= n!, and you get the above
so our goal now is just proving that
2n + 1 <= n * n!
you can think of it as kind of "canceling", because we already know that n^2 <= n!, so we just need to check whether that inequality holds for the other part as well
gritty moat
for all these questions do i need to learn the identities of all the values like (n+1)^2 = n^2 + 2n1 + 1^2 and that (n+1)! = (n+1) * n!
i dont really understand why we need to go from (n+1)^2 to n^2 + 2n1 + 1^2
polar chasm
for all these questions do i need to learn the identities of all the values like...
that's part of algebra knowledge I'd say
polar chasm
i dont really understand why we need to go from (n+1)^2 to n^2 + 2n1 + 1^2
The motivation behind that step is because we're trying to apply the inductive hypothesis n^2 <= n!, so we need n^2 to be somewhere. And simplest way to make n^2 appear is applying that identity
I think that you dont really have trouble with induction, you're quite weak on your algebra though
I think that you should probably revisit algebra
is this for uni / hs / self-training for olympiad or sth else?
gritty moat
is this for uni / hs / self-training for olympiad or sth else?
for uni im doing computer science
polar chasm
I see
If math is part of that, you will definitely need to revise HS algebra
try khanacadmey
gritty moat
okay thx
polar chasm
it takes some time, but it's worth it
gritty moat
i was also wondering The motivation behind that step is because we're trying to apply the inductive hypothesis n^2 <= n!, so we need n^2 to be somewhere. And simplest way to make n^2 appear is applying that identity why do we need n^2 to appear?
polar chasm
i was also wondering The motivation behind that step is because we're trying to ...
because n^2 is part of the inductive hypothesis n^2 <= n!
the idea in induction questions like this one is usually that you try to get the induction hypothesis on both sides like this (left side has n^2, right side has n!. They make n^2 <= n! together) and then "cancel" them
but also, the motivation is also just algebra
most algebra students would spit out n^2 + 2n + 1 immidiately when they look at (n+1)^2
it's pretty much the only thing you can do with it
polar chasm
well, more like it allows you to cancel the n^2 later
and simplifies the goal to
2n + 1 <= n * n!
which is a pretty major simplification
Now 2n+1 <= 3n (we know that n >= 5..) and 3n <= n! n, because 3 <= n! (again, we know n >= 5 and its pretty obvious that n! will be atl east 3 from there)
so 2n+1 <= 3n <= n! n = n! * n, so it's proved and we're done
gritty moat
where does the 3n come from?
polar chasm
i made it up because it made it simpler for me to prove it
there are many other ways to prove it though
2n+1 is kind of difficult to work with algebraically, because it contains both a linear term and a constant term
so to avoid that difficulty, I decided to replace 1 by n, because n is obviously larger than it
so 2n + 1 <= 2n + n = 3n
now i asked myself, does 3n <= n * n! still hold? Intuitively, yes. That n! still grows very very fast
so that simplification probably didn't impact the validity of the claim
and it made it easier to prove, because now both sides are multiples of n
the left side is 3 * n, the right side is n! * n
and since 3 < n!, we know that it's true
polar chasm
you can think of it as kind of "canceling", because we already know that n^2 <= ...
another way to do it would be doing:
near the beginning
gritty moat
in the induction step do i always need to simplify it down like that or can i just leave it at 2n + 1 <= n * n!
polar chasm
in the induction step do i always need to simplify it down like that or can i ju...
you still need to prove that
so you have to prove it somehow
gritty moat
can my proof be like the base case where i just insert n = 5 and solve it that way?
polar chasm
can my proof be like the base case where i just insert n = 5 and solve it that w...
unfortunately not
you need both the base case and the inductive step
is this for a homework or are you studying for an exam or sth?
gritty moat
is this for a homework or are you studying for an exam or sth?
this is homework
polar chasm
I see, what would happen if you didnt submit it? Or submitted it late? I think it'd be much more beneficial for you to learn algebra at this point
i dont think you can do proofs by induction without understanding algebra
gritty moat
I see, what would happen if you didnt submit it? Or submitted it late? I think i...
its not graded so i can do it whenever
polar chasm
If I were you, I'd probably try to catch up on algebra
look at khanacademy and try to do as much of the algebra as you can (you can take the course tests to figure out what you know already, and what you struggle with)
devout snowBOT
gritty moat
i appreciate your time
devout snowBOT
Available help channel!
wary dust
can you help me with linear equations
devout snowBOT
can you help me with linear equations
Asking the actual question right away is more likely to get responses.
Asking "Can I ask...?" or "Does anyone know about...?" doesn't give people enough information to decide whether they can help, and answering can feel like a promise to help with the actual question, which they might find themselves unable to.
wary dust
what is linear equations
trail eagle
A linear equation is some equation where every term is either constant or some degree 1 term with constant coefficient.
stone stump
linear equations are things like 7x+3=4 and 17x-1=23+5x
and not things like 8x^2-3x+7=11 or x^x=13
knotty kraken
think if its in the form ax + b = 0 where a =/ 0 (can you see why a =/ 0)?
stone stump
dont restrict a. unless you want to tell me that 7x+3-2x=4x+2+x+1 shouldnt count as a linear equation
devout snowBOT
@wary dust Has your question been resolved?
knotty kraken
naw, its an identity rather than equation
mild bane
:𝐂𝐥𝐨𝐬𝐞
supple knot
.close
spring stump
devout snowBOT
supple knot
Do you have a non red marked image
spring stump
Do you have a non red marked image
Ye
supple knot
Great show that
supple knot
What else is given in the problem
spring stump
Sadly that is it
supple knot
What are you assuming about the shapes then
spring stump
Oh wait well
It said that its a rectangle
The left shape is semi circle and the right is a quadrant circle
supple knot
Amazing context
spring stump
Real
But we can see thanks to that the radius of the semi circle is half the radius of the quadrant
rotund umbra
hint
spring stump
Youtube live shorts
idle dune
.closed
devout snowBOT
devout snowBOT
Available help channel!
devout snowBOT
Available help channel!
tulip hill
devout snowBOT
tulip hill
can someone tell me if this is the right approach
tall comet
log(a+b) != log a + log b
tulip hill
no
tall comet
can someone tell me if this is the right approach
you can do sinx = t but i dont think it will simplify things here
tulip hill
log(a+b) != log a + log b
OHHHshit thats what i did
tulip hill
you can do sinx = t but i dont think it will simplify things here
hm hmm
tulip hill
you can do sinx = t but i dont think it will simplify things here
hm hmm
so what would be the right way to start
and like what alll approaches can i think of
before directly jus diffrentiating
last parrot
Good handwriting sakura
tall comet
if it looks complicated you can try a substitution but you need to think what will that result in
like here x^sinx * sinx ^ x is not that hard to differentiate on its own
tulip hill
if it looks complicated you can try a substitution but you need to think what wi...
hmm hmm like the sinx= t one was not very effective
tulip hill
like here x^sinx * sinx ^ x is not that hard to differentiate on its own
yea i mean i can by parts
tall comet
hmm hmm like the sinx= t one was not very effective
yeah if it simplifies one term but turn the other terms to shit then its no good
devout snowBOT
tepid stag
can someone tell me if this is the right approach
you were correct with the log step, but i think in the given equation you should differentiate both the terms differently and use log one only for the first term
devout snowBOT
Available help channel!
tulip hill
you were correct with the log step, but i think in the given equation you should...
rightt that would have been simpler thanks
devout snowBOT
rightt that would have been simpler thanks
tulip hill
.close
devout snowBOT
green crypt
sakura from the hit anime "naruto"?
north lodge
sakura from the hit anime "naruto"?
devout snowBOT
Available help channel!
analog lantern
Gng i need help
devout snowBOT
Gng i need help
solid osprey
the general trick for this infinite square roots is that the inner square roots are exactly like x
analog lantern
I have a exam today
rain summit
the general trick for this infinite square roots is that the inner square roots ...
that's an observation you gotta make for those infinite series
analog lantern
So i make it quadratic?
rain summit
but i worry that some of these kinds of series don't have a value or smth
crude niche
but i worry that some of these kinds of series don't have a value or smth
they kinda do?
rain summit
So i make it quadratic?
uhhhh not quite
rain summit
they kinda do?
some
crude niche
So i make it quadratic?
no
analog lantern
Cuz i gort 1+root2
crude niche
Gng i need help
well wait you can do quadratic
solid osprey
So i make it quadratic?
yes basically
crude niche
yes basically
mb 😭
solid osprey
nice
devout snowBOT
rain summit
RAHHHHHHHHHH
worn salmon
Can any infinite real process have an imaginary result 🤔
crude niche
Can any infinite real process have an imaginary result 🤔
if infinite is even
devout snowBOT
Available help channel!
waxen geode
Hello I have no idea how to continue from this
devout snowBOT
Hello I have no idea how to continue from this
waxen geode
.close
devout snowBOT
devout snowBOT
Available help channel!
gray oriole
how they got to dx/dx
devout snowBOT
how they got to dx/dx
gray oriole
but for y it is dy/dx
winter torrent
multivariate chain rule
(it also kinda looks like the product rule which it is, but that's hard to prove)
gray oriole
(it also kinda looks like the product rule which it is, but that's hard to prove...
dyk abt tree diagrams in chain rule
idk if its a thing, our prof explained it to us like that
winter torrent
um idts
winter torrent
winter torrent
if y isn't implicitly a function of x then i'm not sure what they could possibly mean by d/dx f(x, y)
devout snowBOT
@gray oriole Has your question been resolved?
gray oriole
can i denote the derivative as f_r?
void fox
yes
gray oriole
what?
i took dx/dr first then i did df/dx so maybe it makes sense to use f_x_r for this
or f_r_x
void fox
they have defined f as a function of x and y which themselves are functions of r and θ
so you may think of f as a function of r and θ and it makes sense to differentiate f with respect to r
there is no need to include x in this
also if you included x and you would need to include y as well since f depends on both
void fox
it is just not necessary in the notation to keep track of x and y
gray oriole
(the pinned one)
void fox
also f_xy means something else; it means
,tex $ frac{partial^2}{partial x partial y} f $
woven radishBOT
void fox
which is a totally different thing
void fox
btw dyk the asnwer for the prev question
it is just the chain rule
Did hayley not already answer your question
devout snowBOT
@gray oriole Has your question been resolved?
gray oriole
it is just the chain rule
but x and y are not functions in my case, the definition of chain rule she shared was saying that x and y are functions of another variable t
void fox
x and y are functions of r and θ
gray oriole
x and y are functions of r and θ
but it is not mentioned anywhere in the original image i shared
this is just a derivation of the implicit diff formula
void fox
well non rigourous maths tends to be hand wavy like that
Technically what they are doing is this
gray oriole
they take an implicit function and just put it equal to f(x,y)
gray oriole
this is just a derivation of the implicit diff formula
then they do this
void fox
You start with a function f: ℝ² -> ℝ
you are given a coordinate change ψ: ℝ² → ℝ² and now consider the composition fψ
ψ is the thing that translates polar coordinates (r, θ) to Cartesian coordinates (x, y)
gray oriole
ψ is the thing that translates polar coordinates (r, θ) to Cartesian coordinates...
but the proof above does not talk about polar coordinates
how are the related here?
im just asking how they did this
void fox
but the proof above does not talk about polar coordinates
well it is implicit in the setup
gray oriole
well it is implicit in the setup
how
it is implicit in terms of y but then we just took that away from it by setting it equal to z
x and y are both independent now arent they
cuz now z = f(x,y)
void fox
x and y are considered as functions x(r, θ) and y(r, θ)
then f is a function of r and θ given by f = f(x(r, θ), y(r, θ))
vast wyvern
The work that the mass did from picture 1 to 2 is m * g * H, right?
topaz axle
it should be negative I think?
vast wyvern
Why
topaz axle
idk
vast wyvern
Anyways, now I want the maximal height H wrt L, mass and F
Conservation of energy must be useful somehow
tepid stag
True
vast wyvern
Any ideas?
tepid stag
Ill try wait
vast wyvern
Ok well we also know that $$W = \int_0^P \vec F(\vec r) \dd \vec r = \int_0^1 \vec F(\varphi(x)) \dot \varphi(x) \dd x = H \int_0^1 \vec F(Hx) \dd x = H^2 \int_0^H \vec F(u) \dd u$$ and so $$W = H^2 (E_{\text{pot}}(H) - E_{\text{pot}}(0))$$ with $\varphi(x) = Hx$ because its conservative and so it doesnt matter what way we choose
And as path, we can choose the straight line
Since its conservative
And F is constant so we can pull it out
vast wyvern
But yeah I feel like this is the way
topaz axle
work is informally lost energy, the mass gained mgH joules, so it did −mgH
vast wyvern
W = mgH but also this integral with F
vast wyvern
work is informally lost energy, the mass gained mgH joules, so it did −mgH
Ok but if I ask "what work did the wind do to the mass" it might be different than the other way around
vast wyvern
Ok but if I ask "what work did the wind do to the mass" it might be different th...
so if the problem is posed this way
Yeah
So its positive
r = phi, dr = dot phi dx
Now let u = Hx, then du = H dx
tepid stag
Ok wait i think im getting somewhere
How about this
Like when the wind keeps blowing, it will reach a point where like it will be in equillibrium and then it drops down again
Assume that point and try to do stuff maybe?
Equating forces
And i think there would be no centrifugal force that point
woven radishBOT
ILikeMathematics
plucky jolt
Hey guys. I’m trying to learn calc for the first time. Does anyone wanna help me
tepid stag
Hey guys. I’m trying to learn calc for the first time. Does anyone wanna help me
#1021175428326633542 not here
woven radishBOT
ILikeMathematics
tepid stag
**ILikeMathematics**
No idea what this is, im in 10th grade 😭😭😭
tepid stag
Is potential energy kinetic or height here
That sentence feels wrong ;-;
vast wyvern
I mean, is the potential energy in this situation (pendulum) the kinetic energy or the elevation energy
elevation, right?
tepid stag
I mean, is the potential energy in this situation (pendulum) the kinetic energy ...
Potential would just be mgH
tepid stag
Subtract x from L and is that the answer?
Ill be really surprised if this is the answer tho lol
vast wyvern
work is informally lost energy, the mass gained mgH joules, so it did −mgH
Actually, its apparently not mgH at all
W = F * x and we can look at the distance travelled in x-direction which is by Pythagoras sqrt(L^2 - (L - H)^2)
(because the work in direction travelled upwards and not in x-direction will be 0 because F is perpendicular to that)
but why is it not mgH too
Well it probably is both
Anyways thanks
.close
devout snowBOT
vast wyvern
Ill be really surprised if this is the answer tho lol
(in this case we can set F * x = mgH and solve)
(maybe your method works too, though)
tepid stag
(maybe your method works too, though)
gpt says the same method as me but ig im not aupposed to promote that
tepid stag
(in this case we can set F * x = mgH and solve)
oo i see tbh work kind of confuses me sometimes lol
topaz axle
positive doesn't make sense anyway
tepid stag
positive doesn't make sense anyway
Force and displacement are in the same direction, so by definition W=FScostheta work done would be positive..
small inlet
devout snowBOT
deep timber
What
devout snowBOT
Available help channel!
final bronze
Help this is what I've done so far but I don't know what the mapping diagram is and how to do it to do the right table of values
devout snowBOT
Help this is what I've done so far but I don't know what the mapping diagram is ...
@final bronze Has your question been resolved?
final bronze
<@&286206848099549185>
devout snowBOT
final bronze
uhh u still want help on this?
no i figured it out but idk if it's gonna stay in my head till tuesday for my test 😅
devout snowBOT
Available help channel!
devout snowBOT
supple knot
bronze void
what is your ratios? Does Soh Cah Toa ring a bell?
signal sand
Yeah
bronze void
what does it mean
signal sand
I did sohcahtoa but I think I messed up
Sine cos tan
It’s the
Opposite/hypotenuse
Adjacent/hypotenuse
And opposite/adjacent
I think
Wait
bronze void
correct
signal sand
Oh yay
bronze void
so, if we want tan(G) thats Opp/Adj right
signal sand
Yeah
bronze void
so which side lengths would that be in the above triangle
signal sand
32 and 40?
bronze void
yes, and is 32 or 40 the opposite or adjacent ?
signal sand
The opposite
bronze void
hmm, so in a right triangle, the hypotenuse is the longest side right , and we want the opposite and adjacent so should we have 40?
bronze void
indeed, we want 24 and 32
bronze void
adjacent is the side that connects the right angle and the point you want that doesnt include the longest side length
signal sand
adjacent is the side that connects the right angle and the point you want that d...
OH
bronze void
so would it be tan(32/24 ?
yes
signal sand
So Adjacent is never the longest?
bronze void
So Adjacent is never the longest?
only hypotenuse is the longest
devout snowBOT
bronze void
np, any other things you needed assistence with?
signal sand
Nope!
TYSM 😭
The calculator doesn’t need to be in degree form for tan right
Like degree mode
bronze void
depends
bronze void
The calculator doesn’t need to be in degree form for tan right
you should have degree and radian, the calculator is normally used in degree for these types of questions
signal sand
Aha
devout snowBOT
Available help channel!
silk zenith
How do I solve this without using the divisibility rule?
devout snowBOT
How do I solve this without using the divisibility rule?
silk zenith
(The answer is 9) except I had to look up the divisibility rule for 13
tepid stag
(The answer is 9) except I had to look up the divisibility rule for 13
observing a pattern we get 10^5+10^4+10^3+10^2+10^1+10^0 leaves zero reminder when devided by 13
so if we write N=10^49+10^48+10^47...+X(10^25)+10^24..+10^0
tepid stag
observing a pattern we get 10^5+10^4+10^3+10^2+10^1+10^0 leaves zero reminder wh...
i think using this we can say the reminder simplifies down to 10^49+(X-1)10^25
silk zenith
i think using this we can say the reminder simplifies down to 10^49+(X-1)10^25
why a (X-1)?
tepid stag
why a (X-1)?
because we need consecutive powers of 10
tepid stag
so if we write N=10^49+10^48+10^47...+X(10^25)+10^24..+10^0
but 10^25 has X as co eff so we add 1 and subtract the one which we added will contribute to series
do yk binomial theorem?
silk zenith
do yk binomial theorem?
yup
tepid stag
ok
silk zenith
oh that's gonna be a pain
tepid stag
i think using this we can say the reminder simplifies down to 10^49+(X-1)10^25
10^49 is just 13-3 whole power 49
so -3^49 now hmm
i mean i think this might be little better
silk zenith
wait
silk zenith
wait
silk zenith
10^49 is just 13-3 whole power 49
well binomial thm is annoying to deal with so i figured comparing multiples would be easier
tepid stag
oh
silk zenith
10^49+(x-1)^25 = (10^25)(10^24+x-1), therefore I only need to look at 10^24+x-1
tepid stag
10^25 gives 10 remainder btw
silk zenith
10^25 gives 10 remainder btw
? wdym
silk zenith
when divided by 13
yea but that doesnt matter
I know it's not divisible by 13 anyways so I can discard it
tepid stag
ic
silk zenith
I know it's not divisible by 13 anyways so I can discard it
I was thinking about using the little bit of mod I know next
tepid stag
10^24 is 13-3 whole power 24 so 3^24 meaning (3^3)^8 or 27^8 so 1 remainder?
silk zenith
10^24 is 13-3 whole power 24 so 3^24 meaning (3^3)^8 or 27^8 so 1 remainder?
yea but same thing with the binomial thm
I could do it
but it's annoying and has a lot of terms
silk zenith
x is 0 or 13 then gang
x cant be 0 because its not an option and x can not be 13 because it is not a digit
also x is 9(i checked with the divisibility criteria)
how did you end up at 11111/13 is an integer?
silk zenith
111,111 is divisible by 13
yes but calcs arent allowed for this exam, so if there's a shortcut ya used that'd be great
tepid stag
i consided pairs of 6 successive powers of 10
so 10^0-10^5 would be next 10^6-10^11... so last one would be i think 10^42-10^47
ya just consider 10^48 in this i think you will be done
tepid stag
10^49+(x-1)^25 = (10^25)(10^24+x-1), therefore I only need to look at 10^24+x-1
so add 10^48 to this
taking common , i think 10^23 goes inside
100x 10^21 10^21 is just 1 so uhm
9 would be remainder due to that
bro this is ragebait i quit
potent nebula
theres also a way to do this using 1000 = -1 mod 13
silk zenith
theres also a way to do this using 1000 = -1 mod 13
Im not really comfortable with mods all that much(idk inverses etc etc, mod is technically outside the syllabus but i know the very basics)
silk zenith
number theory is just ragebait
ah
silk zenith
so add 10^48 to this
oki ya wanna hear my solution?
tepid stag
sure
frosty crescent
isn't the factorization 1001 = 7 * 11 * 13 famous?
thus 111111 = 1001 * 111 = 3 * 7 * 11 * 13 * 37
silk zenith
using $11111 = 10^5+10^4+10^3+10^2+10^1+10^0$ (Let this number be $\alpha$)
You can whittle the problem down to:
$10^{25}+10^{24}+...+ 10x+1$ is divisible by 13.
Now instead of subtracting $\alpha$, just $10^{\beta}\cdot\alpha$ from it repeatedly(where $\beta + 5 = $ remaining highest power of 10).
At the end of the sequence, you'll be left with 10x+1 is a multiple of 13
and since we know x is a single digit, we get:
$7\cdot13 = 91$, since $9\cdot10+1 = 7\cdot13$
$x = 9$
woven radishBOT
Supernova
silk zenith
isn't the factorization 1001 = 7 * 11 * 13 famous?
idk is it? i've never heard it
tepid stag
hm i see
frosty crescent
idk is it? i've never heard it
it’s a fun factorization to memorize
the product of 3 consecutive primes is 10^3 + 1
woven radishBOT
Supernova
frosty crescent
no sadly
woven radishBOT
frosty crescent
,w factorize 100001
woven radishBOT
silk zenith
My problem with this proof however is that if I didn't know 111111 is divisible by 13, that'd be incredibly hard to find
frosty crescent
,calc 11111/13
woven radishBOT
Result:
854.69230769231
Most of the exams I have don't allow calculators, and im not sure this one does
frosty crescent
My problem with this proof however is that if I didn't know 111111 is divisible ...
it’s not?
frosty crescent
111111
silk zenith
it's a neat proof but its kinda contingent on me knowing a singular number
frosty crescent
you could notice that 111111 = 111 * 1001 for instance
and then 1001 = 10^3 + 1^3 = (10 + 1)(10^2 - 10 + 1)
silk zenith
you could notice that 111111 = 111 * 1001 for instance
If I don't know that I have to look for 111111, then I wouldnt even try that far, no?
frosty crescent
If I don't know that I have to look for 111111, then I wouldnt even try that far...
oh yeah
memorizing the famous factorization is probably the best you could do ig 
silk zenith
memorizing the famous factorization is probably the best you could do ig <:catsh...
or learn modulo so i can do the inverses
😭
silk zenith
modular arithmetic is really handy
ye i know like the very basics, idk the rules for multiplication or division
frosty crescent
multiplication is the same
division is the euclidean algorithm and a few theorems
silk zenith
multiplication is the same
$(p-1)! \equiv -1 \bmod p$
doesnt imply that if im reading that right
woven radishBOT
Supernova
frosty crescent
Bezout’s Lemma. For integers a and b there exist integers c and d such that ac + bd = gcd(a, b).
frosty crescent
**Bezout’s Lemma.** For integers a and b there exist integers c and d such that ...
this is the basis of division in mod
silk zenith
**Bezout’s Lemma.** For integers a and b there exist integers c and d such that ...
I think i'll learn it in college 😭
frosty crescent
**Supernova**
this is formed by pairing inverses
frosty crescent
I think i'll learn it in college 😭
i learned this stuff in 6th ngl
it’s technically elementary number theory
silk zenith
it’s technically elementary number theory
I meant to reply to learning mod
woven radishBOT
The current time for nadat12 is 10:56 PM (PDT) on Sun, 08/03/2026.
silk zenith
aight cya
frosty crescent
cya
devout snowBOT
devout snowBOT
Available help channel!
gritty saddle
How to show that steiners symmetrzation preserves measure? I know it has to be with fubini but fubini isnt useful for the normal basis?
devout snowBOT
How to show that steiners symmetrzation preserves measure? I know it has to be w...
@gritty saddle Has your question been resolved?
devout snowBOT
@gritty saddle Has your question been resolved?
stone stump
-> adv channels
gritty saddle
.close
devout snowBOT
devout snowBOT
Available help channel!
gray oriole
how to do (a)
devout snowBOT
how to do (a)
gray oriole
(the determinant is 0)
so that means the equations are linearly dependent
does that conclude a no unique solution?
sand dove
With a system with n variables and n equations and 0 determinant, no unique solution indeed
When determinant is 0, either there are no solutions or an infinity of them
Hope that helps @gray oriole
gray oriole
With a system with n variables and n equations and 0 determinant, no unique solu...
NO unique solution when number of equations is same as number of unknowns?
isnt that the other way around
sand dove
isnt that the other way around
Edited, reread
gray oriole
With a system with n variables and n equations and 0 determinant, no unique solu...
so how do i figure out which one is (a)
the determinant is 0
tired glade
hmmmm
gray oriole
is unique solution the same as no solution?
sand dove
the determinant is 0
Well the question doesn't ask you that
sand dove
is unique solution the same as no solution?
No
A unique solution = exactly 1 solution
Different from 0 solutions
sand dove
Well the question doesn't ask you that
If you want to know, you probably need to RREF the system
devout snowBOT
@gray oriole Has your question been resolved?
mild sorrel
devout snowBOT
mild sorrel
Let h->0.
Isn't the thickness Rsin(θ+h) - Rsin(θ) = R(sin(θ) cos(h) + sin(h) cos(θ) - sin(θ)) = R(hcos(θ)) = Rhcos(θ)?
devout snowBOT
@mild sorrel Has your question been resolved?
trail eagle
Let h->0.
Isn't the thickness Rsin(θ+h) - Rsin(θ) = R(sin(θ) cos(h) + sin(h) cos...
For small theta cos(theta) = 1 and h = dtheta
So rd(theta)
mild sorrel
Theta itself isn't small; it ranges from 0 to pi/2
h is small
Nvm. I've got it. I just need to consider the discs as frustums, not as cylinders.
trail eagle
Theta itself isn't small; it ranges from 0 to pi/2
Yes you're right.
devout snowBOT
devout snowBOT
Available help channel!
wide elm
What's did I do wrong
devout snowBOT
What's did I do wrong
wide elm
woven radishBOT
Result:
80.4
,calc 75-1.8 * 12 - 15 * 1.8
woven radishBOT
Result:
26.4
@wide elm Has your question been resolved?
restive river
wrong server
.is asking non math q's not allowed?
potent compass
devout snowBOT
tepid stag
-6b when u were getting 2nd equation
potent compass
Huh
potent compass
Tysm
I keep getting it wrong
It’s suppose to be -3
Nvm Ik why
Actually nvm
supple knot
The leading coefficient isn't 1, it's -2
So to get the factored quadratic into (x-root1) * (x - root2) = 0 you divide both sides by -2 first
potent compass
Wait wha
supple knot
-2x^2 + ... = 0
supple knot
So to get the factored quadratic into (x-root1) * (x - root2) = 0 you divide bot...
When you expand this factored form the coefficient multiplying x^2 is 1. But your quadratic doesn't match that
supple knot
,rotate
woven radishBOT
potent compass
I don’t understand
supple knot
I don’t understand
which part
potent compass
did you try factoring 6p^2
.
supple knot
factoring means writing as a product of two or more terms
in this case specifically, it'll be 2 terms
6p^2 = (first term) * (second term)
potent compass
I still don’t get it
supple knot
I still don’t get it
find a way to multiply two terms to get 6p^2
supple knot
6 = 2 * 3 and p^2 = p * p
the answer is basically here
potent compass
I don’t understand why
supple knot
just like all your other problems, you're expanding (x-root1)(x-root2) to get something like x^2 + x(root1 + root2) + root1 * root2 = 0
so you're matching coefficients. 6p^2 is the only term in the problem that doesn't have an x. so you set it equal to root1 * root2
potent compass
I still don’t get it
supple knot
...
well start by expanding (x - p)(x-p) and see what you're not getting in your problem
then slowly change the first (x-p) and second (x-p) to see if you can match what you're given in the problem
crystal iris
The easier the question the more the attention
potent compass
well start by expanding (x - p)(x-p) and see what you're not getting in your pro...
Wht sre we doing this
supple knot
The easier the question the more the attention
!occupied
devout snowBOT
The easier the question the more the attention
Someone else is already using this help channel. If you need help with a question, please open your own help channel/thread (see #❓how-to-get-help for instructions).
potent compass
I don’t understand why
supple knot
the intent is to write the equation as (x-root1)(x-root2) = 0
so you can solve for x
potent compass
Is it Bx it’s x^2
potent compass
what is Bx
Bc
potent compass
what is Bx
Why do we find two x’s
supple knot
the intent is to write the equation as (x-root1)(x-root2) = 0
product of two numbers = 0 implies one OR both numbers equal 0
2 * 0 = 0, 0 * 3 = 0, and 0 * 0 = 0
potent compass
Ok
devout snowBOT
@potent compass Has your question been resolved?
potent compass
I tried the quadratic formula and got it wrong
supple knot
quadratic formula should also give the correct answer
supple knot
find a way to multiply two terms to get 6p^2
this is simpler though
supple knot
I tried the quadratic formula and got it wrong
show
rain summit
what's the equation here may i ask?
potent compass
𝓧-5px+6p^2
supple knot
I tried the quadratic formula and got it wrong
why do you think this is wrong
do you know how to simplify to get two expressions?
potent compass
why do you think this is wrong
Yes
supple knot
Can you help me w completing the square
,tex .cts
woven radishBOT
riemann
potent compass
It’s not working
supple knot
It’s not working
It's faster if you just show your work rather than have me ask for the third time
potent compass
supple knot
yea there's not enough information to understand what you mean by "not working"
all you have is an expression and some algebra
potent compass
yea there's not enough information to understand what you mean by "not working"
I was trying to solve all the dashed lines
Qiestion 15
But I’m going to take a shower
supple knot
,rotate
woven radishBOT
supple knot
you only factored 2 out of x^2, not the rest of the terms
potent compass
you only factored 2 out of x^2, not the rest of the terms
Wdym
potent compass
,, (x+\frac{9}{2})^2 = x^2 + 9x + \frac{81}{4}
I don’t understand this
woven radishBOT
mercy
restive river
so $2(x+\frac{9}{2})^2 = 2x^2 + 18x + \frac{81}{2}$
woven radishBOT
mercy
restive river
,, (x+\frac{9}{2})^2 \
= (x)^2 + 2(x)(\frac{9}{2}) + (\frac{9}{2})^2 \
= x^2 + 9x + \frac{81}{4}
woven radishBOT
mercy
restive river
firstly if you add something to an equation you subtract the same otherwise the equation is changed
so in the second line
when you add 81/16 you subtract it
potent compass
firstly if you add something to an equation you subtract the same otherwise the ...
Huh
O Uk r how
*like how
restive river
you had ax^2 +bx +c
then you add some t to the equation
ax^2 +bx +c +t -t
it remains same, like adding zero
4 +8-8 = 4
potent compass
you had ax^2 +bx +c
then you add some t to the equation
ax^2 +bx +c +t -t
it ...
For this do I factor out two
rain summit
you could do that to simplify things up
restive river
I still don’t get it
hm.. how can i put in different words
rain summit
split -8x into something
rain summit
these 2 numbers should multiply to 12 and add up to -8
restive river
yes factoring out 2 first is correct
potent compass
Not jormal factorization
woven radishBOT
1 divided by 0 equals Infinity
rain summit
completing the square means making a square
a square is just $(a - b)^2 = a^2 - 2ab + b^2$
woven radishBOT
1 divided by 0 equals Infinity
rain summit
here your $a^2$ is $x^2$, $-2ab$ is your $-4x$
woven radishBOT
1 divided by 0 equals Infinity
rain summit
so you can easily figure out that $b$ is $2$
woven radishBOT
1 divided by 0 equals Infinity
rain summit
so you can find what $b^2$ is and split the $3$ accordingly
woven radishBOT
1 divided by 0 equals Infinity
rain summit
this is basically the same idea
potent compass
I don’t ur tests d urs
rain summit
???
potent compass
you should +4 and -4 to keep the expression the same
Oh
Wait why -4
So 3-4?
I thought u gave up on me
Like this pls
Can someone explain uh the two different completing the square tu me pls
The one with quadratic
And the normal one
torpid sun
hmm?
sour furnace
hm
supple knot
Wait why -4
factoring an expression means you have to keep the expression the same
+4 +4 changes the expression
+4 -4 doesn't change the expression because 4 - 4 = 0
and adding 0 doesn't change the expression
thing + 0 = thing
potent compass
H ok
Erm what about quadratic
What’s the difference
And Ty for staying w me for this long
Ik Im stressful
restive river
Im back
supple knot
**riemann**
i also explained that here
restive river
Ig reimann explained it , same goes with prev question
$\
2x^2 + 9x + 3$ \
$\implies 2x^2 + 9x + \frac{81}{16} + 3 + \frac{81}{16}$\
this was wrong because $\frac{81}{16}$ is added twice.
so should be $+81/16$ and $-81/16$ \
adding correctly
$2x^2 + 9x + 3 = 2x^2 + 9x + \frac{81}{16} + 3 - \frac{81}{16}$
woven radishBOT
mercy
potent compass
Tysm
restive river
Second $\frac81/16$ was not correct
If you take
$\
2x^2 + 9x + \frac{81}{16} + 3 - \frac{81}{16} \$
Then factor $2$ from the first three terms
$\
2(x^2)+2(\frac{9}{2}x)+2(\frac{81}{32}+3 - \frac{81}{16}$ \
$\implies 2(x^2 + \frac{9}{2}x + \frac{81}{32}) + 3 - \frac{81}{16}$ \
However, look inside the bracket
$x^2 + \frac{9}{2}x + \frac{81}{32}$ \
Is not a perfect square
woven radishBOT
mercy
