otherwise the question makes no sense because a point cannot be parallel to a line.
#help-39
honest vigil
wooden flare
honest vigil
ah yeah more or less the same thing I said then.
you missed out on the word 'and' in your writing.
basically what you want to do is join P to each point separately. only one of these points will form a line with P such that it's parallel to the z-axis.
honest vigil
well you can prove it algebraically, I hope? if you want to confirm, that is.
wooden flare
wooden flare
well you can prove it algebraically, I hope? if you want to confirm, that is.
yeah... I totally do ehem
.close
pearl pondBOT
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magic ruin
how to guess that a vector is a normal vector
pearl pondBOT
how to guess that a vector is a normal vector
tropic saddle
wdym guess
strange fox
if it is perpendicular to the surface at a given point you mean?
like Az + By + Cz = D?
magic ruin
yes..why do we repressnt it like n =《a,b,c》
verbal whale
Wdym why?
honest vigil
that's just the notation for any vector, yes?
verbal whale
yes..why do we repressnt it like n =《a,b,c》
And also this doesn't have to do with it being normal
magic ruin
please,first clarify me what is a plane
verbal whale
...
magic ruin
then i will follow up with more questions
strange fox
so are you working on a problem or something
cause this is a little confusing to help with no idea what ur trying to do
verbal whale
!1q
pearl pondBOT
Please stick to one question at a time to help others follow the conversation better and prevent confusion/crossed answers. Once it has been answered, you may send your next question.
magic ruin
so are you working on a problem or something
no..just learning theory right noe
strange fox
might want to watch a video first
magic ruin
the more i am moving forward the less i understand
austere tinsel
In 3d space, a plane is a 2d subspace
magic ruin
might want to watch a video first
i was watching one...
strange fox
thats just mathematics in general ngl
verbal whale
i was watching one...
And?
austere tinsel
It's like an infinitely flat piece of paper that never bends
verbal whale
Yes
austere tinsel
A vector is normal to a plane if it is pointing directly away from the plane
magic ruin
and ...then i understood that a normal vector is a 90° vector
verbal whale
What do you mean with "90° vector"?
verbal whale
You don't know what 90° is?
magic ruin
You don't know what 90° is?
i know
magic ruin
https://upload.wikimedia.org/wikipedia/commons/thumb/a/a5/Normal_vectors2.svg/12...
okay..and is this always present in 3d space?
magic ruin
bro..you guys are roasting me
austere tinsel
Well every plane has vectors that are normal to it
But different planes might have different normal vectors
austere tinsel
Sure thing
pearl pondBOT
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trail linden
matrixes, if i have a 4 x 2 and a 2 x 6 matrix, the products will be stored in a 4 x 6 matrix?
(rows) x (columns)
fluid axle
yes indeed
trail linden
wait
for multiplying matrices, is it always row for the first matrix and column for the second matrix?
<@&268886789983436800>
fluid axle
for multiplying matrices, is it always row for the first matrix and column for t...
Yea
Fkin spammers
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viscid shale
flint basalt
ask google for practice problems and if you get stuck then ask for help here
main oriole
listen man i too wanna try to find some logical application for the lagrangian
but this channel isn't for that
flint basalt
!done
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tropic jay
Pls help
tropic jay
Ye?
ivory wasp
they said three girls
shell brook
three girls?
tropic jay
Pls help
shell brook
oh its part of the question
tropic jay
I did some construction too
Plij help guys 
Some help would be appreciated chat
.close
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brisk steeple
Let e be such that e^2 = 0 but e ≠ 0.
e = e^2/e (as e is not 0)
= 0/e = 0 (again, the denominator is non zero)
So e = 0
brisk steeple
Where's the mistake?
wispy rapids
summer imp
The mistake is assuming such an e exists.
tiny egret
are you thinking of dual numbers
flint basalt
if $e \cdot e = 0$ with $e \neq 0$ you can't really cancel terms nor can you divide by arbitrary nonzero numbers since e would be a zero divisor?
jolly parrotBOT
Krish
flint basalt
so e^2/e would also be invalid i believe
pearl pondBOT
@brisk steeple Has your question been resolved?
brisk steeple
are you thinking of dual numbers
of course
tiny egret
what makes you think 0/e=0 implies that e=0?
brisk steeple
Bro it's a chain:
e = e^2/e = 0/e = 0
tiny egret
oh okay. putting it on seperate lines was weird
dusty jungle
Bro it's a chain:
e = e^2/e = 0/e = 0
wait is division by e even defined in the first place
tiny egret
^
brisk steeple
wait is division by e even defined in the first place
e is defined and division is defined
pearl pondBOT
pearl pondBOT
✅ Original question: #help-39 message
dusty jungle
e is defined and division is defined
that doesnt mean division by e is defined especially since dual numbers are an extension of real numbers
brisk steeple
Like how we defined i^2 = -1 and don't define division for it specifically
tiny egret
we define divison of complex numbers by multiplication of an inverse. with dual numbers, multiplicative inverses are not always defined
same with any ring
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fast canyon
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fast canyon
i cant seem to understand why the answer is C. Isn't really a math question but if theres anyone who remembers electricity that'd be helpful.
Now i know since the Light on LDR decreases so the Resistance Increases so more the Voltage it has, which means the fixed resistor is getting lesser voltage, what i dont understand is why moving the slider to the left would make the Ammeter 0 again
valid nova
Now i know since the Light on LDR decreases so the Resistance Increases so more ...
check out #old-network there's a physics server
timid lodge
check out <#406129911992942593> there's a physics server
Aren't physics questions still accepted on this server?
valid nova
Aren't physics questions still accepted on this server?
idk
flint basalt
Aren't physics questions still accepted on this server?
ehh depends, sometimes you'll find someone who knows how to do these problems. but generally we try to keep it mostly math based. if there's some equations or something that the person is stuck on then yes for sure that can be sent here
imo couldn't hurt to try but if it's pure physics go to the physics server
pearl pondBOT
@fast canyon Has your question been resolved?
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lapis lynx
im having trouble visualizing this region for dφdρdθ. the first triple integral covers a hemisphere of radius 3sqrt2, and then im not sure what to do for the second integral
lapis lynx
im not quite sure how to determine the upper bound for the middle integral for the second triple integral but it's 6 somehow
plush bramble
!original
pearl pondBOT
Please show the original problem, exactly as it was stated to you, with the entire original context. A picture or screenshot is best. If the original problem is not in English, then post it anyway! The additional context might still be helpful. Do your best to provide a translation.
lapis lynx
toxic fractal
shouldnt the bounds of 2nd integration be a function of theta, and third be a function of phi and theta (on first) or rho and theta (on second)?
lapis lynx
yea but they can be constant
toxic fractal
they should not be
toxic fractal
yeah, but rho's boundaries definitely depend on the others
pearl pondBOT
@lapis lynx Has your question been resolved?
pearl pondBOT
@lapis lynx Has your question been resolved?
pearl pondBOT
@lapis lynx Has your question been resolved?
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fading nexus
Can I go any further for this one? Number 9 I included the textbook page cus my handwriting sucks
sweet junco
Can I go any further for this one? Number 9 I included the textbook page cus my...
if I'm reading this right, that's a y^3 right
sterile python
Question 4 looks right, could factorise 2 out
molten kraken
Can I go any further for this one? Number 9 I included the textbook page cus my...
no you cant but I think u misread the question
fading nexus
~~if I'm reading this right, that's a y^3 right~~
Whoops yea
that’s an easy fix tho
sterile python
Wait thats number 9 nvm
sterile python
Bro did ,w☕
sweet junco
how do I rotate
sterile python
,rccw
sweet junco
thank you
sterile python
Yeah so basically just change 2 to 3
fading nexus
Also for number 10 is the only thing I can do is make it 4log(1000x)
fading nexus
Yeah so basically just change 2 to 3
Yea
sweet junco
for real values of x, x^2 is always positive, y^3 needs restrictions tho
sweet junco
Also for number 10 is the only thing I can do is make it 4log(1000x)
well, that's not exactly right
sterile python
Also for number 10 is the only thing I can do is make it 4log(1000x)
1000 is 10^3 if you notice
fading nexus
Why not ?
sweet junco
Also for number 10 is the only thing I can do is make it 4log(1000x)
because 4log(1000x)=log((1000x)^4) or log(1000000000000x^4) not log(1000x^4)
hint: log(1000x^4)=log(1000 * x^4)
fading nexus
wait so how would I separate the 4 out?
sterile python
Log1000 here seems familiar since its base 10, what can you do?
sweet junco
wait so how would I separate the 4 out?
we will get to that
fading nexus
I’m not really sure I missed this class yesterday because I was sick
sterile python
$1000x^4 \neq (1000x)^4$
jolly parrotBOT
Minλ
sweet junco
hint: log(1000x^4)=log(1000 * x^4)
do this first and see
sterile python
I’m not really sure I missed this class yesterday because I was sick
No worries
fading nexus
Then why could I separate it for number 9?
sweet junco
Then why could I separate it for number 9?
x^2/y^3 is x^2/y^3, 1000x^4 is not (1000x)^4
you can separate because they are multiplied
but, notice that 1000x^4 is not equal to (1000x)^4
fading nexus
Aren’t they being divided?
sweet junco
or rather not equivalent
sweet junco
Aren’t they being divided?
dividing is just the opposite of multiplying(quite literally)
fading nexus
I just don’t know how I will tell on the test if I can do it or not
sweet junco
I just don’t know how I will tell on the test if I can do it or not
try working backwards from your answer to see if it works/is equal to what the initial given was
fading nexus
Wait so ln(x^2)=ln(x)^2? That doesn’t make sense to me
Sorry I’m just really confused
sterile python
Wait so ln(x^2)=ln(x)^2? That doesn’t make sense to me
No no, ln(x^2)=2lnx
$log(a^b)=b.log(a)$
jolly parrotBOT
Minλ
fading nexus
Oh wait for 10 the ^4 is outside the ()? How would I know that because the () are not written in the given problem
jolly parrotBOT
Minλ
sterile python
Can you work from here?
fading nexus
Wait no I have no idea how you got that
carmine schooner
remember log rules
fading nexus
And I’m confused how I am supposed to know if the ^4 or any ^ is outside or inside the () if they are not written
carmine schooner
as for the exponent, they always only attach to the first thing they stick to
sterile python
Yes csharp is right
carmine schooner
it's poorly written but you wld assume that $\log 1000x^4$ means $\log(1000(x^4))$
jolly parrotBOT
bsharp
sterile python
It is $log(a\times b)=log(a) + log(b)$
fading nexus
But I thought if it’s inside the () then you can make the 4 go on the outside 😭
sterile python
Whereas: $a=1000 and x^4=b$
jolly parrotBOT
Minλ
carmine schooner
this is false.
$\log(ab) = \log(a) + \log(b)$
jolly parrotBOT
bsharp
sterile python
My bad i mistyped🐧☕
fading nexus
Wait how is that false?
jolly parrotBOT
Minλ
sterile python
typoing `+` as `\times`? lol
☕ i was copying \times
carmine schooner
Wait how is that false?
what minh gave at first was just incorrect. it's resolved now
fading nexus
**bsharp**
But if the ^4 is in the inside of the () why can’t we do 4log(1000x)?
Like we did for 9
sterile python
But the brackets for (1000x^4) implies this whole term being applied to the log
carmine schooner
But if the ^4 is in the inside of the () why can’t we do 4log(1000x)?
because it only applies to the x, not the whole 1000x
so we split it first, so then that rule works
fading nexus
Because I’m confused why ln(y^3x) can = 3ln(x) but log(1000x^4) can’t equal 4log(1000x)
sterile python
Wait what, ln(y^3x)?
carmine schooner
Because I’m confused why ln(y^3x) can = 3ln(x) but log(1000x^4) can’t equal 4log...
$ln(y^3x) \neq 3ln(x)$? did you mean $ln(x^3) = 3ln(x)$?
jolly parrotBOT
bsharp
fading nexus
Yea sorry
carmine schooner
when applying the rule $\log(x^a) = a\log(x)$, the $a$ has to be for the whole expression x inside the log
jolly parrotBOT
bsharp
fading nexus
So if the x is larger than 1x fb the can’t apply?
carmine schooner
in the question, $1000x^4$ is actually $1000 \times x^4$ and this doesn't work
jolly parrotBOT
bsharp
fading nexus
Oh
carmine schooner
So if the x is larger than 1x fb the can’t apply?
more nuance than this but you're getting there
fading nexus
Then isn’t it this the then ? The second row
rugged edge
nice :))
carmine schooner
Then isn’t it this the then ? The second row
yes! and then we apply the power rule
so put it all together and boom
fading nexus
carmine schooner
sterile python
Log1000 can be simplied too
fading nexus
And sorry more questions for this one I know I am going to do the rule that is division to subtraction. But I’m not sure how to write it because if the expoinet
sterile python
Note that log here is $log_{10}$
fading nexus
Log1000 can be simplied too
Would it be like divided by 10 so the base is 1
jolly parrotBOT
Minλ
carmine schooner
Would it be like divided by 10 so the base is 1
what's $\log_{10}(1000)$?
jolly parrotBOT
bsharp
fading nexus
Uh 10? i don’t know off the top of my head
carmine schooner
recall the definition of the logarithm
fading nexus
Some multiple of 10 I feel like
sterile python
10^2 is?
sweet junco
Wait no I have no idea how you got that
1000 times x^4 is multiplied, when you have log(a times b), it is equal to loga+logb, whereas log(a divided by b) is equal to loga-logb
fading nexus
100
sterile python
So 10^3?
fading nexus
1000
sterile python
Then, log1000 is?
fading nexus
3?
sterile python
Correct
carmine schooner
(so what's your final answer for question 10?)
fading nexus
1000 times x^4 is multiplied, when you have log(a times b), it is equal to loga+...
Isn’t it the expoinet tho not multiplied
fading nexus
(so what's your final answer for question 10?)
3+4log(x)
carmine schooner
And sorry more questions for this one I know I am going to do the rule that is d...
the exponent is not in the right place here (otherwise you have missed another pair of brackets.)
sweet junco
Isn’t it the expoinet tho not multiplied
x^4 is x times x times x times x, which is just logx+logx+logx+logx=4logx
sweet junco
x^4 is x times x times x times x, which is just logx+logx+logx+logx=4logx
also, it is the exponent of SOLELY x
NOT 1000
carmine schooner
also, it is the exponent of SOLELY x
this was explained above
carmine schooner
you can write $\log(\left(\frac xy\right)^{1/4})$
jolly parrotBOT
bsharp
carmine schooner
and then play with this a little and use log rules
fading nexus
Can I do this?
carmine schooner
the power of 1/4 is still inside the logarithm. you need to clean up the inside a bit first
fading nexus
I’m really confused I just have no idea what to do
fading nexus
Don’t I already have that?
Even with that I have no idea what to do the only thing I know how to do is make it into the subtraction
carmine schooner
recall that we can do $\left(\frac xy\right)^{1/4} = \frac{x^{1/4}}{y^{1/4}}$
jolly parrotBOT
bsharp
carmine schooner
apply this and then it's just log rules from there. this is a nice way to make powers easier to work with
fading nexus
So I would seperate it into subtraction then do like 1/4logx?
carmine schooner
fading nexus
Ty
sterile python
You are a fast learner 
pearl pondBOT
@fading nexus Has your question been resolved?
stoic imp
how can you prove that if gcd(a,b) = 1 then gcd(a^n, b^m) = 1 for n != m
stoic imp
can we maybe use bezouts lemma?
dense jasper
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Available help channel!
fiery oasis
How
∞
Sum
Works?
pearl pondBOT
How
∞
Sum
Works?
fiery oasis
<@&286206848099549185>
I need help
How
∞
Sum
Works?
summer imp
!15m
pearl pondBOT
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.
summer imp
Also, what do you mean? Any particular example of infinite sums?
sage igloo
in general it's just a limit of partial sums
fiery oasis
summer imp
Yes we got that much
The point is what do you want to know about them?
What do you mean by "how do they work"?
fiery oasis
Sum = 1+2+3=6/2
∞sum=?
trim trench
<@&286206848099549185>
I need help
How
∞
Sum
Works?
An infinite sum (∞ sum) works by not actually adding infinitely many numbers at once instead, we look at what happens as we keep adding more and more terms.
fiery oasis
An infinite sum (∞ sum) works by not actually adding infinitely many numbers at ...
So
1+2=3
1+2+3=6
1+2+3+4=10
?
summer imp
Yes, you can keep on adding terms like this.
pearl pondBOT
summer imp
In this case, you'll notice that as you do this, the number gets arbitrarily big.
So this particular infinite sum diverges.
fiery oasis
In this case, you'll notice that as you do this, the number gets arbitrarily big...
Ok
magic bolt
pearl pondBOT
magic bolt
The regex for the dfa im getting is 1*01*00*1(0*1*)*
S has 16 possibilities
But the DFA regex constrains every 1 to be followed by 0
Or if that 1 is skipped then
00
Idk how to progress from here
I hate this problem
summer imp
There's a particular characteristic strings accepted by M have which makes it easy to do this problem
More precisely M accepts strings with a certain substring.
summer imp
Contiguous
summer imp
Say, what's the shortest string M accepts?
magic bolt
001
Between the first and second 0 may or may not be inserted a 1
Or Right
Ohh
F
I got tunnel vision
Right
Of course
It's 001
summer imp
So then you need the number of strings of the form 1__1__1 which contain 001
summer imp
You're counting a string twice
magic bolt
Yeah
So 7
1001(001/111/101/011)
1(111/101/011)001
Damn
Any tips on doing ts kinda problems it's kinda forking me up
Im going into rabbit holes
And tunnel visions
Seeing ghosts
summer imp
I think the regex does help, but usually I think it's good, either from the regex or the automaton, to try to get what kind of strings it accepts more explicitly.
There's patterns you run into often. For instance, those first three states express the fact that to get to the end, you eventually need to write two 0s and a 1.
It's not always easy
You get better at spotting those patterns
magic bolt
Hmmm
I struggled with this problem w bit much
But like even now
Im unsure
Like yea u need 001
But r u sure that dfa isn't mandating something else
Im addition to it
summer imp
Well if you spot something and you want to make sure it characterizes all strings you can try to convince yourself that :
- if a string does not have the pattern, it is not accepted.
- if a string has the pattern, it is accepted.
In this case, it's pretty easy to check that if 001 is a substring, the string is accepted.
magic bolt
Hmm
summer imp
If there is never a 001 in a string, then you can never reach the end state
magic bolt
Hmm that's actually clever
And also really sensible
Nice
Btw
Do u have an intuition for proving 2 stack PDA equivalent to a Turing machine
summer imp
Well the usual construction is how you'd expect : you "run" a 2-stack PDA on a Turing machine by seeing the left and right sides of the tape as the first and second stack>
You similarly "run" a TM on a 2-stack by doing the opposite.
The detail of how you push and pop or move left or right comes kind of naturally from this.
static raft
why is the answer 57.26km and not 57.30km when thats the rounded answer from inputting the true values? are you supposed to round before that point? admittedly, i rounded when doing the bearings which may be why my bearing is correct in this textbook
static raft
so im just wondering i guess to clarify if i did this highlighted part correctly
pearl pondBOT
@static raft Has your question been resolved?
light helm
seems fine
static raft
seems fine
okay thanks 😭 i hate jacplus when its like this
lone panther
Can someone help me with part 3 of this? I know it
1: wants mid-point rectangles
2: wants 3 of them
But i am unsure of how to do it.
lone panther
(the answer you see there is before i realised it wants 3 haha)
sharp vigil
what would be the endpoints of those rectangles?
lone panther
the middle of each i suppose
but im unsure if that means we set 8, 16, and 24 as the middle of their own rectangles?
thats like what i tried at first but it obv didnt work
because like
that would extend the last one
past 24
which we dont want
sharp vigil
just not looking at the table for a minute, if we suppose we wanted to divide the interval [0,24] into three equal rectangles, where would we draw the dividing lines?
lone panther
the dividing lines would be at 8 16 and 24 so each has a width of 8
wait
i think i know what ur saying
ohhhh i found it. the width is the 8 but the height is the middle of that which is the 4 12 and 20
Does this look good?
sharp vigil
yeah
lone panther
Ok nice, i will calculate it now
pearl pondBOT
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Available help channel!
dusky ferry
Q108/ i did can someone please check from a notation stand point if it right or and also in yellow. As i wrote the other notation as gemini did it asw.
dusky ferry
spiral pivot
@dusky ferry I was in your other channel reading
Please don't clopen to boost yourself
dusky ferry
yeah but it was never mine
it was #help-12
the guy posted bfr me so i had to delete as it wasnt mine....
spiral pivot
Oh! Fair enough
Anyway, very clever way to create a telescoping series.
The notation in yellow is not correct though
dusky ferry
i see, thats the gemini notation
spiral pivot
As you surmised
dusky ferry
my work was in the red
spiral pivot
i see, thats the gemini notation
!nogpt
pearl pondBOT
i see, thats the gemini notation
Please do not trust ChatGPT or similar AI tools for mathematical tasks, as they often generate output which "sounds correct" but has numerous factual or logical errors. Use of these AI tools to answer other people's help questions is strictly against server rules (see #rules).
spiral pivot
Good on you for double checking it tho
But maybe don't rely on it
I would say probably the best way to note this is:
(typing up latex one moment)
dusky ferry
my tutor did it this way. (i was also making sure that i understand it so did it again) but he skipped some steps which were kinda of confusing so i did it again
thank you!
dusky ferry
nw thanks again
spiral pivot
\begin{align*}
S_n &= \sum_{i=1}^{n} \frac{1}{i 2^i} - \frac{1}{2^{i+1} (i + 1)} \
&= \sum_{i=1}^{n} \frac{1}{i 2^i} - \sum_{i=1}^{n} \frac{1}{2^{i+1} (i+1)} \
&= \sum_{i=1}^{n} \frac{1}{i 2^i} - \sum_{i=2}^{n+1} \frac{1}{2^{i} (i)} \
&= \frac{1}{2} + \sum_{i=2}^{n} \frac{1}{i 2^i} - \sum_{i=2}^{n} \frac{1}{2^{i} (i)} - \frac{1}{2^{n+1} (n+1)} \
&= \frac{1}{2} - \frac{1}{2^{n+1} (n+1)} \
\end{align*}
jolly parrotBOT
OmnipotentEntity
spiral pivot
@dusky ferry does the above make sense?
spiral pivot
Notice that the index changes from i=1 to i=2 in that step
dusky ferry
yes
dusky ferry
mhmm one sec could you let me know where does urs begin with in my image, maybe it will make things easier to understand
wait from here?
spiral pivot
Essentially:
\begin{align*}
\sum_{i=1}^n \frac{1}{i 2^i} &= \frac{1}{1 \cdot 2^1} + \frac{1}{2 \cdot 2^2} + \ldots + \frac{1}{n \cdot 2^n} \
&= \frac{1}{1 \cdot 2^1} + \qty(\frac{1}{2 \cdot 2^2} + \ldots + \frac{1}{n \cdot 2^n}) \
&= \frac{1}{1 \cdot 2^1} + \sum_{i=2}^n \frac{1}{i 2^i} \
\end{align*}
jolly parrotBOT
OmnipotentEntity
dusky ferry
omg sry, my dumass was reading the first half of the answer
spiral pivot
no worries, because you were worried about the yellow material, I thought I would start from around about the yellow material started.
at least I thought that was were it was intended to start...
it's a little bit confusing because it doesn't actually much sense notationally
Also I changed your variable of summation, sorry if it's a little confusing.
I tend to try to avoid lower vs upper case differences.
dusky ferry
thats fine yeah lol its kinda hard to see the i and n lol but
dusky ferry
**OmnipotentEntity**
does that get "split" in to that?
spiral pivot
no
The red turns into the first two and the blue into the last two
1/2 then the first sum = red
second sum then the last term = blue.
dusky ferry
mhmm i see
tbh the index changes also confuses me alot
but what i also dont understand is how urs went from for the red - i=1 to i=2 in the second summation
why?
versed remnant
why?
trying to get the index same for both the summations
easier to simplify 2 summations that way
versed remnant
ye
dusky ferry
**OmnipotentEntity**
in the first line - i see now that everything is same except the "+1" portion, so essentially what you did was you took summation on both sides so they look the same?
third line - and then removed the "+1" and brough it to the top so its "n+1" and this causes the index to change to i=2? or is it because since it is n+1 , its the second term so thats why we change the index to 2? and then also same thing with the second summation, you added the "n+1" back to the "i" so why did the summation still stay?
third to forth line - my issue is here, once u applied put the value of i in the summation why did u write it again in the forth line, when you already solved it in the line before.
so u kept the summation and the only difference between the 2 were that index so you changed that inherently because + - just cancel out, and u were left with the last line,
versed remnant
in the first line - i see now that everything is same except the "+1" portion, s...
alot of reading D:
versed remnant
in the first line - i see now that everything is same except the "+1" portion, s...
basically used this for 1st line
after that its all simplifying
@dusky ferry for 3rd line , we are just trying to simplify. you can see that these 2 summations look alike with the exception that the 1st one is "i (2)^i" while 2nd one is "(i+1) 2^(i+1)"
now either we can convert the 1st summand (1/i*2^i) into the 2nd summand ...
or what we did rn was to convert the 2nd summand to 1st summand
how?
we let a new variable (say k) k = i+1 , then our summation (which was from i=1 to n) turns to k = 2 to n+1 , and our summation turns to sum from k=2 to n+1 {1/(2^k * k) }
spiral pivot
sorry, I got mod pinged and then sidetracked
pearl pondBOT
@dusky ferry Has your question been resolved?
versed remnant
btw if u understood it a tiny bit , u can try convert the 1st summation thingy into 2nd summation
with the proper index and stuff (as a practice)
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bronze coyote
I'm not sure where to start for this question
pearl pondBOT
I'm not sure where to start for this question
random ermine
diagonalize
bronze coyote
is that eigenvectors?
random ermine
is that eigenvectors?
related
cobalt tundra
or, did you know that if we identify $\begin{pmatrix} x \ y \end{pmatrix}$ as $x+y \mathrm i$, then multipling $\begin{pmatrix} 0 & -1 \ 1 & 0 \end{pmatrix}$ is equivalent to multiplying $\mathrm i$
jolly parrotBOT
1048576Orz
bronze coyote
i haven't learnt eigenvalues and eigenvectors yet
fresh solar
i haven't learnt eigenvalues and eigenvectors yet
Then as someone told earlier, try computing the powers of the matrices
bronze coyote
I calculated up to like A^8 and i got I
so I'm assuming A^2026 is just gonna be I?
jolly parrotBOT
1048576Orz
fresh solar
Yeah the pattern repeats every fourth power
bronze coyote
so its I * A^2
cobalt tundra
yes
bronze coyote
so the answer would be A^2
fresh solar
so its I * A^2
Yes
cobalt tundra
and you have finished the problem, congrats!
pearl pondBOT
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modest tartan
we are asked wether f(x) is odd even or none of the above
f(x)=ln(x^3 + root(1+x^6))
i tried putting x->(-x)
but couldnt solve it further and get an answer in terms of f(x)
rough forge
we are asked wether f(x) is odd even or none of the above
f(x)=ln(x^3 + root(1+...
real
north talon
you can just try subtracting f(x) and f(-x)
rough forge
the ln will be problematic
modest tartan
you can just try subtracting f(x) and f(-x)
like doing the addition and subtraction test
north talon
like doing the addition and subtraction test
no idea what that is but yeah i think so
modest tartan
i havnt tried that yet
modest tartan
no idea what that is but yeah i think so
f(x)+f(-x)=0
or
f(x)-f(-x)=0
north talon
you could maybe suppose for contradiction that f(x) + f(-x) = 0 and etc see if it holds for all x
modest tartan
idk what the actual name is
rough forge
I suppose one counter example for each suffices
modest tartan
I suppose one counter example for each suffices
f(x) + f(-x) becomes 0
i completly forgot we can do this to check
just started doing odd even question
north talon
idk what the actual name is
its just the definition
modest tartan
@north talon @rough forge thanks a lot
modest tartan
its just the definition
?
rough forge
real
modest tartan
i just call it add sub test
north talon
f(x)+f(-x)=0
or
f(x)-f(-x)=0
definition of odd and even function is just this
modest tartan
definition of odd and even function is just this
oh you meant that
yeah its just the definition ig
well ima go do sum more questions
ill prolly be back soon
.close
pearl pondBOT
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static latch
Hey guys, I'm having a little bit of trouble with this. I just got off work, so sorry if I seem a little slow
static latch
So here's what I know so far
I have this theorem here, and how you get it is basically by differentiating f(tx) in 2 ways. One via the standard chain rule, and one via the relation f(tx)=t^a*f(x)
I know what I'm meant to do, which is just differentiate f(tx) twice, plug in t=1, and get the result
I'm just having trouble differentiating $\sum_{j=1}^{n} \frac{\partial f}{\partial x_j}$
jolly parrotBOT
Sir Timmy
static latch
I know I'm making some silly mistake somewhere, but can't quite see it
pearl pondBOT
@static latch Has your question been resolved?
hearty fiber
What is the fundamental period in the expression,
f(2x+3)=f(2x+11)
hearty fiber
I looked the answer online some sites say it's 4 some 8
If you answer 8 what would you say that the time period of the function sin(7x/2) is
By your logic it's 2pi
tardy reef
period of f(x) is 4, however thats different from f(2x+3). For example, in your second question, you need to ask whats the period of g(x) = sin(7x/2). Period of sin(x) is different from the period of sin(7x)
I think your confusion is you tried to find the period of f(2x) instead of the function f(x)
your function f(x) has the special relationship f(2x+3)=f(2x+11) and off of that, you are asked to calcullate the period of f(x)
hearty fiber
Soo the period of f(2x) is coming out to be 8 by this
Hence the period of f(x) should be 16 ???
tardy reef
Did just try plugging in x+4 to the equation
no, use this idea
2x needs to change by 8, so x changes by 4
tardy reef
yea
hearty fiber
Soo the period of f(2x) is coming out to be 8 by this
But can you please tell what's wrong with this, generally isnt this true that if period of f(x) is T then period of f(2x) will be T/2 ?
tardy reef
I think my issue with this question is it does not tell the period of what in the questionWhat is the fundamental period in the expression
they just leave this kinda ambiguous hence you see the answers being 4 and 8
period of the function f(x) wrt x would be 4
period of the specific term f(2x+3) or f(2x+11) as it appears in the equation would be 8 coz there you need to increase the argument by 8 to get the period
hearty fiber
Okk, I'll ask my proffesor if this question is correct or not
Thank you very muchh!
tardy reef
well, its more of the case being unambiguous than wrong
but yea, ask your prof for clarification
hearty fiber
.close
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torn lodge
im not too sure how u would come up with a substitution here
rough stream
The answer is a hint to the sub. Your sub should include some hyperbolic trig function
You'll want the identity:
sinh^2(x) + 1 = cosh^2(x)
torn lodge
Yeah I tried doing x = sinhu but it didnt help simplify
rough forge
Can u show ur W
torn lodge
Can u show ur W
sure! this is what I've done so far
bitter herald
whenever you see sqrt(var^2 + const^2) its usually a good sign that a trig/hyperbolic sub would work
bitter herald
sure! this is what I've done so far
issue is that you are meant to utilise the unit hyperbola identity
and you are kinda not achieving that with what you are doing
torn lodge
yeah I'm more familar with integrals with sqrt(x^2 +1) or smthing in form similar to that where there isnt a coefficient in front of x^2
im not too sure how would u would configure a substitution when there is a coefficient in front of x^2
bitter herald
im not too sure how would u would configure a substitution when there is a coeff...
ok maybe lets work backward
torn lodge
kk
green aurora
try to reduce it to a significant integral, don't make a substitution right away
torn lodge
try to reduce it to a significant integral, don't make a substitution right away
factor out 1/2?
green aurora
yes
bitter herald
you can still do a substitution
torn lodge
you can still do a substitution
with its original form?
bitter herald
<@&268886789983436800>
jolly parrotBOT
bitter herald
i said 9 😭
torn lodge
i said 9 😭
ikik my mind's somewhere else sorry
bitter herald
okok so now
try to equate that lhs expressiion with what u got in the square root
and solve for x
torn lodge
x = 3/2 sinhu?
pearl pondBOT
sharp vigil
sorry we gotta keep the secrets of mr beast's online casino giveaway to ourselves
because we're greedy
cobalt hinge
sorry we gotta keep the secrets of mr beast's online casino giveaway to ourselve...
i want the secret formula 🥺
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rustic gale
pearl pondBOT
random ermine
sqrt(a^2) = |a| for all real numbers a
jolly parrotBOT
#whatdatmean
random ermine
yes that's |a|
rustic gale
|a|<=|a^2+b^2|
rustic gale
|z|^2?
random ermine
no
rustic gale
i got my mistake
random ermine
consider when Re(z) > 0 and < 0
random ermine
Re(z) < |z|
yes and -|z| <=0 <= Re(z) so
-|z| < Re(z) < |z| when Re(z)>=0
pearl pondBOT
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young adder
In Argand diagram, the four points representing complex 12 are concyclic, thes the ratio ((z_{1} - z_{3})(z_{2} - z_{4}))/((z_{4} - z_{4})(z_{2} - z_{3}))
young adder
I am trying to solve this by taking random complex numbers
vague ether
In Argand diagram, the four points representing complex 12 are concyclic, thes t...
z{4}-z{4}??
Do you mean z{4}-z{1}?
tropic mesa
In Argand diagram, the four points representing complex 12 are concyclic, thes t...
Dude just send a picture
vague ether
or z1-z4
Yeah.
I think he means z1,2,3,4 are concylic.
So we have to find that ratio.
vague ether
that's the condition for things to be concylic
?
vague ether
that's the condition for things to be concylic
Dang I didn't know.
I was starting with.
tropic mesa
....
vague ether
All their mod will be 1.
tropic mesa
All their mod will be 1.
yeah that's the unit circle method
vague ether
I attempted 5 days back lmao.
nimble flume
I attempted 5 days back lmao.
oh how did it go?
vague ether
oh how did it go?
Eh.
vague ether
The paper was easy so I'm not expecting much.
vague ether
Cuz cutoff for good %ile will be much.
vague ether
my cousin gave it 4 times and failed all of them
You can't even give 4 times.
tropic mesa
off topic....... guys........
vague ether
Oh nvm twice in 2 years.
nimble flume
You can't even give 4 times.
consecutive years
nimble flume
Oh nvm twice in 2 years.
yes ;-;
vague ether
Hmm.
nimble flume
off topic....... guys........
completely on topic bro jee is nuts
vague ether
Rip, I haven't taken a drop yet, don't plan to.
vague ether
Maharashtra.
vague ether
you can prove it
Send the proof?
vague ether
anyways gtg
Baii.
tropic mesa
Send the proof?
Sure.....
tropic mesa
Ye thanks.
Oh yeah if I may yes I know in the JEE they love unit circles because you can use cube roots of unity
it might not always be the case
we're assuming mod z is one for the unit circle method
let's not do that
(whiteboard incoming)
vague ether
we're assuming mod z is one for the unit circle method
I mean even if you take |z| = k, uppar se and neeche se cat jayega.
buoyant cypress
benzene might not understand it
tropic mesa
I mean even if you take |z| = k, uppar se and neeche se cat jayega.
sure i guess if we need it that way
buoyant cypress
go on, enlighten me.
no i do not know complex numbers yet lol i meant the sentence
tropic mesa
no i do not know complex numbers yet lol i meant the sentence
I see. Well, Hindi is one of the languages I learnt, so, I don't mind
pearl pondBOT
@young adder Has your question been resolved?
young adder
vague ether
Oh ye nice.
vague ether
benzene might not understand it
Well we're from the same country and this is the language there lol.
tropic mesa
answer {0,pi}
tropic mesa
proof
buoyant cypress
Well we're from the same country and this is the language there lol.
Same
buoyant cypress
Oh i see
pearl pondBOT
@young adder Has your question been resolved?
young adder
Which part?
patent vessel
In Argand diagram, the four points representing complex 12 are concyclic, thes t...
Use sum of opposite angles = pi
So arg() + arg() = arg [ ()() ] = pi so the thing inside the arg is real
young adder
Thanks
young adder
Use sum of opposite angles = pi
How do you know they will be opposite angles?
pearl pondBOT
@young adder Has your question been resolved?
junior tusk
Anyone know how to do this?
pearl pondBOT
Anyone know how to do this?
patent vessel
Anyone know how to do this?
Write y = k/x^3 and find k using what's given
junior tusk
ok
pearl pondBOT
@junior tusk Has your question been resolved?
pearl pondBOT
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dire tapir
well that was quick
light helm
natural log
dire tapir
what does ln mean?
huh, why did you delete the message then
toxic bane
huh, why did you delete the message then
it was on accident
light helm
make a new channel if you need to ask more
past perch
please open a new channel
lethal nacelle
its already closed
dire tapir
when you deleted your message, it closes it
past perch
dw about it just open a new channel
dire tapir
but idk why you deleted it when you asked the same question
past perch
is this channel bugged 😭
warped violet
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north talon
Let $\mathbb{Z}{>1}$ be the set of integers greater than $1$. Does there exist a function $f: \mathbb{Z}{>1} \to \mathbb{Z}_{>1}$ such that [ f^{f(n)}(m) = m^n ]
jolly parrotBOT
Copter
north talon
heres what i have so far
omg
$f^{f(a)+f(b)}(m) = (m^b)^a = m^{ab} = f^{f(ab)}(m)$ so $f(a)+f(b) = f(ab)$
jolly parrotBOT
Copter
north talon
from what i already did we have $2f(n) = f(2n) = f(n) + f(2)$ implies $f(n) = f(2)$ contradicting injectivity?
jolly parrotBOT
Copter
plush bramble
https://tenor.com/view/sad-mbappe-gif-3761409877544286013
!memes
pearl pondBOT
https://tenor.com/view/sad-mbappe-gif-3761409877544286013
memes go in #chill
rare scaffold
Let $\mathbb{Z}_{>1}$ be the set of integers greater than $1$. Does there exist ...
are we no longer that interested in finding a function that works, but finding all of them?
soft sinew
is not a meme bro
plush bramble
is not a meme bro
Then stop spamming
soft sinew
Then stop spamming
???
rare scaffold
also are we reading this as $f^{f(n)}(m)={(f(m))}^{f(n)}$?
jolly parrotBOT
Flip
rare scaffold
ah, ok
north talon
are we no longer that interested in finding a function that works, but finding a...
i mean i showed that there exists no such function
north talon
also i have no idea why i let m = p be prime cause i didnt use that at all
twin sigil
the answer is no
rare scaffold
I don't immediately see why f^(2f(n))(p) = f^(f(n))(p^2)
north talon
oops wait
it shouldve been p^n, right?
yikkkeees
since $f(a)+f(b) = f(ab)$ by induction $f(a^k) = kf(a)$ now set $k = f(b)$ $\$ $f(a^{f(b)}) = f(b)f(a)$ but similarly$ f(b^{f(a)}) = f(a)f(b)$ so we get $a^{f(b)} = b^{f(a)}$ clearly false if we pick $a,b$ coprime
is this better?
rare scaffold
since $f(a)+f(b) = f(ab)$ by induction $f(a^k) = kf(a)$ now set $k = f(b)$ $\\$ ...
think the induction says f(a^k) = f(ka)
north talon
wait mb
jolly parrotBOT
Copter
north talon
should be kf(a) now
north talon
**Copter**
^
rare scaffold
that seems fair yeah
as long as that lemma is true
although I get f(a) + f(b) = ab rather than f(ab)
that's just direct though
north talon
p^ab is f^{f(ab)} (p), no?
rare scaffold
it is
ah and you use your lemma
well I guess the neat part about that is ab = f(ab)
but I don't have any issues with what you have now
north talon
wait howd you get p^{f(a) + f(b)} = f^{f(a) + f(b)} p
north talon
aw okay😭
rare scaffold
that would've been f(f(a)+f(b))
north talon
so, all good?
rare scaffold
seems real yeah
pearl pondBOT
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limber nimbus
i'm trying to prove or disprove that a convex, real-valued function $f$ on a compact, convex subset $D$ of $\mathbb{R}^d$ achieves its supremum $s$.
you can create a sequence of points $x_n$ such that $\lim f(x_n) = s$, and you can produce a convergent subsequence $x_{n_i}$ such that $\lim x_{n_i} = z \in D$ since D is compact.
since f is convex on D, the function must be continuous on the interior of D, so if $z$ is in the interior we are done (basically just extreme value theorem)
i'm not sure what to do if $z$ lies on the boundary, and I can't think of a counterexample either
jolly parrotBOT
snowflake
limber nimbus
to clarify, f is only convex on D, not necessarily the rest of R^d
daring grove
Usgjhaibsihoshjsohdohsl
pearl pondBOT
@limber nimbus Has your question been resolved?
tulip ore
to clarify, f is only convex on D, not necessarily the rest of R^d
thats rather misleading to say, you should phrase it as "a real-valued function f on R^d that is convex on a compact subset D"
you can also just show the original problem to make sure you got it written correctly
tulip ore
so its convex on its entire domain?
limber nimbus
treating the domain as D, yes
tulip ore
to clarify, f is only convex on D, not necessarily the rest of R^d
so whats this for?
tulip ore
when you say a thing like that, you imply f is defined on all of R^d
tulip ore
bro you just made a different question
pearl pondBOT
tulip ore
**snowflake**
...also doesnt the proof always work for all z, not just z in the interior
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sudden badger
hi
rough forge
hi
sudden badger
i have this exercise that is making me do vector product
it asks me to do VxU then compare it with UxV
this is just for me to see that order doesnt matter?
rough forge
it does matter tho
sudden badger
does it
honest spear
^
rough forge
ja
sudden badger
bru my teacher said it didnt
honest spear
honest spear
this is the cross product
sudden badger
oh yeah then it is cross product
jolly parrotBOT
rough forge
,w CrossProduct[[3,1,5], [1,2,8]]
honest spear
this is dot product?
this is cross product
visual ruin
this is cross product
honest spear
,w CrossProduct[[3,1,5], [1,2,8]]
jolly parrotBOT
sudden badger
do the other one
rough forge
<@&268886789983436800>
steel dagger
<@&268886789983436800>
honest spear
<@&268886789983436800>
sudden badger
,w CrossProduct[[3,1,5], [1,2,8]]
jolly parrotBOT
honest spear
,w CrossProduct[[1,2,8],[3,1,5]]
jolly parrotBOT
honest spear
notice how swapping the order changed the result
sudden badger
yea
honest spear
yeah
rough forge
<-->
sudden badger
i see it now
visual ruin
for cross product, swapping the order always flips the direction: (v \times u = -(u \times v)). So the two results are opposite vectors, not parallel in the usual sense they’re both perpendicular to the original two vectors, just pointing in opposite directions.
rough forge
real
honest spear
!done
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