But how does the x component of the tension help me find distance to the center of mass?
#help-49
green mesa
vast ginkgo
But how does the x component of the tension help me find distance to the center ...
it doesn't
but in your torque equation you use x
vast ginkgo
yes
green mesa
Wouldn't I use the y component and not the x component?
grand pondBOT
BuilderDolphin
vast ginkgo
(x components)
grand pondBOT
BuilderDolphin
grand pondBOT
BuilderDolphin
green mesa
rght
vast ginkgo
the y component force equation isn't really necessary to consider
you can rearrange the x component equation though
$\frac{T_1}{T_2}=\frac{\sin{\phi}}{\sin{\theta}}$
grand pondBOT
BuilderDolphin
vast ginkgo
keep that equation on the side
now for torques
torque is defined as perpendicular force times r (dist from pivot)
for T_1 r is just x
for T_2, r is L - x
(the remaining length)
green mesa
okay, I'm following
grand pondBOT
BuilderDolphin
green mesa
okay
vast ginkgo
but you can also rearrange this
$\frac{T_1}{T_2}=\frac{\cos(\phi)(L-x)}{\cos(\theta)x}$
grand pondBOT
BuilderDolphin
vast ginkgo
you have 2 expressions equal to $\frac{T_1}{T_2}$ now
grand pondBOT
BuilderDolphin
vast ginkgo
therefore $\frac{\sin{\phi}}{\sin{\theta}}=\frac{\cos(\phi)(L-x)}{\cos(\theta)x}$
wait wrong one
grand pondBOT
BuilderDolphin
green mesa
Let's hope so, I shouldn't be back next semester if I can't
I really appreciate your help, I'm going to favorite this channel and go back to this when I do it
midnight plankBOT
@green mesa Has your question been resolved?
midnight plankBOT
midnight plankBOT
Available help channel!
tidal turret
unique juniper
could just ask what problem you have first?
and perhaps translate the whole thing 
tidal turret
could just ask what problem you have first?
what?
unique juniper
just helping solve it or are there any specifics
and show your work if you have any
tidal turret
and show your work if you have any <:sully:651816820122189834>
why the sully WTF.
unique juniper
um... ignore it 
tidal turret
just helping solve it or are there any specifics
wdym??
unique juniper
like do you require help solving the whole thing or are you stuck in a specific part?
please make that clear for the helpers
tidal turret
!status
midnight plankBOT
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
midnight plankBOT
@tidal turret Has your question been resolved?
grand pondBOT
Banana Steeler
tidal turret
paper inlet
how do you find a point on a plane
paper inlet
oh wait nevermind
the point Q is im pretty sure the projection onto the normal vector of the plane
paper inlet
the distance is 0
how did you get that
tidal turret
I AM UNSURE
paper inlet
I AM UNSURE
😭
the first step is just finding the point to plane distance from P to plane Pi
this is kinda how it would like like
kindaaaaaaaaaa
midnight plankBOT
@tidal turret Has your question been resolved?
tidal turret
i appreciate the help but is late in here 3am already
tidal turret
nice drawing
median ridge
can someone explain whats happening here
midnight plankBOT
can someone explain whats happening here
median ridge
the gauss-jordan elimination method
prime hornet
in step 1, we take the first row, and divide everything in it by 2
in the second step, we take the second row and add -3 times the first row to it
then we take the first row and subtrac the second row
and lastly, we multiply the second row by 2
prime hornet
the strategy is like this
when we want to find an RREF, we're looking to have 1s in our columns in a staircase pattern
so our operations will have this in mind
let me draw something
median ridge
yes the identity matrix
rose lake
not necessarily the identity matrix, thats only if the matrix is square
prime hornet
say we have this matrix
median ridge
not necessarily the identity matrix, thats only if the matrix is square
what would it look like if its not a square matrix
prime hornet
say we have this matrix
if we want our columns to form the identity matrix, we gotta turn these into 0s
and this into a 1
rose lake
what would it look like if its not a square matrix
the non pivot columns will have all zeros after it passes the row of the previous pivot entry
prime hornet
then turn the second column's last row into a 0, followed by scaling the second row into a 1
and then our last column final row is already a 1 in this case, but we could scale it if it wasn't
this gets us our staircase of 1s
so the row operations we perform aren't exactly out of thin air
they're done with this goal in mind
median ridge
but dont oyu have to work row by row??
prime hornet
yes
you do all this one step at a time
first elimnate the 5 by turning it into a 0
then the 3
then scale the first col so the first element is a 1
and then do this for the 2nd col, and so on
after that, you can get rid of all the stuff above the staircase too
median ridge
but you cant just eliminate the 5 like, choosing what operations to use is the weird part for me because it just feels like trial and error
1st row for example, i could subtract 1 from everything
or i could multiply everything by 1/2
prime hornet
well, any row operation is legal
but you want to use row operations that are useful
median ridge
subtracting 1 makes the first entry a 1 which we need
prime hornet
that's not a valid row operation
sharp coral
row operations only let you manipulate the rows you already have
median ridge
multiplication or addition/subtraction of other rows only
prime hornet
the three row operations are
a) swapping two rows
b) scaling a row by a nonzero constant
c) adding a constant multiple of a row to another
prime hornet
let's go back to this example
sharp coral
if you read up on gauss-jordan elimination there actually is a consistent set of steps that you are given to follow. you don't have to follow those steps exactly but you can if there's nothing else better to do
prime hornet
let's go back to this example
your immediate first goal is to make the diagonal entry a 1, so a sensible row operation to perform is to divide the first row by 1/2
it's the simplest one we can do
median ridge
yeah
prime hornet
then the next goal is to eliminate everything under the first diagonal entry
make it all a 0
median ridge
then turn the second column's last row into a 0, followed by scaling the second ...
i did look up the gauss jordan elimination steps and you'd swap the 2nd row with 1st for this right?
lyric charm
^ i like to call that "cleaning out" the column
prime hornet
i did look up the gauss jordan elimination steps and you'd swap the 2nd row with...
that is one method, yes
prime hornet
make it all a 0
does this make sense to you? @median ridge
median ridge
yes
prime hornet
right, so the idea here is as follows
we just turned the first diagonal entry into a 1
so we can just use it to clear out all the entries below it, by subtracting the entry*row 1
in this case, the thing under the 1 is a 3, so we can subtract 3 * row 1 to get rid of it
median ridge
i feel like the example given in class makes more sense and its more easy, it might be bc i already know how to do it for that one, could we perhaps use a different example please?
prime hornet
let me find one...
median ridge
sorry
prime hornet
$\begin{pmatrix}
4 & -3 & 1 & -8 \
-2 & 1 & -3 & -4 \
1 & -1 & 2 & 2
\end{pmatrix}$
grand pondBOT
higher!
prime hornet
sorry, I don't remember how to make augmented matrices in LaTeX 
pretend there's a bar between columns 3 and 4
@median ridge
median ridge
yes
prime hornet
we wanna get a 1 in the first diagonal, so what can we do? 
median ridge
$$
\left[
\begin{array}{ccc|c}
4 & -3 & 1 & -8 \
-2 & 1 & -3 & -3 \
1 & -1 & 2 & 2
\end{array}
\right]$$
grand pondBOT
suds
prime hornet
median ridge
its negative 💀
if we do it in one step, we could do x0.25
or
-3 R3
which will also make -3 -> 0
prime hornet
mhm, the second option sounds nicer 
another option is to swap rows 1 and 3
any of these is valid
so just choose the one you like the most
median ridge
im gonna go with -3 r3 because it makes the second column into a 0 too
median ridge
lol
prime hornet
yes, go ahead and do the operation 
median ridge
im on my laptop so im gonna have to write out the matrices lol
prime hornet
I'll do it then
median ridge
$$\begin{bmatrix}
1 & 0 & -5 \
-2 & 1 & -3 \
1 & -1 & 2
\end{bmatrix}$$
grand pondBOT
suds
prime hornet
oh uh, missing the augmented part
median ridge
okay sure
prime hornet
$$
\left[
\begin{array}{ccc|c}
1 & 0 & -5 & -14 \
-2 & 1 & -3 & -3 \
1 & -1 & 2 & 2
\end{array}
\right]$$
median ridge
$$ \left[
\begin{array}{ccc|c} 1 & 0 & -5 & -8 \
-2 & 1 & -3 & -3 \
1 & -1 & 2 & 2 \end{array} \right]$$
prime hornet
yes, the entire row has to change 
rose lake
the vertical bar is a social construct
median ridge
okay
im just writing this out on overleaf sorry
ok now
we want to clear the column
so
we could do R2 + 2R3?
prime hornet
yes, we could do that
also I'm really really sorry, but I actually need to go to bed soon 
median ridge
thats fine dw about it
prime hornet
I'll ask if anybody can take over this
median ridge
ill just work it through here outloud and i can probably get it on my own
prime hornet
ah, okay
median ridge
which would be 0, -1, 1, 1
if anyone comes across it then they can help out
if i get stuck
prime hornet
once you get that staircase of 1s, you can clear out everything above the 1s by using them (more or less the same thing as clearing things below using the 1s)
and after that process, you'll get your RREF
good luck, and have a wonderful day/night 
median ridge
thank you!!!
midnight plankBOT
@median ridge Has your question been resolved?
median ridge
.close
plain herald
Prove that angle BAD = angle MAC
midnight plankBOT
Prove that angle BAD = angle MAC
rose pilot
Prove that angle BAD = angle MAC
Standard construction of symmedian https://en.wikipedia.org/wiki/Symmedian
midnight plankBOT
@plain herald Has your question been resolved?
plain herald
Im a secondary schooler
I didnt learn that
@rose pilot can u put it into simpler words?
rose pilot
<@522719729836621844> can u put it into simpler words?
If you follow link there is proof\
plain herald
If you follow link there is proof\
I dont get this part
rose pilot
Law of sines was used
plain herald
Law of sines was used
u meant this?
midnight plankBOT
u meant this?
rose pilot
yes
rose pilot
what you dont undertand?
plain herald
i think they exclude AM'
and it becomes (sin BAM' * sin ACM') / (sin ABM' * sin CAM')
but i dont know how they transform sin ACM' into sin ABD
oh wait
rose pilot
just check angles on pic
rose pilot
you cant do that
rose pilot
and? does 1/2 = 3/4?
plain herald
the angles are not equal
but the ratio is
uhhhh its too hard
@rose pilot
I think there is a problem
Their problem is given that two angles are equal, prove M is midpoint of BC
But mine is given that M is midpoint of BC, prove two angles are equal
midnight plankBOT
@plain herald Has your question been resolved?
worldly swan
midnight plankBOT
grand pondBOT
visual tiger
I managed to find these
you have that $(n+2)I_{n+2} \leq \sqrt 2\frac{n+2}{2n+3}$
grand pondBOT
rafilou is not not born in 2003
visual tiger
so $nI_n \leq \sqrt 2 \frac{n}{2n-1}$ for n sufficiently large
grand pondBOT
rafilou is not not born in 2003
visual tiger
and at the same time $n\sqrt 2 \leq (2n+3)nI_n$
grand pondBOT
rafilou is not not born in 2003
visual tiger
so $nI_n \geq \sqrt 2 \frac{n}{2n+3}$
grand pondBOT
rafilou is not not born in 2003
worldly swan
oh
visual tiger
$\sqrt 2 \frac{n}{2n+3} \leq nI_n \leq \sqrt 2 \frac{n}{2n-1}$
grand pondBOT
rafilou is not not born in 2003
visual tiger
squeeze theorem
tribal wharf
How do I know what the corresponding inverse of a function is?
midnight plankBOT
How do I know what the corresponding inverse of a function is?
exotic stratus
!xy
midnight plankBOT
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.
polar star
try finding it in the form of x=f(y)
tribal wharf
once i chose the domain, i did that, but it returned a positive and negative answer and I had to choose. how do i know which is the corresponding inverse for my chosen function domain?
i chose the domain x > 0
so would it be the positive or negative root
midnight plankBOT
@tribal wharf Has your question been resolved?
midnight plankBOT
@tribal wharf Has your question been resolved?
midnight plankBOT
@tribal wharf Has your question been resolved?
quartz hornet
i chose the domain x > 0
your initial domain of f(x) would be the range of f(y) {inverse} and range of f(x) would be the domain of f(y) ,
so maybe try checking if the domain of the new function matches the range of f(x)
midnight plankBOT
@tribal wharf Has your question been resolved?
midnight plankBOT
midnight plankBOT
Available help channel!
thorny folio
Hello great math fellows, need some help(hint) regarding a simple algebra question.
Assume G is an abelian group with order number n, and a_1,a_2,. . . a_n are members of G. Let x = a_1 a_2 . . . a_n, then prove the following statement
If G has only one member b such that b^2 = e, then x = b
So can I tell that since b^2=e, then we can define H=<b>, so |H|=2, and according to the Lagrange theorem, |H| divides |G|, so the order of G is an even 2k number.
So what now? Did I actually start from a proper point?
midnight plankBOT
Hello great math fellows, need some help(hint) regarding a simple algebra questi...
@thorny folio Has your question been resolved?
dawn dagger
Hello great math fellows, need some help(hint) regarding a simple algebra questi...
I think i have an idea, since you figured out it has an even order n = 2k, then consider x^n = b^n
thorny folio
I think i have an idea, since you figured out it has an even order n = 2k, then ...
I came with the following result:
So since |H| = 2, then H is a normal subgroup of G, so for any g in G we have gH = Hg.
Now G has 2k members, one of them is b (the generator of H, and another one is identity. and for the rest, we have 2k-2 members.
Now since for each element g, its inverse is also existed in G, and also G is abelian, then multiplying all 2k-2 together results e, hence b^2 =e, and for any g, the eg=g. So for a_1 a_2 . . . a_n one of them is b, and one is e, so let say they are a_1, and a_2, then a_3 a_4 ... a_n results e, and finally a_1 a_2 e = a_1 where b=a_1, so that's it?
dawn dagger
well my approach is using a_1^n ... a_n^n = (a_1 ... a_n)^n = e but i will look at yours
thorny folio
I think i have an idea, since you figured out it has an even order n = 2k, then ...
I'll dig into it as well. Thank you for your time and consideration. I think my own solution sounds kind of logical too, but not sure if it's always true.
Thank you, really appreciate your help.
Math bless us all
dawn dagger
I came with the following result:
So since `|H| = 2`, then `H` is a normal subgr...
I don't understand the "then multiplying all 2k-2 together results e"
this how i did it
||b^2 = e
b^(2k) = e
b^n = e
b^n=a_1^n ... a_n^n = (a_1 ... a_n)^n
b = a_1 ... a_n =: x
b = x||
thorny folio
I don't understand the "``then multiplying all 2k-2 together results e``"
like in the G, one element is identity, and one is the b, like a_1, and also let a_2=e for instance. So if G has 2k members, then by having a subset of G as K=G\{e,b}, then K has 2k-2 members.
Now multiplying all members in K, so since every element in K which will result e.
Now multiplying the result with e, and b which give b at the end.
dawn dagger
ah i see
thorny folio
Great, I think both solutions must work. Nice solution by you as well. Thanks.
dawn dagger
nice thanks, seems i am not that rusty 
thorny folio
nice thanks, seems i am not that rusty <a:catRoll:720222027125882925>
Thank you for your time and consideration. worth a lot to me, much appreciated.
Math bless us all 🍻 🤘🤓
.close
midnight plankBOT
midnight plankBOT
Available help channel!
fervent burrow
midnight plankBOT
fervent burrow
My every part is correct except it is -5/2 < x < - 4/3
And not 9
Why not 9
Isnt it supposed to be smallest 1st and largest in the end
oak ether
are u suppoesed to solve both equation?
fervent burrow
Yes
fervent burrow
I am
vast ginkgo
write out the interval where the first expression is < 0
fervent burrow
write out the interval where the first expression is < 0
How
Oh -5/2 < x < 5/2
Like this
1st one
midnight plankBOT
@fervent burrow Has your question been resolved?
brave gate
I genuinely do not know where to begin with both questions? someone help me out
vital sinew
I genuinely do not know where to begin with both questions? someone help me out
Do you have access to your calculator on this question
brave gate
yess
vital sinew
Right, cause this is Q2.
Ok, so how do you think we can express the region S
Like visually how does it seem like we could go about finding this
brave gate
its both Q1 and Q2/
usually id do f(x) - g(x) but they didnt give the function g(x), just the area underneath it which is bounded by both axises
vital sinew
its both Q1 and Q2/
usually id do f(x) - g(x) but they didnt give the function...
Well yes, we don't have g(x), so we can't directly calculate int f(x)-g(x)
But what's another way to describe the area R they've given us?
brave gate
wait so is R like the integral of g(x)?
vital sinew
wait so is R like the integral of g(x)?
Right, exactly
vital sinew
so is it like f(x) - R basically? or like we differentiate R to get g(x)
Well R isn't a function
Here, try to write this out properly
What is R equal to
brave gate
integral of g(x)?
vital sinew
Well specifically it's $\int_0^3 g(x)dx$
grand pondBOT
Buzzing Hornet
vital sinew
Ok, so what's the thing we're trying to find?
vital sinew
the area between f(x) and g(x)
And how can we express this?
brave gate
int f(x) - g(x)?
vital sinew
int f(x) - g(x)?
Right, it's $\int_0^3 (f(x)-g(x))dx$
grand pondBOT
Buzzing Hornet
vital sinew
Now what's a slight modification to this we can make
brave gate
oh wait so its like [int f(x) - R] basically then?
brave gate
OHHH 😭
brave gate
So now what's the only thing we need to calculate
the integral of f(x) on the bounds of 0 and 3?
brave gate
And I'm sure you can do this very easily given your calculator
really appreciate it, i just need to know part b of this question
vital sinew
I genuinely do not know where to begin with both questions? someone help me out
Recall that for part a, we basically said "S looks like what we get when we take the area under f and take out R"
Does it seem like we could try something similar for the solid of revolution?
brave gate
wait so like [f(x) - (-2)]^2 - [g(x) - sqrt(2)]^2 or something?
vital sinew
I'm not sure where the -2 and sqrt2 are coming from
brave gate
yea its the second part that im struggling with, i dont how to account y=-3 when were squaring it (i thought it was a -2 not -3 my bad)
vital sinew
yea its the second part that im struggling with, i dont how to account y=-3 when...
Well ok notice that rotating S around y=-3 is the same as rotating [everything between f and y=-3 and taking out the stuff below S]
So try to draw a picture and see what that region looks like
brave gate
so like we get something that looks like a shape that requires us to use the washer method?
vital sinew
so like we get something that looks like a shape that requires us to use the was...
Yeah, basically
brave gate
Yeah, basically
yea i get getting the green part of this question, i just dont know how to do the red part of this question
vital sinew
yea i get getting the green part of this question, i just dont know how to do th...
I think it would really help to draw a picture here
brave gate
I think it would really help to draw a picture here
like this?
vital sinew
like this?
Yeah precisely
Well I think it's -3, not -2
But you've got the idea
Now what does it seem like r(x) is based on this picture
brave gate
ohhh [g(x) - (-2)]^2?
vital sinew
I'm not sure why we're using 2 instead of 3 every time, but yes
vital sinew
yea i get getting the green part of this question, i just dont know how to do th...
Now what do we need to do to calculate this?
brave gate
we use the other graph where y= [g(x)]^2?
vital sinew
we use the other graph where y= [g(x)]^2?
Exactly
And once you realize that it’s just a bunch of calculator work
brave gate
wait how do i formulate it tho since in the formula [g(x) - (-3)]^2 , the square is outside the equation whereas were given [g(x)]^2?
i feel like im struggling with the algebra aspect of this question
vital sinew
wait how do i formulate it tho since in the formula [g(x) - (-3)]^2 , the square...
Well try to simplify (g(x)+3)^2
brave gate
OHHH
g(x)^2 + 9?
wait i reallt appreciate this, i was stumped on this algebra part of this question 🙏🙏🙏
vital sinew
g(x)^2 + 9?
Sometimes the algebra is the hardest part of calc
You might want to check that
brave gate
mb so its [g(x)^2 + 6g(x) + 9] = r?
vital sinew
=r^2 but yeah
brave gate
and that [f(x) - (-3)]^2 - [r^2] basically?
vital sinew
and that [f(x) - (-3)]^2 - [r^2] basically?
Yeah
brave gate
rlly appreciate it 🙏🙏🙏🙏
midnight plankBOT
Available help channel!
steel coyote
Can someone explain how to do this my teacher said you need to "steal" because the dv column requires u-substitution
midnight plankBOT
Can someone explain how to do this my teacher said you need to "steal" because t...
fossil knot
what work have you done so far
steel coyote
I tried to use integration by parts but it doesnt seem to work
u sub doesnt work either
fossil knot
hmmm
flat hollow
Can someone explain how to do this my teacher said you need to "steal" because t...
I think when your teacher said "steal" he prob meant that you had to take part of the x and bring it to the dv term cause it requires u-sub
so like instead of making u = x^5 and dv = sinx^3 dx, try to have dv "steal" part of the x from u to make the u-sub in dv work
main current
Consider:
x⁵ = x³x²
u = x³
midnight plankBOT
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Available help channel!
safe notch
.close
.reopen
can someone explain to me why is gravitational potential energy is negative and also explain what the negative sign hold as any meaning there is
midnight plankBOT
can someone explain to me why is gravitational potential energy is negative and ...
@safe notch Has your question been resolved?
midnight plankBOT
@safe notch Has your question been resolved?
waxen musk
Its because kinetic plus potential must be negative for a “gravitationally bounded field”
But that doesnt mean we consider it a “negative energy” in general
midnight plankBOT
waxen musk
We can also literally say “down means -, + means up”
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Available help channel!
twilit field
midnight plankBOT
grand pondBOT
What a wonderful world !
twilit field
$\pdv{f}{v} = (0,2v,u)$
grand pondBOT
What a wonderful world !
twilit field
We thus want $(x_0,y_0,z_0) + \lambda( \pdv{f}{u} \cross \pdv{f}{v}) \Bigg \vert_{(1,1)}$
grand pondBOT
What a wonderful world !
twilit field
so $(1,1,1)+ \lambda( -2v^2,2u^2,4uv) \Bigg \vert_{(1,1)}$
grand pondBOT
What a wonderful world !
grand pondBOT
What a wonderful world !
twilit field
is that right
This feels very sus
this is the equation of a line
not a plane
nah, I messed up
it's (1,1,1) + span( (2u,0,v) , (0,2v,u)
which is $(1,1,1) + span {( 2,0,1), (0,2,1)}$
grand pondBOT
What a wonderful world !
twilit field
Does that work
midnight plankBOT
Available help channel!
midnight plankBOT
remote spade
Crazy
digital pebble
GET OUT
quasi mauve
<@&286206848099549185> Solve this ?
On a piece of 4x4 grid paper, 4 squares are colored. If 1 set of white 2x1 or 1x2 grid is picker at will and painted with the same color, find the probability that the colored region is a symmetrical figure (Examination test - Semester II test at Secondary School Ngô Sĩ Liên)
midnight plankBOT
<@&286206848099549185> Solve this ?
On a piece of 4x4 grid paper, 4 squares are...
exotic stratus
!15min
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green ibex
Please explain decimals I can't get my head around it
exotic stratus
!occupied
midnight plankBOT
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@quasi mauve Has your question been resolved?
ember tide
a
void pivot
@quasi mauve could you try to state the question more clearly?
quasi mauve
the numbẻ 6
digital pebble
take cases
like first take only half of the grid
and take one block cuz the other one must follow this block
and solve
and then for 2nd case take that the blocks lie in 2nd and 3rd row such that they themselves are symmetrical accross 3rd line
quasi mauve
@digital pebble the goal is find the probability
digital pebble
<@857454174677762058> the goal is find the probability
exatly
so
when u take the cases
there are cases when it wont allign
quasi mauve
umm ok
digital pebble
like in case 1
u have 10 ways to choose 2 adjacent blocks
that means there are 10 x 9 ways of things going wrong (WE HAVENT CONSIDERED CASE 2 YET, WHICH CAN MERGE WITH CASE 1)
digital pebble
aight
so
in case 2
there are 4 ways
to pick a 2x1 block so that each block lies in either side
so there are 3+2+1=6 cases possible
so total legit cases of symmetry is 16
now we want to find total scam cases
in case 1 we already got 90
we want too work with case 2
quasi mauve
ok
digital pebble
i think i am getting 12 cases foor case 2
so total cases is 102+16=118
is answer 9/59?
i think am missing smthing
wait i think i missed some cases
nvm
can u check answer?
quasi mauve
why 9x10 ? This one i makes me confused
digital pebble
look
there are 10 ways to pick a 1x2 or 2x1 block in a 4x2 grid
now
the same thing can be replicated the other side
there is 1 way the other side is replicated
and 9 ways we choose the wrong combination
like if we have a set {1,2,3,4,5,6,7,8,9,10}
there are 10 ways to get a set with 2 elements such that both elements are the same number
but 90 ways to get one with a different number
quasi mauve
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raw vector
From M outside (O), draw MA, MB tangent to (O). Draw diameter AC. MC intersect (O) at D. DK ⊥ AC (K ∈ AC). DK intersect AB at I. Prove that I is the midpoint of DK
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From M outside (O), draw MA, MB tangent to (O). Draw diameter AC. MC intersect (...
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verbal pumice
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verbal pumice
Not sure how i'd begin considering the contour isn't simple
lyric charm
break it into the sum of two simple contours 
or find its winding number around each pole
last slate
is inf/10 the same as inf?
lyric charm
and then do the residue thing
lyric charm
is inf/10 the same as inf?
!occupied
midnight plankBOT
is inf/10 the same as inf?
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@verbal pumice Has your question been resolved?
verbal pumice
break it into the sum of two simple contours <:sotrue:826971355542192148>
that's fine?
lyric charm
yeah why not
verbal pumice
or find its winding number around each pole
i'm not aware of this
verbal pumice
yeah why not
how would I break it? just like normal?
two circles around the poles each?
lyric charm
break it up at the self intersection point
yes
following the orientation as depicted
verbal pumice
Okay but like
lyric charm
i'm not aware of this
as in you don't know what a winding number is?
verbal pumice
I can "jump"?
Oh I guess it only needs to be piece wise continuous
Right thanks
.close
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glossy compass
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glossy compass
im a little confused for D because
it asks for the AMOUNT of unprocessed gravel right
so wouldn't we find where the first derivative (which were given g(t)) changes sign
or like equals zero
but from the answer it seems like we have to find where the second derivative equals zero??
or am i misundurstanding this
wait hold on it got it lol
.close
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upbeat plinth
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vast ginkgo
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do you know how error works for alternating series
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vast ginkgo
the rule for alternating series is that the maximum error is just the next term
(which is probably where you got the n+1 from)
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vast ginkgo
do you know what n you plug in for the error
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vast ginkgo
well the problem uses a third degree taylor polynomial
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vast ginkgo
that means n = 3 (not the one you plug in for error)
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vast ginkgo
yes
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vast ginkgo
yes you're plugging n = 4 into the sequence for error
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vast ginkgo
read the problem
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vast ginkgo
what value of x in f(x) is being estimated here
(if you plugged in 2 the error would be 0)
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vast ginkgo
yes
upbeat plinth
(if you plugged in 2 the error would be 0)
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vast ginkgo
(2 - 2) = 0
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vast ginkgo
if you multiply everything by 0
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vast ginkgo
you get 0\
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vast ginkgo
(a taylor polynomial/series at x=a has 0 error unless it diverges)
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vast ginkgo
just plug in and simplify
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vast ginkgo
yes
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vast ginkgo
so technically the 5th in the sequence
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grand pondBOT
Kenzo
vast ginkgo
yep
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midnight plankBOT
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midnight plankBOT
✅
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vast ginkgo
ok
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grand pondBOT
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vast ginkgo
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do you know taylor's inequality/the lagrange error bound formula
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vast ginkgo
$|R_n(x)| \leq \frac{M|x-a|^{n+1}}{(n+1)!}$
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grand pondBOT
BuilderDolphin
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vast ginkgo
yeah that max part is just M in this
you have the maximum (n+1)th derivative here
just have to identify which one to use
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vast ginkgo
what degree taylor polynomial is this
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vast ginkgo
yes
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vast ginkgo
the n+1th derivative would be the 6th correct
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vast ginkgo
now what's x in this case
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vast ginkgo
(also if this was tripping you up note R(x) = f(x) - P(x))
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vast ginkgo
(since it's the difference between the actual value and the taylor polynomial)
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vast ginkgo
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howd you get that
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vast ginkgo
did you plug in your values
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vast ginkgo
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this equation finds a max value for R(x)
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vast ginkgo
$|R(x)| \leq k$
grand pondBOT
BuilderDolphin
vast ginkgo
you can find R(x)'s maximum value right
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vast ginkgo
if you find the maximum value of R(x), and k has to be greater than or equal to R(x), that means R(x)'s maximum is equal to the minimum value of k
since k can't be any smaller than the highest possible value of R(x)
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vast ginkgo
minimum k = maximum R(x) basically
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vast ginkgo
minimum k = maximum |f(x) - P(x)| yes
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vast ginkgo
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right here
vast ginkgo
but you can find R(x)
(the max value)
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vast ginkgo
$|R_n(x)| \leq \frac{M|x-a|^{n+1}}{(n+1)!}$
grand pondBOT
BuilderDolphin
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vast ginkgo
(M = maximum of n+1th derivative)
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vast ginkgo
wdym
vast ginkgo
**BuilderDolphin**
you're just plugging in values into this to get the maximum remainder
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vast ginkgo
yes since it's a 5th degree taylor polynomial
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vast ginkgo
what's a
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vast ginkgo
read the question
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vast ginkgo
you now have all values you need
vast ginkgo
**BuilderDolphin**
plug them into here
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vast ginkgo
yes
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vast ginkgo
did you calculate it correctly
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vast ginkgo
yep thats what i got
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near tide
Hello, this is a more general question, but my friend has been telling me about complex numbers and I do not understand them. I found two courses, this one which covers Complex Numbers: https://www.coursera.org/learn/basicmathematics#modules and this one which doesn't seem to, but I'm wondering if I should do this one first? https://www.coursera.org/specializations/algebra-elementary-to-advanced
midnight plankBOT
Hello, this is a more general question, but my friend has been telling me about ...
lyric charm
i mean do you want and/or need to learn about complex numbers
near tide
Yes I want to, I don't know if I need to but it can't hurt
near tide
Ok I will do the one that covers Complex Numbers and if it's too hard, start the Elementary to Advanced Algebra one
Khan Academy is very good but I kind of like Coursera's assignment structure it motivates me
lyric charm
if you have any more specific questions related to complex numbers you can and should ask them here
viral dagger
$ABC$ is a triangle with $AB=AC$, let $\ell$ be the tangent of the circumcircle at $C$, $D$ is on the circumcircle and $BD//\ell$, if $E$ is the intersection of $AC$ and $BD$
midnight plankBOT
$ABC$ is a triangle with $AB=AC$, let $\ell$ be the tangent of the circumcircle ...
viral dagger
a. prove that $ABE\cong ACD$
grand pondBOT
skissue.in.a.teacup
viral dagger
b. if $AB=6$ and $BC=4$ find $AE$
grand pondBOT
skissue.in.a.teacup
chrome vessel
a. prove that $ABE\cong ACD$
what have you tried?
viral dagger
ok i did a, using that AE=AD
hard shard
im assuming E is the intersection of AC and BD
viral dagger
oh yea
grand pondBOT
skissue.in.a.teacup
viral dagger
show the work
AB=AC (s)
angle ABD=ACD (a)
(let M be a point on the tangent below C and N be above C)
angle AEB=ACM=180-ACN=180-ABC=ADC (a)
by ASA, they are congruent
BE=BC also cause EBC is simmilar to DAE
chrome vessel
BE=BC also cause EBC is simmilar to DAE
and for part b what have you tried so far?
viral dagger
wait a minute i didnt realize ABC is simmilar to BCE
chrome vessel
wait a minute i didnt realize ABC is simmilar to BCE
that should give you a pretty straight forward way to finding AE
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chrome vessel
.solved alr ty <:pandahugg:513069003120574464>
np, see you again soon
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austere kraken
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austere kraken
I use method in lecture note prove this is true
like this
but for this
I follow the method which is used in lecture note and find that 1/(Sqrt[2]+1)+Sum[-1/(Sqrt[2k-1]-1)+1/(Sqrt[2k]+1),{k,2,n}]=Sum[((-1)^k)/(Sqrt[k]+(-1)^k),{k,2,2n}]
austere kraken
but for this
can I say this diverges?
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@austere kraken Has your question been resolved?
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@austere kraken Has your question been resolved?
lethal path
but for this
use the limit comparison test with this sum and the sum of $\frac{(-1)^k}{\sqrt{k}}$
grand pondBOT
south
austere kraken
use the limit comparison test with this sum and the sum of $\frac{(-1)^k}{\sqrt{...
I dont see how can I use limit comparison test
tribal temple
use the limit comparison test with this sum and the sum of $\frac{(-1)^k}{\sqrt{...
(Doesn’t limit comparison need your terms to be nonnegative 
)
Beat me to it 
lethal path
(Doesn’t limit comparison need your terms to be nonnegative <:SadCat:48166108639...
oh right
well for absolute convergence, taking the absolute value for both, you can now use the limit comparison test
austere kraken
bro this doesnt converge absolutely
lethal path
bro this doesnt converge absolutely
cool, I know
then you should have just said you wanted to check conditional convergence
austere kraken
austere kraken
<@&286206848099549185>
austere kraken
is it possible to say this is ture and say the right hand also diverges?
midnight plankBOT
@austere kraken Has your question been resolved?
austere kraken
like this
my idea is form my lecture note
If S2n and S2n+1 both diverges, then Sn diverges?
blissful rune
my idea is form my lecture note
If S2n and S2n+1 both diverges, then Sn diverges...
But S2k converges
From what i can see
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@austere kraken Has your question been resolved?
shadow schooner
nah i thought i had something but i messed up the direction of this implication
austere kraken
But S2k converges
wait what?
midnight plankBOT
wait what?
austere kraken
how come it converges
frank iris
Have you use absolute convergence?
frank iris
Sum of 1/(sqrtk + 1)? Considering absolute values of (-1)^k?
lament knoll
sum 1/sqrt(k) diverges
frank iris
But 10111 said that the RHS convergence.
austere kraken
But 10111 said that the RHS convergence.
???????when
frank iris
???????when
What converges?
blissful rune
how come it converges
Why wouldn't the rhs converge ?
austere kraken
Why wouldn't the rhs converge ?
If I know the RHS converges why I need to ask questions here?
floral junco
Could pairing up terms help?.
Its a different series, but something for starters
frank iris
Make two separate sums for even and odd k.
Then subtract the sum of the even k from the odd k.
Therefore there is no alternation.
lament knoll
this converges/diverges??
multiplies the numerator and denominator by the conjugate of the denominator
you should have 2 sums
austere kraken
you should have 2 sums
it tends to inf, I cant break them up, I dont see how that can help me
floral junco
Make two separate sums for even and odd k.
doesnt that diverge
frank iris
doesnt that diverge
That's the point.
blissful rune
sum 1/sqrt(k) diverges
now i'm a bit confused myself
looks like i did a bad comparison then
lament knoll
let $a_k=\frac{(-1)^k}{\sqrt{k}+(-1)^k}$and multiply the numerator and denominator by $\\sqrt{k}-(-1)^k$
grand pondBOT
tm
frank iris
one to +inf and one to -inf that dont help in any way
Both sum to infinity it's just at they're subtracted from each other.
lament knoll
after developing you’ll end up with a convergent series + a divergent series
floral junco
right if you can find a formula for the partial sums, that could help
a formula for the even partials and one for the odd partials
lament knoll
which makes a divergent series
blissful rune
the series 1/n^2 converges so i compared 1/sqrt(n) with 1/n^2 , by my comparison it would hold that 1/sqrt(n) ?
floral junco
after developing you’ll end up with a convergent series + a divergent series
oh im dumb 
i assumed they both diverged
lament knoll
the series 1/n^2 converges so i compared 1/sqrt(n) with 1/n^2 , by my comparison...
u did 1/sqrt(n)=o(1/n^2) ?
blissful rune
hm
lament knoll
whats the limit of n^2/sqrt(n)
floral junco
after developing you’ll end up with a convergent series + a divergent series
wait realllyy? 
i dont think so
lament knoll
wait realllyy? <:opencry:586078614865444865>
yes
blissful rune
whats the limit of n^2/sqrt(n)
that will diverge
austere kraken
after developing you’ll end up with a convergent series + a divergent series
what can I do with this
blissful rune
yes it doesn’t converges to 0
Rather confusing , but alright , i guess my comparison just won't work then
lament knoll
what can I do with this
use liebniz rule for the first one
and you should recognize a well-known series for the second
lament knoll
Rather confusing , but alright , i guess my comparison just won't work then
$a_n=o(b_n) \iff \frac{a_n}{b_n} \to 0$ no?
grand pondBOT
tm
frank iris
$a_n=o(b_n) \iff \frac{a_n}{b_n} \to 0$ no?
It does.
austere kraken
I'm not sure whether I can do that, isnt that just treating them as some individual sum? would that fk any thing up?
lament knoll
I'm not sure whether I can do that, isnt that just treating them as some individ...
yeah that’s that
to be more rigorous, k=2 to infinity should not be entered directly but rather as partial sums
austere kraken
yeah that’s that
but isnt that just rearranging the sum? I'm allowed to do that?
lament knoll
to be more rigorous, k=2 to infinity should not be entered directly but rather a...
yes u can do that with partial sums
floral junco
but isnt that just rearranging the sum? I'm allowed to do that?
you can to form a contradiction
which is what u r doing here i think?
"if it converges, its impossible for .."
lament knoll
what’s ur method
lament knoll
Make two separate sums for even and odd k.
this ?
austere kraken
Make two separate sums for even and odd k.
my lecture note didnt taught this any example?
frank iris
my lecture note didnt taught this any example?
It's part of Math Olympiad training.
lament knoll
Yes.
u can but it’s a bit longer, as you’ll have to do several asymptotic developments
frank iris
Define 'asymptotic developments'.
lament knoll
taylor expansions
frank iris
taylor expansions
To prove divergence?
lament knoll
yes
at the end you should end up with the series ~ harmonic series
to nearest coefficient
frank iris
Tolerable.
lament knoll
to nearest coefficient
it should be -3/4
frank iris
Is it convergent?
lament knoll
Is it convergent?
no, with ur method the partial sum is equivalent to -3/4 * sum 1/n
which diverges
frank iris
no, with ur method the partial sum is equivalent to -3/4 * sum 1/n
Did you solve that mentally?
austere kraken
you can to form a contradiction
I'm not sure would this work,because I need this two things both converge so that I can break then up
lament knoll
Did you solve that mentally?
no i have a chalkboard at home lol
floral junco
I'm not sure would this work,because I need this two things both converge so tha...
i get u. i vibe theres gotta be some way to justify
lament knoll
I'm not sure would this work,because I need this two things both converge so tha...
check the alternating series criterion for the first series
floral junco
maybe with sandwitch?
lament knoll
and for the second you should recognize a divergent series
floral junco
I'm not sure would this work,because I need this two things both converge so tha...
sum a < sum(a+b) < sum a + sum b
i think this should be a sufficient justification right? or similar
idk idk not sure at all rofl
otherwise i think tm was talking about partial sums
to justify
lament knoll
instead of working with the series, work with the partial sums
austere kraken
sum a < sum(a+b) < sum a + sum b
i think this should be a sufficient justificat...
this works for +ve sum but now we have -ve things
lament knoll
instead of working with the series, work with the partial sums
it changes nothing, only the notation
austere kraken
it changes nothing, only the notation
so it come back to the things I asked
austere kraken
my idea is form my lecture note
If S2n and S2n+1 both diverges, then Sn diverges...
this
I mean when n->inf