damn
#help-38
azure dagger
vagrant prism
but in general you basically never have to do this
azure dagger
ik
vagrant prism
this is more to convince you that they are the same
azure dagger
i just didnt wanna use quotient rule for smth like this
austere cedar
đââď¸
vagrant prism
ngl i wouldn't have noticed the way you did it
i would've just done quotient rule straight away lol
but yours is def cleaner
azure dagger
ty now i can use this to save some time on the test tmr
since my teacher said we're gonna have a problem with implicit differentiation of some rational function
.close
solemn agate
In regards to factoring cubes, why is stopping at the binomial common?
What do I mean by this? --> a lot of algebra we're trained to simpify simplify so when I get to the binomial why do a lot of books/examples just stop there? Why not simplify further? Does the binomial help mathematicians more?
frail heron
In regards to factoring cubes, why is stopping at the binomial common?
What do ...
It cannot be further simplified
$$(3p - q)^2 = 9p^2 - 6pq + q^2 $$
solid kilnBOT
SkyAndNight
steep knot
In regards to factoring cubes, why is stopping at the binomial common?
What do ...
if you treat p as the variable and the others as constants then you can rewrite the highlighted expression as:
9p² - (3q)p + q²
Then the discriminant will be (-3q)² - 4(9)(q²) = 9q² - 36q² = -27q² which is negative
Therefore there are no real roots and hence cannot be further factorised
trim joltBOT
@solemn agate Has your question been resolved?
azure dagger
no
not all cubics have complex roots
y=x^3 has 3 equal real roots
y=(x-1)(x-2)(x-3) has 3 distinct real roots
solemn agate
Ok then looking at just the abstract idea of a cubic equation
Why do books generally present the final form as something like above?
A monomial and then a binomial
Do we not simplify the binomial further? Or does this monominal * binomial form help us better than the root form?
azure dagger
Why do books generally present the final form as something like above?
idk abt this one
the final form can be represented as a(x-r)(x-m)(x-n) if applicable
it can also be presented as a(x+n)^2(x+m)
trim joltBOT
@solemn agate Has your question been resolved?
trim joltBOT
trim joltBOT
Available help channel!
pliant frost
trim joltBOT
pliant frost
Can anyone explain to me why the angle in the trapezoid on the right is 30 degrees and not 60?
Because isnât that angle In the triangle on the left equal to 120 and 120 divided by 2 is 60
versed jasper
Because isnât that angle In the triangle on the left equal to 120 and 120 divide...
angle ABD is 120 degrees yes but in the trapezoid you split the angle into 30 + 90 = 120 degrees
stuck plover
It's not half
versed jasper
(90 degrees comes from the rectangle)
trim joltBOT
@pliant frost Has your question been resolved?
agile prawn
how did we assume the center point?
trim joltBOT
how did we assume the center point?
main sigil
how did we assume the center point?
Wdym by assumw the center point?
agile prawn
well like it said that n is not the center and they drew the rest of the circle, when the original image gave us no context as to what the length of LN was within the circle
i dont know if im making sense đ
main sigil
well like it said that n is not the center and they drew the rest of the circle,...
It said LN is 2 and that LN is perpendicular bisector of the chord KM
There is a theorem which says that radius is perpendicular bisector of chords
And so LN must lie on some radius
If thats what you mean
agile prawn
wait so --> 2 multiplied by (perpendicular bisector) = radius?
main sigil
wait so --> 2 multiplied by (perpendicular bisector) = radius?
Not necessarily 2 *
The point is that if you were to extend the line LN, then it would pass through the centet
Center
And thats all you need to solve it
agile prawn
oh i guess im just confused as to how we exactly know that the rest of the length needed to be the length of the radius is x-2, when we're told that it is the perpendicular bisector, so i guess im confused about their relationship
if that makes sense đ
main sigil
oh i guess im just confused as to how we exactly know that the rest of the lengt...
Oh i see
Well SL is the radius
Say the radius is x
Oh and let S be the center
But SL consist of SN and NL
And NL is 2
So SN + 2 = x
SN = x - 2
main sigil
ohhh so we have to come up with like an imaginary point in a sense
The center is there, we just draw it and give it a name
agile prawn
and then the remaining value of the rest of the distance from point N to the center point of S is just x-2 because the entire value of the radius is represented by x therefore we subtract 2 to get the rest of the distance?
am i cooking or am i getting cooked
main sigil
and then the remaining value of the rest of the distance from point N to the cen...
Yes, exactly
agile prawn
OKAY OKAY so wait then we use that value and multiply it by 2 to get the diameter
OKAY OKAY
LETS GO
thank you
agile prawn
its very late at night im sorry if this was stupid
main sigil
Np at all, it wasnt stupid
agile prawn
.close
grim sparrow
forms! 
trim joltBOT
forms! <:EB_EeveeExcited:1036363504845856919>
grim sparrow
let me write up my solution
I just need this verified 
we know that $d(f \wedge \omega) = df \wedge \omega + (-1)^k f \wedge d\omega$, but we also know that $d(f \wedge \omega) = 0$ by hypothesis, so we get $df \wedge \omega = (-1)^{k + 1} f \wedge d\omega$. since $f$ is nonvanishing, we solve for $d\omega$ to get $d\omega = (-1)^{k + 1} \frac{1}{f} df \wedge \omega$. now we can write $\omega \wedge d\omega$ as follows:
\begin{align*}
\omega \wedge d\omega & = \omega \wedge (-1)^{k + 1} \frac{1}{f} df \wedge \omega \
& = (-1)^{k + 1} \frac{1}{f} \omega \wedge df \wedge \omega \
& = (-1)^{k + 1} \frac{1}{f} \omega \wedge (-1)^k \omega \wedge df \
& = (-1)^{2k + 1} \frac{1}{f} \omega \wedge \omega \wedge df
\end{align*}
but we know that $\omega \wedge \omega = (-1)^{k^2} \omega \wedge \omega$. because $k$ is odd, $k^2$ is odd, so $\omega \wedge \omega = (-1) \omega \wedge \omega$, which implies that $\omega \wedge \omega = 0$. thus, $(-1)^{2k + 1} \frac{1}{f} \omega \wedge \omega \wedge df = 0$, so $\omega \wedge d\omega = 0$.
solid kilnBOT
higher!
grim sparrow
does this look good?
trim joltBOT
@grim sparrow Has your question been resolved?
trim joltBOT
@grim sparrow Has your question been resolved?
burnt urchin
**higher!**
yea its fine but i wouldve wrote a little bit more working to justify the cancellation step to get that end result
grim sparrow
yea its fine but i wouldve wrote a little bit more working to justify the cancel...
which step?
trim joltBOT
trim joltBOT
Available help channel!
drowsy moss
trim joltBOT
stark bison
Which point confuses you the first?
drowsy moss
The last sentence and the second sentence.
stark bison
They define an even positive integer to be prime if it canât be written as a product of two even positive integers. Itâs the same definition as in positive integers, the difference being that here we consider only positive even integers
trim lichen
I don't understand this sh*t.
you do not have to censor yourself when cursing btw
you are allowed to say fuck and shit just fine
pliant aspen
good for me
stark bison
And the last sentence is just a conclusion of whatâs been said previously
They gave an example of a positive integer with two distinct prime factorizations
trim lichen
The last sentence and the second sentence.
in the second sentence they introduce a notion of primality in this deliberately "weird" set E
to illustrate a point that you should not take uniqueness of prime factorization in N for granted
trim joltBOT
@drowsy moss Has your question been resolved?
trim joltBOT
trim joltBOT
Available help channel!
lean kraken
trim joltBOT
solid kilnBOT
native shuttle
a^2 +b^2 +ab +ba
twilit frost
m^2 will be multiplied with identity matrix
and if you see carefully , A^2 is just -4 times Identity
lean kraken
m^2 will be multiplied with identity matrix
and if you see carefully , A^2 is ju...
Yes
So
m^2I-4n^2I
twilit frost
yeah
twilit frost
take I common and A to the RHS
twilit frost
equate that, 2 var 2 eqn
twilit frost
should i show you my work ?
lean kraken
(m^2 - 4n^2)I=A(I-2mn)
twilit frost
(m^2 - 4n^2)I=A(I-2mn)
yeah
twilit frost
then open identity matrix as
1 0
0 1
And A as
0 2
-2 0
lean kraken
should i show you my work ?
Sure
twilit frost
Still we need m and n
then you can do this
twilit frost
yeah
cuz it says +ve real nos
otherwise m=-1 n=-1/2 is also possible
but did you understand how we did it?
lean kraken
Tq very much
radiant portal
trim joltBOT
radiant portal
I tried getting the derivative of both circles and subbing in 0 for both
It give me x = 7 for c1 and x = 28 for c2 which made me think I was doing something wrong
So yeah not really sure what to do for this one
violet gust
do you need to solve this with derivatives?
Coz theres an easy solution without derivatives
radiant portal
do you need to solve this with derivatives?
No it was just the only thing I could think of trying
violet gust
Oh ok
Do you want non-calc method? Or we could go over where you messed up if you post your solution
radiant portal
My solution is really long and messy but ill post is 1 moment
violet gust
Ok, I'll just drop the non-calc method in spoilers
radiant portal
No need for spoilers I wanna learn this method too hehe
violet gust
tangent is radius distance away from the center of circle. So you can verify that the centers (3,4) and (16,12) are 4 and 12 distance away from x-axis)
pretty trivial if you ask me
solid kilnBOT
radiant portal
First I expanded the brackets, then got the derivative, then subbed in 0 and solved
That gave me x = 7 and x = 28
violet gust
uhh ok, what is your derivative? Like you are differentiating the circle wrt what?
radiant portal
tangent is radius distance away from the center of circle. So you can verify tha...
Oh ya thats 1000x easier đĽ˛
radiant portal
uhh ok, what is your derivative? Like you are differentiating the circle wrt wha...
I was kinda just guessing what to do ngl
Oh... just realized I got the derivative with respect to x and y at the same time đ¤Ą
violet gust
yea
radiant portal
But ok, lets forget the derivatives lol
Im happy with your method
Thank you!!
â¤ď¸
.close
trim joltBOT
trim joltBOT
Available help channel!
slender junco
hello im working on a C program and i need to make a vector from a plane (in 3d context) to be scale around a "center" of this plane and i wondered how i can do that
but to be scaled in that plane only
if you see what i mean
clear cloud
Center of this plane ?
slender junco
like a reference point
clear cloud
Like (0,0) for R² ?
austere cedar
In order to scale a vector (x,y), you multiply both components by your scaling factor.
I suspect that's not what you want though. What am I missing?
clear cloud
You want to make a vector as a unit reference for the plane ?
slender junco
not really
austere cedar
Planes are infinite and we can't scale them
Other than vertical stretches or something like that
slender junco
like a homotetia (if thats the word)
for a vector in a plane with a reference point
and a scalar is stretching space
im not clear sorry ill find a way
.close
trim joltBOT
clear cloud
im not clear sorry ill find a way
Maybe a drawing could help actually but im also wondering how you would do that in C
trim joltBOT
Available help channel!
keen panther
help please
keen panther
help please
<@&286206848099549185>
tired shoal
help please
jee?
tired shoal
help please
the game is to put different values of x to get relation
like put x = 2
u get 0 on one side and simply summation of ar on the other
nvm u dont get 0 u get b0 on other side
trim joltBOT
@keen panther Has your question been resolved?
winged hinge
You make the transformation y = x - 1
Then keep deriving both sides to solve for a1,a2,a3,etc.
This assumes a_r are the same regardless of n
Then you can plug in for n such that the b_r on the right side are only such that r >= n
So for example take n = 1, you have a_0 + a_1y + a_2y^2 on the left side
Derive to get a_1 + 2a_2y = b_1 +2 b_2(y -1)
tired shoal
yes i tried matching coefficiients like that
a1 = 3
a2= -1
b2 is -1
b1 is 1
b0-a0 = 2
winged hinge
Looks right
trim joltBOT
trim joltBOT
Available help channel!
fiery cairn
trim joltBOT
fiery cairn
ill translate the text
4
represent on the real line and on the interval form
its all translated literally so yall are just gonna have to make up the contextđ
lime sphinx
yep
fiery cairn
thing is
whats th other one
i have to give 2 answers
but i didnt understand the other
lime sphinx
i would say they gave you the interval form for a) itâs ]-1,3]
lime sphinx
then idk either for me interval form is the shorthand notation ]-1,3] for the set builder notation {x in R | -1 < x <= 3}
lime sphinx
you have to understand $\cap$ and $\cup$
solid kilnBOT
pola_touche
fiery cairn
that'd be union and intersectionfor me
lime sphinx
yeah so you can compute a) and see if itâs equal to the set they propose.
first to be in (A n B) you need to be both in A and B
then to get (A n B) u C you add to the intersection you just computed all the elements\numbers in C
lime sphinx
fiery cairn
i mean cause theres no final symbol
fiery cairn
i think the teacher like
doesnt make his work[
cause he forgets most of the symbols in every assignment
lime sphinx
imo thatâs a bit harsh like this seem to be the worst typo in there
fiery cairn
imo thatâs a bit harsh like this seem to be the worst typo in there
i mean worksheets
i dont know what americans call them
i just say papers or work
but theres at least 2 or 4 mistakes in them
im just gonna message him and wait for an answer on 6
fiery cairn
well, since this IS a graded assignment
20 bucks he forgot to ctrl c ctrl v the rest of the question
lime sphinx
fiery cairn
im gonna kill mtself i swearđ
lime sphinx
I was talking about the blue, you seem to talk about orange
lime sphinx
im gonna kill mtself i swearđ
happens to anyone
had a friend who forgot to read in a test that you could use a formula without proving it, he proved it and realized after
so what do you think it is for 6?
fiery cairn
b union a intersection c
lime sphinx
huh you have to choose from choice a) to e)?
lime sphinx
you have intersection and union mixed up
lime sphinx
$\cap$ is intersection and $\cup$ is union
solid kilnBOT
pola_touche
lime sphinx
and I would guess - is set difference so A - B is all that is in A but not in B
there is a u at the stat of union and a n at the start of inter, might be a way to remember
but also !noans
trim joltBOT
The purpose of this server is to help you learn, not to hand out answers. Do not ask someone to give you the answer directly.
lime sphinx
yeah but someone gave it out, i mean you seem to compute the thigs so itâs ok
fiery cairn
i think b is the one that makes sense
cause c-b would be like -5 -4 -3 -2
and thats what the extensive form says
7
if a then b:
no what
if a and b then
ok
b again i believe
lime sphinx
translation tough?
lime sphinx
cause c-b would be like -5 -4 -3 -2
entao is then in english ok i see
fiery cairn
yeah
lime sphinx
yep itâs b the other ones donât work because some <= should be < or vice versa in the sets they propose
lime sphinx
if you have a feel for N, Q and R yeah
fiery cairn
A
lime sphinx
but idk whats the question tranlated tough
sharp swift
cause c-b would be like -5 -4 -3 -2
-2 e -5 nĂŁo estĂŁo no intervalo c-b, porque ele ĂŠ aberto. Nesse caso, sĂł estĂŁo os nĂşmeros maiores que -5 e menores que -2
fiery cairn
ah ĂŠ, nem vi o menor ou igual faltando
sharp swift
-2 e -5 nĂŁo estĂŁo no intervalo c-b, porque ele ĂŠ aberto. Nesse caso, sĂł estĂŁo os...
VocĂŞ pode imaginar como os infinitos nĂşmeros entre eles
fiery cairn
but idk whats the question tranlated tough
given r as the real numbers, n the naturals and q the rationals, whats the wrong alternative
read the queston wrong again
did not see i was supposed to choose the incorrect
then
i'd go for d
lime sphinx
i'd go for d
d) is correct imo like any natural number n can be written n/1 so N is a subset of Q
but i see two others in the choices that are really wrong
fiery cairn
i thought that said intersection doesnt it?
the intersection between rationals and naturals are a subset of rationals?
lime sphinx
yes all naturals are rationnals n/1 as i said so they are in N and in Q so n = n/1 is in N inter Q, this true for any n in N
fiery cairn
i dont think i quite get what you think i got-
i know all naturals are rationals
but the intersection between that, what would that be (im very slow)
lime sphinx
no no you have not played with sets enough to see that if $A \subset B$ then $A \cap B = A$
solid kilnBOT
pola_touche
fiery cairn
lemme tell ya i saw this today
forgot as soon as i left class
ok so
forget that
uhhh
what does e say?
lime sphinx
oh i didnât see the $\neq$
solid kilnBOT
pola_touche
lime sphinx
e) asks if there is a number that is both in R and Q
lime sphinx
yeah a) and e) are correct i think
fiery cairn
yea
lime sphinx
are you talking about 8?
yes
eliminate the one we ruled out are true so we can check the remaining
fiery cairn
c
lime sphinx
but do you get why itâs false
fiery cairn
đ
fiery cairn
well what you say is true
what do you mean
solid kilnBOT
PuĐi
royal drift
yes
fiery cairn
q and n only dont make r
lime sphinx
there are numbers in R that are not in Q and therefore not in N, because R is the set of irrational and rational numbers combined.
something like $\pi$ or $\sqrt{2}$ is not in $Q \cup N$
solid kilnBOT
pola_touche
fiery cairn
yeah i made that ouy
lime sphinx
in general itâs good to know : $N \subset Z \subset Q \subset R$
solid kilnBOT
pola_touche
fiery cairn
also I
fiery cairn
irrationals
lime sphinx
**pola\_touche**
k but you cant put I between Q and R here
fiery cairn
yes yes i understand
lime sphinx
all right
fiery cairn
just pointing out that i is also part of r
lime sphinx
homework done ima go
lime sphinx
.close
fiery cairn
.close
trim joltBOT
trim joltBOT
Available help channel!
amber tulip
If f is continuous and bounded, does it necessarily mean it has an antiderivative
fading lantern
I cannot tell if gpt is trolling or not
Idk it's worded weirdly, I just interpreted it as all as if a function is continuous then the derivative of the integral is the function
also try this: https://en.wikipedia.org/wiki/Fundamental_theorem_of_calculus
the fundamental theorem of calculus
The first part of the theorem, the first fundamental theorem of calculus, states that for a continuous function f , an antiderivative or indefinite integral F can be obtained as the integral of f over an interval with a variable upper bound.
trim joltBOT
@amber tulip Has your question been resolved?
amber tulip
It's actually real https://web.ma.utexas.edu/users/m408n/CurrentWeb/LM4-9-2.php#...
i should be allowed to say this then right
zinc ginkgo
what is "this" you're referring to
amber tulip
"because f is continuous, let f = F'"
so im saying because f is continuous, f has a antiderivative F
zinc ginkgo
that is correct
if you've proven that in class, then all you should need to do is cite it
if not, then you need to prove it
amber tulip
that is correct
alr
trim joltBOT
trim joltBOT
Available help channel!
wraith hinge
trim joltBOT
wraith hinge
Well, this question involves a functional equation
But is there any other possible method to do it directly
Like any intuitive method or smthg
hollow cypress
,rccw
solid kilnBOT
wraith hinge
Okay, assume there exists no readymade functional equation for this question, how would u approach it. how could I derive the function equation from it.
dull pilot
Okay, assume there exists no readymade functional equation for this question, ho...
what?
wraith hinge
Let's assume there's no known or pre-existing functional equation for this problem. How would you approach it from scratch? How could you go about deriving the functional equation yourself?
vagrant imp
what?
I think he's asking if its not possible to find the function, what would have been the approach
gusty plank
Let's assume there's no known or pre-existing functional equation for this probl...
you mean if you didn't have the information that f(3x)-f(x)=x ??
willow mortar
Actually there is a sure way to get the answer, are you familiar with the sequential definition of continuity ?
wraith hinge
Yep
willow mortar
Can you look into f(8/3^n) ?
wraith hinge
My question is like this
U given an equal f(x) and f(8)= 7 and u have to find f(14)= ?
Like there is readymade equation, like u know the quadratic equation smthg like that, if u put this into that u get answers. My question is how did u got that readymade equation in the 1st place
willow mortar
You have to Link everything to f(0) By itĂŠrations
wraith hinge
Whats that?
vagrant imp
Like there is readymade equation, like u know the quadratic equation smthg like ...
uh you make use of the fact that f(3x)-f(x)=x, its impossible to tell then nature of the function with just f(8)=7
wraith hinge
I havent learned iterations yet
wraith hinge
uh you make use of the fact that f(3x)-f(x)=x, its impossible to tell then natur...
Yea thts wht i got
But you know, if u manipulate it in the right way
U cn get it right?
willow mortar
Here ...
Now that you found f(0), you can use the same process to find a link between f(14) and f(0)
This is all possible because f is continuous
wraith hinge
But, u see. Imagine I don't know that function equation in the exam hall and how will I solve this
willow mortar
But, u see. Imagine I don't know that function equation in the exam hall and how...
Imagine you donât know what ?
vagrant imp
But, u see. Imagine I don't know that function equation in the exam hall and how...
then you cant bruh, its not possible to solve it
f(8)=7 will not help you to find f(14) directly
willow mortar
I mean, imagine youâre given 0 data about a problem. You canât solve the problem
When you have a problem like this. Try to use continuity or derivability to âlinkâ points to each other
willow mortar
No worries, do you see how to solve it now ?
trim joltBOT
trim joltBOT
Available help channel!
woven garden
can we say this triangle is equilateral
trim joltBOT
can we say this triangle is equilateral
woven garden
<@&268886789983436800>
burnt mulch
can we say this triangle is equilateral
You can cheat your way to the answer by saying that, but you canât get a rigorous solution by doing that
woven garden
yea thats what i was wondering because my math tutor did that and i was confused
balmy hornet
trim joltBOT
balmy hornet
I decided to use integral compare since it looks like it want a u substitution
but I forget how to go from here
zinc ginkgo
,tex .exp quotient
solid kilnBOT
riemann
zinc ginkgo
use chain rule or u sub after that
balmy hornet
so
e^-u
and so it just alternates between postive and negative?
vagrant prism
why alternates
balmy hornet
each derivative i mean
vagrant prism
derivative of what?
balmy hornet
the antiderivative of
- e^-u
is
- e^-u
vagrant prism
yes
trim joltBOT
trim joltBOT
Available help channel!
jovial leaf
hi im kinda confused how to set up this question
trim joltBOT
hi im kinda confused how to set up this question
lime sphinx
Lagrange multiplier or parametrise the circle
dull island
From the sounds of it youâve not encountered lagrange multipliers?
ebon rune
parameterising prob for this then
What experience have you had with finding the maximum value of functions?
cat
gusty bone
there's also probably some way you could use the power mean inequality
lime sphinx
i dont understand
Do you understand geometrically what the problem asks though?
like how can you interpret x^2+y^2 = 1 and x - 2y the function to optimize more geometrically in the plane R^2
jovial leaf
What experience have you had with finding the maximum value of functions?
uh
wdym by experience
ive only done optimization ways
jovial leaf
like how can you interpret x^2+y^2 = 1 and x - 2y the function to optimize more ...
huh?
i thought they wanted the max point of x-2y that intersects the circle
yeah my undertanding is probably incorrect
ebon rune
so like If I told you do find the maximum value of y = -x^2 + 1, would that be doable?
,w plot y = -x^2 + 1
solid kilnBOT
jovial leaf
so like If I told you do find the maximum value of y = -x^2 + 1, would that be d...
oh yes
vertex?
jovial leaf
yes
ebon rune
alright, then can you write the equation you would need to solve to get the maximum Y value (and its corresponding x input value) of y = -x^2 x + 1
lime sphinx
i thought they wanted the max point of x-2y that intersects the circle
no that's ok, f(x,y) = x-2y is a function that takes two numbers and spit out one number, and you want the number on the circle that makes the output the biggest
jovial leaf
alright, then can you write the equation you would need to solve to get the maxi...
sorry where did you get y = -x^2+1 from
jovial leaf
no that's ok, f(x,y) = x-2y is a function that takes two numbers and spit out on...
huh đ..
ebon rune
I just made it up to see how you solve it
as an example
I want to see your work specifically
jovial leaf
alright, then can you write the equation you would need to solve to get the maxi...
wait what is the equation?
-x^2x?
-x^3?
jovial leaf
-x^2+2x?
solid kilnBOT
Yeatte
jovial leaf
oh
okay i would do:
y'=-2x+1
0=-2x+1
x = 1/2
then test values for derivative an it turns out that it's a minimum (verifies) then f(1/2)
ebon rune
yep
in this case, the functions is -x^2 + x + 1, and we have only 1 variable we are putting into the fuction
Whereas in your original question
there's two variables
x and y
ebon rune
yep
jovial leaf
we cant isolate it so that y = 1/2x?
ebon rune
not for that
jovial leaf
oh interesting
ebon rune
instead we can treat the x-2y as its own variable,
so like lets have this be z
just as you did dy/dx = -2x + 1 = 0
we can do dz/dy = 0, dz/dx = 0
with the condition that whatever x or y values that we choose have to satisfy x^2 + y^2 = 1
lime sphinx
For concretness you can plug in f(x,y) any value you want and it will give you a number so for instance it associate -1 to the point (-1,0) on the circle
ebon rune
we can continue to work in terms of x and y, or we can substitute them for another variable t, that would make it easier to work with potentially
jovial leaf
with the condition that whatever x or y values that we choose have to satisfy x^...
im lost here đ
im so sorry
this question is so confusing
jovial leaf
For concretness you can plug in f(x,y) any value you want and it will give you a...
so there's a point from x-2y? which is also on the circle?
ebon rune
there's a point on a circle, where x -2y is the highest it can be
lime sphinx
im lost here đ
it means we have to choose x and y so that x^2 + y^2 = 1 for instance (-1)^2+0^2 = 1 so (-1,0) is a point we can pick to make f(x,y) the biggest we can
jovial leaf
ohhhh
lime sphinx
but there is no garantee (-1,0) is the best (x,y) we can choose on the circle so f is biggest
jovial leaf
Wait so a diagram like this? Weâre trying to find (a,b)?
ebon rune
we are trying to find the maximum value f(a,b), not necessarily the point (a,b) itself
and the function f(x,y) looks like this
,w plot z = x-2y
lime sphinx
Wait so a diagram like this? Weâre trying to find (a,b)?
I'm not too sure about this unfortunately, to truly plot f(x,y) you need a third axis a third dimension to see where the output of f is big or small for any point you pick on R^2
ebon rune
lime sphinx
Wait so a diagram like this? Weâre trying to find (a,b)?
alternatively you can do something similar to the plot you did here using level curves (where x-2y = 0 for instance that would be the curve in R^2 where f = 0 ) to keeps thing 2D
jovial leaf
I'm not too sure about this unfortunately, to truly plot f(x,y) you need a third...
WHAT
this question is 3d...?
ebon rune
yep
lime sphinx
one way to see it
ebon rune
though since x^2 +y^2 = 1, we don't actually have 2 independent variables, its just 1
so it can be reduced to a single-variable calculus problem
a great way to achieve this is to substitute x and y for some variable t
jovial leaf
so it can be reduced to a single-variable calculus problem
huh?
jovial leaf
a great way to achieve this is to substitute x and y for some variable t
so both x and y = t?
ebon rune
not necessarily t, but in terms of t
so if you want, you can have x = t, and y = 2t, though that wouldnt be useful in this particular case
the equation for a circle is x^2 + y^2 = 1, though you can also write it as|| (cos (theta), sin(theta)) for 0 <= theta <= 2pi||
lime sphinx
lime sphinx
so both x and y = t?
But Yeatte gave what goes where the ? is
jovial leaf
so if you want, you can have x = t, and y = 2t, though that wouldnt be useful in...
oh okay
jovial leaf
the equation for a circle is x^2 + y^2 = 1, though you can also write it as|| (c...
đ i dont know the unit circle..
ebon rune
ah
jovial leaf
is it r(theta)?
ebon rune
lemme put it like this
do you remember the equations for sin and cos?
sohcahtoa specifically
also like a^2 + b^2 = c^2
jovial leaf
sohcahtoa specifically
yea
jovial leaf
also like a^2 + b^2 = c^2
pythagorean?
ebon rune
yep
jovial leaf
yea
ebon rune
so let's have the adjacent side be a
and the opposite side be b
then that means that sin(theta) = b / c, and cos(theta) = a/c
or in other terms
b = c * sin(theta), and a = c * cos(theta)
a^2 + b^2 = c^2 = sin^2(theta) * c^2 + cos^2(theta) * c^2
dividing both sides by c^2, we get
cos^2 (theta) + sin^2(theta) = 1
if we instead sub a = x, and b = y
lime sphinx
Basically for the circle or radius one what x^2+y^2 =1 describe in the plane Yeatte just argue we have this picture
jovial leaf
wow i didnt know the circle function came from triangles
ebon rune
x^2 + y^2 = 1 = cos^2 (theta) + sin^2(theta)
jovial leaf
ohh
ebon rune
whenever you see sin and cos, think both circles and trianges, they go hand in hand
because we set x = a, this means that x = a = c * cos(theta), and as c = 1, then x = cos(theta), likewise, y = ?
I want you to do this part on your own
if you want to
jovial leaf
whenever you see sin and cos, think both circles and trianges, they go hand in h...
damn thats so cool i didnt kow tha
jovial leaf
because we set x = a, this means that x = a = c * cos(theta), and as c = 1, then...
y = sinx?
i truly appreciate y'alls patience đđ sorry if im kinda slow lol
ebon rune
we sub y = b in the equation a^2 + b^2 = c^2, where we found that sin (theta) = b / c and cos (theta) = a/c as per sohcahtoa
I want you to put y in terms of theta
just like I did with x= a = c * cos(theta)
jovial leaf
y = b = c*sin(theta) = 1(sin(theta)) = sin(theta)
ebon rune
yep
now, if we write out the point (x,y) what would that look like in terms of the single variable theta? ||x = cos(theta), y = sin(theta)||
lime sphinx
Back to og problem now we want the (x,y) on the circle (x^2+y^2=1) such that f(x,y) is biggest, but we have r(theta) now that associate to a number/angle in [0,2pi[ to all the points on the circle that's roughly what we mean by parametrization. It draws out the circle as theta gets bigger
ha maybe I was too quick
ebon rune
cat?
jovial leaf
now, if we write out the point (x,y) what would that look like in terms of the s...
(cos(theta),sin(theta))
ebon rune
yep
So if we look at the problem now, it says: maximize the function z = cos(theta) - 2 sin(theta) where theta is any angle where cos(theta)^2 + sin(theta) ^2 = 1. but cos(theta)^2 + sin(theta)^2 = 1 is true for all angles theta
so you only need to maximize the z = cos(theta) -2 sin(theta) function now and don't need to worry about choosing specific x or y points like we did before to satisfy x^2 +y^2 =1
jovial leaf
So if we look at the problem now, it says: maximize the function z = cos(theta) ...
ookay so i can just differentiate the z function?
ebon rune
yep
jovial leaf
and find theta?
ebon rune
find the maximum z value
jovial leaf
oh
ebon rune
i dont think they asked for the point
lime sphinx
~~i dont think they asked for the point~~
ha yeah you right
lime sphinx
z = cos(theta)-2(sin(theta))
then yeah just max this and you done
ebon rune
yeppers
ebon rune
arctan(-2) would be exact even if not pretty
jovial leaf
oh
solid kilnBOT
Yeatte
jovial leaf
like
the answr sayd
the questyion says*
oh wait
the maximum value
by value are they talking about theta
or like
a coordinate
ebon rune
the maximum z value
lime sphinx
the (x,y) that produce it is not important just what x-2y = cos(theta)-2sin(theta) is that special theta
trim joltBOT
ebon rune
!rats
trim joltBOT
ebon rune
yw
jovial leaf
wouldnt have thought of it
but like
what branch of math is this
that requires x y z coordinate
!rats
trim joltBOT
lime sphinx
but geometry make sense in 3D
jovial leaf
so like calc 2 -3 ish
is that uni stuff
lime sphinx
you might get things like sphere and planes
and describe them with equation like we did here with the circle
ebon rune
but for this question the main point was realizing that this z=x-2y equation that seemed difficult can be reduced to just a cos(theta), sin(theta) simpler question that becomes a single var and a strong intuition for circles and their equations
I would say that's the most important takeaway here
jovial leaf
and describe them with equation like we did here with the circle
but i still dont get why this is 3d
bc
lime sphinx
making someone understand what sin and cos with the unit cicle is always a big win imo
jovial leaf
x^2+y^2=r^2 isnt that in 2d
jovial leaf
but for this question the main point was realizing that this z=x-2y equation tha...
ohhhhh
jovial leaf
I would say that's the most important takeaway here
thank you
jovial leaf
making someone understand what sin and cos with the unit cicle is always a big w...
ya
ty @ebon rune
ebon rune
yep, but because we asked about a 3rd value, being z = x -3y, it becomes 3d
jovial leaf
yep, but because we asked about a 3rd value, being z = x -3y, it becomes 3d
oh đ
ebon rune
đ
lime sphinx
https://www.desmos.com/3d/p28f3xcybq
yep but you can visualize the value x-2y takes by adding a third axis Yeatte did this here
that's when it gets 3D when you need the graph of f(x,y)
anyhow me go eat food
jovial leaf
anyhow me go eat food
bon appetite and have a great day both of yall
ebon rune
random barn
how to Prove Ore's theorem using Posa's theorem?
trim joltBOT
how to Prove Ore's theorem using Posa's theorem?
@random barn Has your question been resolved?
trim joltBOT
@random barn Has your question been resolved?
trim joltBOT
@random barn Has your question been resolved?
empty prairie
Ore's theorem states that
$$\forall G := \left(V, E, \deg\right) \text{s.t.} E \subseteq V \times V, \abs{V} \ge 3$$
$$\forall u, v \in V \text{s.t.} (u, v), (v, u) \notin E, \deg(u) + \deg(v) \ge \abs{V} \implies G \text{ is a Hamiltonian graph}$$
solid kilnBOT
HitenTandon
random barn
**HitenTandon**
so, how can that prove Posa's theorem?
empty prairie
In the similar vein, Posa's theorem states that
$$\forall n \in \mathbb{N}, n < \infty$$
$$\forall G:= (V_{i =1}^n, E, \deg) \text{s.t.} E \subseteq V \times V, \forall i \le j \le n,\deg(V_i) \le \deg(V_j),$$
$$\forall k \in \mathbb{N}, 1 \le k < \frac n 2, \deg(V_k) < k \implies G \text{ is Hamiltonian}$$
solid kilnBOT
HitenTandon
random barn
so, what is the proof?
trim joltBOT
@random barn Has your question been resolved?
broken axle
How to do a)?
trim joltBOT
How to do a)?
lunar stirrup
How to do a)?
force equations
b has a bigger mass so assume acceleration of B is in downward direction
therefore acceleration of A would be in upper direction
using this, u can frame 2 equations, 1 for A and one for B.
the variables are acceleration and tension so u have 2 variables and 2 equations
should be solvable
broken axle
therefore acceleration of A would be in upper direction
Soooo I need to make suvat equation?
lunar stirrup
Soooo I need to make suvat equation?
what is suvat
sacred sapphire
what is suvat
s for displacement
u initial velocity
v final velocity
a acceleration
t time
trim joltBOT
@broken axle Has your question been resolved?
lunar stirrup
s for displacement
u initial velocity
v final velocity
a acceleration
t time
oh
@broken axle nono not that
the force equations
eg: T-mg = ma
summer haven
Soooo I need to make suvat equation?
no you just need to consider the forces
suvat is about motion. We're not looking at the motion just yet in part a
just need to consider the forces acting on it, i.e. newton's second law
random barn
.reopen
trim joltBOT
broken axle
eg: T-mg = ma
Ohhh how do I know which one comes first like T or the weight
trim joltBOT
Available help channel!
lunar stirrup
Ohhh how do I know which one comes first like T or the weight
that is dependant on direction of acceleration, if the tension is in direction of acceleration then tension comes first if mg is in direction of acceleration then mg comes first
broken axle
that is dependant on direction of acceleration, if the tension is in direction o...
Why is that?
lunar stirrup
Why is that?
F = ma
and this is in vector form
if lets say acceleration is down
if u take T-mg = ma, wouldnt that be implying that the tension is creating the acceleration ? but how can this be when acceleration is downwards and tension is upwards?
sharp hornet
acceleration could be negative or positive, if you resolve up and you have the correct signs it wouldn't matter which way you do it
i normally just look at each system individually and resolve both up
or down, doesnt matter
broken axle
What does 700kg correct to the nearest 50kg mean?
trim joltBOT
What does 700kg correct to the nearest 50kg mean?
inner talon
hi
narrow fox
What does 700kg correct to the nearest 50kg mean?
are u talking about upper and lower bounds?
fossil folio
They probably mean rounding
Which in that case, 700kg is already the closest
650kg, 700kg, 750kg, are increments of 50kg and what we're rounding to
narrow fox
What does 700kg correct to the nearest 50kg mean?
if ur finding bounds the nearest value (50) is divided by 2 which means that the actual weight can either be 25 less/more than 700 so the bounds are: lower bound is 675 and upper bound is 725
trim joltBOT
@broken axle Has your question been resolved?
broken axle
if ur finding bounds the nearest value (50) is divided by 2 which means that the...
Oh okayyy thank you
.close
trim joltBOT
trim joltBOT
Available help channel!
visual blaze
sin 100°=k, what is tan 260° in terms of k?
trim joltBOT
**sin 100°=k, what is tan 260° in terms of k?**
visual blaze
I tried rewriting tan 260° into tan (360°-100°), and got +tan 100°, which is not correct.
twilit frost
360-260 you mean?
and it would be -tan100 right?
hollow cypress
find the basic angles for both
can you tell me what the basic angle of 100 and 260 are
solemn parrot
360-260 you mean?
and it would be -tan100 right?
Yeah and once you have that just split it into sin/cos and use the fact that cos² = 1-sin²
visual blaze
hollow cypress
youre on the right track
solemn parrot
So draw the same thing for -100°
It will be what you drew reflected in the x-axis
visual blaze
tan 260° is tan (90°Ă4-θ), with θ=100°, which should be -tan (θ). and -tan (100°) is -[-â(1-k²)/k].
I should treat the -100° as acute angle and then substitute the real value in later.
.close
trim joltBOT
hollow cypress
tan 260° is tan (90°Ă4-θ), with θ=100°, which should be -tan (θ). and -tan (100°...
uh no wouldnt it be -k/-(sqrt(1-k^2)) ? therefore giving an overall positive?
trim joltBOT
Available help channel!
wild zodiac
Can someone give me hand solving a non linear 3rd order ODE numerically? The equation I want to solve is an adimensional equation characterizing the profile of a polymer over non-uniform topography on a wafer. Essentially, I want to find H(X), given S(X) as in the picture, subject to the boundary conditions of H=1 and dH/dX = 0 at X=+/-infinity. So far this is my code:
c
I cannot get this to return me a sensible solution. I have no idea what I am missing
wild zodiac
Sorry for the image, but discord does not allow me to paste so much code in a single message
trim joltBOT
@wild zodiac Has your question been resolved?
trim joltBOT
@wild zodiac Has your question been resolved?
zinc ginkgo
trim joltBOT
@wild zodiac Has your question been resolved?
trim joltBOT
@wild zodiac Has your question been resolved?
stark heart
Is there a way to solve this with the common factor way
trim joltBOT
Is there a way to solve this with the common factor way
left oriole
what do you mean by common factor way?
stark heart
U take the biggest common factor
left oriole
unlikely to be fruitful here
vagrant prism
try it
left oriole
try seeing this as a difference of square roots
and do what you should do 99.999% of the time in that case
stark heart
Ain't working
stark heart
and do what you should do 99.999% of the time in that case
U multiply by the conjugate
vagrant prism
that you can either do with l'hopital (overkill here) or do as bungo said
left oriole
(multiply and divide)
stark heart
that you can either do with l'hopital (overkill here) or do as bungo said
I haven't taken that yet
vagrant prism
great then ignore what i said
stark heart
But I know how to use it
But my teacher says I'm not allowed to use it
I thought it was only used for 0/0 and inf/inf
left oriole
you could probably manipulate this into one of those forms (using log/exp) but the conjugate trick will work nicely here
vagrant prism
I thought it was only used for 0/0 and inf/inf
you can transform it to be either of those but you don't need it here
stark heart
I solved with conjugate
stark heart
But this is the only limit I've seen that needs to use the conjugate
stark heart
what did you get?
.5
vagrant prism
correct
vagrant prism
But this is the only limit I've seen that needs to use the conjugate
oh there's plenty
left oriole
But this is the only limit I've seen that needs to use the conjugate
you'll probably see more in the future, it's a standard technique
stark heart
Our textbook has 0đ
vagrant prism
it's useful every time you have a difference of two things where one of them is under square root
trim joltBOT
trim joltBOT
Available help channel!
lunar stirrup
trim joltBOT
lunar stirrup
where does my teacher find these questions bro đ
vapid grove
is that just a matrix
empty orchid
where does my teacher find these questions bro đ
||From his ass||
lunar stirrup
is that just a matrix
determinant
lunar stirrup
||From his ass||
on god bro
empty orchid
You can easily eliminate one of these options
empty orchid
Ah, shit
vapid grove
determinant
ohh ok just write out the expression for the determinant then 
empty orchid
No, you must cheese
lunar stirrup
Well, try the ||constant term|| first
put x = 0?
lunar stirrup
ohh ok just write out the expression for the determinant then <:verycool:8842702...
;/
empty orchid
put x = 0?
Yeah
lunar stirrup
,w det{{(1+x)^a,(1+2x)^b,1},{1,(1+x)^a,(1+2x)^b},{(1+2x)^b,1,(1+x)^a}}
solid kilnBOT
lunar stirrup
Bro?!
just wanted to see what would come ofit
now what do i do
@empty orchid
come back
empty orchid
Am I blind
lunar stirrup
Am I blind
no i mean lets do it without seeing what came out of that determinant
trim joltBOT
@lunar stirrup Has your question been resolved?
lunar stirrup
<@&286206848099549185>
empty orchid
@lunar stirrup Back
empty orchid
no i mean lets do it without seeing what came out of that determinant
What do you mean?
I already suggested the method of finding the constant term, and there is definitely an option you can eliminate (and one you can pick) from our observation of f(0)
lime sphinx
Maybe Newton's binomial formula to expand these but this will be an intense computation
would be much nicer if there was a trick
lunar stirrup
I already suggested the method of finding the constant term, and there is defini...
no but how do u know the only constant term will come from f(0)?
empty orchid
(Itâs the JEE, there is always a trick)
empty orchid
no but how do u know the only constant term will come from f(0)?
Because this is a polynomial?
lime sphinx
plug in 0 for x seems a pretty good move tbh
lunar stirrup
Because this is a polynomial?
wait could u tell me if there is a difference in plugging x=0 before expanding and after expanding
empty orchid
wait could u tell me if there is a difference in plugging x=0 before expanding a...
There isnât, itâs the same function
lime sphinx
there should be no difference
empty orchid
I donât get what youâre talking about
lunar stirrup
There isnât, itâs the same function
oh ok then i get it
empty orchid
Okay
lunar stirrup
so f(0) = 0
D is correct?
lunar stirrup
a,c
empty orchid
Okay, I have no clue how to approach B to be honest
lime sphinx
me neiter maybe differentiate the thing then evaluate at 0 but that's a bit much
empty orchid
Oh, that works
lunar stirrup
me neiter maybe differentiate the thing then evaluate at 0 but that's a bit much
u want to differentiate that?
huh wait
lime sphinx
no but then evaluating the thing at 0 you will get the coeff in front of x of the og polynomial
empty orchid
Ah screw it, use the ambiguos method (let a=1, b=2)
lunar stirrup
Ah screw it, use the ambiguos method (let a=1, b=2)
can we put 0
empty orchid
âPositiveâ
empty orchid
âDistinctâ
lunar stirrup
.
empty orchid
Wtf?
lunar stirrup
đ
lunar stirrup
this is worksheet of determinant + matrices
empty orchid
|A^n|=|A|^n right?
lunar stirrup
yes
empty orchid
Belch
lunar stirrup
but that is a matrix not a determinant
lime sphinx
yeah again just bluntly trying to compute this thing seems hopeless
lunar stirrup
yeah again just bluntly trying to compute this thing seems hopeless
thats usually how it works
there is only 1 fast method
and if u dont find it, dont bother attempting the question
empty orchid
how did u know i was writing jee
Iâve been in this community for a while
You recognize questions
carmine spade
ok ill close this now
Did you get clear on b in the end, it's unclear from what was said above
I can help you with it if you like
lunar stirrup
Did you get clear on b in the end, it's unclear from what was said above
I can h...
yea can u help
carmine spade
Are you familiar with row/column operations?
lunar stirrup
Are you familiar with row/column operations?
yes
carmine spade
Ok cool