when you multiply H*G^T you are doing H*c^T for all rows of G
#help-49
inland horizon
when you multiply H*G^T you are doing H\*c^T for all rows of G
What do you mean by saying for all rows of G? I thought that c is a codeword in C, not G?
inland horizon
the rows of G are in particular also codewords
From C?
runic hamlet
from where else
inland horizon
I don't get it. How would that even work?
runic hamlet
how would what work
the rows from G are a basis of the subspace
in particular they are in the subspace
inland horizon
the rows from G are a basis of the subspace
Do you mean that it works for all rows from G and all rows from C meaning that for all g's and all c's such that you have H * g^T = 0 for all rows of G and H * c^T = 0 for all rows from C?
runic hamlet
if H*g^T=0 for all rows g of G, then because the g's are a basis of C it automatically follows that H*c^T=0 for all c in C
inland horizon
if H*g^T=0 for all rows g of G, then because the g's are a basis of C it automat...
Because G is a subspace from C, so C inherits every codeword from G and adds more on top?
runic hamlet
G is a matrix. its not even a subset of C, so it cant be a subspace
again, the rows of G are a basis of C
inland horizon
G is a matrix. its not even a subset of C, so it cant be a subspace
The rows from G (also known as basis vectors) span the code C?
runic hamlet
what do you think the word basis means
inland horizon
what do you think the word basis means
I don't know. I'm confused. I know that C spans the basis over the row vectors from G?
runic hamlet
your linear algebra basics are lacking. you cant do coding theory like this
the rows of G are a linearly independent set that spans the subspace C
inland horizon
the rows of G are a linearly independent set that spans the subspace C
So this implies that the basis vectors are sufficient to span the subspace C?
runic hamlet
yes thats what I said
inland horizon
So I shouldn't mix subset or subset in relation with G and C?
runic hamlet
you shouldnt mix up things in general
inland horizon
you shouldnt mix up things in general
C and F are spaces, whereas G (generator matrix) and H (parity check matrix) are sets?
runic hamlet
while F is technically a space you should think about it as a field
you already called G and H as matrices
matrices are not sets
inland horizon
matrices are not sets
Is a matrix a rectangular array?
runic hamlet
jesus dude
have you taken any linear algebra course?
yes a matrix is basically a 2 dim array
inland horizon
have you taken any linear algebra course?
Yes, I did
runic hamlet
you cant do coding theory like this
inland horizon
you cant do coding theory like this
That's why I ask π£
inland horizon
you cant do coding theory like this
Why does it have to be a -P and not just -? What does the -P do differently?
inland horizon
what should that even mean
I don't know. I know it as P, not -P
runic hamlet
again, over F_2 it doesnt make a difference
inland horizon
again, over F_2 it doesnt make a difference
Because -1 % 2 = 1 and 1 % 2 = 1?
runic hamlet
yes
runic hamlet
very much
a vector is an element of a vector space
vector spaces can be lots of things
inland horizon
Is F a different kind of space? It is a field, right?
runic hamlet
its a space and a field
shut canyon
@shy fern @lyric charm please look at this solution I found for the length of edges of the inflated, or spherical cube.
midnight plankBOT
<@522731522319581184> <@183668144404037632> please look at this solution I found...
shut canyon
lyric charm
how do you know the angle you claim to be pi/6 really is pi/6?
shut canyon
shut canyon
how do you know the angle you claim to be pi/6 really is pi/6?
I'm not sure now, need to check again.
lyric charm
alternatively: how do you know this rectangle has aspect ratio 1 : sqrt(3)?
shut canyon
Looking at this 3d drawing the portions look different.
lyric charm
ok but you understand that the angle you claimed to be pi/6 actually satisfies sin(theta) = 1/sqrt(3) and not 1/2, yes?
shut canyon
ok but you understand that the angle you claimed to be pi/6 actually satisfies s...
Yeah, I took a rectangle with side lengths β3 and 1, when this diagonal rectangle of the cube has β2, and 1
lyric charm
yup.
shut canyon
Thank you
lyric charm
why is pi/6 still happening?
shut canyon
why is pi/6 still happening?
I think because I thought 30 was known, I am trying try to find the angle, and ratios
shut canyon
And I can't calculate this I'm trying to make the side β2/2 be 1, and so 1/2 would change accordingly and that would show how much of the circle is the edge length
midnight plankBOT
@shut canyon Has your question been resolved?
twilit field
midnight plankBOT
twilit field
I'm really confused here
sharp coral
have you gotten an expression for the tangent plane at any given point?
twilit field
My question is how is this a function of one variable anymore
twilit field
have you gotten an expression for the tangent plane at any given point?
That isn't too hard
hard shard
y/x equals some value
wheat nymph
My question is how is this a function of one variable anymore
It's not
It's a surface, so a function of 2 variables
sharp coral
My question is how is this a function of one variable anymore
f is a function R -> R which has been composed with a function R^2 -> R
the fact that the inside function is a function of 2 variables doesn't change anything about f itself
twilit field
$z = af(\frac{b}{a})+ (f(b/a) + \frac{-b}{a}f'(b/a))(x-a)+ f'(b/a)(y-b)$
solid iris
,w partial xf(y/x)
twilit field
,w partial xf(y/x)
Is this better
solid iris
yes
twilit field
setting x=a and y=b, we find every plane passes via $(a,b a f(\frac{b}{a}))$
grand pondBOT
What a wonderful world !
twilit field
a,b are arbitrary points...
hmm
solid iris
(a,b) is point of tangency so ur common point must be indep of a,b
twilit field
I don't follow
solid iris
common point must be constant
twilit field
common point must be constant
so what should I do
solid iris
so what should I do
play with the tangent plane til u can see a constant point satisfying it
twilit field
hmm
twilit field
play with the tangent plane til u can see a constant point satisfying it
b=a may work
solid iris
i just said the point must be constant
twilit field
I can't seem to find oneπ
solid iris
take more time to play with the tangent plane eqn
twilit field
I have no idea
twilit field
take more time to play with the tangent plane eqn
I'm tempted to say (0,0). But then f(0/0) isn't defined
twilit field
yea
twilit field
a,b center of tangency and x,y variables of the plane
I get that
solid iris
just work with a!=0 to avoid /0
twilit field
solid iris
I'm tempted to say (0,0)
why?
grand pondBOT
The current time for math_rocks is 11:39 PM (IST) on Mon, 10/03/2025.
twilit field
My curfew is in 20 minutes too
solid iris
pls answer my q before u sleep
twilit field
sorry
twilit field
> I'm tempted to say (0,0)
why?
That would get rid of many of the terms
solid iris
when u say (0,0) do u mean x=0, y=0
solid iris
ok nope
ill try to make it clearer and u think about it tmr
a,b are arbitrary (and a!=0), u cant pick specific ones
u have to find a common point x,y independent of a,b
and the hint for that is keep playing with the plane eqn
midnight plankBOT
twilit field
Sorry, I'm quite tired
solid iris
np
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floral junco
floral junco
https://garticphone.com/en/?c=0220797a36
these help channels are for math help
not for gamerooms
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steel tulip
not for gamerooms
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@steel tulip Has your question been resolved?
round parcel
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keen bear
Help please
midnight plankBOT
Help please
deep vine
i'm pretty sure that to be a multiplicative group, n and m must be prime
that's my hint for you
keen bear
Ok thanks
jaunty canopy
i'm pretty sure that to be a multiplicative group, n and m must be prime
i think when it says multiplicative group it means the group of units
so m and n can be anything
jaunty canopy
Help please
uhh rn i'm a little tired
but by CRT you can classify group of units mod n for n a prime power
if you know a little bit of NT then you can classify these groups completely
jaunty canopy
it's past midnight for me so i'm gonna be heading to bed soon
but basically it depends on how much number theory you know
by the chinese remainder theorem, it suffices to classify what the group of units mod n looks like for n a prime power
this can be done but idk how much NT u've done
(in fact, units mod p^k is cyclic for p >= 3)
if u still need more help, generally uni questions get more traction in like #groups-rings-fields
i'm gonna be heading to bed now
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quiet moss
Ok so I got Part A) i) question wrong, I though it was g(f(-1)) = g(2) = 0.610
ii) I got the second question right, which was x = 2
Part B) i) I said there are no real zeros, which was wrong
ii) I got wrong, the limit I put was lim g(x) -> - infinity (this part is below it: x -> - infinity)
Part C) i) I said yes, which was correct
ii) I got this one wrong, I said "The input values have a single unique output value. f(-2) = -1 and f(1) = 2
Help me understand what I got wrong and what are the right answers.
midnight plankBOT
Ok so I got Part A) i) question wrong, I though it was g(f(-1)) = g(2) = 0.610
i...
wicked mauve
Ok so I got Part A) i) question wrong, I though it was g(f(-1)) = g(2) = 0.610
i...
for part A.i
did you mean g(2)? or g(-2) ?
quiet moss
for part A.i
I'm looking at what I wrote and it seems like I didn't even specify myself. the equation that I wrote is exactly what I typed in but I'm guessing what I meant was g(2)
wicked mauve
I'm looking at what I wrote and it seems like I didn't even specify myself. the...
it should be g(-2) then
quiet moss
Is that why its wrong? I put in the wrong number?
quiet moss
Alright thanks
For the first part should the equation look like g(f(-1)) = g(-2) = ____ and that should get me the right answer?
wicked mauve
For the first part should the equation look like g(f(-1)) = g(-2) = ____ and tha...
yes
quiet moss
you good, give me a sec
So the answer you got which was 2.1331, that indicates that there are real zeros?
wait I meant 2.7730
quiet moss
I thought there were no real zeros, thanks
wicked mauve
because g(0)= a real value
wicked mauve
I thought there were no real zeros, thanks
are you also able to graph this ?
thats just the work if youre unable to
quiet moss
Ok so how come my limit state is wrong?
wicked mauve
but if you graph it you can see it is a zero
quiet moss
are you also able to graph this ?
no not really
wicked mauve
Ok so how come my limit state is wrong?
for B.ii ?
quiet moss
alright
wicked mauve
why - ?
quiet moss
because of the -2
wicked mauve
what -2 are you referring to
quiet moss
wait nvm, I really don't know why I wrote that
I guess it was the decreases without bound part
wicked mauve
oh okay
x decreases so -inf
makes sense
and then
and x -> -inf the limit is 5.4
quiet moss
how would that look like?
quiet moss
you used the equation from B.i and put it into your limit function to get the answer?
I've never ever seen that
wicked mauve
you used the equation from B.i and put it into your limit function to get the an...
wdym
that is the function of g(x)
it is asking for the end behavior of g(x) as x decreases, no ???
quiet moss
Yeah your right
wicked mauve
does this make sense though ?
quiet moss
I thought as x decreases without bound, I would just put an infinity symbol instead
quiet moss
Its just what I would do normally for school work. This is what I've seen
something like that
I thought it was that'
no numbers or anything
wicked mauve
ohh okay
i see
ohhhh
for this function, it would be D
sorry i forgot about end behaviors using limit notation
but it would be like D
because as x -> -inf, g(x) is positive and "increasing"
quiet moss
ok i see, i still think your right though
wicked mauve
and as x -> +inf, g(x) is negative and decreasing
wicked mauve
ok i see, i still think your right though
well the limit as x->-inf is correct
quiet moss
oh the top portion?
quiet moss
ok
final question for C) ii) how come that was wrong? shouldn't there only be one unqiue output?
quiet moss
-5, -1, 2, 5
-4, -2, 1, 3
quiet moss
Ok i see it now
wicked mauve
final question for C) ii) how come that was wrong? shouldn't there only be one ...
looking at your answer, your statement is true yes
but the values are wrong
did you write f^-1(-2) =-1 ?
or f(-2) =-1
wicked mauve
that is most likely the reason why
quiet moss
I was looking at my table
wicked mauve
quiet moss
so f(-2) = -1
wicked mauve
couldve been because you didnt put it in the correct notation, because otherwise it looks correct to me
wicked mauve
so f(-2) = -1
no
quiet moss
I think I got f(-2) = -1 and f(1) = 2 wrong, it couldn't prove that they have a single unique output. that's the part i'm stuck on
quiet moss
sorry I'm not that ahead, in AP precalc
wicked mauve
I got it dude, thanks for helping and sorry for being quite slow.
what is it
cause ngl you seem correct aside from the notation π
quiet moss
Yeah i think i was right, I just don't know how to explain it very well
wicked mauve
i mean i can see how there is a way to prove this but gawsh diddly, ive long forgotten this
quiet moss
Aside from the notation
they all got a unique single output, i believe I put in the wrong X and Y values.
Probably why it got counted wrong
wicked mauve
yeah, when i read what you put, i thought you either did the wrong notation
or the wrong values
like swapping them
which, would be the inverse haha
quiet moss
yeah
wicked mauve
dont worry too much for the frq, you did great regardless
you get points for your work
so even if its wrong
quiet moss
thanks, i'll keep on trying to improve
wicked mauve
show your steps and youll get something
midnight plankBOT
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meager ore
Show that if ( (x_n) ) is unbounded, then there exists a subsequence ( (x_{n_k}) ) of ( (x_n) ) such that
[
\lim_{k \to \infty} \frac{1}{x_{n_k}} = 0.
]
midnight plankBOT
Show that if \( (x_n) \) is unbounded, then there exists a subsequence \( (x_{n_...
grand pondBOT
Halex
meager ore
How do I construct that subsequence
I know it must be increasing, but still no clue what to do
Since $x_n$ is unbounded then for all $M$, there exists $n \in \mathbb{N}$ s.t $|x_n| > M$
grand pondBOT
Halex
tribal temple
All M, you say? And you want a subsequence indexed by k, you say?
meager ore
All M, you say? And you want a subsequence indexed by k, you say?
$\forall M \in \mathbb{R}$
grand pondBOT
Halex
tribal temple
**Halex**
I'm kinda hinting about thinking about the definition here: for whatever number I can think of, you can find some term of the sequence such that the absolute value of said term is greater than the number I thought of
meager ore
yes
meager ore
I'm kinda hinting about thinking about the definition here: for whatever number ...
do you know a way to actually prove this? I've been getting headaches with this problem
tribal temple
Maybe if I give a slightly different version of the problem, could you create a subsequence x_n_k such that its absolute value diverges to infinity?
meager ore
Are you talking about this Theorem that states that an unbounded sequence has a monotone subsequence?
tribal temple
It might be that one, depending on how it's stated 
meager ore
It might be that one, depending on how it's stated <:catThink:519997422194393108...
In this case, an unbounded above sequence has an increasing subsequence?
tribal temple
It would, and said subsequence would go to +infty (similar if you were not bounded below)
tribal temple
Cool, what if I said to set M to 2?
meager ore
|x_n| > 2
tribal temple
Sure, do you see where I'm trying to go?
meager ore
yes, |x_n| > k for k in N
tribal temple
Sure, of course, you could start indexing those n's in a certain way, set n_1 such that |x_n_1| > 1, so on, n_k such that |x_n_k| > k (and that x_n_k of course actually forms a subsequence of x_n)
meager ore
yes, but how can we justify we can actually choose |x_n_k| > k subsequence?
meager ore
but the definition states there exists
I'm guessing since $n$ is just an index, or a name, we can just say there exists n_k?
grand pondBOT
Halex
I'm guessing since $n$ is just an index, or a name, we can just say *there exists* n_k?
```Compilation error:```! Missing $ inserted.
<inserted text>
$
l.49 ... a name, we can just say *there exists* n_
k?
I've inserted a begin-math/end-math symbol since I think
you left one out. Proceed, with fingers crossed.```
tribal temple
the n_k needs to be in the dollar signs
And well, the n is just the index of the sequence, the idea is that you pick some n such that |x_n| > 1, and label that one as n_1
Then do the same, find some n such that |x_n| > 2, and label that one as n_2, so on and so forth
tribal temple
Alright, let's pretend that we do the sequence out, and that I see that x3 is such that |x3| > 1, then you choose n_1 = 3
Let's then pretend that now, I find that x5 is such that |x5| > 2, then we choose n_2 = 5
And I go along now, the next one, it's x8 such that I have |x8| > 3, at that point you choose n_3 = 8
You're choosing the terms that correspond to the inequality like that, if you see?
meager ore
in math terms how would you express it?
|x_n_1| > 1?
we're just making an increasing sequence by finding indices that satisfies the equalities we need?
|x_n_2| > 2?
tribal temple
Pretty much, we're making the subsequence that way, from where each of the terms in the subsequence are such that |x_n_k| > k
meager ore
for M = 2 we can find some n s.t |x_n| > 2 and we call that n as n_2?
meager ore
Pretty much, we're making the subsequence that way, from where each of the terms...
Is this correct then?
could you spot the mistakes, please?
tribal temple
That's pretty much the idea yep, I would reword the first part that to create the subsequence $(x_{n_k})$ such that $n_k$ is (...) such that $\abs{x_{n_k}} > k$ and $n_k > n_{k - 1}$, removes a bit of redundancy
grand pondBOT
@tribal temple
meager ore
That's pretty much the idea yep, I would reword the first part that to create th...
I didn't do this proof, so I'm not sure why $n_k > n_{k - 1}$ is justified
grand pondBOT
Halex
meager ore
if I'm not mistaken, it ensures the increasingness
tribal temple
You could e.g. add the additional restriction while picking your $n_k$ such that you have both $\abs{x_{n_k}} > k$ and also $\abs{x_{n_k}} > \abs{x_{n_{k - 1}}}$
grand pondBOT
@tribal temple
meager ore
You could e.g. add the additional restriction while picking your $n_k$ such that...
it is just a restriction? will it work in the end with no further explaination?
tribal temple
**Halex**
That comes about from the fact that you have this definition here, so you can of course, set the "M" at each stage to be the larger of k and |x_n_{k-1}|
meager ore
That comes about from the fact that you have this definition here, so you can of...
Sorry if I sound dumb, I'm a bit confused but what do you mean by the larger of k and |x_n_{k-1}|?
tribal temple
Sorry if I sound dumb, I'm a bit confused but what do you mean by the larger of ...
Take the maximum, if k is larger than |x_n_{k - 1}|, then you get |x_n_k| > k >= |x_n_{k - 1}|, similar argument if it's the other way around
tribal temple
"if |x_n_{k - 1}| is larger than k, then you get |x_n_k| > |x_n_{k - 1}| >= k"
meager ore
Ohh so the = changes a lot
So to clear it out, for the proof can I just add the restriction and it will work?
midnight plankBOT
@meager ore Has your question been resolved?
gray zenith
$\sin\left(\frac{x}{2}\right) \cos(nx) = \sin\left(\frac{x}{2} + nx\right) + \sin\left(\frac{x}{2} - nx\right)$
$\because \sin(a)\cos(b) = \frac{1}{2} \left[ \sin(a+b) + \sin(a-b) \right]$
midnight plankBOT
$\sin\left(\frac{x}{2}\right) \cos(nx) = \sin\left(\frac{x}{2} + nx\right) + \si...
grand pondBOT
WhatWhereWhen?
gray zenith
is this correct? my book uses cosA sinB instead and I wasn't sure if they're the same
delicate sage
**WhatWhereWhen?**
You missed the 1/2 in the first identity
gray zenith
oh thanks. kinda getting used to latex.
gray zenith
You missed the 1/2 in the first identity
but other thn that is it ok to use sinAcosB instead of the cosAsinB?
delicate sage
but other thn that is it ok to use sinAcosB instead of the cosAsinB?
Yeah it is
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molten bay
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molten bay
,rotate
grand pondBOT
molten bay
Should I use expansion if e^x,sinx
lime agate
molten bay
So how can i check lhs rhs
last slate
Continuity is usually checked by left and right limits I think
So check limit f(x) as x-->0+ and f(x) as x-->0-, check if they're equal
molten bay
Continuity is usually checked by left and right limits I think
So check limit f(...
Yes. But i am asking which method would be best?
molten bay
Should I use expansion if e^x,sinx
..m
last slate
Should I use expansion if e^x,sinx
I don't think you need any expansions, just substitute x for -x first and see what equation you get
last slate
-1<=sin(f(x))<=1
molten bay
But i want to know more
last slate
so at both limits sin value will be -1 and 1 respectively
Also when you substitute -x in the equation you just get the same equation with sin(1/-x)
so it'll be -sin(1/x)
I think that should make it discontinuous but I may be missing something but the values on both sides go towards infinity
midnight plankBOT
rose geyser
it is similar to pythagoras iirc
im gonna search
wait why are they perpendicular
nvm my bad
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primal saffron
How does halting behavior of a Turing machine relate to polynomials having integer solutions
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How does halting behavior of a Turing machine relate to polynomials having integ...
@primal saffron Has your question been resolved?
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jaunty wren
how do i solve 43 by βrationalizingβ
midnight plankBOT
how do i solve 43 by βrationalizingβ
dusk pier
how do i solve 43 by βrationalizingβ
What do you mean "rationalizing"?
jaunty wren
it says you have to find the solution by making a substitution that makes the integrand be a polynomial over a polynomial
dusk pier
it says you have to find the solution by making a substitution that makes the in...
So do it. Let u^3=x^2+1
jaunty wren
im confused on what to do after that though
dusk pier
im confused on what to do after that though
Simplify it.
jaunty wren
where did i fail
fallow scarab
where did i fail
why do you think this is wrong
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wicked mauve
what
wicked mauve
why did they divide the og series by the p-series
sharp coral
to perform the limit comparison test
wicked mauve
limit comparison is dividing them ??
how is this different from the ratio test then
lavish venture
brother review your definitions
wicked mauve
π
lavish venture
this is comparing two different sequences that are being summed whereas the ratio test is concerned with the ratio of successive terms
from a single sequence being summed
lavish venture
π€
wicked mauve
:(
lavish venture
itβs comparing another series to another this is not the same
brazen stream
why did they divide the og series by the p-series
They divided the given series by the p -series to apply the Limit Comparison Test (LCT) for series convergence.
wicked mauve
okay i see
heavy falcon
what
if this way is bothering you then you can use the fact that 7n^(7/2)+6>=7n^(7/2)+6n^(7/2)=13n^(7/2) for all n in N*
then comparison test
wicked mauve
im just very confused by series in general
heavy falcon
well you need to practice
wicked mauve
i can usually see if a series diverges or converges
but
whats the point if you cant prove it
heavy falcon
to get used to all of the convergence tests and when to apply each one etc
wicked mauve
well you need to practice
i know
wicked mauve
but i dont find this part of calc very fun
heavy falcon
yea limit comparison sucks ass tbh
wicked mauve
so its difficult
heavy falcon
yea limit comparison sucks ass tbh
but maybe sometimes it is good
lavish venture
did yβall see that
wicked mauve
meow meow
heavy falcon
did yβall see that
no i didnt
lavish venture
brother got insta banned
lavish venture
to get a good grade
yea but is it really a proof
youβre probably just saying like
by ratio test L < 1
wicked mauve
like how do i prove this converges
wicked mauve
by ratio test L < 1
yeah
heavy falcon
it seems that the "β¦.." was some nasty word 
wicked mauve
but dead ass i cant even do that
lavish venture
like how do i prove this converges
alternating series test sir
lavish venture
it seems that the "β¦.." was some nasty word <:opencry:586078614865444865>
indeed
starting with n
lament knoll
if $a_n$ is positive, decreasing and converges to 0, then $\sum (-1)^n a_n$ converges
grand pondBOT
tm
wicked mauve
like how do i prove this converges
like how tf am i supposed to know this is 1/3n!
lavish venture
like how tf am i supposed to know this is 1/3n!
3 * 2 =6 , 6 * 3 =18, 18 * 4 =72, 72 * 5 =360
lavish venture
3 * 2 =6 , 6 * 3 =18, 18 * 4 =72, 72 * 5 =360
once you see this itβs obvious
heavy falcon
like how tf am i supposed to know this is 1/3n!
because you are supposed to notice that every denominator has 3 as a factor for example
wicked mauve
because you are supposed to notice that every denominator has 3 as a factor for ...
yes i can see that
im just not built like yall
im slower
heavy falcon
yes i can see that
then take 1/3 common factor and you will see the result
heavy falcon
im just not built like yall
well i didnt spawn with the ability to notice patterns a bit quickly
this ability improved with practice
wicked mauve
its not that i cant do this
its just i dont like series
so its making it 100 times harder
lavish venture
you donβt like series because you donβt understand it
wicked mauve
its a neverending cycle
lament knoll
im just not built like yall
youβll win against series bro, donβt let them get you down π«‘
wicked mauve
they feed off each other tho
lavish venture
,av tm_29495
grand pondBOT
lavish venture
fire
heavy falcon
you can hate it but whatever you feel about it you still need to study it
lament knoll
,av tm_29495
larry will touch you tonight
lavish venture
been waiting for you to say that
wicked mauve
youβll win against series bro, donβt let them get you down π«‘
je l'espère
bahhabaha
lament knoll
blud is not ready for that
heavy falcon
been waiting for you to say that
premove?
lament knoll
je l'espère
bien sΓ»r
wicked mauve
okay so this is convergent
heavy falcon
it turns out that knief is hikaru π₯
wicked mauve
i got this correct
but its absolutely convergent because its being compared to the absolute value
???
lavish venture
but its absolutely convergent because its being compared to the absolute value
wdym compared to the absolute value
wicked mauve
the left value
lavish venture
itβs absolutely convergent because $\sum |a_n|$ converges
grand pondBOT
knief
wicked mauve
yes
dusty portal
i will end it all
Don't, it's just that the wording made no sense
wicked mauve
oh bruh i got this right off of luck
i though the left function was greater than 1/n^3
dusty portal
Wait, are you just saying that $\sum_{n\ge 1}\frac{\sin(7n^2)}{n^3}$ converges?
grand pondBOT
;(
lavish venture
okay so this is convergent
do you understand where the bound came from
wicked mauve
do you understand where the bound came from
what bound
heavy falcon
Don't, it's just that the wording made no sense
what if he meant that he will finish calculus tonight
lavish venture
what bound
how they got that itβs <= 1/n^3 mate
dusty portal
The inequality given ther
dusty portal
i got 1/n[3
How?
wicked mauve
because
lament knoll
let him cook
lament knoll
maybe because $|sin(n)| \leq 1$
grand pondBOT
tm
lavish venture
\
lavish venture
what?
dusty portal
but i thought the sin would be greater
lavish venture
what are you learning from
lavish venture
how?
wicked mauve
like i can get those correct
lavish venture
you act like you want to prove these convergence tests
wicked mauve
but when i walk into my quiz im cooked
i was on kahn earlier
but my hw assignment has slightly more difficult questions that i js dont understand
heavy falcon
but i thought the sin would be greater
well something like this isnt an issue with series
lavish venture
these arenβt as difficult as the ones on khan academy i donβt think
heavy falcon
you need to review some facts about some functions
lavish venture
you just asked before how the limit comparison test is different from the ratio test
wicked mauve
πΉ
heavy falcon
especially if they have bounds, etc... because these are needed for comparison tests for example
lavish venture
you can find good exercises in an introductory analysis text too
some have chapters that arenβt strictly about proving things beyond your scope
but a standard calculus book is enough
go find something like stewart or larson or whatever online
every calc book is the same
openstax is enough rrally
wicked mauve
yeah ive used openstax and such
and wtv the wolfram stuff is
it helps ofc
i just dont know what im doing
heavy falcon
you canβt skim through a math book
imagine skimming a math book
lavish venture
been there
then you get to the exercises and realize you didnβt learn shit
and have to read it again
heavy falcon
in this case i am constantly there too
stone snow
how would you do d?
midnight plankBOT
how would you do d?
old plume
how would you do d?
start by finding an expression in terms of t that gives the distance between particle P and particle R at time t. integrate this over the range, and divide by the width of the range
stone snow
so for the distance would you just subtract them ?
old plume
remember that distance is always positive
stone snow
absolute value the difference?
old plume
yes
stone snow
ok so would it be integral of 1 to 3 |p(t)-r(t)|?
old plume
dont forget to divide by the width of the range
stone snow
whats the width of the range
old plume
the range is from t=1 to t=3
old plume
yes, average value of f(x) on interval [a,b] is equal to integral from a to b of f(x) dx over (b-a)
old plume
urw
stone snow
.close
midnight plankBOT
midnight plankBOT
Available help channel!
marsh gale
||trying to make this as brief as possible|| I want to push my math skills with a focus for learning differential equations for analog computing. Finished highschool at algebra 2. What paths would be recommended for self directed learning or relatively cheap courses?
midnight plankBOT
||trying to make this as brief as possible|| I want to push my math skills with ...
violet dune
start by perfecting your algebra skills
then try going on khanacademy or similar websites to learn precalculus, calculus, etc. there should be some shown path of what usually comes first
the path that i had after algebra 2 was precalculus, calculus I, calculus II, calculus III (i completed these with AP calculus, so maybe look into AP Calc BC), discrete math I, discrete math II, multivariable calculus I, linear algebra, differential equations
discrete math helps moreso with programming, it covers set theory/graph theory, logic, probability, and combinatorics
you can take multivariable calculus before discrete, it doesnt really make a difference
lavish venture
||trying to make this as brief as possible|| I want to push my math skills with ...
donβt buy courses
youtube is your friend
violet dune
theres so much information online that you dont have to pay for
you can learn extremely advanced level mathematics without paying for books and lessons
marsh gale
diff equations is quite further along than I thought but I am still determined. Thank you @lavish venture and @violet dune for the advice, and I hope you day is well.
.close
midnight plankBOT
lavish venture
diff equations is quite further along than I thought but I am still determined. ...
itβs not as far as you might think
tacit rose
diff equations is quite further along than I thought but I am still determined. ...
OpenCourseWare from mit is free too https://ocw.mit.edu/ for university level
lavish venture
if youβve taken algebra 2 depending on what that actually means youβve learned you can probably jump into calc 1 now
calc 1/2 are like the same thing and wonβt take that long
then itβs just calc 3 and some intro linear algebra before differential equations
depending on how dedicated you are you can do that in a year for sure
tacit rose
i would look into ap calc bc
Mainly calc 1 and some basic integral techniques
Not calc 3
lavish venture
calc 3 is multivariable calculus and vector calculus
maybe some schools do vector calc as calc 4 idk
marsh gale
Def concepts I will look more into later, thank you two again
It does make me happy that math information is much more open source than expected
lavish venture
It does make me happy that math information is much more open source than expect...
oh definitely
midnight plankBOT
Available help channel!
fringe pine
midnight plankBOT
supple pilot
i have a feeling i solved b instead of a
i think you almost solved b
a is easier than that, you can compute line tangent to the graph from derivative of function right?
supple pilot
no, you want to approximate the f(1.2) with the tangent line, so you need the tangent line first
and then you plug 1.2 into the line equation
supple pilot
no, you want to approximate the f(1.2) with the tangent line, so you need the ta...
but as i said you only need derivative for getting the tangent line, and derivative is given by the diffeq
fringe pine
i have a feeling iβm overthinking it
fringe pine
but as i said you only need derivative for getting the tangent line, and derivat...
iβm still lost π π
supple pilot
iβm still lost π π
the slope of a line tangent to graph at a point is given by dy/dx at this point, right?
and you have the dy/dx supplied, so you just need to plug (1, 0) into it
which gives you a in ax+b, you just need to solve for b
fringe pine
so i did the right steps
i think
or am i supposed to plug (1,0) into dy/dx
wait
that only solves for slope of a point
right?
wait i actually started thinking
then i find the antiderivative of dy/dx via separation of variables
and plug 1.2 into the anti derivative to find b
supple pilot
and plug 1.2 into the anti derivative to find b
i'm not sure what you're getting at but "approximate with a tangent line" usually means plugging the 1.2 into line equation
fringe pine
i thought i was cooking π
supple pilot
if you find the antiderivative it's not really an approximation... it's exact solution xd
fringe pine
which equation? the y=mx+b or the derivative
fringe pine
if you find the antiderivative it's not really an approximation... it's exact so...
ur right π
supple pilot
which equation? the y=mx+b or the derivative
line equation as in mx+b, where m is the slope obtained from derivative
fringe pine
and i got that right?
supple pilot
and i got that right?
uh sorry, where?
fringe pine
one sec
fringe pine
i hate this unit
supple pilot
m will be equal to this evaluated at (1, 0)
which is -3 if i'm not mistaken
fringe pine
m will be equal to this evaluated at (1, 0)
so plug the point in right?
supple pilot
yup
fringe pine
THATS WHAT I MEANT THE ENTIRE TIME π π π
supple pilot
sowwy then...
fringe pine
wait no
nvm
i said something different
π π
but it was in the back of my mind
fringe pine
sowwy then...
it wants me to approximate the value at 1.2 right?
supple pilot
it wants me to approximate the value at 1.2 right?
yep
fringe pine
using an equation i made?
supple pilot
the mx+b one, yes
fringe pine
i think i did it wrong or something
supple pilot
show we'll see
fringe pine
now you find b
i should plug in (1,0) for that right?
supple pilot
i should plug in (1,0) for that right?
yea, because the line should pass through that point
supple pilot
what i got so far is b=3
so you have the approximation of y=-3x + 3 right
supple pilot
naow you plug 1.2 into x
supple pilot
i was gonna say plug the same point again
nah that would give you 0 xd
or more like, the line passes through that point so there's no point in plugging it after you found the equation
wait lemme draw it, maybe it will make it more clear
fringe pine
f(1.2)=-3(1.2)+3
supple pilot
f(1.2)=-3(1.2)+3
yup
supple pilot
wait lemme draw it, maybe it will make it more clear
fringe pine
time for b
supple pilot
this is the situation
fringe pine
i see
supple pilot
time for b
you already almost solved b, it's the separation of variables you did
so you found some antiderivative and C constant
the only thing that's left is writing it as y(x) = ...
supple pilot
probably not
fringe pine
i'll use what im familiar with then
supple pilot
doesn't matter if you name it f or y, as long as you solve for it
because you want it as a function of x
fringe pine
the next steps have something to do with (1,0) right?
supple pilot
the next steps have something to do with (1,0) right?
you did this
which doesn't involve derivatives already, now you have to 1. find C (from the (1, 0) point) 2. solve for y
brb, i have a phone call
supple pilot
okay i'm back
supple pilot
-1/(e^0) is -1, not 0 though
fringe pine
c=1
supple pilot
ye
supple pilot
how should i solve for y
you have this equation to solve for y
fringe pine
thats it?
supple pilot
yeah
you want y as a function of x, so if you'll have y at left side and no y at right side that'll suffice
supple pilot
ah, but you cannot plug anything into x
you have to solve the equation with x that can be anything
supple pilot
you have this equation to solve for y
this one, move e^y onto one side and rest to the other side and take logarithm
supple pilot
yep
supple pilot
completely isolate the y
did you get it?
supple pilot
js woke up from a "2 minute power nap" π
my 2 minute power naps turn into 2 hours power naps usually
supple pilot
seems correct
fringe pine
finally 
test tmr and i only get 20 minutes
took an hour for two parts i might be cooked
supple pilot
just do more exercises until you get the hang of it
midnight plankBOT
@fringe pine Has your question been resolved?
fringe pine
b
fringe pine
ik the slope is actually positive 3/2 because of the given graph, hypothetically, if i wasn't given a graph, how would i know my slope for point (1,2) is + or -?
(taking a 5 minute nap while waiting for response
old rampart
Right here youβre taking sqrt(9) to be both 3 and -3
But in general, we use the principle root of this which is just positive 3
fringe pine
But in general, we use the principle root of this which is just positive 3
what are cases where i cant assume the positive option?
long dagger
But in general, we use the principle root of this which is just positive 3
sqrt(x) is just defined as the positive root
fringe pine
also for c can i just justify that its an underestimate bc it's cc up or do i have to add more info
long dagger
In no case will you ever make a sqrt negative
fringe pine
In no case will you ever make a sqrt negative
noted
how should i justify c?
do i just say its cc up?
5 minute nap
midnight plankBOT
@fringe pine Has your question been resolved?
fringe pine
midnight plankBOT
midnight plankBOT
fringe pine
too tired to study
midnight plankBOT
Available help channel!
signal rock
Is ans a (5)
midnight plankBOT
Is ans a (5)
signal rock
COuldn't find any sols online sadly
smoky walrus
Is ans a (5)
that's wrong, show your steps
last slate
Is ans a (5)
that isn't correct unfortunately
Make sure you are keeping the polarity of the graph in mind π
flat spire
Is ans a (5)
HSC trauma
signal rock
is middle 13? or is it 7
last slate
is middle 13? or is it 7
13
signal rock
see where i went wrong
last slate
But do you understand why?
signal rock
i did 10 - 3 instead of x - 3 = 10
last slate
Ahh okay
last slate
Make sure you are recognising the integral calculated between intervals where the graph is in the negatives would yield a negative value
signal rock
Make sure you are recognising the integral calculated between intervals where th...
negative y values right?
last slate
negative y values right?
yup!
flat spire
negative y values right?
r u yr12?
flat spire
Doing 3u?
signal rock
yeh but this is 2u
signal rock
how do yk
signal rock
Ur 2u exam is gonna be really easy this year
2u isn't too difficult, mainly making silly mistakes mostly though
flat spire
Because I sat 2024 2u exam last year and nesa made it hell
like everyone complained about its difficulty
signal rock
did you do 3u?
flat spire
No but I learnt 3u and 4u stuff
signal rock
why?
i did 4u for like a week
too much hw and cbb learning the harder stuff so dropped
flat spire
U shouldβve kept 4u then u can skip stats Lol
signal rock
sack im too dumb for it
plus ill do bad in Ext 1 (like not put as much study in it), coz its only 1 unit towards the ATAR
I just need for prerequisites
flat spire
Nice
signal rock
if u dont mind me asking, what were ur subjects in HSC
flat spire
adv eng 2u math phys ancient his and legal
flat spire
nawwwww I hated legal lmaoo
signal rock
same was debating whether to drop or nah
but gonna probs hold on since I don't think my ranks are that bad
what do u study then?
flat spire
physics
midnight plankBOT
flat spire
same was debating whether to drop or nah
Only drop if you know you can do well in all ur other units otherwise if ur not sure keep it just in case
Cya
midnight plankBOT
Available help channel!
twilit night
Hi so for all I have done is draw out hte graph but im not sure where to go to set up this integral Ice tried chat gpt but its not making sense could I perhaps get some help or hint
midnight plankBOT
Hi so for all I have done is draw out hte graph but im not sure where to go to s...
@twilit night Has your question been resolved?
sharp patio
Hi so for all I have done is draw out hte graph but im not sure where to go to s...
integral of root x from 0 to 1 take away Integral of x^2 from x=0 to 1 so you can just do integral of root x - x^2 from 0 to 1
midnight plankBOT
@twilit night Has your question been resolved?
twilit night
ohh no
like i have to do it fro mlike
double integral way
so like i have ste limits from a dy dx
but yes I get that but its training me for harder irregular shapes
sharp patio
oh mb, do integral of 1 dy where y goes between x^2 and root x, then do the integral of x from zero to 1?
twilit night
oh mb, do integral of 1 dy where y goes between x^2 and root x, then do the inte...
Nah ur good and get the answer should like that and is good to know to get if im on the right track
sharp patio
I think its like ur adding up very small areas over the region
icl i learnt this like 2 weeks ago lol
twilit night
oh gotya gng
strong storm
Question
Consider the differential equation y' = y.
Solve the differential equation (i.e., find y(x)), supposing that y(1.57) = 2.12. Evaluate that solution at the x-coordinates in the table
below. Fill out the table with your answers.
Answer
x-coordinate
x1 = -1.35
x2 = 2.80
x3 = 4.04
Solution Value
y(x1) =
y(xβ) =
y(x3) =
Enter as many decimal places as your calculator allows (8 to 10). Your answer must be within +0.005 of the correct answer to be considered correct
midnight plankBOT
Question
Consider the differential equation y' = y.
Solve the differential equat...
strong storm
I heard you change it into this notation: dy/dx = y
dy/y = dx
and then integrate it from there but i'm not sure how to integrate that exactly from there and i need some help
gray prism
I heard you change it into this notation: dy/dx = y
dy/y = dx
Yes that is correct: $\int \frac 1y \dd{y} = \int \dd{x}$
grand pondBOT
Ο=βg
strong storm
so x on the right side?
gray prism
Yep
strong storm
and log(y)?
strong storm
ohhh +c and then sub the initial conditions?
gray prism
Yep
strong storm
ok thank you very much
gray prism
Or isolate y to make things a bit easier
midnight plankBOT
strong storm
Consider the 1st order linear differential equation
y ' + x ^5 y = g ( x )
for some function g ( x ) . Use the integrating factor method to find the integrating factor for this differential equation (recall that I ( x ) = e β« p ( x ) d x ).
Evaluate that integrating factor at the x -coordinates in the table below. Fill out the table with your answers. Answer x -coordinate Integrating Factor
x 1 = β 0.93 I ( x 1 ) =
x 2 = -0.28 I ( x 2 ) =
x 3 = 0.6 I ( x 3 ) =
.close
midnight plankBOT
Available help channel!
civic stratus
Give the function h:x -> 4x + m and h^-1 -> 2kx + 5/8, m and k are constant
midnight plankBOT
Give the function h:x -> 4x + m and h^-1 -> 2kx + 5/8, m and k are constant
civic stratus
how do u even do this
olive matrix
you know that h^-1 [ h(x) ] = x
olive matrix
if you replace h and h^-1 with their definitions then you should get an equation
civic stratus
huh
with their defination
sorry im a bit confuse
can u show how u would do it, so i can see
stiff pollen
you know that h^-1 [ h(x) ] = x
||does h stand for haley||
olive matrix
so like
h^-1 [ h(x) ] = x
h^-1 [ 4x + m ] = x
and then you apply the definition of h^-1
2k [ 4x + m ] + 5/8 = x
and then you simplify the left side
to get
(stuff)x + (other stuff) = x
civic stratus
ok but how do u get the constant