exactly
#help-0
slow sleet
but this IS true for normal distributions then?
slow sleet
sort of
i still dont get your descirption of it being like the actual value?
this confuses me
jaunty wharf
you'll get it when you solve problems with actual data, dont worry
it's pretty straightforward once you start using data
slow sleet
i mean i am
print(st.t.interval(0.9, len(data)-1, loc=data.mean(), scale=st.sem(data)))
but i also have to do st.sem
and i dont understand at all what that is
its standard error of the mean or something
but idk what that is
orrrr is that like standarddeviation but with the mean instead of median
slow sleet
OHHH
jaunty wharf
yep
slow sleet
my teacher never said that 😭
jaunty wharf
yeah here theyre basically saying that for 95 percent of the individuals their mean value is in the 95 % CI
its the graph u drew
slow sleet
thing is a random value would make sense to me
but the mean doesnt
because the mean is already calculated
jaunty wharf
um think of it this way
slow sleet
you are putting the mean into the equation
jaunty wharf
the mean also has a distribution of its own
so it in itself is also random
or or
think of it this way
you draw a sample right, and you calculate their mean
that mean will lie in the interval 95 percent of the time
such an interval is the 95 % C.I.
jaunty wharf
we say usually higher the number of samples, the better the result
slow sleet
It’s still kinda confusing
Cuz like you say
If you do the mean again
it depends on the size of the set
so then wahts teh size
is it the same size?
jaunty wharf
yes
you fix a sample size
and you get multiple say 1000 samples
you will see 95 percent of the time the mean is in the 95 % C.I.
slow sleet
mhm interesting
jaunty wharf
it is
hearty hazel
<@&268886789983436800>
twilit dome
wtf? go away
lone heartBOT
lean depot
.close
lone heartBOT
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verbal swallow
Hi guys,
let a,b be real numbers with 0 < a < b and sides of a rectangle. I am to show a < H(a,b) < G(a,b) < A(a,b) < b
with A being the arithmetic mean, G geometric mean and H the harmonic mean.
How can I show this? Where do I start?
lone heartBOT
@verbal swallow Has your question been resolved?
frail dagger
What is this question?
By the AM-GM-HM inequality, AM > GM > HM if number_1 > number_2
b > a
2b > a + b
b > AM
verbal swallow
"I am to show a < H(a,b) < G(a,b) < A(a,b) < b" but I dont know where to start with that
frail dagger
Do you aware of the AM-GM-HM inequality?
verbal swallow
b > a
2b > a + b
b > AM
this makes sense, but what does it give us?
verbal swallow
Do you aware of the AM-GM-HM inequality?
No
and I am not sure whether we can use it
it wasnt introduced in the lectures yet
verbal swallow
Hm, cant say I understand what I am looking at tbh :S
lavish cave
probably algebraically is the way to go then
show that $\sqrt{ab} \le \frac{a + b}{2}$: this is quite famous, so you can start from $a^2 - 2ab + b^2 \ge 0$ and then add $4ab$ to both sides
(the equality case of this inequality is when a = b)
ocean sealBOT
south
lavish cave
likewise, to prove $HM < GM$ you need to show $\frac{2p^2 q^2}{p^2 + q^2} < pq$
ocean sealBOT
south
lavish cave
(2ab)/(a + b) comes from the definition
lavish cave
likewise, to prove $HM < GM$ you need to show $\frac{2p^2 q^2}{p^2 + q^2} < pq$
the first step in the proof is (p - q)^2 > 0, since p is not equal to q
lone heartBOT
@verbal swallow Has your question been resolved?
lone heartBOT
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grizzled verge
guys is this valid as a proof? id want to prove that for a polynomial f(x) of degree n with rational coefficient not all equal to 0, then f(pi) is always a trascendental number. let's assume for the sake of contradiciton that f(pi)=A where A is an algebraic number, thus we can move A to the other side of the equation creating a new polynomial g(x)=f(x)-A. so if pi is the root of g(x), g(pi)=0 then it would make pi an algebraic number itself, but this contradicts the fact the pi is trascendental thus making the initial assumption wrong, so f(pi) must be trascendental itself.
quasi vector
guys is this valid as a proof? id want to prove that for a polynomial f(x) of de...
g(x) doesn't have all coefficients be rational numbers because A may or may not be rational
but you can say that if f(pi) = A where A is algebraic, then there exists some polynomial g with rational coefficients such that g(A) = gof(pi) = 0, and since gof is another polynomial with rational coefficients, pi must be algebraic
grizzled verge
but you can say that if f(pi) = A where A is algebraic, then there exists some p...
so if i correct my statement it does hold true this way right?
earnest lance
can someone tell me if 1 or 4 is right?
lone heartBOT
can someone tell me if 1 or 4 is right?
digital magnet
No
digital magnet
Neither of them are correct
digital magnet
Nope
digital magnet
We know that the lines inside of the circle are the same length right
That’s what the problem tells us
earnest lance
yteag
digital magnet
So we also know that the arc ST will be the same as the arc SR
and SR is x
And we also know that the total arc measure in a circle is 360
So set up an equation using that information to find x
Lmk what you get for problem 1
earnest lance
148?
digital magnet
Yep
earnest lance
and is problem 4 135?
digital magnet
Ye its 135
digital magnet
Anytime 🙂
lone heartBOT
Available help channel!
civic obsidian
I'm not sure where to go for number 8. My workbook doesn't have any examples similar to this question
civic obsidian
Okay cool, so it says I have to find the values.. do I just do the b² - 4ac formula again with that now?
edgy hemlock
You’re supposed to get values of k right?
civic obsidian
Yeah
civic obsidian
Do you know how?
No 😓
edgy hemlock
Well here’s how, it will go a long way so it will be worth it if you note it down
First look at 20 and -12
Find me two numbers that when multiplied gets you 20
and when added together get you -12
Try to look for the 20 first, then see if the numbers you find sum to -12
civic obsidian
Something with 10 and 2?.. one has to be negative but I'm never really able to figure out which
edgy hemlock
Well actually
unreal carbon
the product has to be +20, not -20
edgy hemlock
Since the 20 is positive, they both have the same sign
civic obsidian
OOH 🤦🏾♀️🤦🏾♀️
edgy hemlock
Yup, so what do you think, 10 & 2 or -10 & -2?
civic obsidian
both minus
edgy hemlock
Because?
civic obsidian
Do I have to switch the signs ?
edgy hemlock
No no
civic obsidian
edgy hemlock
Well
Yeah that’s your answer
But you do know why both minus right? Cuz they sum -12
civic obsidian
yes !!
royal star
you could have used the discriminant formula to factorize it as well btw
civic obsidian
But you do know why both minus right? Cuz they sum -12
it's been a while since I've done factorization
civic obsidian
you could have used the discriminant formula to factorize it as well btw
b² - 4ac?
royal star
yeah
edgy hemlock
you could have used the discriminant formula to factorize it as well btw
I think this one with bit more practice can be done quicker
royal star
sometimes you won't get nice numbers
edgy hemlock
sometimes you won't get nice numbers
True
civic obsidian
yeah
It said to find the set of values and that would have only given me one value I think
royal star
it would have given you both
edgy hemlock
No it would give you two
civic obsidian
oh!
royal star
remember the plus minus
civic obsidian
remember the plus minus
Oh!!!!!
royal star
$k = \frac{-b \pm \sqrt(discriminant)}{2a}$
ocean sealBOT
asm
civic obsidian
isn't that the quadratic formula?
royal star
yea, it needs the discriminant as well though
edgy hemlock
It is
civic obsidian
oh okay
royal star
it's also nice to know that
if b is an even number in
ax^2 + bx + c
you can use the following formula
edgy hemlock
Well that’s new to me
royal star
you can still use the ordinary formula, but it might make things a bit faster
edgy hemlock
And is your question finished btw?
shadow fjord
Once you have the quadratic factored, I recommend using a sign chart to find the values of k where the expression is > 0.
edgy hemlock
Like i feel the answer might use more specification
edgy hemlock
Once you have the quadratic factored, I recommend using a sign chart to find the...
Yeah could work
civic obsidian
Another question... How do I transition from the equal sign to the greater than?
civic obsidian
Once you have the quadratic factored, I recommend using a sign chart to find the...
Can you explain what you mean by a sign chart?
shadow fjord
Can you explain what you mean by a sign chart?
You draw a number line and test values where the quadratic isn't =0.
In the problem above, the quadratic is =0 at k=2 and k=10. So you basically test k<2, 2<k<10, and k>10.
civic obsidian
You draw a number line and test values where the quadratic isn't =0.
oh oh like when you draw a graph and do the shading thing? like linear inequalities?
shadow fjord
oh oh like when you draw a graph and do the shading thing? like linear inequalit...
You don't actually have to draw the graph, but drawing a graph would give the answer.
Part of the point is with the sign chart you don't have to draw the graph.
For example, if you test k=0, you have (0-2)(0-10) = (-2)(-10) = 20>0. So k<2 works.
shadow fjord
But if you test k=4, you have (4-2)(4-10)=(2)(-6)=-12 <0, so 2<k<10 doesn't work.
edgy hemlock
oh oh like when you draw a graph and do the shading thing? like linear inequalit...
you can directly eliminate the numbers in between the two numbers you found, the numbe line and graph help you see them, but if you want only the answer this way should be enough
royal star
do you understand the signs by the way
edgy hemlock
excuse my intervals
Pretty clear to me
civic obsidian
you can directly eliminate the numbers in between the two numbers you found, the...
Okay makes sense
royal star
my group mates make fun of my triangular shape thing
edgy hemlock
Yeah, if you look here 2 and 10 make the answer 0 and anything between them will make the values less than 0 which you dont want, you can try some numbers and see for yourself
civic obsidian
Yeah, if you look here 2 and 10 make the answer 0 and anything between them will...
OOOOOHHHHHH I understand now
royal star
depending on the roots you get, you try out the numbers
for example
$(k - 10)(k - 2)$
ocean sealBOT
asm
royal star
when k is 0, what is the sign?
edgy hemlock
my group mates make fun of my triangular shape thing
I like it, it sounds like a jumping rabbit passed by
civic obsidian
Okay wait I think I just confused myself.. where would k land on this ? the + parts?
royal star
Another question... How do I transition from the equal sign to the greater than?
put the roots on the
interval
royal star
Okay wait I think I just confused myself.. where would k land on this ? the + pa...
depends on what the question is asking
we're looking for greater than 0
edgy hemlock
OOOOOHHHHHH I understand now
For your final answer you can write it like this:
R/10=>k=>2 ( i don’t recommend)
or k>10 U k<2 ( i recommend)
civic obsidian
we're looking for greater than 0
okay I fully get it now
edgy hemlock
Interval is the space between any two numbers in the number line, when you put your roots, 6 and -2 in this case, you will have the interval from -2 to 6
royal star
civic obsidian
darn so I used the wrong numbers?
civic obsidian
Just get each bracket alone and equate each to 0 and see what values you get
ah 😭
edgy hemlock
Yup
edgy hemlock
Yup
Made a funny moment in class in my end, 10 mins of discussion and the doc wouldn’t tell us why and laughed it off
civic obsidian
royal star
Made a funny moment in class in my end, 10 mins of discussion and the doc wouldn...
are you studying mathematics in uni?
edgy hemlock
are you studying mathematics in uni?
Engineering
royal star
if it has a equal sign as well, you have to include aka just bracket it
royal star
Engineering<:catwhy:608492030489919508>
nice, you live in the USA?
civic obsidian
if it has a equal sign as well, you have to include aka just bracket it
?? 💔
royal star
so
if you had something like
(k - 6)(k + 2) > 0
you would do
(-inf; -2) U (6;+inf)
edgy hemlock
nice, you live in the USA?
Nah, somewhere else
royal star
we don't want to use brackets in this case
royal star
Nah, somewhere else
are the entrance exams tough in your country?
the uni entrance exams
civic obsidian
you would do
(-inf; -2) U (6;+inf)
what does inf mean?
civic obsidian
oh
royal star
$((-infty) , -2) U (6, + \infty)$
ocean sealBOT
asm
Compile Error! Click the 
reaction for more information.
(You may edit your message to recompile.)
edgy hemlock
?? 💔
You have to specify your interval, the place where the result is bigger than 0, since you got 6 and -2 you want to include all the number line except the numbers between them, so to do that you do this:
(-inf, -2) U (6, +inf)
This way you get all numbers and eliminate the numbers between your roots
i DID NOT put closed parentheses on inf
civic obsidian
So I found the answer that I did to this question a while back and I just want to know which version is correct.. Because Although it has a tick my answer key might be wrong
royal star
edgy hemlock
the uni entrance exams
Well i dont think we have this system, just SAT, SAT II and highschool grades
royal star
i DID NOT put closed parentheses on inf<:catglasses:1037770587235434567>
me about to do that on the exam
royal star
Well i dont think we have this system, just SAT, SAT II and highschool grades
I see
by the way, is geometry used in engineering?
except for trigonometry and whatnot
royal star
so, coordinate geometry
edgy hemlock
Yeah
civic obsidian
Okay thank you both for your help :)
royal star
🥲
royal star
Okay thank you both for your help :)
np 👍
if you do not have any questions, you could close this channel via .close
edgy hemlock
So I found the answer that I did to this question a while back and I just want t...
This interval is correct
civic obsidian
if you do not have any questions, you could close this channel via .close
👍👍
edgy hemlock
Okay thank you both for your help :)
Anytime
lone heartBOT
edgy hemlock
okay tyy
Actually not correct since you put bigger/less than or equal to, you strictly want results bigger than 0 not just less than 0
So remove “equal to”
lone heartBOT
Available help channel!
lone heartBOT
hushed locust
well you would get a shape with area pir^2 if you had a rectangle with base pir and height r
and your shape essentially "bends" that shape, making it have a different area
still prism
two different shapes can have the same area
fierce narwhal
do you know calculus?
lone heartBOT
@pure fjord Has your question been resolved?
lone heartBOT
lone heartBOT
@forest raft Has your question been resolved?
patent vale
i dont think i can see the whole problem
lone heartBOT
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Available help channel!
alpine sable
lone heartBOT
alpine sable
help me
I assume that the sentence he gets the initial score is true or false, and then divide the case of the number of sentences True or false
lone heartBOT
@alpine sable Has your question been resolved?
alpine sable
no
lone heartBOT
@alpine sable Has your question been resolved?
alpine sable
<@&286206848099549185>
lone heartBOT
@alpine sable Has your question been resolved?
lone heartBOT
alpine sable
.reopen
lone heartBOT
✅
lone heartBOT
@alpine sable Has your question been resolved?
scarlet ibex
how do you integrate 1/x^(sqrt(x)). if it's not possible explain why
scarlet ibex
vale wigeon
i would be very surprised if this had a remotely clean antiderivative tbh
forest marsh
Ah so nvm
scarlet ibex
Write it as x^...
scarlet ibex
you're SOL on this sorry <:kekehands:1130985995102322719>
SOL?
winter light
,w integrate 1/x^√x
ocean sealBOT
winter light
how do you integrate 1/x^(sqrt(x)). if it's not possible explain why
!original
lone heartBOT
how do you integrate 1/x^(sqrt(x)). if it's not possible explain why
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.
scarlet ibex
Please show the original problem, **exactly as it was stated to you**, with the ...
Well it was a challenge to find the antiderivative of f(x) = 1/x^(sqrt(x)) and also to find the area under it
winter light
A challenge? Given by the teacher?
scarlet ibex
A challenge? Given by the teacher?
from a fellow student. encouraged by the teacher
winter light
I see
Yeah your fellow (and ideally your teacher as well) should have tried it before giving it to you 😅
scarlet ibex
Yeah your fellow (and ideally your teacher as well) should have tried it before ...
you're saying it's unsolvable
vale wigeon
yup
winter light
@scarlet ibex
scarlet ibex
but why?
winter light
Because no composition of elementary function gives you that, after differentiating it
vale wigeon
there is not really an easy answer to that most of the time
something something risch algorithm
scarlet ibex
so there is no way to solve this at all?
winter light
Indeed
zinc bolt
so there is no way to solve this at all?
Not a standard one at least
You can try reverse-engineering
tho
vale wigeon
...
scarlet ibex
is the area possible to find?
winter light
The area can be found quite precisely with numeric approximation, yeah
With GeoGebra or Wolframalpha, for instance
But you need the bounds
zinc bolt
Well it was a challenge to find the antiderivative of f(x) = 1/x^(sqrt(x)) and a...
This is the closest thing I got
$$f(x) = x^{\sqrt{x}} + f(x) \cdot \sum_{i = 1}^{\infty} -1^{i}\frac{d^i}{dx^i}\left(\frac{2\cdot \ln\abs{x} - 1}{(\ln\abs{x})^2}\right) + c$$
and I don't even think this would work tbvh
@zinc bolt
zinc bolt
Oh I made a mistake typing it whoops
This is the closest thing I got
$$f(x) = x^{\sqrt{x}} - f(x) \cdot \sum_{i = 0}^{\infty} -1^{i}\frac{d^i}{dx^i}\left(\frac{2\cdot \ln\abs{x} - 1}{(\ln\abs{x})^2}\right) + c$$
and I don't even think this would work tbvh
@zinc bolt
zinc bolt
.close
mossy mason
As you know square root of 16 is plus minus 4 .
lone heartBOT
As you know square root of 16 is plus minus 4 .
mossy mason
I want to know in which cases we put plus minus?? And in which cases there is plus only.?
woeful zenith
I want to know in which cases we put plus minus?? And in which cases there is pl...
Okay so the set of square roots of a number contains both positive and negative, but the radical symbol $\sqrt{\cdot}$ always represents the positive value, called the principal root
tiny sky
As you know square root of 16 is plus minus 4 .
ok you haven't put a level of education as a role so idk what sort of level ur asking this at
but yeah if ur talking about the real numbers then yh by convention we take sqrt to be positive so we have a continuous single valued square root
if we work in the complex numbers, then to define a square root we need to take a branch cut of the logarithm
depending on which cut of the logarithm you take you might get sqrt(16)=4 or sqrt(16)=-4
mossy mason
(4*10^-3)^1/3 = ? , I want to know the 1/3 power applies to both ??
4 and 10^-3
Applied to these individually??
(4)^1/3 * (10^-3)^1/3 is this right??
??
@tiny sky
tiny sky
(4)^1/3 * (10^-3)^1/3 is this right??
yh
tiny sky
Ok...the 3 root of number contains both positive and negative??or positive only?
it's a real number
it's the number you get on your calculator
mossy mason
it's a real number
No I mean like any number to the power of 1/3 ...for example. 7^1/3
tiny sky
No I mean like any number to the power of 1/3 ...for example. 7^1/3
3^3 = 27 so cuberoot(27) = 3
(-3)^3 = -27 so cuberoot(-27) = -3
the cube root of a positive number is positive, the cube root of a negative number is negative
mossy mason
.close
thin zinc
lone heartBOT
thin zinc
how do i understand what the shape of the region is?
$\omega ^2= -\frac{1}{2} -i \frac{\sqrt{3}}{2}$
ocean sealBOT
∮Ē.dĀ = Qₑₙ꜀/ε₀
pliant charm
this is conjugate
ocean sealBOT
∮Ē.dĀ = Qₑₙ꜀/ε₀
thin zinc
idk how to proceed from here
pliant charm
put w and w²
thin zinc
so $a- \frac{b+c}{2} -i\sqrt{3}(\frac{b+c}{2})$
vale wigeon
not gonna be worthwhile to do it this way
ocean sealBOT
∮Ē.dĀ = Qₑₙ꜀/ε₀
thin zinc
not gonna be worthwhile to do it this way
how else then?
vale wigeon
i didn't do it fully but this is the idea
if you just consider the a term you get a line segment from 0 to 1
if you also include the bw term you kinda get it dragged linearly through the vector joining 0 to w
and get that blue parallelogram region
thin zinc
so when i add c w^2 i get 2/3rds of a hexagon?
thin zinc
how
lone heartBOT
lone heartBOT
Available help channel!
alpine sable
I assume that the sentence he gets the initial score is true or false, and then divide the case of the number of sentences True or false
thin zinc
theres 4 ways for nam to know the correct answer to a question to. it can either be q1, q2, q3 or q4
now 3 questions remain where nam has to answer at least 2 correctly
lone heartBOT
@alpine sable Has your question been resolved?
alpine sable
theres 4 ways for nam to know the correct answer to a question to. it can either...
exactly
lone heartBOT
@alpine sable Has your question been resolved?
prime badge
it's confusing, it shouldn't matter that they can't be all false
prime badge
.close
lone heartBOT
lunar saddle
what
harsh swallow
I need some help but i think it might be too esoteric for general help. I need to prove that $|\mathbb{G}^n| = 2^n$ where $\mathbb{G}^n$ is the geometric algebra corresponding to $\mathbb{R}^n$. i know that i can construct $\mathbb{G}^n$ from $\mathbb{G}^{n - 1}$ by taking 2 versions of them taking the geometric product $\mathbf{e}_n \mathbb{G}^{n - 1}$ and unioning with the second one $\mathbf{e}_n \mathbb{G}^{n - 1} \cup \mathbb{G}^{n - 1}$.
ocean sealBOT
Katharine
harsh swallow
I don't know how to use this in a proof
lone heartBOT
Available help channel!
verbal timber
sqrt{x} = (1/2)(x-3)
lone heartBOT
sqrt{x} = (1/2)(x-3)
verbal timber
why squaring both sides doesn't work?
vale wigeon
what exactly do you mean by "doesn't work"
vale wigeon
1.9?
like one and nine tenths??
what does the dot mean
i do not understand your notation
vale wigeon
and what's with this i?
vale wigeon
but the actual answer would be (i,9)
here
verbal timber
i, sqrt(-1)
vale wigeon
x=i is not a solution to your equation
(i-3)/2 is not equal to either of the two square roots of i
verbal timber
oh, i am sorry, that's some other part, but why does 1 appear when we square both sides/
vale wigeon
it is an extraneous solution
any real solutions to the original equation must satisfy (x-3)/2 >= 0 in order to make sense
x=1 is rejected on those grounds
vale wigeon
so you are wrong to say it doesn't work. it does work, you just need to take precautions at the beginning and end
verbal timber
thank you, miss
vale wigeon
you're welcome
jaunty wharf
.close
lone heartBOT
Available help channel!
spring turret
how to calculate maximum likelihood estimator on uniform distribution.
ebon stream
how to calculate maximum likelihood estimator on uniform distribution.
doesnt uniform distribution mean theres equal likeliness everywhere
spring turret
yeahh i think so
ebon stream
lmao
spring turret
vale wigeon
0 > 0
spring turret
its [0, theta]
and theta > 0
sorry i just directly google translated it. thats why it looks weird
on last homework we calculated ML estimator on uniform distribution
obtuse moth
Can anyone solve this with full process
spring turret
Can anyone solve this with full process
bruhh this channel is already occupied
quartz forum
obtuse moth
Can you help me how to create new channel
obtuse moth
Ok
spring turret
Suppose X1, ..., Xn are generally uniformly distributed on [0, theta] with theta> 0. Derive the maximum likelihood estimator for theta
.close
lone heartBOT
lone heartBOT
Available help channel!
rocky hearth
triple integral
lone heartBOT
triple integral
tacit arch
dk where i went wrong
gotta show the question and work if you want help
rocky hearth
gotta show the question and work if you want help
sorry internet so slow it just loaded
ik the trippe int setup is correct just solving it i made a mistake
lone heartBOT
@rocky hearth Has your question been resolved?
lone heartBOT
@rocky hearth Has your question been resolved?
craggy gulch
what does this f|_{[c,d]} mean
lone heartBOT
what does this f|_{[c,d]} mean
craggy gulch
craggy gulch
like restricting the domain to [c,d]?
grave matrix
Yeah
Essentially, the bit of f that acts on [c,d] specifically
(which has a valid meaning seeing as [c,d] is a part of [a,b], which is what the whole function f works on)
craggy gulch
i see
what does that mean for this proof though? since both [a,b] and [c,d] are arbitrary, and f is riemann integrable on the arbitrary [a,b], couldnt you just say its also riemann integrable on the arbitrary [c,d]?
grave matrix
You have to show it's Riemann integrable via the definition of Riemann integrability
i.e.
lone heartBOT
@craggy gulch Has your question been resolved?
craggy gulch
does this work?
Since $f\in \mathcal{R}([a,b])$, for every $\epsilon>0$, there exists a partition $P$ of $[a,b]$ such that $U(P,f)-L(P,f)< \epsilon$. Let $Q=P\cup{c,d}$ be a refinement of $P$. Now, since refinements can only make the difference between the lower and upper sums smaller (or stay the same), we have $U(Q,f)-L(Q,f)\leq U(P,f)-L(P,f)< \epsilon$. Now, let $R=Q\cap[c,d]$ partition $[c,d]$ so that $U(R,f|{[c,d]})- L(R,f|{[c,d]}) = \sum\limits_{j\in R}(M_j-m_j)\Delta x_j$. Since $R$ is a restriction of $Q$, it now follows that $U(R,f|{[c,d]})- L(R,f|{[c,d]})\leq U(Q,f)-L(Q,f)< \epsilon$. Thus,$f|_{[c,d]} \in \mathcal{R}([c,d])$.
ocean sealBOT
swisher
grave matrix
Yh that works afaik
lone heartBOT
@craggy gulch Has your question been resolved?
lone heartBOT
lone heartBOT
Available help channel!
wind atlas
,,2sin(2x)=-1
lone heartBOT
,,2sin(2x)=-1
ocean sealBOT
∫ cj ☠
wind atlas
Oops
wide mirage
Mb
ocean sealBOT
∫ cj ☠
wind atlas
im not sure how im suppoed to do it without taking the inverse of both sides
forest marsh
Sin(X) = sin(a)
X = a[2pi] or X = pi - a [2pi]
wind atlas
Sin(X) = sin(a)
X = a[2pi] or X = pi - a [2pi]
how did you get that
fair crypt
,,2sin(2x)=-1
if you know the graph of sin(x) i was thinking of pushing sin(30*) = 1/2 onto the "negative" section of the sine wave which comes after pi = 180 deg but it doesnt seem to work
wind atlas
if you know the graph of sin(x) i was thinking of pushing sin(30*) = 1/2 onto th...
so its not possible?
fair crypt
or maybe it does work?
oh wait
pi = 180*
30 = 180/6
we gonna have
sin(210)
sin(pi + pi/6) = sin(7/6pi)
works
perfect
fair crypt
so its not possible?
i made a mistake my brain was tripping a little
i wrote 4/3 pi instead of 7/6 pi
but calculations clearly show it 7/6 pi and it works according to desmos
so now you gonna have
wind atlas
but like how do you get those values
forest marsh
how did you get that
Rules of solving sin equations actually
wind atlas
im so confused
like how am i supposed to know if i need to take the inverse of both sides or not
fair crypt
$sin(2x) = sin(\frac{7}{6} \pi) \implies 2x = \frac{7}{6} \pi \implies x = \frac{7}{12} \pi \implies sin(2t) = - \frac{1}{2}: t \in { (\frac{7}{12} \pi + 2 \pi k): k \in \bZ }$
ocean sealBOT
Prelude to archbishop
fair crypt
might have made a mistake on the last part where i was writing out the set so correct it yourself
but the previous parts are correct
fair crypt
like how am i supposed to know if i need to take the inverse of both sides or no...
no
you don't use cyclometric function sin^-1 at all
you need a theorem
that states that sin(30 deg) = sin(pi/6) = 1/2
fair crypt
if you know the graph of sin(x) i was thinking of pushing sin(30*) = 1/2 onto th...
knowing that theorem you do this:
you push sin(30*) = 1/2 onto negative part of the sine wave by adding pi in the domain
forest marsh
**Prelude to archbishop**
The mod 2pi appear when the first implies occurs which leads to having +kpi instead of 2kpi, and you are forgetting half solutions actually, it works for 11pi/12
fair crypt
The mod 2pi appear when the first implies occurs which leads to having +kpi inst...
ye
wanted to give em a little exercise tho so i didnt correct myself
forest marsh
?
wind atlas
wait how would you do
fair crypt
wait how would you do
?
the sin(30 deg) = 1/2 ?
It's a theorem you need to know it from elementary trig where you had sin(angle) = opposite/hypotenuse
but we can go over the proof if you want i guess
wind atlas
,,sin(\frac{2x}{3})=-\frac{\sqrt{2}}{2}
fair crypt
?
ocean sealBOT
∫ cj ☠
forest marsh
So
forest marsh
we have that sin(5pi/4) = sin(7pi/4) = -sqrt(2)/2
fair crypt
because sqrt(2)/2 = 1/sqrt(2)
forest marsh
And so by the theory, sin(X) = sin(A) => X = A[2pi] or X = pi-A[2pi]
fair crypt
we have that sin(5pi/4) = sin(7pi/4) = -sqrt(2)/2
wont it be 4/3 pi ?
forest marsh
wont it be 4/3 pi ?
Sin(4pi/3) = -sqrt(3)/2
fair crypt
we have that sin(5pi/4) = sin(7pi/4) = -sqrt(2)/2
this good
forest marsh
my teacher never taught this
What did they taught you ?
fair crypt
What did they taught you ?
prob nothing productive 🤣
wind atlas
What did they taught you ?
only inverses and just like
idk she only really taught inverses and the unit circle
fair crypt
inverses?
wind atlas
yeah
fair crypt
What would that mean
forest marsh
You would do arcsin for this ?
fair crypt
Can you like show me simplest exercise that you remember that was solved using inverses metohd?
or like shortest one*
wind atlas
She said not to use arcsin if it gives it to you in sin
because then you loose a solution
fair crypt
???
well the domain of sin^-1(x) is only for angles between -90 and 90
so perhaps that could be related to that
it's easy to see why the domain of sin^-1(x) needs to be like this if you look at the graph of sin^-1(x)
wind atlas
idk what is happening
forest marsh
You can use arcsin but its not as easy as using the sinX = sinA
wind atlas
ima just ask her tmr
forest marsh
Cuz for any x
arcsin(sin(x)) = (-1)^k(x-pi*k) where k = floor(x/pi + 1/2)
So yeah
lone heartBOT
forest marsh
Yw
lone heartBOT
Available help channel!
charred hemlock
hi its an optimization task that involves geometry and derivatives:
there is a right triangle with a given angle a and a given hypotenuse c. we're supposed to draw rectangles inside of it in a way that one side is contained in the hypotenuse c. which rectangle has the largest area?
charred hemlock
I'll send a drawing and what I have so far in a minute
so it looks like this
and I've noticed several right triangles which are similar (proportions)
below I also have some equations but they're honestly kinda useless and don't lead anywhere
such as this
and yeah I just don't know where to go with it besides the triangles I noticed
I named some sides d, e, a-d, etc etc but I worry it's just giving me more unnecessary variables lol
opal jolt
you only need a single variable
charred hemlock
yeah but the issue is I don't know which one
tho today I started thinking about using the angle as a variable
opal jolt
call point P a generic point on AC
call x the distance AP
you can now draw your rectangle, and obtain its area based on x
charred hemlock
alright I'll try that and get back to you
opal jolt
ive also noticed that you have not used any of the trigonometric functions in your first try. you might wanna look into that as well
charred hemlock
yeah that's on me idk why that didn't occur to me when I was doing it a couple days ago. we had a similar-ish task with a circle explained by a teacher today and some variables were for example x=rsina where r is given so that's what gave me the idea now
however a friend of mine managed to solve it with just the pythagorean theorem (not the one I showed, it was the one with the circle) tho it was chaotic as hell but I may have been influenced by that too also I had to catch up with some schoolwork lately ig anyway excuses excuses lmao
opal jolt
i guess you can, but its gonna be ass to differentiate
lone heartBOT
@charred hemlock Has your question been resolved?
charred hemlock
I'm sorry I'm nearly falling asleep while doing this task
I'm exhausted bruh
well anyway I managed to get something ig?
that's all my sleep deprived brain can achieve rn and i NEED to go now so I'm gonna close the channel and feel free to give me a clue in dms if there's something I'm missing so I'll see it in the morning
@opal jolt and anyone else is welcome to give their input
.close
crude shadow
i ve never studied trigonometry fully we stopped at basic things in school not arctan and the like however in this physics question i think i need it, i ve gotta find the value of theta using the 2 other angles that i have
crude shadow
please just help me find it and if you have time to explain to me i d be glad
tacit arch
,rotate
ocean sealBOT
keen idol
!xy
lone heartBOT
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.
keen idol
perhaps there are more constraints given
keen idol
is this not the expression you need/
crude shadow
i cant replace it with the values yet
ocean sealBOT
mia
crude shadow
its in french
summer dirge
$\sin(2x) = 2 \sin x \cos x$ might be a useful identity for you
ocean sealBOT
higher!
keen idol
wait im trying to deduce the relation
crude shadow
okeee
keen idol
could you provide 5.1 and the context
crude shadow
this is a physics exercice
a projectile
there are different parts but let me send u the fifth one 1sec
anyways the distance D is equal to the coordinates of xc that s why i replaced D and made an equality
but yeah i gotta find theta now
keen idol
**mia**
this is the most simplified form you can get
unless you want to use
$\sin X+\sin Y=2\sin!\frac{X+Y}{2}\cos!\frac{X-Y}{2}$
ocean sealBOT
mia
keen idol
but i dont think it matters much
crude shadow
so i cant just isolate it
keen idol
do you want $\arcsin\left( \frac{\sin(x) + \sin(y)}{2} \right)$
you have that theta = this divided by 2
that is your final isolated form
crude shadow
but i dont think it matters much
that s not what my handsome(psycho) teacher would say 🫠
keen idol
whats his @ by any chance
crude shadow
he s 60 lol
keen idol
better
crude shadow
doesnt have social media
keen idol
sad
summer dirge
this is the most simplified form you can get
wait, why can't we use sin(2x) = 2sin x cos x on everything in Penelope's latest equation
crude shadow
i got his whatsapp but its 12am i dont wanna disturb him:(
summer dirge
the RHS should be nicer, no?
keen idol
wait, why can't we use sin(2x) = 2sin x cos x on everything in Penelope's latest...
you would get 2sin(theta)cos(theta) = (sin(x)+sin(y))/2
harder to isolate
summer dirge
just don't do the LHS 
keen idol
the RHS is a consequence of the derivation from the physics
it is not an identity but an equation u need to solve
summer dirge
pretty much
lone heartBOT
lone heartBOT
Available help channel!
upper rock
lone heartBOT
upper rock
not sure what to do next
tacit arch
expand and simplify
upper rock
expand and simplify <:thumbsupanimegirl:1203420374243610665>
is (rsintheta)^2 just r^2sin^2theta
summer dirge
is (rsintheta)^2 just r^2sin^2theta
yes
upper rock
oh i thought u had to distribute it to (rsintheta)(rsintheta)
upper rock
but that expression also ends up as r^2 sin^2(theta) <:thumbsupanimegirl:1203420...
what happens to the rsintheta(rsin theta )
from distributing
summer dirge
,, (r \sin \theta)(r \sin \theta) = r \sin \theta \cdot r \sin \theta = r \cdot r \cdot \sin \theta \cdot \sin \theta = r^2 \sin^2 \theta
ocean sealBOT
higher!
upper rock
ohh
i see
i thought r was separate
alright
How do I solve this integral
oh oops forgot to put the 1 in from the trig identity
lone heartBOT
@upper rock Has your question been resolved?
upper rock
@summer dirge
When I put 1/2 cos 2pi - 1/2cos0 in my calculator it comes out to 0
nvm
i put it outside
supposed to be inside
.close
lone heartBOT
lone heartBOT
Available help channel!
obtuse berry
lone heartBOT
obtuse berry
for b)
is the equation for total cost given by
C(v) = (v/400 + 12/v)(1000) + 20(1000/v)
the fuel costs for going 1000 km + wage cost for going 1000 km
tiny fable
I believe that the total amount you'd spend should be the total wages minus the cost of fuel
or i guess the other way around, oops
lone heartBOT
@obtuse berry Has your question been resolved?
alpine sable
can someone help me think i got the b wrong
lone heartBOT
can someone help me think i got the b wrong
alpine sable
lone heartBOT
@alpine sable Has your question been resolved?
alpine sable
<@&286206848099549185>
winged pawn
where are the verticies? first and foremost
and where is the centre?
then you can find the semi transverse and semi conjugate axis lengths or whatever they call it
alpine sable
i just need help with b
alpine sable
then you can find the semi transverse and semi conjugate axis lengths or whateve...
idk why my answer was marked wrong
i already know i got everything else right
so im guessing 9 is wrong
i got it thanks anyways it was 36
.close
golden sparrow
Hi can someone explain the factoring process for this (it's mathematical induction btw)
lone heartBOT
lone heartBOT
Available help channel!
carmine topaz
how do i solve this?
lone heartBOT
how do i solve this?
merry iris
how do i solve this?
get the first 3 non zero terms first?
carmine topaz
i got it lnx = (x-1) - (x-1)^2/2 + (x-1)^3/3 +....
merry iris
turn the abs into piecewise function
carmine topaz
how?
merry iris
so find all critical points of ln x - (...)
carmine topaz
x=1
merry iris
ok so when x < 1 is f(x) positive? what about when x > 1?
carmine topaz
x<1 positive x>1 negative
merry iris
x<1 positive x>1 negative
can u show ur work
merry iris
wel still show your work cuz i got smth different
carmine topaz
f(2) = 1 -4/2 + 8/3 for f(0) = -1 -1/2 - 8/3 so for x>1 its gotta be positive and x<1 its gotta be negative
lone heartBOT
@carmine topaz Has your question been resolved?
obtuse berry
lone heartBOT
hushed locust
so we want to optimize such a tank based on the cost, and we're told that the cost is based on the area (reasonable since it's a hollow tank)
so can you come up with a formula for the surface area of the tank?
obtuse berry
so can you come up with a formula for the surface area of the tank?
S(r) = 2(pi)rh + 4(pi)r^2
and h = 200/(pi)r^2 - 4/3r
obtuse berry
S(r) = 2(pi)rh + 4(pi)r^2
the surface area of the cylinder + the hemispheres
hushed locust
S(r) = 2(pi)rh + 4(pi)r^2
that seems reasonable
hushed locust
and h = 200/(pi)r^2 - 4/3r
how did you come up with this?
obtuse berry
it tells us that the area is 200
so the area of the cylinder + area of himespheres = 200
right
hushed locust
do you mean the volume?
obtuse berry
oh yea sorry
obtuse berry
do you mean the volume?
so i got
(pi)r^2(h) + 4/3(pi)r^3 = 200
hushed locust
ok that seems reasonable
obtuse berry
ok that seems reasonable
but i have a question
when i do (pi)r^2(h)
is that right
for the cylinder
hushed locust
that's the correct formula for the volume of a cylinder
obtuse berry
that's the correct formula for the volume of a cylinder
but can i plug h = 16 from here
the question says the max length is 16
hushed locust
that's the max length from end to end
hushed locust
but i would interpret that as an endpoint for your optimization
so you find the critical points and test those, but also test the endpoints
obtuse berry
so i got
(pi)r^2(h) + 4/3(pi)r^3 = 200
ok im confused what to do after this step
like after finding the surfance area formula
hushed locust
well now you can substitute your formula for h into that, so that you have one formula in terms of r
obtuse berry
well now you can substitute your formula for h into that, so that you have one f...
yea
this is what i get
hushed locust
actually if we go back for a minute, we did make an error in our problem setup
we assumed we should find the surface area, but really we are trying to find the cost function
obtuse berry
we assumed we should find the surface area, but really we are trying to find the...
but wouldnt the lowest surface area give us the lowest cost?
hushed locust
if all parts cost the same amount
obtuse berry
oh is it cuz the himsepheres are more expensive
i see
so can we introduce a new variable x and let it = the price for the cylinder part per unit area
and so 2x would be the price for the himspheres
hushed locust
that would work
hushed locust
well not really, because x is just a constant which can be factored out of the entire cost function
obtuse berry
well not really, because x is just a constant which can be factored out of the e...
can't the same be said for r?
oh
wait i see
so now we need to model a cost equation?
hushed locust
yes
hushed locust
why is it 4x on the second term?
hushed locust
yes
hushed locust
yes
obtuse berry
stable sonnet
Yo tui
hushed locust
since x doesn't change or depend on the level geometry at all, we can just focus on optimizing the function multiplied by x
obtuse berry
since x doesn't change or depend on the level geometry at all, we can just focus...
so just differentiate ?
hushed locust
well yes, that's how we'll optimize our function
obtuse berry
well yes, that's how we'll optimize our function
hmm ok
then equate to 0
and solve for r?
hushed locust
yes, that will give us our critical points
and we know the max/min values will be either at critical points or at end points
obtuse berry
yes, that will give us our critical points
but u'll get r in terms of x no?
so how would u get a number
hushed locust
you'll get (x) * (derivative of that) = 0
and since x is not 0 you just have the derivative = 0
obtuse berry
oooh
obtuse berry
and we know the max/min values will be either at critical points or at end point...
also one more thing, what would the endpoints be in this case?
the domain is from 0 to 16?
hushed locust
the smallest and largest possible values of r
the smallest r would be when it's long and thin, so we would want to find that from the max length. the biggest r would be from when it's short and fat, no cylinder at all
obtuse berry
the smallest r would be when it's long and thin, so we would want to find that f...
so the smallest r would be 0 correct?
or close to 0
hushed locust
no, it would be finite
because the tank can only be 16 m long
so that limits how thin it can get
hushed locust
relate the end-to-end length to h and r
obtuse berry
h + 2(something)
idk how to figure the something out
the height of a hemisphere?
wait that would just be radius
so h+2r?
hushed locust
yes
hushed locust
yes
hushed locust
so that's your left endpoint for r then
obtuse berry
so that's your left endpoint for r then
how would i figure out the right?
hushed locust
well the largest r would be when it's short and fat
hushed locust
yes
hushed locust
it's not the maximum length in this scenario
obtuse berry
2r>=0
hushed locust
you still have an equation relating h to r which is always true, not just for max length
obtuse berry
you still have an equation relating h to r which is always true, not just for ma...
i dont get it
so h + 2r >= something?
hushed locust
no, that was only true when considering length, which is no longer the limiting factor
obtuse berry
so just 2r>= something
hushed locust
remember you originally related h and r using volume, which must always be true
obtuse berry
right
the volume must be 200
oh wait
do i just solve for when h>=0
@hushed locust
hushed locust
wdym solve for when h >= 0?
obtuse berry
okok
so i get the domain is re[2.08, 3.63]
so then i just check C(the critical point) and C(2.08) and C(3.63)
hushed locust
yes
obtuse berry
yes
wait how do i evaluate for C(something) if i dont know x?
would i just keep the constant
and see based on whichever leading coefficient is greater
hushed locust
whichever has the highest coefficient of x, yes
obtuse berry
whichever has the highest coefficient of x, yes
okay thank u so much
i really appreciate u walking me thru the steps
lone heartBOT
@obtuse berry Has your question been resolved?
sleek briar
Help
sleek briar
Oopsies also part 1 here
sleek briar
So far here are what are of my understanding
(a) and (b)
Could you also check these just in case they are wrong
I have no idea how to do C or D?
cunning coyote
(a) and (b)
Could you also check these just in case they are wrong
(a) looks good
sleek briar
No mistakes?
cunning coyote
this is an interesting problem
cunning coyote
No mistakes?
i didnt spot any
sleek briar
what about B?
cunning coyote
looks good too
sleek briar
can you help me with C and D?
merry iris
can you help me with C and D?
draw CDCCC once, then apply it again, possibly in a different colour
what do you notice
