#help-0
keen plinth
you do need a minus sign
torpid kestrel
yup
torpid kestrel
I'm sorry for asking, but depending on the formulation, I might have convergence issues?
In my case x_1 and x_2 are between 0 and 1, so maybe a constraint will help?
harsh swallow
the two ones with 0 =
keen plinth
it just comes down to solving
[
\dot r_f \parallel \dot r_g, \quad (r_f - r_g) \times \dot r_f = 0
]
where $r_f(t) = (t, f(t))$, $r_g(t) = (t, g(t))$
ocean sealBOT
keen plinth
those are the conditions required in general for two curves to come closest to each other
keen plinth
I'm sorry for asking, but depending on the formulation, I might have convergence...
if you're using a numerical solver, whats wrong with just minimising (x - y)^2 + (f(x) - g(y))^2
keen plinth
why dot with r_f and not r_g?
they'll be parallel, so it doesnt matter
torpid kestrel
if you're using a numerical solver, whats wrong with just minimising (x - y)^2 +...
idk, I'm just trying to make it fast
keen plinth
hm
torpid kestrel
Easy fix, I'll try both
keen plinth
torpid kestrel
the two ones with 0 =
ty very much @keen plinth and @harsh swallow
I'll close this
.close
torn isle
lone heartBOT
torn isle
May I know how to get the limit for the red parts?
upbeat hornet
What limit
torn isle
What limit
0 till a, 0 till t, then a till t
I only know for Heaviside function, greater or less than a will be 0
upbeat hornet
I only know for Heaviside function, greater or less than a will be 0
What is the value that u steps up at
vague coral
$U(t-a) * e^{2t} = \int_{-\infty}^{+\infty} U(x-a) e^{2(t-x)} dx$
ocean sealBOT
herels
vague coral
now using the definition of Heaviside function :
if x-a<0, then U(x-a) = 0
if x-a>0, then U(x-a) = 1
so the integral is just :
$U(t-a) * e^{2t} = \int_{a}^{+\infty} e^{2(t-x)} dx$
ocean sealBOT
herels
vague coral
$= e^{2t} \int_a^{+\infty} e^{-2x} dx$
ocean sealBOT
herels
vague coral
$= e^{2t} \times \frac{1}{-2} (-e^{-2a})$
ocean sealBOT
herels
torn isle
$U(t-a) * e^{2t} = \int_{-\infty}^{+\infty} U(x-a) e^{2(t-x)} dx$
Why the limit is not 0 to t from the beginning?
vague coral
$=\frac{e^{2t} e^{-2a}}{2}$
ocean sealBOT
herels
torn isle
Because I read from textbook it starts from there
vague coral
Why the limit is not 0 to t from the beginning?
look at the condition on heaviside
torn isle
$=\frac{e^{2t} e^{-2a}}{2}$
I know how to express this, just don’t really understand the limit for Heaviside
vague coral
You dont calculate the limit of Heaviside
$U(x) = \begin{cases} 1 & x>0 \ 0 & x<0 \end{cases}$
ocean sealBOT
herels
vague coral
now using the definition of Heaviside function :
if x-a<0, then U(x-a) = 0
if x-a>0, then U(x-a) = 1
vague coral
you can look for yourself
they are calculating laplace transform here i dont know what u mean
torn isle
Because the limit here starts from 0
vague coral
what are you calling limits
torn isle
And the answer given also from 0
torn isle
what are you calling limits
The integral limits
vague coral
The integral limits
its a because of the conditions on Heaviside
its pretty much logic, when x-a<0, H(x-a) = 0, so the integral from -infinity to a is just 0
we are left with the integral from a to +infinity
I just used the definition of Heaviside Function
torn isle
we are left with the integral from a to +infinity
Then what does it meant by if t greater than a, it’s also 0?
vague coral
$U(t-a) * e^{2t} = \int_{-\infty}^{+\infty} U(x-a) e^{2(t-x)} dx$
ocean sealBOT
herels
vague coral
we are integrating wrt x
so if x-a>0, U(x-a) = 1
if x-a<0, U(x-a) = 0
$\int_{-\infty}^a 0 \times e^{2(t-x)} dx + \int_a^{+\infty} 1 \times e^{2(t-x)} dx$
torn isle
Yea, I understand this
ocean sealBOT
herels
vague coral
is it clear now ?
vague coral
they are wrong because they didnt even write the definition of convolution
torn isle
Also here you put a till infinity, but it’s the same for a till t or is it different
torn isle
they are wrong because they didnt even write the definition of convolution
Oh, okay
vague coral
I have no idea why they wrote the integral goes from 0 to t
or they limited the domain of the function
torn isle
So I should write a till infinity instead of a till t?
vague coral
convolution is always defined like that :
$$(f*g)(x) = \int_{-\infty}^{+\infty} f(x-t)g(t)dt$$
ocean sealBOT
herels
vague coral
its also a commutative operation
$(gf)(x) = \int_{-\infty}^{+\infty} g(x-t)f(t) dt = (fg)(x)$
ocean sealBOT
herels
torn isle
convolution is always defined like that :
$$(f*g)(x) = \int_{-\infty}^{+\infty}...
My convolution theorem, my lecturer use 0 till t as well
vague coral
in my course about convolution, ive never seen a definition where we go from 0 to t
unless they are some hypothesis he used
torn isle
torn isle
in my course about convolution, ive never seen a definition where we go from 0 t...
Or I misunderstood something from this?
vague coral
Is it you that wrote everything ?
torn isle
Is it you that wrote everything ?
You mean the slides?
vague coral
yes
torn isle
No, it’s the annotated version by my lecturer
Nvm, I’ll ask him to clear my doubt why it starts from 0 to t instead of -infinity till +infinity
vague coral
why the integral goes to 0 to t
vague coral
yea
torn isle
vague coral
hmm wdym
torn isle
hmm wdym
I wanna ask how do solve this question
Because I asked in another platform, the answer given is using Fourier sine transform to solve
But, from what I learnt, we should use Fourier cosine transform
And i actually did, just that I don’t know how to express in the signup function
vague coral
welp, i dont know how to start lol
lone heartBOT
@torn isle Has your question been resolved?
torn isle
welp, i dont know how to start lol
Okay, nvm
Btw, do you know how to change to sgn function?
Cause I’m stuck there XD
torn isle
Okay, then mvm
peak stag
i want to express this using logic symbol. did I do it correctly?
peak stag
vale wigeon
yes
molten pivot
bitter iris
can i get help on determinants
lone heartBOT
can i get help on determinants
vale wigeon
bitter iris
how can i dtermine the inconstituent matrix with 0 as solution
vale wigeon
what's an "inconstituent matrix"...?
bitter iris
1 min
vale wigeon
if you are looking at a problem, you should share that problem with us too. that way we have a better idea of the context behind your questions.
bitter iris
ok
eg 28
i don't understand the solution
@vale wigeon hlo ?
and also i have another doubt
vale wigeon
sorry, was/am a little busy.
bitter iris
ok
kindred anchor
Do you mean "Inconsistent system of linear equations"?
vale wigeon
i don't understand the solution
the example is there, but you haven't yet shown the solution...
bitter iris
Do you mean "Inconsistent system of linear equations"?
ye
and is there a simple way to find the determinant of 3X3 matrix
vale wigeon
,rccw
ocean sealBOT
vale wigeon
i think you're only showing the back half of the solution...?
bitter iris
pardon
vale wigeon
but it looks like they are just calculating det(A) [offscreen] and then A^-1 and writing it all out in full detail
what's there not to understand?
bitter iris
it's the steps
here they are taking 1/17 out and how does it became -ve
and also where are the cofactors
therefore it become complicated
.close
nocturne robin
why the equation above deduce this?
lone heartBOT
why the equation above deduce this?
rustic coral
why the equation above deduce this?
First part comes from $4x^2=1 \longrightarrow x^2=\frac{1}{4}$
ocean sealBOT
civil_service_pigeon
rustic coral
It should say x>0, not x>4
tranquil badger
hi, so i'm confused on how to start with this equation because i can't tell the actual plots of the graph.
gray isle
approximate/draw the line of best fit
wary stream
.close
teal ember
lone heartBOT
teal ember
<@&286206848099549185>
Help me please ..I did it already.. I just want to confirm the answer
Will the Answer be π/4 ?
lone heartBOT
Please only use the <@&286206848099549185> ping once if your question has not been answered for 15 minutes. Please do not ping or DM individual users about your question.
teal ember
please help me.. it's urgent to me
left escarp
,w integral of 1/(1+sqrt(tan(x))) dx from pi/6 to pi/3
ocean sealBOT
left escarp
looks smaller than pi/4
analog sand
looks smaller than pi/4
Look like it's π/12
teal ember
thanks man
weary wyvern
ocean sealBOT
lone heartBOT
@teal ember Has your question been resolved?
weary quartz
hi
median oar
,w 2sin(7pi/6)+1
ocean sealBOT
ocean sealBOT
median oar
How did you get to 5pi/6 and 7pi/6
weary quartz
weary quartz
How did you get to 5pi/6 and 7pi/6
pi - (-pi/6)
and pi + (-pi/6)
oh wait i’ve found one error already
median oar
But sin θ ≠ sin 2pi - θ
weary quartz
err
median oar
,calc sin(1)
ocean sealBOT
Result:
0.8414709848079
,calc sin(2 pi -1)
ocean sealBOT
Result:
-0.8414709848079
Not the same
median oar
I use a unit circle method
So here I’m just drawing out the triangle that corresponds to the question
Then I draw this
This tells me which angles I’m looking for
Do you follow?
lone heartBOT
@weary quartz Has your question been resolved?
heady finch
How do I prove this
$tan:72^{\circ }=tan18^{\circ }+2tan54^{\circ }$
ocean sealBOT
krishiv_
heady finch
so far I could only convert the RHS to cot72 + 2cot46
slender gull
It would be 36
And you can prove it for a more general value than these numhers.
I'm using x = 72
(Degrees)
cot(x) - tan(x) = 2cot(2x)
median oar
There’s \tan
heady finch
cot(x) - tan(x) = 2cot(2x)
oh yeah there's that
yeah thanks i forgot about that identity lol
.close
eternal ore
How would i solve this? No L’Hopital allowed
lone heartBOT
How would i solve this? No L’Hopital allowed
eternal ore
i get 0/0 indeterminate form so i know i can cancel it out somewhere
But i really dont know where to start on simplifying
echo socket
Are you allowed to use the definition of a derivative and implicit differentiation?
eternal ore
Isnt the definition of a derivative just a derivative
Im not too sure though the only stipulation she gave was that you arent allowed to use l’hopitals rule
echo socket
Hmm
eternal ore
Not yet
wind cloak
What if you let x = 1/t
eternal ore
Would t refer to another variable
wind cloak
$\lim_{t \to \pm \infty} \frac{\arctan e^{\frac 1t } - \pi / 4}{\frac 1t}$
ocean sealBOT
neonperseus
slender gull
pi/4 = arctan(1), you'll have arctan(a) - arctan(b) you can simplify this to get arctan(c) then you can actually use series expansion, that ought to work.
wind cloak
ah
eternal ore
Am i supposed to know series expansion for calc 1? i wasnt taught it yet
echo socket
Here is my method (involves stuff that I mentioned):
After the substitution $t = e^x$ we get [ \lim_{t\to1}\frac{\arctan{t} - \frac{\pi}4}{\ln{t}}] or [ \lim_{t\to1}\frac{\arctan{t} - \arctan1}{t - 1}\cdot(\frac{\ln{t}-\ln1}{t-1})^{-1}]
The first term in the product is the derivative of $\arctan{t}$ at $t = 1$, we may use implicit differentiation to differentiate that:
\begin{align*}
y &= \arctan{t} \
\tan{y} &= t \
\sec^2{y}\cdot\dv{y}{t} &= 1 \
\dv{y}{t} &= \cos^2{y} = (\cos{\arctan{t}})^2 = \frac1{1 + t^2} \
\eval{\dv{y}{t}}_{t = 1} &= \frac12
\end{align*}
The other term can be figured out in a similar fashion, ultimately yielding that the limit is equal to [ \frac12] Which is wrong.
Hold on let me notice the mistake
Oh I see
Wait the ln's derivative should be inverted woops
eternal ore
How did you go from lim x-> 0 to lim as t -> 1?
echo socket
How did you go from lim x-> 0 to lim as t -> 1?
t = e^x
x -> 0, t -> e^0 = 1
eternal ore
Ah okay
echo socket
Fixing mistakes, my apologies
eternal ore
All good
ocean sealBOT
alonelybean
echo socket
Okay this is it
eternal ore
Okay i think i still need a little bit more work on these kinda problems
Ill review the solution over thank you guys
.close
brittle wigeon
When doing partial fraction decomposition and roots of x_0 = 1, x_1&2 = -2. Do i have A + B or A + B + C, because of x_1 being the same as x_2 ?
brittle wigeon
for context: its about this
i started off finding t_0 = 1 just by inserting. Then polynomal division and got to t_1 and t_2 = -2
Afterwards i went with this but didnt get the right solution. Do i need to have C also because of t_2 ?
waxen flame
,w factor t^3 + 3t^2 -4
ocean sealBOT
waxen flame
There should be a third term, C/(t+2)^2.
waxen flame
No, it will be (t+2)^2. All I can say is give it a try and you will see the math works out.
brittle wigeon
wait this would be A/(t-1) and B/(t+2)^2, if i would insert a C here, then i would need to split and have B/(t+2) and C/(t+2) no?
brittle wigeon
Can you give me an explanation why the third term is (t+2)^2 ?
waxen flame
The degree in the numerator is alway one less than the degree of the numerator. If you were to substitute t+2 with y, you could combine the second and third term to get
The second term is one that you should be familiar with for denominators that have a degree of 2 or higher.
dim drift
How would one approach this puzzle, the goal is to find the angle theta, you are only given that figure, I've tried trying to calculate the length of the red part of the figure by using systems of equations, but a solution has evaded me, I also thought about revolving the figure such that theta * some constant = 360, but that doesn't do me any good if I don't know theta
lone heartBOT
How would one approach this puzzle, the goal is to find the angle theta, you are...
@dim drift Has your question been resolved?
dim drift
<@&286206848099549185>
warped vector
start with the bottom inside triangle
dim drift
start with the bottom inside triangle
I should note that the entire line segment is x, not just ending where they intersect
dim drift
What you want
well to know where I should start
cursive lion
Yeah
dim drift
I already know the answer but I want to know how to arrive at it
cursive lion
What we find
cursive lion
I already know the answer but I want to know how to arrive at it
Okay
what are we looking for
you want to know x or theta
What
dim drift
theta
cursive lion
Oh okay
dim drift
It is
cursive lion
You solved?
dim drift
You solved?
No but I know the answer
dim drift
.close
placid swallow
trying to help my lil cousin w his hw and I got kinda stuck with this. I'm able to prove that this DNE by using series but he hasn't done em yet, can somebody help me figure out a dumb way to say that it doesn't exist?
placid swallow
(it doesn't exist fr)
waxen flame
trying to help my lil cousin w his hw and I got kinda stuck with this. I'm able ...
Perhaps use the Product Rule to split the limit and show the sin(x) is DNE.
mortal trellis
just give two subsequences that obviously dont converge to the same value?
placid swallow
Perhaps use the Product Rule to split the limit and show the sin(x) is DNE.
doesn't the product rule imply that both limits exist?
I can't split into lim sin(x) * lim e^x/2x
cause lim sin(x) doesn't exist
placid swallow
just give two subsequences that obviously dont converge to the same value?
he doesn't know about that
placid swallow
only thing he can use is de l'hopital and basic limits algebra
mortal trellis
what definition of limit is he using
the subsequence idea could still be applied to that
mortal trellis
you could definitely make that idea rigorous. but thats also probably not expected of them
placid swallow
there's gotta be some dumb way to solve this
mortal trellis
"hurr durr, sin oscillates and it obviously doesnt go to 0"
lone heartBOT
@placid swallow Has your question been resolved?
lone heartBOT
lone heartBOT
Available help channel!
alpine sable
guys how to solve this? help 🥲
lone heartBOT
guys how to solve this? help 🥲
granite rapids
this answer is fairly easy to put into photomath
tardy stag
you just kinda work from the outside in inside out simplifying each fraction
alpine sable
can you show me how to do it?
tardy stag
like start with $1 + \frac38$ and rewrite it as one fraction
ocean sealBOT
Hayley
tardy stag
by rewriting $1$ as $\frac88$ and doing the addition
ocean sealBOT
Hayley
alpine sable
okay what next
tardy stag
ok now do the next layer, this is under a 1/ right?
tardy stag
so you have $\frac{1}{11/8}$?
ocean sealBOT
Hayley
alpine sable
yes right
tardy stag
remember how to divide fractions? multiply by the reciprocal
so you have $\frac11$ divided by $\frac{11}8$
ocean sealBOT
Hayley
tardy stag
meaning it's the same as $\frac11\cdot\frac{8}{11}$
ocean sealBOT
Hayley
tardy stag
now just do that like five more times 
void swift
I'm wondering how to set up bounds for triple integration (volume of solid) when bounds for a certain axis are not given?
tacit arch
solve for intersection points
void swift
Ex I have a solid bounded by z=-y^2+1, z=y^2-1, x=-1, x=1
void swift
solve for intersection points
In theory as long as the y bounds exceed the intersection points though, shouldn't we get the same results?
Like I picture that if you set bounds y=-5, y=5, it should be the same as y=-50, y=50
Because the actual volume enclosed is still the same no?
lone heartBOT
@void swift Has your question been resolved?
lone heartBOT
@void swift Has your question been resolved?
tardy stag
Because the actual volume enclosed is still the same no?
if you integrate naively beyond the boundaries you'll get negative area
vapid shuttle
Okay, so let ${x_n}$ and ${y_n}$ be bounded sequences. Then I want to show that $$\liminf{x_n}+\liminf{y_n} \leq \liminf{x_n+y_n}$$
vapid shuttle
the problem has a hint
but I'm not sure what it does for me
I was hoping I could get a bit of a hint about the hint
:p
ocean sealBOT
austinu
vapid shuttle
Why would finding such sequences help me with this proof?
worn fox
presumably to do with the fact that lim inf is the smallest limit value of a convergent subsequence
but the hint is weird, why do both xs have m_i as subscript
ocean sealBOT
austinu
vapid shuttle
instead
Reposting this
I really don't know what to do
they both are bounded, so they both have convergent subsequences
but other than that I need help with what to do
rose sigil
i think the point is whatever x_n_m_i converges to is greater than or equal to liminf x_n
can you post a screenshot of the hint?
vapid shuttle
ye
rose sigil
i'm a little lost on what it was with the corrections
rose sigil
i think the point is whatever x_n_m_i converges to is greater than or equal to l...
idk if the proof i have in mind is what the hint is alluding to but it might be... i was thinking after this, {y_n_m_i} has a subsequence {y_n_m_i_j} that converges
converges say to y
and {x_n_m_i_j} converges say to x
then take a look at {x_n_m_i_j + y_n_m_i_j}
lament forge
(what kind of notation is x_n_m_i)
lament forge
$x_{n_{m_{i_j}}}$
ocean sealBOT
bee [it/its]
lament forge
well the bot can handle it apparently
...tbh i don't really understand what the hint is supposed to be about
rose sigil
i think it's suggesting what i said
vapid shuttle
I'm glad I'm not the only one
lament forge
i also don't understand what you're talking about though
vapid shuttle
no offense Layla becuase I am completely lost by the problem to even begin with, but neither do I
rose sigil
lol what do they say there
rose sigil
can i see?
vapid shuttle
rose sigil
oh rip i was hoping it would be in the spirit of the hint
full cairn
why is u x (v x w) not equal to (u x v) x w
lone heartBOT
why is u x (v x w) not equal to (u x v) x w
gray isle
who's saying it isn't
outer kite
why is u x (v x w) not equal to (u x v) x w
I'm sorry is there any problem associated with that question
maybe your attachment didn't send ?
full cairn
oh no I was just making sure, thanks then
gleaming ridge
is this about the cross product?
outer kite
|U V| does equal |U| |V| right
yes
gleaming ridge
Just one counter example can show that the cross product is not associative
outer kite
and what about the derivative of them? Like U x V = U* x V*
the derivatives multiplied together does not equal the origional terms multiplied together
also uh I'm pretty sure derivative is denoted by '
I could be wrong
outer kite
yw
vapid shuttle
lone heartBOT
vapid shuttle
Okay so I have an answer for this I was just hoping someone could check my reasoning
If the middle limit exists (being 0 it does) then the sequence {x_n} is convergent
then the liminf must equal the limsup (and also must equal the limit)
but -1 doesn't equal 1
so such a sequence can't exist?
carmine reef
Yeah that's correct
rose sigil
correct
carmine reef
Assuming you don't need to go into more detail about why that is
carmine reef
Then yeah good
tacit bobcat
Is proving oscillation = 0 iff convergent hard
(This is the fact relied upon here so)
carmine reef
oscillation is sup-inf?
naive valley
it's not hard, you can show it for example by proving the useful characterization that the limsup is the supremum of all subsequential limits, and the liminf is the infimum
carmine reef
That doesn't really seem like a simplification unless you already have that result though
naive valley
i think austin may already have it? i recall he proved that there's a subsequence that converges to the limsup, that's part of the battle; just show that no subsequence can converge to a larger number
vapid shuttle
It can only go down from here
vapid shuttle
i think austin may already have it? i recall he proved that there's a subsequenc...
I did prove this
carmine reef
ah maybe
vapid shuttle
wait what are we talking about?
carmine reef
well he has the whole thing too
vapid shuttle
is this about my question or the other guys
carmine reef
other guys
vapid shuttle
ok
cool
Final question
I am feeling confident for this one
Okay
here is my idea
vapid shuttle
❔ 
tardy stag
something seems off about that proof
vapid shuttle
tardy stag
it is i professional bubble burster
vapid shuttle
tardy stag
anyway you suppose that at some point (where?) the differences between successive terms increase for one pair of pairs of terms [ x(n-1) - xn is smaller than x(n+1) - xn ]
vapid shuttle
yes
vapid shuttle
**austinu**
right here
tardy stag
and then you choose epsilon to be that difference, and you claim that means the sequence isn't Cauchy
examine the sequence 1, 1, 0, 0, 0, 0, 0, 0, ...
vapid shuttle
not to be the difference
but if the difference is positive
then there exists a positive element in between the two
tardy stag
sure absolute value of difference
vapid shuttle
which we can call epsilon
which is a positive element less than one of the absolute value things
which we said had to be greater than all positive epsilon
but we just found one that it isn't greater than?
naive valley
examine the sequence 1, 1, 0, 0, 0, 0, 0, 0, ...
^ this is a good hint in general for thinking about this problem
tardy stag
examine the sequence 1, 1, 0, 0, 0, 0, 0, 0, ...
this sequence has absolute gaps of 0, then 1, then 0 forever
which would seem to mean it's not Cauchy per your proof if I understand it correctly
but I think you'll agree it's Cauchy
naive valley
that sequence is cauchy as f---
tardy stag
Cauchtastic
naive valley
you can easily envision extending this idea
(hey three e words in a row)
so that your sequence would have more absolute gaps of 0 in it
perhaps many more
vapid shuttle
what part of the proof is wrong then
tardy stag
your argument doesn't... er... show anything? about the sequence's cauchy-ness?
remember what cauchy means - gaps between elements are getting smaller / have a limit of 0
naive valley
why need your rational have the form 1/n, where n is the same n as in the two differences? that's one issue that stands out
raven haven
**austinu**
this part
vapid shuttle
why need your rational have the form 1/n, where n is the same n as in the two di...
no not the same n
just in between any two real numbers
we can always find that
I can call it 1/m
naive valley
well make it a different letter then haha
tardy stag
(there's also no need to call it 1/m if you're just going to use it as 1/m later... like, the standard is difference/2 idk)
vapid shuttle
No, I am just saying if 1/z is in between the two, and is positive, then neither of them should be larger than it (since it is positive)
naive valley
why need that be true
vapid shuttle
because by the definition we get from the sequence being cauchy
both of those should be less than any positive epsilon
if epsilon was to be 1/z, then one of them would not be less than it
naive valley
but the n's are allowed to depend on epsilon
the n's for which the cauchy criterion holds
vapid shuttle
oh
tardy stag
$\forall \varepsilon: \exists M: \forall i>M: |x_{i+1} - x_i| < \varepsilon$
vapid shuttle
so the big M would just change
ocean sealBOT
Hayley
tardy stag
idk what symbols you prefer to use
vapid shuttle
if we are choosing symbols
I use n and k
inside the absolute values
not that it matters
XD
naive valley
there's nothing in that criterion that says that the differences |x_{i+1} - x_i| need to be a decreasing sequence in i
naive valley
think again about the sequence 1,1,0,0,0,0...
vapid shuttle
well I can see that it is a counterexample
the difference goes from 0 to 1
increases
but it is still cauchy
because let M=2
naive valley
well it's not a counterexample to the question itself because the question only requires the condition to hold for all N past some point
vapid shuttle
then anything larger the difference between any two terms is less than positive epsilon
naive valley
but it gives you an idea how such a counterexample could be formed
naive valley
suppose that instead of immediately going to 0,0,0,0,0 you instead did some more shenanigans
vapid shuttle
1,1,0,0,1,1,0,0,.....
tardy stag
remember it has to be Cauchy (which btw means convergent in the reals)
vapid shuttle
grrrr
tardy stag
vapid shuttle
0,0,1/3,1/4,0,0,1/7,1/8,0,0,.....
naive valley
yep that'll work
vapid shuttle
thought of it all on my own
tardy stag
was very confused by the slashes
naive valley
nicely done
vapid shuttle
didn't even need you guys for a second
naive valley
haha
tardy stag
fun problem
naive valley
fyi the one i had in mind was: 1,1,1/2,1/2,1/3,1/3,1/4,1/4,...
vapid shuttle
mine is clearly better
naive valley
obviously
vapid shuttle
XD
tardy stag
think i was just going to do 0, 0, 1/2, 1/2, 0, 0, 1/4, 1/4, ...
naive valley
the one you think of yourself is always best in math
tardy stag
you seemed to misunderstand what Cauchy meant
vapid shuttle
why
tardy stag
you didn't have anything about M
vapid shuttle
well I had that n,k > M
tardy stag
oh right
i guess i mean - you had epsilon being chosen last instead of first
lone heartBOT
lone heartBOT
Available help channel!
shy skiff
is 1/2sin(x) the same as -sin(x)?
lone heartBOT
is 1/2sin(x) the same as -sin(x)?
shy skiff
or -xsin^x the same as 1/2xsin^2x?
original question was f(x)= xcos(x)sin(x)
I took product rule of xcosx (f) and sinx (g), then did product rule of prime (xcosx) so (x) is (f) and cosx would be (g) near the front
lone heartBOT
@shy skiff Has your question been resolved?
remote heron
is 1/2sin(x) the same as -sin(x)?
no
remote heron
or -xsin^x the same as 1/2xsin^2x?
no
shy skiff
😦
shy skiff
How do you go about thr original equation?
remote heron
differentiating xcosxsinx?
shy skiff
yuh
remote heron
i would first apply an identity
maybe thats what you were shooting at?
$\frac12 \sin (2x) = \sin x \cos x$
ocean sealBOT
janniku
shy skiff
Honestly I don't think that's the objective for this problem
oh wait
well shit maybe it is
remote heron
Honestly I don't think that's the objective for this problem
it simplifies the application of the product rule
shy skiff
cus I know the answer lol
remote heron
its not necessary
shy skiff
how would you do it without simplifying it?
remote heron
i guess you could think of it as nested product rule
treat it however you like, say
(fgh)' = (fg)'h + (fg)h' = [(f'g + fg')]h + (fg)h'
shy skiff
aight ima try that real quick
shy skiff
yeah but I gotta do it like that unfortunately
remote heron
(fgh)' = (fg)'h + (fg)h' = [(f'g + fg')]h + (fg)h'
was going to say, if you want to check this, you can always rewrite, say $x^3 = x \cdot x \cdot x$
ocean sealBOT
janniku
remote heron
check you get what you expect
shy skiff
I got the same thing
lone heartBOT
shy skiff
.reopen
lone heartBOT
✅
shy skiff
cosx-xsin^2x+xcos^2x
remote heron
cosx-xsin^2x+xcos^2x
whats the answer youre expecting
shy skiff
1/2sin(2x)+xcos(2x)
remote heron
,w derivative xsinxcosx
ocean sealBOT
shy skiff
wut de fawk
remote heron
youre better of just apply it before you differentiate
remote heron
yes
tardy stag
those are all valid
remote heron
factor out x from the first two terms
shy skiff
what about sin^2x=1-cos(2x)/2
shy skiff
but its got the 1/2 which is part of the answer
remote heron
but its got the 1/2 which is part of the answer
this comes from the other term
this here fella
shy skiff
thats an identity?
remote heron
it is
shy skiff
jesus christ what theres more I thought there was 14
and why not just do cos^2x+sin^2x=1 identity?
oh wait nvm
shy skiff
so thers more than 14 identities?
remote heron
they are numerous
shy skiff
oh lordy
remote heron
some are more important than others
keen plinth
what 14 are you talking about?
shy skiff
I got a sheet from my teacher with 14 identities on it
remote heron
sum and difference are really important
keen plinth
show them
shy skiff
keen plinth
yeah so it's 10
remote heron
these can a bunch be derived from others, too
shy skiff
this here fella
also this is if sinx is times cos x, we dont have that here I think
remote heron
you do
keen plinth
yeah so it's 10
10
remote heron
10 in one 14 in another
remote heron
Im at cosx+x(cos^2x-sin^2x)
this looks wrong
shy skiff
well fawk no wonder
remote heron
look here
shy skiff
I thought we established it was cosx-xsin^2x+xcos^2x
remote heron
it is not
remote heron
(fgh)' = (fg)'h + (fg)h' = [(f'g + fg')]h + (fg)h'
think back to this
shy skiff
remote heron
say h=x
shy skiff
thats that isnt it?
shy skiff
aight ill go again one sec
remote heron
you can see from the final term, (fg)h' will be sinxcosx
you should anticipate that term coming out
remote heron
this is wrong
if you write it that way, you dont have a product of 3 functions anymore
remote heron
no
shy skiff
could h = sin?
remote heron
im not sure i follow
you can distribute the thing i wrote before to (fgh)' = f'gh + fg'h + fgh'
shy skiff
I cant do that I gotta do it how you wrote it before
And I did, and end up with this so im still doing something wrong
remote heron
And I did, and end up with this so im still doing something wrong
theyre the same
remote heron
idk what you mean
remote heron
you can distribute the thing i wrote before to (fgh)' = f'gh + fg'h + fgh'
just do this
shy skiff
At this point I have warped any foundation I understood about the rules
remote heron
or you could take a break
shy skiff
cant. I got 200 more problems to do
Not graded but so I can get them right lol
exam...
remote heron
well, we have the thing that makes it real easy
shy skiff
Im going to do it for the fourth time, as you wrote it
remote heron
if you can differentiate each function individually, youll be fine
remote heron
oops
well i replied to the right post
look, you got the right answer
all thats left to do is apply the two identities
and youre done
derivative is correct
remote heron
shy skiff
.close
daring sinew
hi, need to know solving this
A = {1,2,3,4,5}, B={4,5,6,7}
solve
A⊕B
this topic is sets\
surreal meadow
!status
lone heartBOT
What step are you on?
1. I don't know where to begin
2. I have begun but got stuck midway
3. I got an answer but I'm told it's wrong
4. I got an answer and would like my work checked
5. I have a question about someone else's worked solution
6. None of the above
1
surreal meadow
what does $\oplus$ mean here
ocean sealBOT
maximofs
daring sinew
direct sum
surreal meadow
how did you define the direct sum of 2 sets?
daring sinew
direct sum
that's all i know
surreal meadow
do you have a picture of the full question
surreal meadow
ok i suggest you check your notes for the definition of direct sum of sets
daring sinew
ok i suggest you check your notes for the definition of direct sum of sets
:/
is that hard?
surreal meadow
i just haven’t seen it before
and it would benefit you to know what the definition is to do the problem
otherwise you just can’t do it
lone heartBOT
lone heartBOT
Available help channel!
alpine sable
35^75/37, remainder=?
lone heartBOT
35^75/37, remainder=?
vale wigeon
!status
lone heartBOT
What step are you on?
1. I don't know where to begin
2. I have begun but got stuck midway
3. I got an answer but I'm told it's wrong
4. I got an answer and would like my work checked
5. I have a question about someone else's worked solution
6. None of the above
@alpine sable Has your question been resolved?
lone heartBOT
lone heartBOT
Available help channel!
unreal epoch
"Prove that there is no positive rational number a such that a^2 = 3. You
may assume that a positive integer can be written in one of the forms
3k, 3k + 1, 3k + 2 for some integer k. Prove that if the square of a
positive integer is divisible by 3, then so is the integer. Then use a similar
proof as for square root of the 2."
I have been stumped by these 2 questions for a while now, can anyone guide me to answer both of the problems? These 2 are a problem from Serge Lang's Basic Mathematics
vale wigeon
for the proof of "a^2 divisible by 3 => so is a itself":
if a=3k, what is a^2? what is its remainder mod 3?
same question for if a=3k+1 and for if a=3k+2.
unreal epoch
I can square 3k into 9k^2 but I don't get the "what is its remainder mod 3"
mortal trellis
"remainder after division by 3"
lone heartBOT
@unreal epoch Has your question been resolved?
unreal epoch
so is it 3k^2?
lone heartBOT
Available help channel!
barren hazel
Hi, I need to prove that if the function has a maximum point at x=0 then it necessarily has 2 other minimum points.
I took out a common factor and then the function looks like [x^2(c)] and I don't understand how x^2 times something can have more than one extreme point
barren hazel
chilly crag
Max at 0 means second derivative at 0 is negative
So 1-b<0
We can then look at what happens for small x greater than 0
barren hazel
I didnt learn about second derivative is there a way u can explain it without that
chilly crag
derivative of f will be negative there
barren hazel
?
chilly crag
How did you introduce maximum and minimum then?
Second derivative is doing derivative twice
barren hazel
with first derivative you can find max and min
barren hazel
Second derivative is doing derivative twice
Ik but I didnt learn its meaning
chilly crag
It is the change in the derivative
barren hazel
ok
hoary hinge
with first derivative you can find max and min
yea u can use either the first or second derivative test
for local max/min
barren hazel
yea u can use either the first or second derivative test
then what
chilly crag
We have a max in 0. That means f(0+d)<f(0)=0 for small d. But we know that for large x, the x^4 dominates so the derivative will be positive for large x. Thus x will diverge to infinity and have to go through 0 again. To do so, it needs a minimum between these 2 roots
And the other min s because symmetry
barren hazel
why do we know we know that for large x, the x^4 dominates
chilly crag
Larger exponent
f'=2x^3+2(1-b)x
For any fixxed b, we can find a X such that f'(x)>0 for x>X
Near 0, lower exponents dominate. For large x, large exponents dominate
If you had limits before, that is related
barren hazel
f'=2x^3+2(1-b)x
chilly crag
Oh yeah, my bad. But the argument still holds
chilly crag
Second derivative is negative for local max. Negative second derivative say that first derivative is decreasing at 0. That together with f'(0)=0 tells us that f is decreasing after x=0
So f gets negative
lone heartBOT
@barren hazel Has your question been resolved?
cunning trout
what is the relationship between the sum of the powers of each term and the power of the binomial?
cunning trout
so for example (a+b)^3
worn fox
Just calculate an example and it should become clear
cunning trout
Just calculate an example and it should become clear
is it the same?
worn fox
What?
echo socket
Sum of the powers in each term
echo socket
Yes, they are equal (to 3, in your example)
cunning trout
but does that mean it’s the same for other binomials?
worn fox
What does the binomial theorem say
cunning trout
the nth power of the sum of two numbers a and b is expressed in the form n+1
‘can be’
worn fox
Eh?
worn fox
Like what's the general formula for (a+b)^n
cunning trout
isn’t that just the binomial expansion formula?
worn fox
Yes what is it
cunning trout
it’s like nc0a^n b^0 +nc1a^n-1b^1 …
worn fox
So you can directly see what the powers add up to
cunning trout
the powers decrease by 1 right?
worn fox
"The powers"?
cunning trout
the power for each term
cunning trout
i don’t know how to phrase this
so I don’t understand what the question means by ‘sum of the powers of each term’
worn fox
What are the powers in for example nc0 * a^n * b^0
cunning trout
n and 0?
cunning trout
i see
so if we consider for example i dunno (a+b) ^3
the sum of the powers for each term is the same
as the power of the binomial
worn fox
Yes, because of the binomial expansion formula
grim berry
hola
gray isle
how did you get that
gray isle
do it on paper
grim berry
why
pale kestrel
This server isn't here to be your answer checker
We can check if your steps to get to the answer are correct.
grim berry
bruh
pale kestrel
We're not here to be "yes" "no" to your homework / test / quiz
grim berry
all you had to say was yes or no
vale wigeon
do not do things in your head unless you are absolutely, without any doubt, 100% certain you got it correct
gray isle
if you just want the answer checked,
shove it into a bot
gray isle
because you've asked "is this correct" for the past 20 questions here
grim berry
because you've asked "is this correct" for the past 20 questions here
yes
he had no issue with it
thorn tapir
grim berry
gray isle
ideally you want you to have the tools to be able to check your own work
grim berry
Don't treat us as checker bots. https://discord.com/channels/268882317391429632/...
stop making assumptions
gray isle
which isn't really possible if you're doing al the "work" in your head
grim berry
u randomly popped up in the ticket making a big ordeal
pale kestrel
Would you like me to call a moderator over to clarify how this server functions then?
vale wigeon
!volunteers