How do I prove that the mapping V -> (V^top, O_V), f -> f define an isomorphism between the category of prevarieties over k and the category of quasi-compact functionally structured spaces which are locally isomorphic to locally-closed subspaces of affine spaces.
#help-0
rigid pine
lone heartBOT
How do I prove that the mapping `V -> (V^top, O_V), f -> f` define an isomorphis...
@rigid pine Has your question been resolved?
mortal trellis
above the lvl of help channels. try the advanced channel for category theory
knotty thunder
lone heartBOT
knotty thunder
my reasoning
unique = {I, T, E, R, A, V}
6C2 = 15
is this correct or am i clueless
lone heartBOT
@knotty thunder Has your question been resolved?
prime badge
yes
knotty thunder
yes
how about these ( still using ITERATIVE )
my reasoning for Q2.
6C2 + 3
my reasoning for Q3.
6P2 + 3
prime badge
sure
lone heartBOT
@knotty thunder Has your question been resolved?
violet ore
How do they know the partial derivative of f with respect to x at (2, 2) is -2
lone heartBOT
@violet ore Has your question been resolved?
spare canopy
How do i approach this?
lone heartBOT
lone heartBOT
Available help channel!
barren pelican
hello people of maths! my name is Utsudere but you can call me Adrian. I have a math inquiry for you all! you see I am developing a little project where I wish to make an accurate simulation of the ocean. I understand most of the maths behind it except one thing. Integrals. I have been attempting to figure out what these are and how I would go about calculating them. I understand that its something to do with an antiderivative but I don't know how to calculate an "antiderivative" either so I've been at this for a little while now and I'm hoping someone could teach me! please and thanks. (here is an example of how the integral will be used)
mellow grail
hello people of maths! my name is Utsudere but you can call me Adrian. I have a ...
Well actually many times its really touch to integrate something and find a closed form, i.e. something that doesnt involve integrals. In some cases, its basically impossible.
So if you want to make a simulation you are better off numerically computing an integral, through something called riemann sums
lone heartBOT
mellow grail
riiiiight ok ill check that out then thanks!
Btw i would first check for a closed through wolfarm alpha by finding the improper integral (the antiderivative) of g(t) * e^(-2pi * i * f * t)
barren pelican
Btw i would first check for a closed through wolfarm alpha by finding the improp...
🫡
lone heartBOT
Available help channel!
summer ermine
Hey guys,
I am currently struggling with the following problem:
I have the Central matrix H = I_n - 1/n 1_n 1_n^T
I need to figure out the eigenvalues of H
I have already proved in a previous exercise that 0 is an eigenvalue of H
What I have already done is calculating the trace tr(H) = n - 1 and I now that the sum of the eigenvalues is equal to the trace, so that i know
λ1 + ... + λn-1 = n - 1
I am stuck here. How can I show that all these eigenvalues are equal to 1? I have no clue...
Thanks in advance ❤️
mortal trellis
you could try finding the eigenvectors
summer ermine
you could try finding the eigenvectors
but how would I do that in this case?
<@&286206848099549185>
lone heartBOT
@summer ermine Has your question been resolved?
mortal trellis
you want that Hv=v if v is one of those eigenvectors
you have Hv=Iv +1/n 1_n 1_n^T v
compare
what does v need to satisfy
mortal trellis
or the sum of the components of v should be 0?
yes
languid prawn
1+3+5...+(2n-1)
What is the phrase and explain it please?
mortal trellis
or if you know this terminology, v needs to be in the orthogonal complement of 1_n
languid prawn
What is the phrase and explain it please?
...
summer ermine
or if you know this terminology, v needs to be in the orthogonal complement of 1...
but how does that help me
summer ermine
are you sure?
no i believe it's n^2
summer ermine
i have now 12m to find it guys
https://www.cuemath.com/sum-of-arithmetic-sequence-formula/ check this out
mortal trellis
now you know that every vector in that space is an eigenvector of H
now you can try finding the dimension of that space. or in other words, linearly independent eigenvectors
wet vector
hi i need some help
lone heartBOT
hi i need some help
wet vector
how do i compute e^-sA of this matrix exponential?
mortal trellis
plug the matrix into the series
mortal trellis
how did you define the matrix exponential
wet vector
how did you define the matrix exponential
would finding the inverse of the matrix above
also work?
since we know that e^-tA = (e^tA)^-1
mortal trellis
you messed up some signs there
either way you need to calculate e^some matrix
at some point
lone heartBOT
@wet vector Has your question been resolved?
ashen trail
lone heartBOT
lofty gorge
you apply chain rule to the whole of (y_u + y_v)
which gives the second equality
(I'm using subscript for partial derivative wrt whatever because I cannot be bothered typing the 'fraction' out)
lofty gorge
chain rule says that $w_x = w_uu_x + w_vv_x$ (in 'fraction' form below), and here we have $w=y_x=y_u+y_v$
\begin{align*}{\partial \over \partial x}(w) &= {\partial w\over \partial u}{\partial u\over \partial x}+{\partial w\over \partial v}{\partial v \over \partial x} \&= {\partial \over \partial u}\left({\partial y\over \partial u}+{\partial y\over \partial v}\right){\partial u\over \partial x}+{\partial \over \partial v}\left({\partial y\over \partial u}+{\partial y\over \partial v}\right){\partial v \over \partial x}\end{align*}
ocean sealBOT
Edward II
lofty gorge
(small note partial derivatives are not fractions but I have no idea how else to call this notation, other than Leibniz notation maybe?)
first equality is the chain rule in general, second one is substituting for w
lone heartBOT
@ashen trail Has your question been resolved?
spare canopy
can anyone help me with this?
lone heartBOT
can anyone help me with this?
lone heartBOT
@spare canopy Has your question been resolved?
spare canopy
.close
lone heartBOT
raw fractal
.close
river bloom
hello! this is more of a physics problem but i keep getting imaginary numbers for 11b - can anyone figure out where i'm going wrong?
snow cobalt
hello! this is more of a physics problem but i keep getting imaginary numbers fo...
work done by the spring is always negative
because the spring (here) is applying force outwards and your object is going inwards
so -1/2 * k * x^2
I suppose that's where the error is
raw fractal
the spring removes energy from the block rather than adding to it, so yeah
hardy plover
Prove the following statements:
(a)
(b) (c)
Let (R, +, ·) be a ring without zero divisors, i.e., (R, +, ·) is a ring and for all a, b ∈ R: a·b=0 ⇒ (a=0 ∨ b=0). (1)
Then the cancellation law holds, i.e.,
∀a, b, c ∈ R: (ab=ac ∧ a ≠ 0) ⇒ b=c.
Let (K, +, ·) be a field. Then, for all a ∈ K: 0·a=a·0=0. Every field is without zero divisors, thus satisfying (1)."
lone heartBOT
Prove the following statements:
(a)
(b) (c)
Let (R, +, ·) be a ring without zero...
@hardy plover Has your question been resolved?
lone heartBOT
@hardy plover Has your question been resolved?
celest shell
Prove the following statements:
(a)
(b) (c)
Let (R, +, ·) be a ring without zero...
Want short form or full explanation
hardy plover
Full explanation would be great!
celest shell
Ok wait I’m going to get my book, to refresh my memory. I haven’t don’t this in a while.
(a) To prove the cancellation law, suppose that ab = ac and a ≠ 0. We want to show that b = c.
Since a ≠ 0, we know that a has a multiplicative inverse in the ring R. Let's denote this inverse as a^(-1). Then we can multiply both sides of the equation ab = ac by a^(-1) to get:
(a^(-1))(ab) = (a^(-1))(ac)
Using the associative property of multiplication, we can rearrange this equation as:
((a^(-1))a)b = ((a^(-1))a)c
Since (a^(-1))a = 1 (the multiplicative identity), we have:
1b = 1c
Which simplifies to:
b = c
Therefore, we have proven that if ab = ac and a ≠ 0, then b = c, which is the cancellation law.
(b) To prove that 0·a = a·0 = 0 for all a ∈ K, where K is a field, we can use the properties of a field.
By definition, a field is a commutative ring with a multiplicative identity where every nonzero element has a multiplicative inverse.
First, let's prove that 0·a = 0 for all a ∈ K.
Since 0 is the additive identity element in K, we have:
0 + 0 = 0
Multiplying both sides of this equation by a, we get:
(a)(0 + 0) = (a)(0)
Using the distributive property, we can simplify the left side of the equation:
(a)(0) + (a)(0) = (a)(0)
Since (a)(0) + (a)(0) = 2(a)(0), we can rewrite the equation as:
2(a)(0) = (a)(0)
Now, using the cancellation law from part (a), we can cancel (a)(0) from both sides of the equation, since a ≠ 0:
2 = 1
This is a contradiction, since 2 ≠ 1. Therefore, (a)(0) must equal 0.
Next, let's prove that a·0 = 0 for all a ∈ K.
Using the commutative property of multiplication, we can rewrite this as 0·a = 0.
We have already proven this in the previous step, so we can conclude that 0·a =
devout egret
raven pike
Question D doesn't make much sense to me, what is it exactly asking me?
lone heartBOT
@raven pike Has your question been resolved?
raven pike
.close
cobalt needle
I dont know how to start this one
lone heartBOT
I dont know how to start this one
cobalt needle
also what an honour being in help channel 0!
I dont want a solution I just want to know where to approach this question from and maybe the first step if I dont understand
ocean hawk
if (x - 4) is a factor, then what do you know happens if you divide f(x) by that factor
cobalt needle
if (x - 4) is a factor, then what do you know happens if you divide f(x) by that...
i tried that but it was super messy and didnt seem to help but is that what i was supposed to do
i got $x^2+px+3x-4p-6$
ocean sealBOT
sealpup321
ocean hawk
alternatively, it tells you that f(4) = 0 right
cobalt needle
aka super messy
cobalt needle
alternatively, it tells you that f(4) = 0 right
ahhhh
thanks ill try that 😄
yup that worked thanks i was trying that for ages @ocean hawk 😄
.close
magic tiger
lone heartBOT
magic tiger
dont really get how to go from max point of one to another
lone heartBOT
@magic tiger Has your question been resolved?
tacit arch
dont really get how to go from max point of one to another
,tex .transformation rules
ocean sealBOT
riemann
granite storm
I need help with how to solve probablitys
lone heartBOT
I need help with how to solve probablitys
dire tiger
User
Hank needs to compare two expressions 1/2 (10x+6) and -(3x+10)+8(x+1)He needs to know if one expression is greater than the other for all values of x. Which statement accurately describes the relationship between the two expressions?
granite storm
.close
zenith fog
I got a quiz on this in a couple mins lmao
lone heartBOT
I got a quiz on this in a couple mins lmao
zenith fog
I just need to know that is the speed at ii different from the speed at the top of the hill(right before it goes on the circle)
<@&286206848099549185>
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@zenith fog Has your question been resolved?
wet vector
i need some help with this ode system
my idea is to reduce it down to a first order system
but need some help
i was thinking let u = x', and v = y'?
limpid turret
You don't really need u=x' imo
limpid turret
you don't need it because there is no second order derivative for x
you keep v=y' tho
wet vector
you don't need it because there is no second order derivative for x
i see
wet vector
you keep v=y' tho
ok
wet vector
you keep v=y' tho
so just v = y'?
alpine sable
lone heartBOT
alpine sable
What am I doing wrong? Do I not understand the question
tacit arch
What am I doing wrong? Do I not understand the question
Doesn't look like you solved for possible values of k
e.g, k=0 is one possible value
alpine sable
Doesn't look like you solved for possible values of k
I don’t even know
alpine sable
e.g, k=0 is one possible value
What do I write then
It’s for a range of possible values
tacit arch
Plot some circles for different k
alpine sable
Wym?
tacit arch
Do you know how to plot circles?
Can you answer this
tacit arch
Plot some circles for different k
Then do this
Along with your line
And see which values intersect and which don't
Then you'll get a clue for what the algebra should be
lone heartBOT
@alpine sable Has your question been resolved?
thorn beacon
Could someone check my answers please
lone heartBOT
Could someone check my answers please
coral moss
Could someone check my answers please
just use wolfram alpha
rocky aspen
rocky aspen
what I though I closed themmmm
ocean whale
what I though I closed themmmm
Use .close to close the channels
But please don't keep opening and closing channels if your question does not get answered, just wait patiently for help
thorn beacon
140+125
rocky aspen
thank you very much William have a wonderful life
lone heartBOT
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faint monolith
A gourmet coffee shop has two different types of coffee. Batch A costs $24/kg and Batch B costs $32/kg. Anna purchased a 3-kg mixture of the batches and was charged $83.20. What mass of each batch was purchased?
faint monolith
I did this:
24x + 32(3-x) = 3 * 83.2
but it doesnt work bc i get a negative number as x
in the solutions they did 24x + 32(3-x) = 83.2 (Without the 3)
why did they not multiply it by 3
faint monolith
but i multiplied the cost by x (the mass of the coffee)
so shouldnt itb e equal to the mass multiplied by the cost
because if its just 83.2 it would just be 24a + 32b = 83x2
placid zinc
"it"?
So let's take a look at the left. You are multiplying ($/kg)(kg) to produce a cost
The right should be a cost as well
faint monolith
I assumed it was similar to this question: Chocolate sultanas cost $12/kg and chocolate peanuts cost $16/kg. How many kg of each item should be used by a confectionery distrubutor if he wants to make a 20kg mix which will cost him $15/kg
what is the difference
bc for this questino i did =20*15
placid zinc
The difference is the unit of the cost value
In your original question, the batch should cost $83.20. Doesn't matter that it's 3kg, the entire batch costs $83.20
In this new question, the batch costs $15/kg. The weight matters.
You're working with a slightly different unit for each question
faint monolith
oh so 20 * 15 is the total cost because it says it costs 15/kg and since its 20 kg thats the total
and the other one gives u the total straight away
.close
magic snow
lone heartBOT
golden chasm
How can I find the linearization of f at x=0 if f'(x)=cos(x^2) without knowledge of integrals?
naive valley
well you don't really have enough info to answer fully
unless there's info that you did not include above
do you know anything about f(x)?
particularly f(0)
junior vigil
How can I find the linearization of f at x=0 if f'(x)=cos(x^2) without knowledge...
linearization is just the linear approximation (namely, at x=0).
golden chasm
yeah
golden chasm
particularly f(0)
f(0)=1
naive valley
ah good, that helps
junior vigil
this is just applying the fact that the derivative is the gradient of the function at every point, and so in a small interval around x=0, this linearization will approximate the function relatively well.
golden chasm
awesome
naive valley
so it'd be y-f(0)=f'(0)(x-0)
y-f(0)
summer quarry
Given a for fuction f(x) =
when x < 1
When x ≥ 1
Continuous at x =1
lone heartBOT
Given a for fuction f(x) =
when x < 1
When x ≥ 1
Continuous at x =1
@summer quarry Has your question been resolved?
forest arrow
What is the LCM of 24 and 18
lone heartBOT
What is the LCM of 24 and 18
forest arrow
What is the LCM of 24 and 18
foggy pecan
What is the LCM of 24 and 18
$24=2\cdot 2\cdot 2\cdot 3\text{ }\text{ and }\text{ }18=2\cdot 3\cdot 3$
ocean sealBOT
Joanna Angel
foggy pecan
so ?
foggy glacier
for each prime you take the max power so here it's $2^3x3^2$
ocean sealBOT
moi
foggy glacier
so 72
lone heartBOT
@forest arrow Has your question been resolved?
silent pewter
can someone explain to me how this was simplified?
lone heartBOT
can someone explain to me how this was simplified?
ionic jewel
im wrong
silent pewter
can you elaborate pls
ionic jewel
$\frac{-1.75(9/\sqrt{n}) + 30 - 28}{(9/\sqrt{n})}$
ocean sealBOT
ionic jewel
$=\frac{-1.75(9/\sqrt{n}) +2}{(9/\sqrt{n})}$
ocean sealBOT
ionic jewel
$=\frac{-1.75(9/\sqrt{n})}{(9/\sqrt{n})} + \frac{2}{(9/\sqrt{n})}$
ocean sealBOT
ionic jewel
$=-1.75 + \frac{2}{(9/\sqrt{n})}$
ocean sealBOT
ionic jewel
$=-1.75 + \frac{2\sqrt{n}}{9}$
ocean sealBOT
agile cloak
I have this question and im just looking for some insight into how I would even go about starting part i)?
tacit arch
!15m
lone heartBOT
Please only use the <@&286206848099549185> ping once if your question has not been answered for 15 minutes. Please do not ping or DM individual users about your question.
agile cloak
<@&286206848099549185>
lone heartBOT
@agile cloak Has your question been resolved?
obtuse folio
lone heartBOT
obtuse folio
not sure how a graph of R composite identity relation on A would look like??
the picture i drew is what i think it is..
although, isn’t the circled arrow describing reflexive?
redrew it
should be right
warm tree
v(x)=u(Mx) where M is rotational matrix and u is a harmonic function. we write this to prove that laplace operator is invariant. But my point is since u is harmonic in Rn and any rotation to vector will be inside Rn only so as lap(u) = 0 regardless of what we put in place of x in Rn what is the point in proving it is invariant?
lone heartBOT
v(x)=u(Mx) where M is rotational matrix and u is a harmonic function. we write t...
@warm tree Has your question been resolved?
warm tree
No
lament mantle
What's extrapolate and interpolate?
lone heartBOT
What's extrapolate and interpolate?
fallen verge
lone heartBOT
lone heartBOT
Available help channel!
summer quarry
Given the ABCD tetrahedron. Call I, J, K in turn is the midpoint of the edges AC, CD, BD.
a) find the intersection of two planes (ABC) and (AKJ).
B) Find E is the intersection of the BI and plane line (AKJ).
C) find G is the intersection of the line Ek and the plane (ACD).
d) prove that AB / / (CDE).
ocean sealBOT
summer quarry
Solved the a)
The line d passes through common point A of 2 planes and parallel to 2 edges KJ, BC
Stuck with b and c
<@&286206848099549185>
lone heartBOT
@summer quarry Has your question been resolved?
summer quarry
primal dagger
i won't pretend like i know how to answer this question but i would like to say that your lines are very neat
sorry i can't be more helpful 😅
summer quarry
i won't pretend like i know how to answer this question but i would like to say ...
Ohhh 😆
summer quarry
Can BI cut through d like this? Idk if they are on the same surface
d is the intersection of (AJK) and (ABC) so d would be on the same surface with both of these
And BI is in (ABC)
So
BI would be able to cut d?
It doesn't feel so right idk
lone heartBOT
@summer quarry Has your question been resolved?
alpine sable
lone heartBOT
alpine sable
I dont understand how they got to that
lone heartBOT
@alpine sable Has your question been resolved?
meager shadow
I used Matric A and C, multiplied each row with each column and then found the sum, but I still got it wrong. I'm not sure why.
glad echo
I used Matric A and C, multiplied each row with each column and then found the s...
Tbh you are right about multiplying A and C cause 3x2 matrix * 2x4 matrix is indeed possible, matrix calculators give me a different answer also so i don't think it's on you buddy. Any possibility the answer is just wrong?
meager shadow
Tbh you are right about multiplying A and C cause 3x2 matrix * 2x4 matrix is ind...
it might be a format problem?
thats how they want the answer
glad echo
oh wait im sorry i thought the box was the final answer
that is indeed not the right answer
is that what you typed yourself?
meager shadow
i typed this in
glad echo
oooh
alright
row 1
first element = (-7 * 9) + (-4 * 5) = -83
second element = (-7 * 3) + (-4 * -2) = -13
third element = (-7 * 3) + (-4 * -2) = -13 (which is already different)
fourth element = (-7 * 6) + (-4 * -4) = -26
row 2
glad echo
first element = (10 x 9) + (-2 x 5) = 80
second element = (10 x 3) + (-2 x -2) = 34
third element = 34 aswell (also different
yup yup
same for the 47
its a 7 aswell
but ye you got it
shit happens 😂
meager shadow
i realized my mistake
i made
the -2 in matrix c (third column) positive in my work for some reason
okay sweet
thank you so much
glad echo
the -2 in matrix c (third column) positive in my work for some reason
oh dangg
no problem !
you can close this chat* with .close 🙃
meager shadow
.close
thin flicker
lone heartBOT
thin flicker
Can anyone help me with this? The work shown is wrong I believe
<@&286206848099549185>
glad echo
did you write the ³ yourself?
glad echo
oh ye you cant do that because its the same as ( entire equation )³
thin flicker
Yea
glad echo
which means you just multiplied it by itself, which is not allowed randomly yk
thin flicker
Yes I realized that
thin flicker
this is what i meant lol
Do we do that to the top and bottom
glad echo
yes to both the 3 root thingy
i do math in my native language so im not sure what the correct name is
thin flicker
That’s okay
glad echo
ngl i feel like youre like there already?
cause like simplify to me, as a non native speaker, just means to get the simplest form?
or i mean you could
maybe
thin flicker
This is not the simplest form.
glad echo
any english native speakers that can comment on what "simplify" exactly means?
thin flicker
You know I gotta go
glad echo
normally you'd find what 3^(1/3) / 7^(1/3) but it has a lot of decimals
so thats kinda difficult haha
ah damn alr
have a great night !
thin flicker
Can’t do nothin with nothing
glad echo
mind doing .close?
thin flicker
.close
rugged dove
A grazing paddock is to be fenced on all sides. It is rectangular in shape, with a length of 244 m and a
breadth of 178 m. If fencing costs $20 per metre, find the cost of fencing required. (2 marks)
lone heartBOT
rugged dove
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lone heartBOT
Available help channel!
proven tendon
lone heartBOT
proven tendon
in the second last line
how did it turn from two integerals added together
to ln|1+u| - ln|1-u|
where did the minus come from
shouldnt it be adition
tall topaz
[\wrb{\int \f{f’(x)}{f(x)};\dd x = \ln\abs{f(x)}+C}]
ocean sealBOT
Pure
tall topaz
You basically need to use ‘reverse chain rule’
cobalt needle
I have done part a and gotten 10 however part b has me confused
lament wyvern
I have done part a and gotten 10 however part b has me confused
lemme see
cobalt needle
Is it just rearranging or is there something I am missing?
lament wyvern
it's almost certainly just rearranging
cobalt needle
Strange I guess I’m just terrible at rearranging then XD
Thanks I’ll give it a few more goes and let you know if I get it now 🙂
lament wyvern
yeah if you expand that bracket you'll get a polynomial in p^2
and then you want to complete the square or whatever the kids are calling it nowadays to get (p-C)^2 + some constant factor
cobalt needle
and then you want to complete the square or whatever the kids are calling it now...
Ah cool thanks I haven’t tried that one yet
lament wyvern
the other way is to work backwards; if you expand the A-B(p-C)^2 mess, you'll have a weird polynomial in p with A+B=stuff and BC=stuff and so on, and you can work out the values that way
cobalt needle
the other way is to work backwards; if you expand the A-B(p-C)^2 mess, you'll ha...
Ah that’s smart that way seems more reliable to me
lament wyvern
Which is basically what completing the square does anyway
cobalt needle
Btw in the completing the square part why is it 13225 and not -13225?
lament wyvern
- * - = +
cobalt needle
But it’s always negative in completing the square ;-;
Anyway thanks for your help 😄
.close
alpine sable
how do i calculate the mass of a goldfish?
zealous lichen
just measure it
prime badge
google denisty of human body and multiply by volume of goldfish
you would have to guess the volume
lone heartBOT
@alpine sable Has your question been resolved?
buoyant thunder
lone heartBOT
@buoyant thunder Has your question been resolved?
white rampart
I can’t see
What is ur question?
buoyant thunder
What is ur question?
what is gradient
white rampart
what is gradient
If ur question is what m is, take the triangle from (-5,-3) to (0,1)
What’s the difference in x and what’s the difference in y here?
That’s what u need to find m
final viper
couldn't reopen so opening again
lone heartBOT
couldn't reopen so opening again
final viper
@uneven wave yeah so to answer your question , becacuse i want to introduce gamma function
so i need the e^t thing
@alpine sable i tried feyman technique but that does not help
uneven wave
yeah, i guess i don't know what iota is
final viper
it just produces cosx/x^{p+1}
final viper
yeah, i guess i don't know what iota is
square root of -1
uneven wave
i mean yeah i call that i so that's why i was confused
I'm trying to solve it from the right side first
final viper
i mean yeah i call that `i` so that's why i was confused
ah
final viper
I'm trying to solve it from the right side first
wow
i wonder how the gamma came in the denominator
chat gpt says that
which lowkey makes sense
the complex infinity
uneven wave
Please for the love of god don't trust chatgpt
final viper
uneven wave
it says a whole lot of bullshit
final viper
Please for the love of god don't trust chatgpt
i know
i had to refine the answer like 4 times
lmao
lone heartBOT
@final viper Has your question been resolved?
versed moss
Chain rule expression. I’m stuck on (ii) and (iii) as I’ve gotten different answers and I can’t verify them . For (ii) I’m left with x and y variables and for (iii) I’m left with t variables as it asks me to express f directly in terms of t. I don’t understand how I can check my answers. I think I’m missing a step or I’m doing a step incorrectly
arctic lintel
wait can you tell me what f(x(t),y(t)) is?
upbeat hornet
wait can you tell me what f(x(t),y(t)) is?
That’s f(x,y)
x and y are functions of t
upbeat hornet
Chain rule expression. I’m stuck on (ii) and (iii) as I’ve gotten different answ...
From the question, x = t^2 and y = t^3 - t
arctic lintel
yeah i was asking for the expression with x(t) and y(t) subbed in
foggy pecan
Chain rule expression. I’m stuck on (ii) and (iii) as I’ve gotten different answ...
a hint for you:
$\frac{\partial
f}{\partial
t}=\frac{\partial
f}{\partial
x}\cdot \frac{dx}{dt}+\frac{\partial
f}{\partial
y}\cdot \frac{dy}{dt}$
ocean sealBOT
Joanna Angel
versed moss
Yea I got that
Sure lemme
Give me a min
This is what I got for df/dt
For part (ii) I subbed in t=-1
Into that but idk if that’s right
foggy pecan
i give you mine and you can compare:
$\frac{\partial
f}{\partial
t}=\frac{\partial
f}{\partial
x}\cdot \frac{dx}{dt}+\frac{\partial
f}{\partial
y}\cdot \frac{dy}{dt}=\=\left( 6xy-5y^{4} \right)\cdot 2t+\left( 3x^{2}-20xy^{3} \right)\cdot \left( 3t^{2}-1 \right)=\=$
ocean sealBOT
Joanna Angel
foggy pecan
when you put the words in order, which unfortunately takes time, put the appropriate functions x(t) and y(t) in place of x and y
versed moss
Oh so for the answer I got in (i) I put the functions of x(t) and y(t) in place of x and y and then I replace all the t’s for -1?
foggy pecan
correct
i) can be even left with parenthesis, but in iii) they want to get final form
versed moss
I see
foggy pecan
the most important is iii) and then you can compue ii) that is main meaning of yoru task
versed moss
So in the end both (ii) and (iii) should have the same answer?
Oh
So I can do (iii) first ?
upbeat hornet
Yes
upbeat hornet
So in the end both (ii) and (iii) should have the same answer?
They shouldn’t have the same answer, you’re evaluating it at a special case in part (ii), and doing the general case in part (iii)
and then in part (iii) you use the general case to find it at t = -1
versed moss
So for both I have to replace the x and y with the functions x(t) and y(t) and change the t to -1. If I’m understanding correctly?
upbeat hornet
So for both I have to replace the x and y with the functions x(t) and y(t) and c...
yes, while in part (i) you substituted t = -1 first
harsh condor
some questions regarding parametric curves
lone heartBOT
some questions regarding parametric curves
harsh condor
slope of parametric curve given by : m = dy/dt / dx/dt
is the operation dy/dt / dx/dt a regular algebraic one or does it imply the use of the quotient rule
(i have already solved for the derivatives dy/dt and dx/dt)
upbeat hornet
is the operation ``dy/dt / dx/dt`` a regular algebraic one or does it imply the ...
What do you mean by regular algebraic one
upbeat hornet
is the operation ``dy/dt / dx/dt`` a regular algebraic one or does it imply the ...
You take the quantities dy/dt and dx/dt and divide them
upbeat hornet
Yes
harsh condor
okay cool
upbeat hornet
(i have already solved for the derivatives dy/dt and dx/dt)
What are the derivatives?
harsh condor
dy/dt = 2-2t^(2) / 1+2t^(2)+t^(3) dx/dt = -4t / 1+2t^(2)+t^(3)
upbeat hornet
dy/dt = ``2-2t^(2) / 1+2t^(2)+t^(3)`` dx/dt = ``-4t / 1+2t^(2)+t^(3)``
You just need to divide them as rational functions
dim matrix
How can I solve it further?
lone heartBOT
How can I solve it further?
dim matrix
upbeat hornet
How can I solve it further?
what is the question?
upbeat hornet
you can collect all like terms and complete the square, since it’s a quadratic
dim matrix
I want to write the whole equation precisely so that I can get the value of x in f(x) when I assume any value of x. @upbeat hornet
lone heartBOT
@dim matrix Has your question been resolved?
upbeat hornet
Then collect like terms in the second picture you sent and complete the square, so that it gets simplified
dim matrix
Then collect like terms in the second picture you sent and complete the square, ...
I tried, but seems incorrect.
upbeat hornet
I tried, but seems incorrect.
Where did you get the function from?
dim matrix
Where did you get the function from?
Lagrange's interpolation
hushed shell
Need help
lament wyvern
!da2a
lone heartBOT
No need to ask “Can I ask…?” or “Does anyone know about…?”—it’s faster for everyone if you just ask your question! See https://dontasktoask.com/
hushed shell
How would the equation be?
hushed shell
This is mid segment find the value of x
rare granite
wouldn't that be the value of x, 64?
since the angle is the same and the x-axis doubles, the y-axis doubles aswell
so x/2=32
if you want to do it backwardss
hushed shell
Wouldnt it be 32 because it like the same angle but it's bigger?
rare granite
well
x is the length
and x and 32 aren't as long
and since we go out 25*2 on the x-axis, we do it on the y-axis aswell
rare granite
yes
lone heartBOT
@hushed shell Has your question been resolved?
south cliff
i need help
lone heartBOT
i need help
south cliff
1234 -> [4,2,1]
1243 -> [4,2,0] -> replace last with 1 before last
1324 -> [4,1,1] -> replace last with 1 before last, replace 1 before last with 2 before last
1342 -> [4,1,0] -> replace last with 1 before last
1423 -> [4,0,1] -> replace last with 1 before last, replace 1 before last with 2 before last
1432 -> [4,0,0] -> replace last with 1 before last
2134 -> [3,2,1] -> ?
trying to implement permutations in this way
would appreciate any help
lone heartBOT
@south cliff Has your question been resolved?
south cliff
no
alpine sable
Hi Brian
hushed shell
Hi
lone heartBOT
@south cliff Has your question been resolved?
lone heartBOT
lone heartBOT
Available help channel!
hollow hill
If anyone is familiar with Formal Logic help would be appreciated.
This is an mock-exam preparation exercise, and I would like assistance in solving this or an example.
Construct and write an argument in natural language, which would contain 5 distinct simple propositions.
Formalize it in propositional logic (constants meaning required). Prove that the argument is valid by natural deduction.
The proof should contain no less than 5 different rules of inference or equivalence.
(1) I'm unsure of how to construct an argument in natural
All humans require oxygen to survive (Premise 1). John is a human (Premise 2), and oxygen is essential for cellular respiration (Premise 3). If someone lacks oxygen, they will experience respiratory distress (Premise 4). Therefore, John needs oxygen to sustain his life (Conclusion)
Premise 1: All humans require oxygen to survive.
Premise 2: John is a human.
Premise 3: Oxygen is essential for cellular respiration.
Premise 4: If someone lacks oxygen, they will experience respiratory distress.
Conclusion: Therefore, John needs oxygen to sustain his life.
(2)
lone heartBOT
If anyone is familiar with Formal Logic help would be appreciated.
This is an mo...
@hollow hill Has your question been resolved?
lone heartBOT
@hollow hill Has your question been resolved?
south cliff
can you help me to implement:
1234 -> [4,2,1]
1243 -> [4,2,0] -> replace 1 before last with last
1324 -> [4,1,1] -> replace 1 before last with last, replace 2 before last with 1 before last
1342 -> [4,1,0] -> replace 1 before last with last
1423 -> [4,0,1] -> replace 1 before last with last, replace 2 before last with 1 before last
1432 -> [4,0,0] -> replace 1 before last with last
2134 -> [3,2,1] -> replace 1 before last with last, replace 2 before last with 1 before last, replace 1 before last with last, replace 3 before last with 2 before last
permutations of 1234
raven nebula
I know how to do the unit, but the expression confuses me, and i know how to find slope and y-intercept but the graphing and interval confuses me
lone heartBOT
I know how to do the unit, but the expression confuses me, and i know how to fin...
@raven nebula Has your question been resolved?
Available help channel!
delicate karma
lone heartBOT
delicate karma
is this
for a) False, True, False
and for b)ii. when i do my orthogonal projection and then find v - proj_{W} v
i then just do the same approach
or is there a shortcut there
K 💀
?
r u trolling
do u mean add them 💀
do u know linear algebra g
tacit arch
@alpine sable don't flood other people's channels with your question
lone heartBOT
@delicate karma Has your question been resolved?
lone heartBOT
@delicate karma Has your question been resolved?
frigid terrace
How can I prove that 10! + 1 is divisible by 11 using modulo congruence?
frigid terrace
I have this image but I do not understand why did it go with 1 * 10 and then 2 * 6 and so on
.close
lavish shell
lone heartBOT
lavish shell
just want some examples to these if possible
lavish shell
x=root d
limpid turret
You're forgetting a solution
lavish shell
negative root d?
limpid turret
yup
lavish shell
so what would apply to B
limpid turret
For what $d$ does $+\sqrt{d}=-\sqrt{d}$?
ocean sealBOT
SWR
lavish shell
real numbers?
limpid turret
All of them? 🤔
lavish shell
natural numbers then?
lavish shell
0 is it?
limpid turret
yes
lavish shell
and im guessing C is negative numbers?
limpid turret
you would be guessing correctly
alpine sable
how do i differentiate tanhx
lone heartBOT
how do i differentiate tanhx
runic mist
sec^2x is the derivative of tan
do you mean for there to be an h, if so is it a constant or a variable
alpine sable
do you mean for there to be an h, if so is it a constant or a variable
isnt tanh x = sinhx / coshx
im looking for tanhx not tanx
runic mist
ok I don't actually know what that is then lol
alpine sable
ok I don't actually know what that is then lol
oh its fine 🙂
thanks for trying to help
junior vigil
how do i differentiate tanhx
quotient rule with sinh(x) and cosh(x)?
derivatives of sinh(x) and cosh(x) are both respectively nice (each other! exactly!)
alpine sable
oh yhhh
junior vigil
$\sinh(x)=\frac{e^{x}-e^{-x}}{2}$
ocean sealBOT
Moosey
junior vigil
$\cosh(x)=\frac{e^{x}+e^{-x}}{2}$
ocean sealBOT
Moosey
junior vigil
you can verify that $\frac{d}{dx}\left(\cosh(x)\right)=\sinh(x)$ and $\frac{d}{dx}\left(\sinh(x)\right)=\cosh(x)$
ocean sealBOT
Moosey
junior vigil
and you also have an identity with cosh^2(x)-sinh^2(x)
you can verify this using definition of cosh and sinh but
$(\cosh(x))^{2}-(\sinh(x))^{2}=1$
ocean sealBOT
Moosey
junior vigil
so derivative of tanh(x) is just (sech(x))^2 :)
junior vigil
nice parallel to derivative of tan(x) being sec^2(x) yes
alpine sable
nice parallel to derivative of tan(x) being sec^2(x) yes
😉
junior vigil
you can also derive it with complex identity, isinh(x)=sin(ix), cosh(x)=cos(ix)
(this is where the parallel with trig functions comes from :) )
lone heartBOT
Available help channel!
untold gulch
i need help calculating this integral
lone heartBOT
i need help calculating this integral
sour verge
What is the range of the radius r on the region that is shown?
Say, what is the smallest r value you could have on the annulus?
sour verge
An annulus is just a shape. Kind of like a flat donut, like the shaded picture you have.
untold gulch
got it
sour verge
but i think the smallest r = 1
Indeed. the smallest is r=1.
Then what is the largest r will be?
untold gulch
2
untold gulch
from pi/2 to 0?
sour verge
That sort of works, but that would just compute the integral over the upper right quarter of the annulus, clockwise.
untold gulch
so the whole circle minus that?
untold gulch
ok yeah that make sense
sour verge
But I think it won't make a difference.
untold gulch
so how do i set up the integral then
sour verge
Well then you have your function of r and theta, and the bounds of integration.
untold gulch
so the outer integral is from 1 to 2 and the inner one is from pi/2 to 2pi?
untold gulch
and then the function inside is r^4cos(theta)dA?
untold gulch
You can have it in any order you want, it doesn't matter.
word
sour verge
dA becomes drd(theta)
sour verge
Actually, I'm pretty sure dA becomes r*dr*d(theta).
untold gulch
how come?
so like this but rdrdtheta?
wait do i need to flip the integrals if i have dr before dtheta
untold gulch
heard u
sour verge
wait do i need to flip the integrals if i have dr before dtheta
You can do either. When you're in the inside integral, the "outer" variable is just like a constant.
So in this case, cos(theta) is constant in the inside integral with respect to r.
untold gulch
ok, and then i can just evaluate this to get R?
sour verge
Yes
You evaluate the inside integral, just like you would usually, then integrate the result again with respect to theta.
sour verge
,rccw
ocean sealBOT
untold gulch
i just realised it doesnt make sense i got a negative number
sour verge
Right, thing number 1 is that in the first part of the second line, you wrote 5cos(t)/5 when it's just cos(t)/5. Not a big mistake.
untold gulch
okok
sour verge
i just realised it doesnt make sense i got a negative number
This is sort of predictable, because the value you're integrating is negative on the whole left part of the annulus (due to cos(theta) being negative there)
untold gulch
ooh ok
sour verge
Now if you have a textbook, I would check the answer. If it's still not right, then it means you actually do need the r*drd(theta) part.
untold gulch
yeah i recalculated it with 31sin(theta)/5 at the end and it still says my answer is wrong
sour verge
I'm just not sure anymore, but I think if it was drdtheta straight away in the integral, you're ok. Since it's dA, you can have a factor of r that pops out to really make this an area unit.
It's not too big of a mistake though, since it will just make the power of r go up 1, and the denominators go up 1 as well.
untold gulch
ok lemme try fixing that quick
sour verge
But that was mb sorry.
untold gulch
no youre all good!
hmm i think i evaluated it wrong because it still says i have the wrong answer
sour verge
You did 64/6 - 1/6 = 65/6 but that doesn't work.
zealous juniper
Hello can anyone help me pls
lone heartBOT
@zealous juniper Has your question been resolved?
steep delta
steep delta
<@&286206848099549185> Primarly this, the other one has been resolved
steep delta
This is a biit f a pain though
alpine sable
!status
lone heartBOT
What step are you on?
1. I don't know where to begin.
2. I have begun but got stuck midway.
3. I got an answer but I was told that it's wrong.
4. I got an answer and would like my work checked.
5. I have a question about someone else's work/solution.
6. I have completed the problem and don't need help anymore. Thank you.
7. None of the above
I got an answer but I was told I was wrong
alpine sable
Okay, is that the R=5?
steep delta
Okay, is that the R=5?
That was correct, i just needed to add x to the terms
steep delta
I'll try the problem agaiin and share my workings afterward
alpine sable
okay, dm me because i gtg for like 5 mins
steep delta
<@&286206848099549185>
steep delta
<@&286206848099549185>
steep delta
I have not the slightest idea on hoow to do this
<@&286206848099549185> since it has been 23 minutes
So I suppose it's fair
serene vault
so
serene vault
do i need to distribute that negative sign?
if so how would it look?
just b gets the negative? or so does c?
i am trying to isolate b
is it even possible to do so?
serene junco
no, don't distribute over multiplication
weird that they're even using an explicit multiplication sign
just think of it as (bc)
and likewise for the last term, bc^2
serene vault
hmm ok
thats cause i wrote that equation
can someone verify that this is what b ='s? $$ 0.25/c*(c-1.0) $$
ocean sealBOT
B
serene vault
when isolated
serene junco
$\frac{0.25}{c(c-1)}$ like this?
ocean sealBOT
tatpoj
serene junco
lol
serene vault
after isolating a and b
i substituted them into a and c for p_3
am i just going down a never ending rabit hole
serene junco
You still have a b in your bottom equation
If you substitute that, you'll have only c
serene vault
I HAVE TO PLUG IN B TWICE?
serene junco
and then it seems possible to solve for c
well, you replaced a with something that still involves b
serene junco
what lmao that was fast
serene vault
oh shit
serene junco
What are these meant to represent?
serene vault
solving (b)
serene junco
What is Pn?
ok wait but still, what is Pn? The probability of n heads out of 2n coin flips?
serene junco
ok but what is Pn generally?
The probability of at least one head in n flips?
or what? what does it represent
that would make sense for P1 and P2 but then P3 would be 7/8, not 5/8
serene vault
huh
serene junco
Can you show the very beginning of the problem?
lone heartBOT
@serene vault Has your question been resolved?
midnight berry
an infinite geometric sequence of first term a and 2log(x-3). to make the sequence convergent, the x value should satisfy ...
(a) 3 < x < 4
(b) 3 < x < 5
(c) 5/2 < x < 5
(d) 7/2 < x < 5
(e) 4 < x < 5
lone heartBOT
@midnight berry Has your question been resolved?
waxen turtle
lone heartBOT
waxen turtle
I think the first one is C