ok maybe a good way to start would be naming all relevant pts
#help-13
tropic oxide
here O is the center of the octagon, which is also the midpoint of the diagonal AE
OA = OB = OE = (1+x)/2
angle AOB is 1/8 of a full circle, thus 45°
you can use this and look at triangle OKB
@fiery dew also general tip: if you dont know what to begin with but you have a diagram, name all relevant points on it
fiery dew
angle OKB= 90*
tropic oxide
angle OKB is 90° because it's marked as such, yeah
fiery dew
from here, how do I find x
tropic oxide
look at triangle OKB
think about what lengths you can find, given what i've told you
fiery dew
Im not sure what to do next
tropic oxide
ok let's look at triangle OKB
this is not a calculation question but rather a "did you read and understand my messages" question:
what lengths in triangle OKB do you already know (in terms of x)?
(do not go off to calculate things)
@fiery dew
quiet ledge
OA = OB = OE = (1+x)/2
^
tropic oxide
yes that is part of what i said.
find the length of OK (again in terms of x).
use the diagram + my messages.
fiery dew
OK=(1+x)/2-1
tropic oxide
can be simplified.
fiery dew
1+x
wraith daggerBOT
Ann
fiery dew
(x-1)/2
tropic oxide
right.
now look at this
i have temporarily removed everything else from the diagram to focus you on only the relevant stuff.
do you see a way forward?
fiery dew
OB+OK=x
tropic oxide
formally correct but goes nowhere.
what other relation can you give me between OB and OK based on the other stuff i drew in the diagram?
this is a right triangle, and one with a 45° angle to boot -- this is very important.
fiery dew
Is OK=BK
fiery dew
Im not sure
tropic oxide
what is the ratio of hypotenuse to leg in a 45-45-90 triangle?
fiery dew
sqrt2
fiery dew
What do I do next?
quiet ledge
Set up an equation
fiery dew
Can I have the answer so that I can see if its correct and then continue the solution process? I just want to make sure its correct.
queen stirrup
Can I have the answer so that I can see if its correct and then continue the sol...
whats another way to write the ratio of the hypotenuse to the leg in triangle OBK
cedar kilnBOT
@fiery dew Has your question been resolved?
cedar kilnBOT
cedar kilnBOT
Available help channel!
crude brook
I need help with a pde question if someone knows how I can solve the following:
flat mica
I need help with a pde question if someone knows how I can solve the following:
well it['s a heat equation
do you know the trick to assume it's a product of two functions of one variable each?
flint cape
-# AKA separation of variables
crude brook
it's the u(x,0)=1-x i have a problem with if i recall correctly
and tbh i haven't comprehended these types of exercises completely, i have a methodology written but i need to have it complete and memorize it for the exams
i know that i must first do $u(x,t)=X(x)*T(t)= not 0$
wraith daggerBOT
MichaelRafto
crude brook
then i end up with $T'-lT=0, X''-lX=0$
wraith daggerBOT
MichaelRafto
crude brook
now i try to assume what happens for l>0 ,l=0 and l<0
wow these are no joke
you know how to help with these, right?
<@&286206848099549185> these really are no joke because no one is helping me yet 😅
paper edge
I don't know much about DEs but in case some more time passes and no one is able to help you, posting in #odes-and-pdes may be of use
People in specialized channels, naturally, have much more knowledge about the topic in question
cedar kilnBOT
cedar kilnBOT
Available help channel!
timid field
there is a question asking this, but is
-420, -60, 300, 660 deg not the right answer? its saying its wrong
opal schooner
did you try excluding the original angle from the answer
timid field
well i only have 2 attempts and i used one of them with the answer choices i mentioned earlier
and 300 should technically be an answer right?
opal schooner
technically yes 300 deg is coterminal to 300 deg
but I would ask your instructor if that's an option
so as to not waste an attempt checking if that's the reason
gritty viper
I mean those are the right angles otherwise
So unless it's a formatting issue it's gotta be removing 300
timid field
i just risked it and did it without 300 and its correct now
but for reference, 300 should be a valid answer?
i just want to make sure before i ask my teacher about it
opal schooner
it is valid, it would be unnecessary to say that two angles of the exact same value are coterminal but I don't see why that would be penalized anywhere by any rational person
cedar kilnBOT
@timid field Has your question been resolved?
magic solar
Does integrating in the complex plane along a path mean that you are summing up the vectors (complex numbers) along the path?
worldly chasm
magic solar
Oh right the outputs, since the plane would be referring to inputs
hollow trail
when we integrate along a path in the complex plane we sum the outputs of the function and also multiply by the 'tangent vector' of the path (sort of like how integrating a path integral involves a dot product with the tangent vector)
upper abyss
Integration always means the same thing:
First, Break the object up into a lot of parts
For each part:
- Evaluate the function somewhere
- Multiply by "the size of that part"
- Then sum this for all parts
magic solar
Is the size referring to the differential operator
hollow trail
or how integraing along an interval of the real number line involves multiplying by the 'length of each subinterval' dx
magic solar
So if I were to integrate across a semicircle contour for 1/1+x^2, I would be summing the outputs for f(z)=1/1+z^2
stiff totem
geometrically, yes
but in practice you would parametrise your path
and integrate f(r(t))r'(t), where r is your parametrisation
magic solar
parametrise by perhaps re^ix from let’s say 0 to pi for the top half?
stiff totem
sure
hollow trail
the way you would go about actually evaluating the integral would be to parameterize the contour as some function $\gamma(t)$ for $t$ in some interval $[a,b]$, and then your integral is
[ \int_C f(z) \odif z = \int_a^b f(\gamma(t)) \gamma'(t) \odif t ]
wraith daggerBOT
cloud
hollow trail
what if what is not a function?
magic solar
f(z) implies function so ig nvm
What paramtrisatiom would you use
Ofc it would be case specific
But in general
stiff totem
re^ix works for your example
flat mica
What paramtrisatiom would you use
for the integral to be defined it should be parameterization-independent
stiff totem
in general, pick something that makes the function easier to integrate when composed with the parametrisation
flat mica
maybe i9 misunderstood the question
stiff totem
unit speed parametrisation if possible also helps
flat mica
are you asking practically or in theory
magic solar
Practically
magic solar
unit speed parametrisation if possible also helps
What’s that?
magic solar
As in the differential operator is 1 so your simply measuring the change multiplied by 1?
flat mica
most of the time the contour will be piecewise defined as pieces of circles and lines
magic solar
Yeah I’ve seen integrals be split up
flat mica
for circles use rexp(it), for lines use a+tz, t in R
stiff totem
As in the differential operator is 1 so your simply measuring the change multipl...
as in r'(t) = 1
magic solar
for circles use rexp(it), for lines use a+tz, t in R
How about horizontal lines
keen panther
prove that there are infinitely many primes in the form 4n + 1
I have read the proof and I am kinda confused..
- Idk what a quadratic residue is
- videos are really too fast lol
- any other proofs?
keen panther
ig i'll do proof by contradiction
surreal cave
cedar kilnBOT
cedar kilnBOT
Available help channel!
proper marsh
how can i find a equation with 2 points
cedar kilnBOT
how can i find a equation with 2 points
proper marsh
i have (-2, 8) and (4,2)
i need to find the equation for a line that passe those points
slope is -1 but idk how to do the other thing
pine valley
$y - y_0 = m(x - x_0)$
dusk goblet
y - y_1 = m(x - x_1)
wraith daggerBOT
raxdiusid
proper marsh
y - y_1 = m(x - x_1)
could you explain this more 🤧
proper marsh
thats uh y= m(x-x1) + y right
dusk goblet
it follows from the slope formula where you let one of the points be arbitrary (x, y)
dusk goblet
thats uh y= m(x-x1) + y right
sure
dusk goblet
given some point (x_1, y_1) on a line with slope m, all other points (x, y) must satisfy (y - y_1)/(x - x_1) = m since the slope is constant
proper marsh
y=-1x+6
dusk goblet
so just multiply through by x - x_1
dusk goblet
y - y_1 = m(x - x_1)
and you get this
dusk goblet
is my formula same thing
yes just add y_1 over
proper marsh
heh okay thanks, one more lowk
its "three less than the quotient of a number and eight is seven"
dusk goblet
did you manage to translate this into an equation
dusk goblet
yes
proper marsh
oh heck yes
dusk goblet
solve for x i assume
proper marsh
yes, do you mind one more, last one
dusk goblet
go ahead
proper marsh
its "an ostrich that is 9 feet tall is 20 inches taller than 4 times the height of a kiwi. Whats the height of kiwi in inches"
do i need variables
proper marsh
yes 108
proper marsh
108=4x-20?
dusk goblet
you mean +
proper marsh
wait what
dusk goblet
20 inches taller than 4 times the height is 4 times the height + 20
dusk goblet
the 20 has to go on the kiwi side right
imagine the kiwi was 5 inches
then 4 times it’s height is 20
20 inches taller than 4 times it’s height is 40
flint cape
imagine the kiwi was 5 inches
-# Knief casually insulting NZers
dusk goblet
lol
proper marsh
okay i think i get it
ima solve
hm okay 22 inches
i assumed it would be opposite
because like
idek
since it said the ostrich is 20 inches taller
okay thank you, hopefully i wont open another one in like 5 minutes
.close
opaque granite
What do these mean?
cedar kilnBOT
What do these mean?
cedar kilnBOT
wraith daggerBOT
brandon
blazing dune
For d = 2 doesn't this have like an infinite number of solutions, while it is square free
?
So you are only taking solution such that x,y < d
wraith daggerBOT
brandon
blazing dune
Ok
blazing dune
**brandon**
Oh I think you meant
$$1 \leq x,y < d$$
wraith daggerBOT
Sherif Player
cedar kilnBOT
@strong marten Has your question been resolved?
ebon stump
Okay, so it's just being weird and shifting the interval to include d and not include 0
Very non-standard, but will still work fine
I guess it might not be standard for number theorists who don't want 0 to be a natural number...
Now I just need to solve the problem lol
Just out of interest, what level of course is this problem from?
It feels like something that might be in my range but also like it might be above me
Okay, you're an undergraduate, so it should be at least possible
lyric plank
i think CRT lends itself more strongly here
we know that any solution should also satisfy the congruences mod each prime, and these are basically like “independent components” to our overall solution
I think if we can find p unique solutions for x^2 = y^3 mod p for general prime p, then that proves the result
ebon stump
Was thinking that was probably gonna be the route.
lyric plank
And now we’ve boiled the problem down to groups instead of rings which are a lot easier to work with I think
okay yay we’re all on the same page
ebon stump
Oh yea, we're just working over a cyclic group now right?
ebon stump
Well yea I guess
lyric plank
Sorry that’s obvious LOL
but yeah with 0 gone we’re working under a p-1 cyclic group
ebon stump
So the problem is now:
show $|{(x,y) \in C_{p-1}: x^3 = y^2}| = p-1$?
lyric plank
right in terms of solutions to the original equation
wraith daggerBOT
Haine
ebon stump
Yea, sorry left out that lol
But that just means 3x = 2y mod p-1
In additive notation
wraith daggerBOT
brandon
ebon stump
Now struggling to remember basic modular arithmetic to solve this simple congruence lol
How exactly?
Neither 2 nor 3 are necessarily units indeed, for almost all p, 2 isn't a unit
Though, for any number that 2b can hit, there are 2 elements that can hit it
blazing dune
By the way, did you try using Hensel's Lemma?
ebon stump
I guess it's gonna be that sort of thing
ebon stump
Oh, yea, that's pretty clear. What's more relevant is the number of solutions
lyric plank
I think I just bashed it bc idk ring theory 😭
if 3 doesn’t divide p-1 then this is pretty easy, the cube side creates unique residues and squaring will give us 2 copies of each even order number, giving us precisely 2 solutions per square residue
if 3 does divide p-1 then… nvm I thought I had it but I didn’t finish 💔
ebon stump
I think we might need to case on whether 2 and 3 are units and then point out that if they aren't, then for any value m that can be written as 3x, there are exactly 3 values of x which achieve m
lyric plank
Well actually we can just finish that thought process can’t we, we have the even powers and the multiple of 3 powers and just want the intersection of these sets
lyric plank
I think we might need to case on whether 2 and 3 are units and then point out th...
this sounds more right 😭
I guess just to finish my process: if 3 divides p-1 then necessarily only the powers of 6 will contribute solutions, and there are 3 ways to pick the cube and 2 ways to pick the square giving us 6 ordered pairs per (p-1)/6 possible 6th powers
ebon stump
I wish we didn't have to split by that, but at least the argument is completely trivial either way. Also... this kinda suggests to me that this can be extended to:
x^p ≡ y^q which is nice
lyric plank
and the reason we know there’s 2/3 ways is just by taking the kernels of the respective maps
lyric plank
I wish we didn't have to split by that, but at least the argument is completely ...
yeahhh it does
ebon stump
Oh, actually not quite
lyric plank
oop :(
ebon stump
It's the fact that 2 and 3 are small and hence don't get reduced that's relevant I think
lyric plank
hm that makes sense
idk if this is related but I was heavily using the fact that 2 necessarily divides p-1 and that sounds a little bit uglier when you throw other powers into the mix
ebon stump
Hm... actually, if they did get reduced, then that means that they were coprume to begin with since we were modulo by something smaller than them
Guess it does work
lyric plank
wait actually the cases are still fine you just work through them all
ebon stump
@strong marten did you get the actual important bit of the argument? I've got a bit sidetracked thinking about extending the problem lol
Okay, I'll try to put it out in words a bit better
I'll handle one of the cases and you can figure out the rest since they're all basically the same
Suppose that our prime p satisfies 2|(p-1) but not 3|(p-1). Then the subgroup of the cyclic group modulo p-1 generated by 2 has index 2, and by the orbit stabiliser theorem, every element x in that subgroup has 2 values g ∈ C_{p-1} such that g^2 = x. On the other hand, since 3 is a unit modulo p-1, there is exactly one element h such that h^3 = x. No other element of C_{p-1} can be expressed as both a square and a cube, so we have all our solutions.
Hm... orbit-stabiliser might actually just solve the problem in general lol
Does that idea make sense?
Okay, I'm gonna be going now, so I wish you good luck with it and the rest of your courses
blazing dune
About using Hensel's Lemma
If we let f(x,y) = x^3 - y^2
Let's find the non-singular solutions of f(x,y) = 0 mod p
That happens when one of the 2 partial derivatives of the function is not 0
So
fx(x,y) = 3x^2 mod p
Which happens to be non-zero for any value of x in (1 to p-1) for any value p ≠ 3
fy(x,y) = -2y mod p
Which happens to be non-zero for all values of y in (1, p-1) for any p ≠ 2
Which means for any prime number p
There are p-1 non-singular solutions that according to Hensel's Lemma lift to be exactly one solution to mod p^k
Now for singular solutions which would happen when
Both partial derivatives of f(x,y) are zero
Which doesn't happen except for
x = p or 0 mod(p)
And
y = p or 0 mod(p)
Which means that they are multiple of p, so
x = ap
y = bp
We want to solve for
x^3 = y^2 mod (p^k)
So
a^3p^3=b^2p^2 mod p^k
Idk but i feel that there would be multiple solutions that would lift from
ebon stump
Isn't Hensel's lemma specifically for univariate polynomials?
I mean, I don't really know, just thought I should point it out in case that's relevant
Also looks like you're aiming for solutions modulo prime powers given a solution to the prime? Isn't that like explicitly what the problem isn't asking for?
Since prime powers aren't square free except for the prime itself
blazing dune
I am trying to prove that the lifted solutions from p to p^k is larger than p^k+1
If so then by CRT it doesn't work for non-squarefree d
blazing dune
Isn't Hensel's lemma specifically for univariate polynomials?
According to wiki, you can generalize the Hensel's Lemma to more variables by the use of the Jacobian matrix
Although I am not sure if what I did is the correct way to generalize it
ebon stump
I am trying to prove that the lifted solutions from p to p^k is larger than p^k+...
Oh, that's cool if it works
I'm not really prepared to delve into hensel's lemma rn though lol
Oh, what's the operation for (if you're okay talking about it)?
Also, I sent a dm request if you'd prefer to keep it to maths talk here
Sounds like you aren't too worried about it, but I hope everything goes well with it.
I understand how it feels since I fractured my spine and was out from university for a significant portion of my final year. Was frustrating not being able to keep up as well as I'd like to.
lyric plank
Suppose that our prime p satisfies 2|(p-1) but not 3|(p-1). Then the subgroup of...
wait i just realized the first isomorphism theorem is basically just a special case of the orbit stabilizer theorem I think
hm im not sure but they feel a lot more related than i thought before 😭
ebon stump
wait i just realized the first isomorphism theorem is basically just a special c...
I think it adds a bit extra structure, but it's very related
lyric plank
I understand how it feels since I fractured my spine and was out from university...
oh my gosh are you okay
ebon stump
oh my gosh are you okay
Yea, that's over and I'm back to throwing myself around like a madman bouldering
lyric plank
oh my gosh are you okay 😭 i hope everything goes well
wraith daggerBOT
brandon
lyric plank
**brandon**
i think this is similar to writing the group in terms of a universal generator
ebon stump
Yea, feels highly related
lyric plank
and the generator was the only way i was able to wrap my head around this LOL
ebon stump
Hm... I think it will work, it's just that it might be difficult to show that those are the only solutions like that.
But please make sure to respond to my wave with a wave I just got reminded how satisfying it is
cedar kilnBOT
cedar kilnBOT
Available help channel!
sturdy escarp
okay there's too much context to this problem
cedar kilnBOT
okay there's too much context to this problem
sturdy escarp
and I'm too lazy to explain it all
but I'm basically trying to calculate error for some taylor polynomial
so f^3(z) = sin(z)
I managed to figure out 0<=z<=1/2. where 0=c (given center value) and 1/2 is the x value from previous context
basically I plugged everything into Rn(2) = [f^3(z) (x-c)^3]/3! properly
except only because I knew what sin(z) looks like
how would I know what value of z to choose if I didn't know what sinz looked like?
feels like a stupid question ngl but idk
nvm
.close
cedar kilnBOT
cedar kilnBOT
Available help channel!
lunar spade
I differentiated the function using leibniz theorem.
cedar kilnBOT
I differentiated the function using leibniz theorem.
lunar spade
and then what i did was differentiate the options which would of course be zero. like $f'(pi/6) = 0$
wraith daggerBOT
Prathamesh
lunar spade
since f(pi/6) =1
when i differentiated f(x) i got $f'(x) = cosx/(1+(sinx)^2 -2sinx)$
wraith daggerBOT
Prathamesh
lunar spade
then i put pi/6 in this and it is not equal to zero so did i prove that the first option is false?
tropic oxide
$\frac{\cos(x)}{1 + \sin^2(x) - 2 \sin(x)}$
wraith daggerBOT
Ann
tropic oxide
did you mean this? if you're gonna use latex you should use it properly and type fractions in it like this
\frac{}{}
lunar spade
ohh wow. thanks, Im new here so I'm still learning this.
btw can somebody help me with my problem?
<@&286206848099549185>
tropic oxide
well assuming your derivative is correct it can be expressed as cos(x)/(sin(x)-1)^2
so that should let you find an expression for f(x) itself by integration
lunar spade
ok lemme try
but when i integrate it i would have to add +c cuz it's an indefinite integration
i got $\frac{1}{1-sinx} + c$
wraith daggerBOT
Prathamesh
wispy cedar
Hi, I am facing some problems in Integration manipulations, and also unable to solve Complex Graphs in Area under curve , any advice ??
fossil dawn
for future helpers: please show your question and work
mighty shuttle
Hi, I am facing some problems in Integration manipulations, and also unable to s...
!xy
cedar kilnBOT
Hi, I am facing some problems in Integration manipulations, and also unable to s...
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.
mighty shuttle
Show us an example
plush creek
Hi, I am facing some problems in Integration manipulations, and also unable to s...
give example
cedar kilnBOT
@wispy cedar Has your question been resolved?
cedar kilnBOT
cedar kilnBOT
Available help channel!
vale kettle
cedar kilnBOT
proven summit
,rotate
wraith daggerBOT
vale kettle
Is it complete?
fossil dawn
technically speaking no. your final matrix (if it is this one) is not in row echelon form. second row's pivot is not 1
vale kettle
technically speaking no. your final matrix (if it is this one) is not in row ech...
Oh,then what i do?
fossil dawn
you can scale row 2 to get its pivot to a 1, but first, what did the question ask you to do?
vale kettle
you can scale row 2 to get its pivot to a 1, but first, what did the question as...
Gausss eliminati9n
fossil dawn
just to confirm, you have learnt about row echelon, correct?
vale kettle
just to confirm, you have learnt about row echelon, correct?
Yes
fossil dawn
right, then you'll need to scale row 2
vale kettle
right, then you'll need to scale row 2
I dont get it
fossil dawn
your middle row's pivot (leading element) is not 1
multiply that row by something to make that element a 1
vale kettle
multiply that row by something to make that element a 1
Oh ok
vale kettle
multiply that row by something to make that element a 1
There is no element such as that
fossil dawn
wdym
vale kettle
and what's wrong with that currently being a -3? you still need to make it a 1
With 3*3
-3*3
?
rapid mauve
-3*3 = -9 though
vale kettle
-3*3 = -9 though
Oh right
vale kettle
-3 multiplied by what gives you 1?
-(1/3)
vale kettle
fossil dawn
you didn't multiply the last column
it's still -9
also might wanna write the matrix bigger
vale kettle
vale kettle
also might wanna write the matrix bigger
.
fossil dawn
-9 * (-1/3) shouldn't remain negative
fossil dawn
...better remember to remove it on the paper itself
wait why is the pivot -1
-3 * (-1/3) also shouldn't be negative anymore
vale kettle
-3 * (-1/3) also shouldn't be negative anymore
My bad
fossil dawn
you didn't write a negative sign here, how did it show up
vale kettle
you didn't write a negative sign here, how did it show up
I wrote
I gtg close caan i keep going later? @fossil dawn
And sorry for the ping
fossil dawn
just close the channel when you're done
but you don't have to specifically ping me when you come back
there will be other helpers around
if i'm still around i'll take a look
vale kettle
fossil dawn
,rcw
wraith daggerBOT
fossil dawn
assuming you didn't make any other mistake in your earlier calculations (i just skimmed past those) this looks ok
vale kettle
assuming you didn't make any other mistake in your earlier calculations (i just ...
Ok thks
Close
.close
frail citrus
if a quantity's differential is zero, then that quantity is a constant? or can it be zero too? (i learnt that it is zero as well as a constant of some sort, if it is wrong elaborate further by your opinions)
floral arrow
Zero is a constant
solid oasis
A truth
long swan
Zero is a constant
Proof?
long swan
Flat line
slender ginkgo
Proof?
“In mathematics, a constant function is a function whose (output) value is the same for every input value.”
cobalt star
if a quantity's differential is zero, then that quantity is a constant? or can i...
0 at a point?
slender ginkgo
What😭
slender ginkgo
Zero is a constant
cobalt star
should backread a bit
slender ginkgo
As nel has stated
cobalt star
whats your understanding of a differential?
frail citrus
Zero is a constant
every dam numeral is
slender ginkgo
Then what’s the issue
dreamy void
But not every constant is 0
floral arrow
If you take any specific point of a function, that value is a constant
But that's useless
frail citrus
whats your understanding of a differential?
its the infinitesmal change in qty with respect to another quantity
floral arrow
What you're probably trying to ask is this:
if a function f has for derivative f'(x) = 0 for all x, then is f a constant function?
to which the answer is yes
frail citrus
What you're probably trying to ask is this:
> if a function f has for derivative...
no that is true indeed
that wasnt my point
my doubt was kind of linked with physics
cobalt star
its the infinitesmal change in qty with respect to another quantity
that might be a sufficient enough understanding to answer this
if there is no infinitesmal change in the quantity with respect to the other quantity, then the function itself is nonchanging
floral arrow
If you have a quantity x that changes over time t but dx/dt = 0, then x is constant
That's a "physicsy" way of saying the same thing
cobalt star
it kind of implies the differential equations df/dx = 0, whose solution is f(x)=c
slender ginkgo
my doubt was kind of linked with physics
How so
frail citrus
if dB=0, then tell the nature of B
paper edge
Following your reasoning, given some magnitude A that satisfies dA=0
Then there exists no "immediate change" in it, that is, between two chosen "arbitrarily close" points the difference will be 0
From there it easily follows the magnitude A must have a constant value, else this breaks down
If you want a simple justification, recall two functions with the same derivative are equal up to an additive constant so dB=0->B=∫0dx +C, which immediately gives B=C
cedar kilnBOT
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exotic hare
can someone tutor me on how to do this, my prof just sent yt vids about this topic but i am not able to relate the samples on the videos on these questions <@&286206848099549185>
rapid mauve
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exotic hare
<@&286206848099549185>
storm zealot
If you put your two feet on the chair, place your hands on your knees and lick your thumb. Then Watari will come to help you
fossil dawn
it's only been 10 minutes but i'll look at it
always start with the individual A and B truth values
then do ~A and ~B
then slowly expand to ~AB and ~BA
basically, do it group by group
exotic hare
how about the +
fossil dawn
inclusive OR
exotic hare
what do i do about the constant at the end
fossil dawn
1 is TRUE
exotic hare
fossil dawn
nice start
exotic hare
Whats the conditions to get ~AB
exotic hare
nice start
Thanks
fossil dawn
if two letters are together, it's an AND
exotic hare
and its the same for ~BA right?
fossil dawn
yup
but remember to look at the correct columns when doing this AND
only one letter in each pair is a NOT
fossil dawn
the order doesn't matter as long as you remember which letter has the NOT, but ofc follow the question when you can
you can write B~A too (with the bar on top of A)
but since the question gave ~AB, just follow it
exotic hare
fossil dawn
why is ~BA all zeroes
exotic hare
i missed it the 3rd row is supposed to be 1
and that makes ~ab + ~ba 1 too on the 3rd row
thanks for the help so far, can you also guide me on how to make it into a logic diagram
fossil dawn
you mean these things?
exotic hare
yes
fossil dawn
i'll guide you through the first
exotic hare
I just write 1 above A? And connect it? Do i use a crescent gate?
fossil dawn
show what you've planned
fossil dawn
I just write 1 above A? And connect it? Do i use a crescent gate?
an OR gate is the last one
exotic hare
fossil dawn
looks ok, but the NOTs are a bit small
still fine though. here's how i would have done it
exotic hare
on no. 2 how do i do this
fossil dawn
that is a low constant (always FALSE)
supple falcon
right?
exotic hare
yep already have it
wraith daggerBOT
1 divided by 0 equals Infinity
supple falcon
it just means $A$ ANDING the value you got for $A + B$
wraith daggerBOT
1 divided by 0 equals Infinity
supple falcon
idk how to describe it but you get the idea
exotic hare
nope 😭 sorry
supple falcon
want me to do an example for one of the rows?
exotic hare
sure
supple falcon
pick one of the rows
exotic hare
or like maybe just the condition to get it
exotic hare
yep i have that
exotic hare
what if its 1.0, 0.1, 0.0
exotic hare
is it the same principle with +?
supple falcon
so A(A + B) should give you 1.1
supple falcon
is it the same principle with +?
it's just substituting the values in
just like how you would do it in normal math
exotic hare
For the ~øA since hana said that its always false is it just the opposite of A?
0 = 1
1 = 0?
fossil dawn
if you're confused about ~(FA) (F to mean FALSE, i'm not typing that null set thingy) do FA first
exotic hare
wraith daggerBOT
exotic hare
I see
fossil dawn
because think about it
F is always 0
0 AND anything is always 0
then the NOT comes in to flip all of the 0s to 1s
if you ever don't understand the interaction between any two elements, that's a sign you need an extra column for them
exotic hare
Sorry im trying hard to understand but i dont get it
fossil dawn
which part
exotic hare
~(FA) should be all 1s
i dont want to just copy and paste the asnwer you said i also want to understand
fossil dawn
sure
as we said, F is always 0
FA is F AND A
can you draw a truth table for that?
exotic hare
fossil dawn
there we go
cedar kilnBOT
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exotic hare
<@&286206848099549185> not sure if i did everything right by myself please check it out
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exotic hare
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steady epoch
Hi
steady epoch
Why is a-11b+10c+d have those specific restrictions
Thank you
<@&286206848099549185>
Sorry
raven shard
Why is a-11b+10c+d have those specific restrictions
it is minimized at (1, 9, 0, 0) which gives -98 and maximized at (9, 0, 9, 9) which gives 108
sacred iron
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steady epoch
You mean for abcd respectfully
steady epoch
it is minimized at (1, 9, 0, 0) which gives -98 and maximized at (9, 0, 9, 9) wh...
You mean for abcd respectfully in the tuple right
drowsy grove
a2 + b2 = c2
raven shard
You mean for abcd respectfully in the tuple right
sure, yes
clever cloud
so like the limits on a - 11b + 10c + d just come from the fact a b c d are digits in a 4 digit number a goes 1 to 9 b c d go 0 to 9 so if you wanna make it as small as possible you make a small b big c small d small so thats 1 - 99 = -98 and if you wanna make it big you make a big b small c big d big so thats 9 + 90 + 9 = 108 so it can only be between -98 and 108
raven shard
(the word is respectively not respectfully)
raven shard
nw
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ashen spire
Im I able to use this room?
cedar kilnBOT
Im I able to use this room?
ashen spire
Ok ty
sacred iron
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ashen spire
Im trying to understand how the answer is this
I get where they got the numbers of -3 and -1 from
But i dont understand how it's x<-3 and not x > -3
proven summit
or, -1, any value x >- 3
what exactly do you get?
ashen spire
X = 1 and x = -1
austere hull
But i dont understand how it's x<-3 and not x > -3
bc for values just greater than -3 but less than -1 (like -2) the product is negative as shown above
austere hull
X = 1 and x = -1
do u mind show the original problem bc im thinking its just $(x+3)(x+1) > 0$
wraith daggerBOT
#1 shitmiss hater
proven summit
Okay maybe using substitute explanation is a bit shit
ashen spire
This is the original question
proven summit
let me draw a diagram a sec
austere hull
bruh
proven summit
<@&268886789983436800> 
ashen spire
Bruh
austere hull
This is the original question
ok so its just (x+3)(x+1) > 0
where are u getting x=1 from
austere hull
he did? where?
austere hull
ashen spire
Or im i being an idiot 🤣
proven summit
This one
okay forget about my other explaintion, notice the highlights of the graph right?
(it's kinda a bad highlight)
proven summit
right and our quadratic is x^2 + 4x + 3 > 0
can you understand why we have it as x <-3? looking from the diagram
austere hull
better doing it that way $-3 < x < -1$ if it helps
ashen spire
can you understand why we have it as x <-3? looking from the diagram
I dont sadly
wraith daggerBOT
#1 shitmiss hater
austere hull
what about this?
ashen spire
better doing it that way $-3 < x < -1$ if it helps
Originally I did that but its the answer on there that confuses me
austere hull
its coming from $(x + 3)(x + 1) > 0$
wraith daggerBOT
#1 shitmiss hater
proven summit
I dont sadly
okay think about what part of the graph is the below the y axis
austere hull
the roots are -3 and -1
ashen spire
Yes
austere hull
so would u agree the solution is $x < -3 \quad \text{or} \quad x > -1$?
wraith daggerBOT
#1 shitmiss hater
ashen spire
okay think about what part of the graph is the below the y axis
Are they below zero?
ashen spire
The curve at the bottom right?
proven summit
The curve at the bottom right?
yes but more specfically, can you write it as an inequality
proven summit
So it will be something like -3 < x <-1 right?
correct 
now we said that's for below our axis right?
ashen spire
Yes
proven summit
smaller than 0
proven summit
As in the number os smaller than 0 correct?
okay since the interval will be -3 < x < -1 for x^2 + 4x + 3 < 0, can you now identify what it would be for > 0?
ashen spire
okay since the interval will be -3 < x < -1 for x^2 + 4x + 3 < 0, can you now i...
1 and increasing numbers
Like 2 3 4
Etc
proven summit
1 and increasing numbers
yes but like we did before, can you put that as an inequality
ashen spire
1 < x < 2?
proven summit
no
proven summit
okay forget about my other explaintion, notice the highlights of the graph right...
look back to our diagram
ashen spire
Yes
proven summit
we agreed the ones for below the axis, meaning x^2 + 4x + 3 < 0, has an interval of -3 < x < -1
ashen spire
Yes?
proven summit
but this time we want x^2 + 4x + 3 > 0
look at the highlights
what two inequalities hold for this to happen?
ashen spire
-1 and -3?
proven summit
it does involve the roots -1 and -3
but we want two inequalities involving those two roots
limber dawn
inequalities to describe x
proven summit
Oh so -3 < x < -1 and -1 > x > -3?
okay i'll give a hint. What do you notice about the inequality x < -3 in the graph
limber dawn
Oh so -3 < x < -1 and -1 > x > -3?
these are equivalent intervals (two ways of writing it)
ashen spire
okay i'll give a hint. What do you notice about the inequality x < -3 in the gra...
You mean the intervals that i gave or the graph that u drew?
proven summit
You mean the intervals that i gave or the graph that u drew?
the one I drew
ashen spire
If its mine x is always greater than -3
proven summit
with the red higlights
proven summit
I mean it's the same thing, I just did the highlights more clearer
ashen spire
Does the line go through below zero whilst when the line at -1 goes above zero?
proven summit
Does the line go through below zero whilst when the line at -1 goes above zero?
?
I'm not sure what you mean by that
I'm asking, what do you notice about the inequality x < -3 in the graph I drew?
is it above or below the y axis
ashen spire
Oh below the y axis
limber dawn
like what is y, at x = -10? @ashen spire positive or negative
ashen spire
like what is y, at x = -10? <@1012483324607987805> positive or negative
It's a horizontal line through the x axis which is to the left of the x axis
limber dawn
It's a horizontal line through the x axis which is to the left of the x axis
nu, i didn't say y = -10
limber dawn
i'm asking, in this graph, is y(-10) positive or negative? it's a single value
limber dawn
are you getting that from the graph?
limber dawn
the y axis is the vertical one right
x axis is horizontal
so going to x = -10 means going far to the left
ashen spire
Yes
ashen spire
U want me to hop on a call with u?
limber dawn
i can't voice chat rn sorry
the y value is positive if the point is above the horizontal (x) axis
ashen spire
I see
limber dawn
so is it making sense that y is positive for all x < -3
ashen spire
That bit i kind off understand now
frank acorn
Hi, im trying to prove the thing at the top but im not really understanding where to go next
frank acorn
<@&286206848099549185>
cedar kilnBOT
@frank acorn Has your question been resolved?
stiff totem
it isn't that there exists a j such that x is not in A_j; this is true for all j in J
x is not in the union of A_j, so it's not in any of the A_j
so for all $j\in J, x\in B\wedge x\notin A_j$, or equivalently, for all $j\in J,x\in B\setminus A_j$
wraith daggerBOT
Desync
stiff totem
which is the same as $x\in\bigcap_{j\in J}B\setminus A_j$
wraith daggerBOT
Desync
stiff totem
all of these steps are if and only ifs, so you don't need to show the other inclusion separately
\begin{align*}
x\in B\setminus\bigcap_{j\in J} A_j&\Longleftrightarrow x\in B\wedge x\notin\bigcup_{j\in J}A_j\
&\Longleftrightarrow x\in B\wedge\forall j\in J, x\notin A_j\
&\Longleftrightarrow\forall j\in J, x\in B\wedge x\notin A_j\
&\Longleftrightarrow\forall j\in J, x\in B\setminus A_j\
&\Longleftrightarrow x\in\bigcap_{j\in J}B\setminus A_j
\end{align*}
wraith daggerBOT
Desync
stiff totem
no, because it's the same proof
stiff totem
**Desync**
as in, every step here is a biconditional
if one was forward only, you would then have to prove the reverse containment separately
but this proof shows x is in the first set if and only if it is in the second, i.e., both containments simultaneously
stiff totem
well, you've written "then" where you could write "so equivalently" (and same with all the other "implication words" like "hence" and "therefore")
like, you've just replaced a <=> (biconditional) with an => (implication)
it's not wrong
frank acorn
Ohhh
ok that’s fair I think I’ve only ever done these as doing both ends
but it makes sense
thank uuu
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limber turtle
so woudnt it be |3-x|
cedar kilnBOT
so woudnt it be |3-x|
slender ginkgo
No
${|3-x| = \sqrt{(3-x)^2} = \sqrt{3^2 - 2(3)(x) + x^2} = \sqrt{x^2 - 6x + 9} \not\equiv \sqrt{9-x^2}}$
wraith daggerBOT
k
slender ginkgo
@limber turtle
limber turtle
eh wtf
i thought
if
sqrt(something**2) = |something|
i thought a root cancels out a square
tropic oxide
$9-x^2 \neq (3-x)^2$
wraith daggerBOT
Ann
limber turtle
thta is also weird to me
slender ginkgo
[ (a-b)^2 = a^2 - {\color{red} 2ab +} b^2]
wraith daggerBOT
k
tropic oxide
you should use ^ in discord (and latex)
slender ginkgo
Python user?
limber turtle
nvm i can just oogle htis
limber turtle
Python user?
how canu tell
oh
my use of **
lmao
muscle memory
python brianrot
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ember heath
cedar kilnBOT
ember heath
i first wrote sin 2x = 2 sin x cos x
then i split the integral and then i took sin^2x = t
2 sin x cos x dx = dt
and then i got stuck
modest fern
i first wrote sin 2x = 2 sin x cos x
then i split the integral and then i took ...
sin2x.e^((sinx)^2)dx is d(e^(sinx)^2)
modest fern
but what about the extra cos x term
you have to do parts by parts
royal finch
It's not really the most satisfying technique, but if I were you I would just take an Ansatz and use undetermined coefficients on this.
ember heath
It's not really the most satisfying technique, but if I were you I would just ta...
ansatz?
ember heath
you can just apply the formula
huh which formula
royal finch
An educated guess that you use as the basis for the solution of the problem
In this case, you would guess the antiderivative would be of the form Ae^(sin^2(x))cos(x) + B e^(sin^2(x))sin(x) and solve for A and B.
modest fern
huh which formula
probably the answer should be e^g(x) f(x) + c
ember heath
In this case, you would guess the antiderivative would be of the form Ae^(sin^2(...
oh yes i know this method
modest fern
e^((sinx)^2)cosx + c
ember heath
probably the answer should be e^g(x) f(x) + c
thats not really helping me
modest fern
In this case, you would guess the antiderivative would be of the form Ae^(sin^2(...
this makes things lengthy
modest fern
thats not really helping me
dude that's the answer
modest fern
i dont want the answer..
can you do integration by parts?
modest fern
how did u recognize that
integral of e^x(f(x)+f'(x)) wrt x is e^xf(x) + c it's provable
following that e^g(x)(g'(x)f(x)+f'(x)) should be e^g(x) f(x) + c
ember heath
integral of e^x(f(x)+f'(x)) wrt x is e^xf(x) + c it's provable
following that e...
oh i know the first one
i did not know the second formula
thats pretty cool
got it
thanks
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cedar kilnBOT
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zenith grail
i have a question "The second hand on Mr. Smith's watch is 0.25 inches long. How fast is the tip of the second hand moving?", and im not sure where to go from here
proven summit
i have a question "The second hand on Mr. Smith's watch is 0.25 inches long. How...
Could you show a screenshot of the question?
the word problem doesn't really have enough info and sounds confusing to me
zenith grail
proven summit
I'm gonna assume it's circular motion though
pulsar lily
it'd make the most sense if the second hand were moving at a constant rate
I imagined it as a ticking clock at first
zenith grail
and currently we are learning angular and linear speed so the question is in that context
proven summit
ah okay, yeah this sounds a bit like circular motion
zenith grail
so how would i go about doing this? im just not sure where to start
tropic oxide
@zenith grail how long does it take for the second hand to rotate through one full turn
zenith grail
i mean i guess 60?
tropic oxide
60 what
zenith grail
seconds
tropic oxide
and why do you guess?
have you seen a clock with hands before?
and specifically one with a second hand?
zenith grail
yeah but the wording just threw me off
its 60 seconds because the second hand completes a full rotation every minute
winter ravine
What is the cain rule
cedar kilnBOT
What is the cain rule
iron wren
Chain rule?
proven summit
you mean chain rule?
winter ravine
Yes
bright fiber
What is the cain rule
"thou shall not kill"
long swan
"thou shall not kill"
"Thou shalt also learn to conjugate"
iron wren
$\frac{d}{dx}f(g(x))=f'(g(x))g'(x)$
ashen shard
wraith daggerBOT
ImOakley
ashen shard
yeah that
cedar kilnBOT
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Available help channel!
lunar spade
somebody please help me with the question 11.
cedar kilnBOT
somebody please help me with the question 11.
upper ruin
Again? 🤔
lunar spade
Like I thought I'll do it myself
after you explained it
but then I got stuck
I wrote the equation for tangent line $y= 3a^2x - 2a^3$
wraith daggerBOT
Prathmesh
upper ruin
Awesome
Now this line intersects y = x³ at point B
So you now have to find the coordinates of B
Or, better, write an equation but do not solve it
upper ruin
Or, better, write an equation but do not solve it
Then exploit the fact that the gradient/slope at B is K times greater than at point A
You'll have a system of 2 equations in two unknowns
cedar kilnBOT
@lunar spade Has your question been resolved?
lunar spade
So you now have to find the coordinates of B
how do I find it? By putting Y= x^3 in the equation i found?
upper ruin
Now this line intersects y = x³ at point B
.
tropic oxide
you disappeared and let your channel time out...
lunar spade
umm sorry I got disconnected
tropic oxide
show what happened when you did that
lunar spade
i got the equation $x^3 - 3a^2x + 2a^3 = 0$
wraith daggerBOT
Prathmesh
tropic oxide
ok what did you do afterwards
lunar spade
I tried to find a value of x satisfying this equation
which is x=a
then I was like I assumed the coordinates at point B as (a, a^3) and again I got x=a?
<@&286206848099549185>
tropic oxide
then I was like I assumed the coordinates at point B as (a, a^3) and again I got...
(a, a^3) is A not B
yes x=a satisfies this equation, that's by design
factorize this equation further to look for its other roots
lunar spade
ok
lunar spade
factorize this equation further to look for its other roots
how do I factorize it?
lunar spade
yup i got $(x-a)^2 (x+2a)$
wraith daggerBOT
Prathmesh
upper ruin
You've been given:
the gradient at B is K times greater than the gradient at point A
upper ruin
ok so the gradient at b will be 3(2a)^2
3•(-2a)² but yeah
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dusk oriole
dusk oriole
<@&268886789983436800>
cedar kilnBOT
@dusk oriole Has your question been resolved?
dusk oriole
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