I KNOW IT SAYS INTEGRATE but would this be the correct d/dx ?
#help-43
dusty harbor
compact pewterBOT
I KNOW IT SAYS INTEGRATE but would this be the correct d/dx ?
dusty harbor
my answer in the box
potent berry
..what-?
meager dragon
Hi
dusty harbor
wuold that be the correct d/dx of f(x)
kind viper
...
potent berry
i dont think so.
meager dragon
Should I go to pre uni
potent berry
im so confused
potent berry
Should I go to pre uni
the channel? no
meager dragon
What channel should I go to for thia?
dusty harbor
it was wrong either way
kind viper
see #❓how-to-get-help for instructions @meager dragon
potent berry
What channel should I go to for thia?
!occupied, an available one
compact pewterBOT
What channel should I go to for thia?
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dusty harbor
what would f,x be then
dusty harbor
f'(x)* 😭
le même breuh
its term by term
thats my answer i want to know if its correct
potent berry
its term by term
what?
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winged lion
have you tried ftc
fair raft
fundamental theorem of calculus i think
boreal girderBOT
winged lion
and then you apply the chain rule
it's a very fundamental theorem you must have heard of when introduced to integrals
boreal girderBOT
fair raft
quick moderators
winged lion
seems to be right yea
compact pewterBOT
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hidden mesa
,rotate
boreal girderBOT
hidden mesa
,rotate
boreal girderBOT
hidden mesa
How’d they get from first to second I’m lost
tame laurel
compact pewterBOT
@hidden mesa Has your question been resolved?
worn loom
How’d they get from first to second I’m lost
Wazup!
Do you know how partial differentiation works?
hidden mesa
I think I’ve forgotten😭
worn loom
The process is actually really simple
You differentiate like normal with respect to the variable but everything else is treated as a constant
compact pewterBOT
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hidden mesa
What have they held constant
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@hidden mesa Has your question been resolved?
wheat pasture
What have they held constant
Everything except for ln(y) 
compact pewterBOT
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delicate musk
To prove that any homomorphism from G->H carries the identity, why can I not just show this:
$\phi(eg) = \phi(e) \phi(g) =\phi(g)$ as $\phi(eg) =\phi(g) \forall g \in G$?
compact pewterBOT
To prove that any homomorphism from G->H carries the identity, why can I not jus...
delicate musk
its not usually proved this way so i was curios
boreal girderBOT
BOSS
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tough tartan
its not usually proved this way so i was curios
looks fine, see https://proofwiki.org/wiki/Group_Homomorphism_Preserves_Identity
signal pecan
Hello
vague pond
Hi
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limpid badge
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thorny urchin
can you show the intended solution
acoustic yarrow
^
limpid badge
Is this one given is incorrect?
acoustic yarrow
oh intended solution
forest token
take out 1/sin^2 x and then take LCM
acoustic yarrow
yeah answer is incorrect
forest token
no its wrong
eternal pulsar
Is this one given is incorrect?
Yes, it is.
limpid badge
Successfully wasted 1 hour omg
eternal pulsar
Why the fuck am I gray now?!
proud halo
gandalf before he became white:
compact pewterBOT
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dense berry
can someone help with question c pls i dont rlly get the worked example
compact pewterBOT
can someone help with question c pls i dont rlly get the worked example
dense berry
ngl i haventcoz i dont get what the question is asking either
potent berry
ngl i haventcoz i dont get what the question is asking either
i wonder if that has something to do with it
it's asking the values of $x$ for which $h(x)=6$ or whatever it was
boreal girderBOT
Percy
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boreal girderBOT
mq
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@copper loom Has your question been resolved?
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@copper loom Has your question been resolved?
pliant beacon
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balmy hamlet
can someone help me on these 2 questions and tell me the answer
compact pewterBOT
can someone help me on these 2 questions and tell me the answer
kind viper
can someone help me on these 2 questions and tell me the answer
!noans
compact pewterBOT
can someone help me on these 2 questions and tell me the answer
The purpose of this server is to help you learn, not to hand out answers. Do not ask someone to give you the answer directly.
serene viper
What… exactly is the question even?
compact pewterBOT
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weak nacelle
what variable name could i use for the area enclosed between the square and circle?
compact pewterBOT
what variable name could i use for the area enclosed between the square and circ...
sudden sparrow
anything you want? you can name variables anything you like
weak nacelle
anything you want? you can name variables anything you like
i know
i was just wondering if there is any standard symbol
or one you recommend
plain quail
No
sudden sparrow
literally doesn't matter what you choose, just as long as you signpost
plain quail
Just don't choose anything too absurd
compact pewterBOT
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kindred sleet
can someone help me
compact pewterBOT
can someone help me
orchid moat
make x the subject of the first equation
kindred sleet
idk about the bottom but for the top I can move the 2x to the left
orchid moat
why would you do that?
kindred sleet
idk I feel like I need to isolate the constant cause that's how it is on the bottom
kindred sleet
make x the subject of the first equation
idk what this means
orchid moat
idk I feel like I need to isolate the constant cause that's how it is on the bot...
you have to isolate the x
and then you can substitute that into the other equation
kindred sleet
oh so -4y ?
orchid moat
no the x is already on the opposite side to the y
kindred sleet
no the x is already on the opposite side to the y
add 4 to both sides ?
kindred sleet
now you have 2x = 4y + 4
do I divide by 2 on both sides ?
orchid moat
yep
orchid moat
yeah now replace the x in the second equation with 2y+2
kindred sleet
2y+2+y=4
i can subtract 2 on both sides and then 2y + y is just 3y
y=2/3
am I done ?
kindred sleet
what how is that ? I thought x was 2y+2
kindred sleet
and then x = 6
didimisssomething?
compact pewterBOT
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pure spear
It says some of the answers are incorrect
Can anyone help me find which one
I'm pretty sure my answers are right
pure spear
2nd one should be ?
Why are 2 and 4 wrong?
gleaming latch
Neither of them are series where you can apply AST
zealous sphinx
i could be wrong abt this i havent done series in a while, so this is more of a question for my own sanity
isnt the 5th one yes?
dusty harbor
Why are 2 and 4 wrong?
it tells you which ones are correct ?
nvm i just learned how to read
zealous sphinx
for the 5th one,
$b_k = \frac{\ln (k+1)}{k}$
$\ln (k)$ grows, by definition, exponentially slower than $k$ so $b_{k+1} < b_k$ for all $k$ and similarly the limit is 0
and the series follows the form $(-1)^{k+1}b_k$
boreal girderBOT
Karma
zealous sphinx
so it should be yes
pure spear
Shouldn't it decrease by each k
zealous sphinx
it does, thats what its supposed to do 
pure spear
Didn't you say it grows but exponentially slower
zealous sphinx
if it increased the series would diverge to infinity
zealous sphinx
Didn't you say it grows but exponentially slower
it does, which means the denominator gets bigger faster than the numerator
pure spear
So it decreases?
zealous sphinx
1/10, 1/100, 1/1000, 1/10000
these numbers get smaller
because the denominator is bigger
zealous sphinx
2 doesnt alternate so the answer should be no
4 i lowkey dont know limits with trig functions are one of the things i forgot how to do. but, my instinct tells me that the answer should be yes, not no
because for negative values of cos it will tend towards -0 and for positive it will tend towards +0
so the limit is 0, probably
gleaming latch
2 doesnt alternate so the answer should be no
4 i lowkey dont know limits with t...
4 also doesnt alternate
zealous sphinx
cos doesnt flip signs alternately when you increment k by 1 but (-1)^k does so i figured it would alternate
pure spear
How about this
I try both
Because I have many attempts left
And let you guys know
gleaming latch
cos doesnt flip signs alternately when you increment k by 1 but (-1)^k does so i...
Cos(kpi)=(-1)^k, for k in N
And obviously (-1)^k*(-1)^k=1
Thus doesnt alternate
compact pewterBOT
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@pure spear Has your question been resolved?
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pine barn
i need to know if this integral can be done doing a simple u-sub (multiplying by 1/3 to make the sub work). That's what ChatGPT did...i'm just wondering because I solved it and used polynomial division, then partial fractions and finally completed the square. It would be good to know if a simple trick could save me all those steps
compact pewterBOT
i need to know if this integral can be done doing a simple u-sub (multiplying by...
Someone else is already using this help channel. If you need help with a question, please open your own help channel/thread (see #❓how-to-get-help for instructions).
thick comet
I think whatever ChatGPT did doesn't work, you kinda need the partial fractions
pine barn
I guess because if I wanted to do this, the 3 would also need to multiply the -10? still not making the derivative
thick comet
Yeah that's why it fails in reality
pine barn
damn, I will drop diff. equations if i get an integral like that in my exam. I already had to make the substitution of a homogeneous differential equation and now use 4 integration methods
thank you for the help
.close
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timber latch
How do I take the derivative of 2-2cos2bx
compact pewterBOT
How do I take the derivative of 2-2cos2bx
potent berry
How do I take the derivative of 2-2cos2bx
by taking the derivative
some chain rule is involved
thats all
timber latch
Ik it comes out to 4bsin(sbx) but isk hiw
potent berry
chain rule.
kind viper
do you know chain rule
timber latch
Yes
kind viper
also just to confirm
kind viper
you're taking the derivative of $2 - 2\cos(2bx)$, correct?
boreal girderBOT
Ann
kind viper
,rccw
boreal girderBOT
timber latch
see ann's thing
?
kind viper
not good on two counts
potent berry
i wish you hadnt used $\times$ there but that's just aesthetic
boreal girderBOT
Percy
kind viper
- left hand side should say [f(g(x))]'
- don't use the same symbol for the letter x and for multiplication
potent berry
who even uses x for multiplication?
potent berry
1. left hand side should say **[**f(g(x))**]'**
2. don't use the same symbol for...
ah i missed 1
knotty basin
Ik it comes out to 4bsin(sbx) but isk hiw
is that even correct
kind viper
well, the 2 is clearly a constant term and so disappears.
timber latch
I write in arabic and no symbol looks like a x
timber latch
is that even correct
I went on a derivative calculator
potent berry
I went on a derivative calculator
they should explain stuff pretty well tbh
timber latch
what is your inner function and what is your outer function
Inner is 2bx
potent berry
but okay
potent berry
There isn't any explanation
use https://www.derivative-calculator.net/ from next time
potent berry
IT GIVES STEPS-
kind viper
Inner is 2bx
yes, and the outer is the cosine.
timber latch
IT GIVES STEPS-
I no understand it
kind viper
f(t) = -2cos(t) and g(x) = 2bx.
can you take the derivatives of these two in isolation
timber latch
How'd we get 4b
kind viper
can you take the derivatives of these two in isolation
ok
let me phrase that as an instruction
knotty basin
did you take the derivatives yes that
kind viper
find f'(t). then, find g'(x).
timber latch
did you take the derivatives yes that
No
kind viper
find these both, then show us what you get.
timber latch
kind viper
,rccw
boreal girderBOT
kind viper
right.
so you left the -2 coefficient alone.
so once you put all that together, you will get: $$-2 \cdot -\sin(2bx) \cdot 2b$$
boreal girderBOT
Ann
timber latch
Yes it's a constant so that goes away
kind viper
does this now make you see where the 4b shows up
kind viper
Yes it's a constant so that goes away
no, it's a constant multiplier so it stays.
the other 2 that was just added is what goes away.
timber latch
Is there any way to dumb that down for me
I'm bad with trig derivatives
Oh oh ohh
I see it
kind viper
you literally did all of the trig stuff yourself
scarlet moat
how would I write this interms of Integrals over the real line?
compact pewterBOT
how would I write this interms of Integrals over the real line?
scarlet moat
I mean is the d^2z just dxdy or is it something else
compact pewterBOT
@scarlet moat Has your question been resolved?
scarlet moat
<@&286206848099549185>
quartz yoke
??
fair raft
I mean is the d^2z just dxdy or is it something else
i think yes
next latch
If we want to keep this in exact form, what would the answer be? I don't get how it is 2(pi)^2
compact pewterBOT
If we want to keep this in exact form, what would the answer be? I don't get how...
next latch
I thought it would be -2(pi)^2
thorny urchin
the ^2 applies to the -
you can't factor the - out of that term being squared like that
(because order of operations)
you can view this as
$$2 \times (-\pi) \times (-\pi)$$
boreal girderBOT
ℝαμOmeganato5
thorny urchin
the - cancel
next latch
oooo
I see
ok thank you!
that makes sense, its better to look at it the way you showed
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compact pewterBOT
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compact pewterBOT
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next latch
I am finding the derivative for a trig function, and the last step I got was
-x^2cosx - 2xsinx - 2xsinx + 2cosx
I am trying to simplify this. Is it possible to join the two - 2xsinx - 2xsinx together, and you get -4xsinx?
regal storm
yes
next latch
Farhan it's open already
limber quartz
by the way can anyone please help me
Im in grade 7
my exam after a month
and i dont know most of the things
regal storm
You can look at it like this:
$-2x \sin(x) - 2x \sin(x) = x \sin(x)(-2-2) = x \sin(x)(-4)=-4x \sin(x)$
Open your own help channel
but $sin(x)=4$ isn't possible for real numbers
regal storm
No worries good luck
next latch
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compact pewterBOT
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late current
Need help with bayes theorem
compact pewterBOT
Need help with bayes theorem
late current
this is what it says in my one sheet, but I know it as:
most places ive seen this its:
P(A | B) = P(A) x P(B | A)
---------------
P(B)
I'm really looking for an explanation on why it could be listed this way (if the first way is also correct)
proud halo
it's the law of total probability
proud halo
most places ive seen this its:
P(A | B) = P(A) x P(B | A)
---...
applied on the denominator here
late current
meaning?
late current
yes
and i understand bayes rule as a specific kind of conditional probability
"probability of x event given that y has occurred"
proud halo
yeah so basically the law of total probability says that the probability of an event is the sum of its probabilities on a countably many pairwise mutually exclusive events
it's kind of common sense if you think about it
late current
yes
proud halo
it's often expressed this way because it's easier application wise
more of a "plug and chug" ordeal
instead of having students weeping about what P(B) is
late current
oh wow omg
that's weird, i feel you
and so x here is the posterior
in this first version i posted
proud halo
"posterior"? sorry i'm not familiar with that terminology
late current
ahhh gotcha
the independent first probability that you're effectively setting A occurring against
actually i got it
tysm!
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zinc kelp
compact pewterBOT
zinc kelp
can someone explain why the bottom expression isn't centered?
\center Let's do the left term first
\begin{gather*}
-\frac{1}{n}\sum_i \partialderiv{w} [t_i log(\sigma(w^Tx_i)) + (1 - t_i)log(1 - \sigma(w^Tx_i))] \\
= -\frac{1}{n}\sum_i [t_i \frac{1}{\sigma(w^Tx_i)}\sigma'(w^Tx_i) + (1 - t_i) \frac{1}{1 - \sigma(w^Tx_i)}(1 - \sigma'(w^Tx_i))] \\
= -\frac{1}{n}\sum_i [t_i \frac{1}{\sigma(w^Tx_i)}\sigmoidderiv{w^Tx_i} x_i - (1 - t_i) \frac{1}{1 - \sigma(w^Tx_i)} \sigmoidderiv{w^Tx_i}] x_i + \frac{\lambda}{2}\|w\|_2^2 \\
\end{gather*}
all of my expressions are inside the same gather block
im genuinely so confused
dusk apex
compact pewterBOT
dusk apex
HELP!
proud halo
is it 6! - 5! btw?
compact pewterBOT
@dusk apex Has your question been resolved?
dusk apex
is it 6! - 5! btw?
yeah
compact pewterBOT
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acoustic dome
i was doing the derivative of y = 5^(sin5x) and i got the wrong answer on my first try. on my second try i got the correct answer but i dont know why my first solution didnt work. does anyone see what i did wrong?
compact pewterBOT
i was doing the derivative of y = 5^(sin5x) and i got the wrong answer on my fir...
delicate musk
this is a example within group actions under conjugation, can anyone explain why it holds or why its important?
compact pewterBOT
this is a example within group actions under conjugation, can anyone explain why...
@delicate musk Has your question been resolved?
floral terrace
conjugating a permutation is the same as relabelling the numbers
if you think about the components of tau
first you apply g^-1, which is relabeling
then you apply sigma to the relabeled numbers
then you apply g to go back to the original labeling
delicate musk
@floral terrace sorry i just got back
okok wait let me slow this down
they dont have the same decomp tho
(14)(23) is not the same as (12)(34) right
floral terrace
same decomp means
if you split it into cycles
and take the lengths of all the cycles
you get the same list of numbers
delicate musk
so just two cycles
delicate musk
why is this important tho
like waht does it add to the question
i dont think this is needed to prove congigation is an action right?
floral terrace
ohh
this is more of an aside
he is using permutations as an example to show the structure of the conjugation action
but this isn't related to why G acts on itself by conjugation per say
delicate musk
he is using permutations as an example to show the structure of the conjugation ...
ok back to this
what does this mean by structure
like why is it important that they need to have the same decomp to have the same structure
floral terrace
it is an example of how conjugation preserves the "structure" of an element
if i gave you the permutation (12) or permutation (13)
they are basically the same thing
swapping 2 numbers
delicate musk
ok
floral terrace
just different labels
boreal girderBOT
BOSS
delicate musk
then $\tau = (13)(12)(34)(31)$ as $(13) \in S_4$
boreal girderBOT
BOSS
delicate musk
then we can see $\tau = (14)(23)$ right
boreal girderBOT
BOSS
floral terrace
is this for g = (13)?
delicate musk
yes
delicate musk
hmmm ok
my question is
how is this preserving hte structure
is it just because it is swapping the same num of elements
like what do i notice here about group actions?
floral terrace
conjugation and group actions are kind of two separate points here
the connection is just
conjugation is an example of a group action
and then permutations are used as an example of conjugation
delicate musk
hmm
ok ill go a but further in and come back to it, im still confused as to why knowing that is important
floral terrace
conjugation is an important concept on its own
have you learned about orbits of group actions?
delicate musk
yes but im revising
I understand conjugation
i dont understand why the mermutation part was worth noting
floral terrace
i dont understand why the mermutation part was worth noting
i guess it is just another example
if it doesn't make sense then don't worry about it too much right now
delicate musk
@floral terrace whenever we say a group acting on itsself
do we always mean congigation?
also, in the previous example, was that (13) acting on $\sigma$
boreal girderBOT
BOSS
delicate musk
oh
@floral terrace groups with the same cycle decomp are an orbit
thats why its important
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quaint galleon
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quaint galleon
Have I graphed this vector correctly?
Or atleast tried to
the points are on the top right, P1,P2, and P3
regal storm
No
P3 maybe depednng on the scales , P1 and P2 no
P1 should be at least negative on the Z axis (underneath the origin) and P2 should at least be negative on the X axis (to the right of the origin)
quaint galleon
Do you know any website that can graph or show visuals for these?
regal storm
No sorry
quaint galleon
Its fine, I will try graphing it again then
regal storm
I'd start by just plotting the horizontal and vertical coordinates, then the 3rd one
with this layout that'd mean plotting [x,z] and then 'shifting' it by y
quaint galleon
So I plot X,Z like im plotting X,Y in a 2D grpah but when I plot Y I just shift up/down?
regal storm
just shift along the actual line Y
so diagonally
In this example i'm plotting [1,2,3] - First I plot [1,3] as if it's a regular 2D graph, so 1 on the horizontal axis and 3 on the vertical axis. Then I shift it along the diagonal 2 points in the positive (up-right) directino to get [1,2,3]
quaint galleon
Hm I somewhat get the idea(?) when you "shift" does it like go x to the right and y up by 2?
compact pewterBOT
@quaint galleon Has your question been resolved?
quaint galleon
Here is my second attempt
Is this more accurate?
Actually on a second thought, this cant create a triangle something is wrong with it
I think p2 should be on the other side
like this:
regal storm
How have you gotten p2 there?
just the xz coordinates are [-11,87]. So it should be very high up on the Z axis
regal storm
Hm I somewhat get the idea(?) when you "shift" does it like go x to the right an...
stop thinking x and y. In a 2D graph, when you move something from say [1, 2] to [1,4] you make a shift of positive 2 in the y direction. you do this by, for example, placing your pen on the point and moving it directly upwards 2 units in the y direction.
similarly here, if you want to shift along the y direction you can place your pen on the point and move it however many units in the appropriate direction. that direction will just be a diagonal rather than directly vertical or horizontal
quaint galleon
Yeah thats why I put the white annotation there
regal storm
the white annotation is even worse
quaint galleon
Oh
regal storm
as Z increases, it moves upwards. not downwards
quaint galleon
what about p1? Is it any better than before?
quaint galleon
Alright I will do a third attempt real quick
I think I pixed p2 now, and I made an error on p1 I will fix that now
I put an X on what I think is wrong and put an arrow to indicate where i replaced it to
Nvm its messed up
atleast I think it is
I think this should be "alright" or something like that
I used a new graph thing Im not too sure if the result is the same or any improvements
Imma just post this in threads
compact pewterBOT
@quaint galleon Has your question been resolved?
hexed dock
Hello
compact pewterBOT
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
runic iron
Let's start at 2 numbers where n is the start of the sequence:
Then 1225 = n+n+1=2n+1
So n=612
4 numbers:
4n+(1+2+3)=1225
4n=1225-(1+2+3)
4n=1225-(6)=1219 which is not divisible by 4
6 numbers:
6n+(1+2+3+4+5)=1225
and so on u get the idea
now generalize this to some even 2k numbers for k in natural numbers
(2k)n+(the sum from 1 to 2k-1)=1225
(2k)n=1225-(1/2)(2k-1)(2k)
how many integer solutions does this equation have
compact pewterBOT
@hexed dock Has your question been resolved?
hexed dock
Let's start at 2 numbers where n is the start of the sequence:
Then 1225 = n+n+1...
So k is 1,5,7 and 25?
compact pewterBOT
@hexed dock Has your question been resolved?
runic iron
So k is 1,5,7 and 25?
You know that $2n=\frac{1225}{2}-(2k-1) \geq 0\$ is even
So you have an odd number minus an odd number. So $\frac{1225}{k}$ is odd.\
\
Write out the prime factorization of 1225: $1225=5^2*7^2\$
The candidates for k are $1,5,7, 5*7, 5*7^2, 5^2,5^2*7,5^2*7^2, 7^2\$
\
For each candidate ask urself:
a) is 1225/k odd?
b) is 1225/k greater than or equal to 2k-1
\
1: Yes and yes\
5: Yes and yes\
7: Yes and yes\
\
now note that $35^2=1225$. so for any $k \geq 35$:\
So $1225/k$ will be less than or equal to k which means it cant be greater than $2k-1$\
\
so the only k's that work are 1,5,7 (not 25)
really the first question is redundant because all of the factors are odd lol
so if u wanna simplify this process, ur just looking for when 2k-1 is greater than 1225/k
hope this makes sense
boreal girderBOT
1013
runic iron
i wasnt sure if you guys have been introduced to the sum of the first k natural numbers formula yet or not
lemme know if u need the solution without relying on that
hexed dock
So the ans is just 3?
runic iron
So the ans is just 3?
yes
runic iron
ofc
runic iron
For your help
actually there is a flaw
for k=25
it's true that 1225/25 >= 2(25)-1
still u get that n=0
and the sum of the first 50 numbers is not 1225
so it doesnt change the answer but my reasoning i posted was flawed
compact pewterBOT
@hexed dock Has your question been resolved?
compact pewterBOT
hexed dock
Can you check if the way I wrote it is correct or not
hexed dock
.
hexed dock
2 10 14
hexed dock
I mean what is k
1 5 7
runic iron
I mean what is k
can do it without k:
mn+m(m+1)/2 = 1225
in this case m=2k, k positive int
compact pewterBOT
@hexed dock Has your question been resolved?
compact pewterBOT
zenith moat
No, you need to check for convergence first before integrating normally.
compact pewterBOT
@spark ermine Has your question been resolved?
slender star
hi
slender star
.close
compact pewterBOT
slender star
nvm
quartz yoke
<@&268886789983436800> check john text logs
winged lion
pretending to be a mod too lol
worn loom
Username
winged lion
i referred to Johnny
zenith moat
Before integrating, check for convergence by examining the limits and behavior of the function. If the limits are infinite or the function has singularities, test if the integral leads to a finite value using limits. If the result is finite, the integral converges; if not, it diverges.
does that help?
compact pewterBOT
Available help channel!
carmine garden
\textbf{Q} Let's call a sequence $(x_n)$ zero-heavy if there exists $M \in\N$ such that for all $N \in \N$, there exists $n$ satisfying $N \leq n \leq N+M$ where $x_n=0$
\
\begin{enumerate}
\item Is the seqeunce $(0,1,0,1,0,1,\dots)$ zero -heavy?
\item If a sequence is zero-heavy does it necessarily contain an infinite number of zeros? If not provide a counter example.
\item If a sequence contains an inifnite number of zeros, is it necessarily zero-heavy? If not provide a counterexample.
\item Form the Logical negation of the above definition.
\end{enumerate}
\textbf{Answers}
\
\begin{enumerate}
\item The given sequence is zero-heavy. $M=2$
\
\t proof: The given sequence can be defined as $x_{2n-1}=0, x_{2n}=0$. Let $N$ be odd. Adding $2$ to $N$. gives the successive odd number. However, there's an even number between them . Now let $N$ be even , adding $2$ to $N$ gives an even number. there are 2N(s)( N, N+2) for which $(x_n)=0$. Therefore , the sequence is zero-heavy.
\item No, no such sequence exists
\
\t Proof: We procced by contradiction. Let there be a finite number of zeros. Let the last time '$0$' appears be at $N= \alpha$. Then note that for all $N > \alpha$, there doesn't exist a $M \in \N$, such that for $\N \leq n \leq N+M$, $a_n=0$
\item Yes. It's necessarily zero-heavy.
\
\t Proof:As the sequence contains an infinite number of $0$(s), for any $0$, at say position $M$, there is a $0$, at some finitely distant ( say distance M) units from it. Thus making it zero-havy
\end{enumerate}
compact pewterBOT
\textbf{Q} Let's call a sequence $(x_n)$ zero-heavy if there exists $M \in\N$ su...
carmine garden
I hit enter by mistake, sorry
timber cipher
Yeah those two answers track for me
boreal girderBOT
What a wonderful world !
timid lodge
proof for (2) looks good
timid lodge
i'm not sure what an RA course is but the wording seems fine to me
carmine garden
i'm not sure what an RA course is but the wording seems fine to me
Real analysis
brazen quiver
Looks good to me
timid lodge
Real analysis
yeah I'd say that proof is good, the proof for 1 is maybe a lil janky tho
brazen quiver
Well, you don't actually show the contradiction explicitly.
carmine garden
I ask because people have told me I'm not using enough quantifiers
brazen quiver
But it's pretty easy to infer.
carmine garden
Well, you don't actually show the contradiction explicitly.
Hmm?
brazen quiver
As in you don't explicitly use the fact that you are have a contradiction. In fact, you probably don't even need it
Let (a_n) be some sequence with finitely many zeroes. We can prove that this sequence is affirmatively not zero-heavy.
Therefore by contrapositive if a sequence is zero heavy then it must have infinite zeroes
Cleaner
(Note this is a proof using the contrapositive, not a proof by contradiction)
carmine garden
Ah
yea
thanks
Yes. It's necessarily zero-heavy.
\
\t Proof:As the sequence contains an infinite number of $0$(s), for any $0$, at say position $M$, there is a $0$, at some finitely distant ( say distance M) units from it. Thus making it zero-heavy.
boreal girderBOT
What a wonderful world !
carmine garden
Proof for the third sub part
carmine garden
The key is that M is some finite fixed value
ah
brazen quiver
If we let (a_n) = 0 whenever n is a square, and (a_n) = 1 otherwise, then this sequence is not zero heavy because for any M we choose we can find two successive squares such that their difference is larger than M
carmine garden
If we let (a_n) = 0 whenever n is a square, and (a_n) = 1 otherwise, then this s...
So M is fixed for the entire seqeunce
brazen quiver
Yes
carmine garden
okie
thanks
last sub question now
A sequence $(x_n)$ is not zero-heavy , if there exists $N \in \N$ , for all $M \in \N$, for all $n \in \N$, such that $N \leq n \leq N+M$, where $x_n \neq 0$
carmine garden
for all n too?
boreal girderBOT
What a wonderful world !
brazen quiver
Or I guess
For each M, there exists an N such that for all n
Because different M will need different N
carmine garden
a sequence $(x_n)$ is zero-heavy if there exists $M \in\N$ such that for all $N \in \N$, there exists $n$ satisfying $N \leq n \leq N+M$ where $xn=0$
boreal girderBOT
What a wonderful world !
brazen quiver
A sequence $(x_n)$ is not zero-heavy if for each $M \in \bN$ there exists an $N \in \bN$ such that for all $n$ satisfying $N \le n \le N+M$, we have $x_n = 0$.
boreal girderBOT
OmnipotentEntity
brazen quiver
(This is how I would write it.)
The idea is that you can pick any M
And give it to me
And I can give you back some N that satisfies this condition
And if I can do that, it's not zero-heavy
carmine garden
I think I should have written it in its logical form before translating it tbh
compact pewterBOT
compact pewterBOT
Available help channel!
past coral
If (a,b) = 1 and n is a prime number, prove that (a^n + b^n)/(a+b) and a + b have no common factor unless a + b is a multiple of n.
compact pewterBOT
If (a,b) = 1 and n is a prime number, prove that (a^n + b^n)/(a+b) and a + b hav...
kind viper
and n is a prime number
any reason why you called it n and not p
past coral
kind viper
hm.
so that's how your book did it.
im still kind of itching to rename it to p anyway tbh
cause it's introduced as a prime lol
knotty basin
😭
kind viper
anyway so it says essentially to prove two things:
- if (a,b) = 1 and a+b isn't a multiple of p, then (a^p+b^p)/(a+b) and a+b are coprime
- if (a,b) = 1 and a+b is a multiple of p, then (a^p+b^p)/(a+b) and a+b are not coprime
have you made any progress on this
compact pewterBOT
compact pewterBOT
Available help channel!
carmine garden
\bf{Let $x_n \geq 0 \forall n \in \N$}
\
Show
(a) if $(x_n) \to 0$, show that $\sqrt{x_n} \to 0$
\
(b) If $(x_n) \to x$, show that $\sqrt{x_n} \to \sqrt{x}$
\
\bf{Answers}
(a)
\t proof: As $(x_n)$ converges to $0$, it follows that
\
$\abs{x_n}< \varepsilon^2$
or $(\sqrt{x_n})^2<\varepsilon^2$
\
Thus $\sqrt{x_n}<\varepsilon$.
\
(b)
\t Proof: We have $(\sqrt{x_n} - \sqrt{x})( \sqrt{x_n+ \sqrt{x}) < \varepsilon$
\
$(\sqrt{x_n} -\sqrt{x})< \frac{ \varepsilon}{\sqrt{x_n} + \sqrt{x}}$
compact pewterBOT
\bf{Let $x_n \geq 0 \forall n \in \N$}
\\
Show
(a) if $(x_n) \to 0$, show that $...
carmine garden
I FORBID YOU FROM ASKING
huh?
bleak dock
rules of the server
inland nacelle
...
boreal girderBOT
What a wonderful world !
Compile Error! Click the 
reaction for more information.
(You may edit your message to recompile.)
inland nacelle
\bf{Let $x_n \geq 0 \forall n \in \N$}
\\
Show
(a) if $(x_n) \to 0$, show that $...
At this time?
inland nacelle
Theres only one other chnnel
flint wing
dont be sorry wai u must be better
inland nacelle
$Womp^2$
boreal girderBOT
Carbonite
carmine garden
Should I close this
\bf{Let $x_n \geq 0\ \forall n \in \mathbb{N}$}
\
Show:
(a) If $(x_n) \to 0$, show that $\sqrt{x_n} \to 0$
\
(b) If $(x_n) \to x$, show that $\sqrt{x_n} \to \sqrt{x}$
\
\bf{Answers}
(a)
\quad Proof: As $(x_n)$ converges to $0$, it follows that
\
$\lvert x_n \rvert < \varepsilon^2$
or $(\sqrt{x_n})^2 < \varepsilon^2$
\
Thus $\sqrt{x_n} < \varepsilon$.
\
(b)
\quad Proof: We have $(\sqrt{x_n} - \sqrt{x})(\sqrt{x_n} + \sqrt{x}) < \varepsilon$
\
So, $(\sqrt{x_n} - \sqrt{x}) < \frac{ \varepsilon }{ \sqrt{x_n} + \sqrt{x} }< \frac{\varepsilon}{\sqrt{x}}< \varepsilon$
\
Therefore
\
$\abs{\sqrt{x} - \sqrt{x_n}}< \varepsilon$
\
Thus it converges to $\sqrt{x_n}$
I've got to bound $\sqrt{x_n} + \sqrt{x}$ now
compact pewterBOT
compact pewterBOT
Available help channel!
urban glen
how do I solve this? I'm kinda stuck
compact pewterBOT
how do I solve this? I'm kinda stuck
potent berry
pseudo rapids
the thing in blue is wrong
kind viper
you tried to do implicit differentiation but messed it up quite a lot
the $2yy'$ bit is correct, but the derivative of $(xy+1)^3$ is nowhere near $3(y'+1)^2$
boreal girderBOT
Ann
urban glen
how do i do it right?
potent berry
how do i do it right?
by using chain rule 👀
kind viper
use chain and product rules properly
urban glen
which one should be done first?
kind viper
here's the first step, using chain rule with the cube as the outer function: $$[(xy+1)^3]' = 3(xy+1)^2 [xy+1]'$$
boreal girderBOT
Ann
kind viper
(note i defer the calculation of the derivative of (xy+1) to the next step)
urban glen
is it 3(xy+1)^2 (xy' + y)
urban glen
3(xy+1)^2 <-- this part is differentiated and i differentiated (xy+1) to get (xy' + y)
potent berry
3(xy+1)^2 <-- this part is differentiated and i differentiated (xy+1) to get (xy...
yeah
how,
oh wait
WHOOPS
SORRY
THATS FINE
certified brainfog moment
you're good
compact pewterBOT
compact pewterBOT
Available help channel!
forest token
compact pewterBOT
potent berry
forest token
can someone tell me what does A into A transpose = I signify
potent berry
can someone tell me what does A into A transpose = I signify
Orthogonal matrix.
forest token
yeah but does it help in the question ?
like do we actually do the multiplication ?
forest token
i hate matrix multiplication
so i try to avoid it whenever possible
but in this i cant ig
okay ty
.close
compact pewterBOT
short ferry
but in this i cant ig
there is a pretty nice pattern in this case
compact pewterBOT
✅
forest token
oh in this case A transpose = A
short ferry
idk what you mean but the multiplication is gonna end up being extremely nice
short ferry
oh in this case A transpose = A
thats also true
forest token
idk what you mean but the multiplication is gonna end up being extremely nice
i meant that in determinant form this would be = -(a^3 + b^3 + c^3 - 3abc)
short ferry
i meant that in determinant form this would be = -(a^3 + b^3 + c^3 - 3abc)
Yeah, that should be the determinant of A
-2 + 3abc
potent berry
btw @forest token if you need some fancy method, the determinant of this form is pretty much something you end up memorising for mains
short ferry
idk how helpful that is tho, im honestly very rusty with matrices
potent berry
so you can treat A transpose as A inverse
and do adj(A)/det(A) nonsense
not worth it tho
forest token
oh ok i got it
a^2+b^2+c^2 = 1
ab+bc+ca = 0
i can take it from here
forest token
btw <@728482160927047742> if you *need* some fancy method, the determinant of th...
yeah i remember..btw how was ur mains?
forest token
and do adj(A)/det(A) nonsense
wont that make it harder? ill have to find adj(A)
potent berry
wont that make it harder? ill have to find adj(A)
yeah not worth it lol
forest token
im sorry man
drifting loom
(I think i translated the question to english right) how do you rewrite sin(3x) to be expressed with sinx?
compact pewterBOT
(I think i translated the question to english right) how do you rewrite sin(3x) ...
potent berry
(I think i translated the question to english right) how do you rewrite sin(3x) ...
do you know what sin(x+y) is
thorny urchin
it'll be one of the first steps
drifting loom
yeah realized i learne about it on second touht. just didnt click with me first beacuse im used to using the expression u+v
carmine garden
compact pewterBOT
carmine garden
I first re-write this as $\int_{C^{+}} (x+y^2,2xy) \cdot (dx,dy)$
boreal girderBOT
What a wonderful world !
solid bison
omg its what a wonderful world progressively Understanding Analysis
carmine garden
omg its what a wonderful world progressively Understanding Analysis
Unfortunately this isn't analysis 😭
potent mesa
omg its what a wonderful world progressively Understanding Analysis
this is more clac 3
proud halo
if the capitalisation is a nod to anything...
carmine garden
if the capitalisation is a nod to anything...
ah
abbott
❤️
I love abbott
hmm, using green's theorm
I'm wondering if I consider an arbitrary closed path
Consider a closed path compsed of two paths $C_1. C_2$
boreal girderBOT
What a wonderful world !
proud halo
sure that works
carmine garden
$\int_{C_1} (x+y^2) dx + 2xy dy = \iint_{D} 2y - 2y dA = \iint_{D} 0 dA =0$, as the integral along a closed path is $0$, it follows that the line integral is path independent
boreal girderBOT
What a wonderful world !
proud halo
your notation is funky
but alright
you still haven't made the conclusion yet
the first integral is over your combined closed path consisting of both C1 and C2
also see: parentheses
carmine garden
the first integral is over your combined closed path consisting of both C1 and C...
Yea, but there's a theorm( I don't think it has a name), states that if the integral along all closed paths between two points is 0, then the line integral is path independent
proud halo
**What a wonderful world !**
and besides you're meant to use Green's theorem
carmine garden
and besides you're meant to use Green's theorem
I did?
I converted it to a double integral using green's theorm
proud halo
Yea, but there's a theorm( I don't think it has a name), states that if the inte...
this doesn't use it
also spoilers that statement comes from Green's theorem
proud halo
well you can derive it from Green's theorem
pliant quarry
also spoilers that statement comes from Green's theorem
||dont we do it by saying integral along path 1 + integral along path 2 = 0 => integral path = -integral path 2||
proud halo
@carmine garden Consider a closed path C comprising of C^+ and C_1 which runs back from the end points of C^+
So by Green's theorem you have this integral turns out to be 0
And split the line integral over these two paths
carmine garden
But what my point was that as the line integral is 0 around all closed curves ( which was ascertained using green's, we can conclude that the line integral is path independent
proud halo
But what my point was that as the line integral is 0 around all closed curves ( ...
all you proved is that the line integral going from 0,0 to a,b and BACK to 0,0 is 0
carmine garden
all you proved is that the line integral going from 0,0 to a,b and BACK to 0,0 i...
yea
proud halo
you did not prove any path independence; C^+ never came up
carmine garden
right
proud halo
And split the line integral over these two paths
tell me if you're able to develop this, if not i'll help
carmine garden
And split the line integral over these two paths
so the line integral along the foward path is equal to that alng the backward path
carmine garden
tell me if you're able to develop this, if not i'll help
that I can do
proud halo
so the line integral along the foward path is equal to that alng the backward pa...
it's equal to the NEGATIVE of the back path
which is equivalent to going back from 0,0 to a,b
proud halo
||dont we do it by saying integral along path 1 + integral along path 2 = 0 => i...
read this
proud halo
And split the line integral over these two paths
which I reiterated
proud halo
what do you have now then?
carmine garden
$\int_{C^+} f+ \int_{C^{-}} f=0$
boreal girderBOT
What a wonderful world !
proud halo
Is C- a different arbitrary path from a,b to 0,0?
carmine garden
yea
boreal girderBOT
dyxn
carmine garden
ooh
proud halo
I hate your textbook's notation
carmine garden
okay
proud halo
😄
proud halo
When I took the course last semester, my university had me use Thomas
carmine garden
I didnn't know thoma's coverd all this imo
anyway, thanks a ton again!
Can I close this now, got to eat
proud halo
I didnn't know thoma's coverd all this imo
its got everything from defining what a function is to the very end :P
compact pewterBOT
Available help channel!
timber latch
How do I take the common factor of x from this:
×^(2÷3) + ײ
compact pewterBOT
How do I take the common factor of x from this:
×^(2÷3) + ײ
native shard
smallest power
timber latch
Let's say I wanna take x as the common factor
alpine canyon
been a while since i saw a division symbol
timber latch
Whats x^2÷3 become
kind viper
×
did you put a multiplication sign instead of the letter x
timber latch
been a while since i saw a division symbol
I don't have roots on my keyboard
small mason
just write sqrt
kind viper
did you mean $x^{2/3} + x^2$
timber latch
> ×
did you put a multiplication sign instead of the letter x
I think so
boreal girderBOT
Ann
alpine canyon
use /
kind viper
you can factor out x^{2/3} much more easily than you can x^1.
small mason
,,x^{2\over3} + x^2
boreal girderBOT
boreal girderBOT
timber latch
you're dividing this by x correct?
Yes
small mason
so first you distribute, then you need to know how to divide the first term
timber latch
(2÷3)-3÷3
small mason
,,\frac{x^{2\over3}}{x} + x
boreal girderBOT
timber latch
Whats wrong
small mason
(2÷3)-3÷3
yes, that's one way to do it
the answer isn't 1/3 like you said a while ago
timber latch
-1/3
small mason
yes
timber latch
Thanks
small mason
,,=x^{{2\over3} - 1} + x
boreal girderBOT
timber latch
.close
compact pewterBOT
compact pewterBOT
Available help channel!
next latch
I am confused with the highlighted parts
compact pewterBOT
I am confused with the highlighted parts
boreal girderBOT
Ann
next latch
Ohh right right
kind viper
$\ln(e^a) = a$
boreal girderBOT
Ann
kind viper
log_e(e^a)
next latch
So then the two es just cancel out with each other?
And we are left with a?
So like
native shard
So then the two es just cancel out with each other?
please don't tell me you think of _e as /e
next latch
log_e(e^a)
The e in the subscript and the e in the base or whatever that's called
native shard
or whatever you're thinking
next latch
Noo but don't then cancel out?
native shard
log_e(e^a)
The e in the subscript and the e in the base or whatever that's call...
$\log_b(x) = y \iff b^y = x$