yes, it's the identity matrix in 3 dimensions
#help-19
weary pelican
ok, there is also 4x4 matrixs?
the same method will be applied with 4x4, only the identity matrix will have an additional row
lament walrus
u know the "de morgan's rules" ?
proper quail
no
weary pelican
how do you want me to do that exactly?
Do you know the elementary "de Morgan" operations? Permuting, Dilating and Transvecting?
weary pelican
Ok so
weary pelican
You have three rows here, row 1 is "a b c" and can be written as R1
row 2 is d e f, written as R2
etc...
is that alright so far?
proper quail
ok i understand that
weary pelican
- Permuting : you are allowed to switch rows between them.
For example, switching rows 1 and 2 will be indicated by "R1 <-> R2".
If the original matrix is:
a b c
d e f
g h i
Then after "R1 <-> R2", the new matrix will be :
d e f
a b c
g h i
proper quail
Can I use any of them?
lament walrus
1 2 0 | 1 0 0
0 -4 1 | -3 1 0
4 1 1 | 0 0 1
1 2 0 | 1 0 0
0 -4 1 | -3 1 0
0 9 1 | 4 0 1
1 2 0 | 1 0 0
0 1 -1/4 | 3/4 -1/4 0
0 9 1 | 4 0 1
1 2 0 | 1 0 0
0 1 -1/4 | 3/4 -1/4 0
0 0 13/4 | -9/4 9/4 1
1 2 0 | 1 0 0
0 1 -1/4 | 3/4 -1/4 0
0 0 1 | -9/13 9/13 4/13
1 2 0 | 1 0 0
0 1 0 | 0 1/13 1/13
0 0 1 | -9/13 9/13 4/13
1 0 0 | 1/13 -18/13 -8/13
0 1 0 | 0 1/13 1/13
0 0 1 | -9/13 9/13 4/13
idk if understand
weary pelican
1 2 0 | 1 0 0
0 -4 1 | -3 1 0
4 1 1 | 0 0 1
1 2 0 | 1 0 0
0 -4 ...
let's not throw this in yet
weary pelican
Can I use any of them?
Yes you are allowed to use all 3 of them in any order
proper quail
I can use only one at the time right?
weary pelican
yep
proper quail
ok I see now, continue please
weary pelican
so do you know Dilating and transvecting or should I recall them to you?
proper quail
I don't know that
weary pelican
- Dilating : you are allowed to scale a single row by a factor. This factor can be any NON NULL number.
For example, Dilating row 3 by a factor of 2 will be indicated by "R3 <- 2R3".
If the original matrix is:
a b c
d e f
g h i
Then after "R3 <- 2R3", the new matrix will be :
a b c
d e f
2g 2h 2i
proper quail
Ok but why I have to do that?
weary pelican
Ok but why I have to do that?
Your question was to find the inverse of a matrix, yes? Those operations will lead you to the inverse, we will find out how
proper quail
ok
weary pelican
- Transvecting : you are allowed to add to a row ANOTHER row, and by a factor. This factor can be any number (including 0).
For example, Transvecting row 2 by -3 times row 1 will be indicated by "R2 <- R2 - 3R1".
If the original matrix is:
a b c
d e f
g h i
Then after "R2 <- R2 - 3R1", the new matrix will be :
a b c
(d-3a) (e-3b) (f-3c)
g h i
(LaTeX version below)
proper quail
which one should I use?
weary pelican
If the original matrix is $\begin{pmatrix}a&b&c\d&e&f\g&h&i\end{pmatrix}$, then after "$R_2 \leftarrow R_2 - 3R_1$", the new matrix will be $\begin{pmatrix}a&b&c\d-3a&e-3b&f-3c\g&h&i\end{pmatrix}$
clever fjordBOT
rafilou2003
weary pelican
which one should I use?
you will use all three of them in a method that I will describe after giving you the objective
The RULES :
- You are allowed to use any of those 3 operations in any order, one at a time.
- Every operation that you do on the first matrix (on the left), you have to do on the second matrix as well. (on the right)
Your GOAL : - Using the operations, make the matrix on the left be the identity matrix (1s on the diagonal, 0 elsewhere). The inverse of your original matrix will be the final matrix obtained on the right
Example :
if we want the inverse of the following matrix :
1 8 0
0 1 0
0 0 1
proper quail
so that's how my result should look like right?
weary pelican
We write it next to the identity matrix :
1 8 0 | 1 0 0
0 1 0 | 0 1 0
0 0 1 | 0 0 1
in order to make the matrix on the left become the identity matrix, we need to remove the factor "8"
So we transvect using "R1 <- R1 - 8R2"
The matrix on the left will be
1 0 0
0 1 0
0 0 1
As the rules state, we have to do the same operation to the matrix on the right, so it will become:
1 -8 0
0 1 0
0 0 1
So now the two matrices look like :
1 0 0 | 1 -8 0
0 1 0 | 0 1 0
0 0 1 | 0 0 1
Since the matrix on the left is the identity matrix, we have reached our goal. The inverse of the original matrix is thus the matrix on the right:
The inverse of $\begin{pmatrix}1&8&0\0&1&0\0&0&1\end{pmatrix}$ is $\begin{pmatrix}1&-8&0\0&1&0\0&0&1\end{pmatrix}$
clever fjordBOT
rafilou2003
weary pelican
Ok so the method :
1st step : Make the matrix on the left triangular, with only "one" coefficients on the diagonal
The method is quite lengthy to explain here, so I redirect you to the internet as to "How to reduce a matrix to RREF (reduced row echelon form)"
proper quail
Ok cool
odd edgeBOT
Available help channel!
upper hinge
how can you figure out what f(x) is when given that f(x+1) - f(x) = x and f(2) = 1
fresh ocean
how can you figure out what f(x) is when given that f(x+1) - f(x) = x and f(2) =...
There is anything else?
upper hinge
no just that
i guess there is also the fact that it's a real function
no imaginary parts
cold sage
f(0)-f(-1)=-1 f(-1)=-1
f(1)-f(0)=0 f(0)=0
[f(1+1)-f(1)=1 f(1)=0
f(2)=1
f(1+2)-f(2)=2 f(3)=3
f(1+3)-f(3)=3 f(4)=6
f(1+4)-f(4)=4 f(5)=10, etc] focus on inside the [] since its even
should be able to derive something
fresh ocean
Its domain is real numbers?
fresh ocean
But then you need know at least
[2,3) what are function value
On this interval
Or there is other condition
upper hinge
no other conditions
fresh ocean
f(0)-f(-1)=-1 f(-1)=-1
f(1)-f(0)=0 f(0)=0
[f(1+1)-f(1)=1 f(1)=0
f(2)=1
f(1+2)-f(...
Ok try to find it
upper hinge
wolfram alpha says it's this function but i'm curious about how it got to that
"recurrence equation solution" what's that
odd edgeBOT
@upper hinge Has your question been resolved?
upper hinge
no 😢
mystic saffron
try summing these equations: $$f(x)-f(x-1)=x-1, ; f(x-1)-f(x-2)=x-2, ..., f(3)-f(2)=2$$
clever fjordBOT
Maria
mystic saffron
@upper hinge
upper hinge
f(x) - f(2) = sum k=2 to x-1 of k
mystic saffron
so f(x) = 1 + 2 + ... + x - 1 = x(x-1)/2
rocky salmon
how to ask for help here? I need help studying Physics since it's our exam on Monday
honest turtle
Post your question, and someone whose able will be helping.
wooden python
how to ask for help here? I need help studying Physics since it's our exam on Mo...
!da2a
odd edgeBOT
how to ask for help here? I need help studying Physics since it's our exam on Mo...
No need to ask “Can I ask…?” or “Does anyone know about…?”—it’s faster for everyone if you just ask your question! See https://dontasktoask.com/
wooden python
!help
odd edgeBOT
To ask for mathematics help on this server, please open your own help channel or help thread. See #❓how-to-get-help for instructions.
@rocky salmon Has your question been resolved?
sharp hill
hello
sharp hill
can anyone help me with solving this problem?
so i know I need to find the value of f' between critical values
however, i'm not quite sure about how exactly to do that
how do i plug in x-values within the intervals between these values?
i guess i pick points on the unit circle that i can evaluate?
odd edgeBOT
@sharp hill Has your question been resolved?
tough galleon
!status
odd edgeBOT
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
tall laurel
what is the x and y coords of the red point
odd edgeBOT
what is the x and y coords of the red point
tall laurel
side length of small hex are 16 units
where the two axis meet is (0,0)
<@&286206848099549185>
quasi sparrow
Break the hexagon down into triangles
quasi sparrow
It's a trig thing
tall laurel
yea I can usualy do trig
but my brain is like fried
man I can't do this
done
riemann noooo
Come back
@quasi sparrow
mi mi mi mi 😭
I'll close
.close
echo ginkgo
nah [-1 3] is indeed an eigenvector for lambda=-3
you can just multiply that vector by the matrix if you're not sure
yeah
you have a whole subspace of eigenvectors
not just one
well (-1, 3) is just -1 * (1, -3)
they're in the same space
tired drift
anyone can tell me what this topic is called?
odd edgeBOT
anyone can tell me what this topic is called?
mild finch
Algebra I think
mild finch
Multiplication? Idk what you mean
mild finch
Or do you mean how you're getting rid of the denominator?
mild finch
Are you asking why you need to do those multiplications?
tired drift
nah just wanted to watch some youtube videos on how this works but i didnt know what the topic is called
tired drift
ty
mild finch
You're welcome
sly timber
i did not know what i should do i n this question plz help me
ou want to make an 80° angle by marking an arc on the perimeter of a 12-in.-diameter disk and drawing lines from the ends of
the arc to the disk’s center. To the nearest tenth of an inch, how
long should the arc be?
sly timber
okay so i use to find the arc only i do not need the 12 in right ?
sly timber
ohh okay 12 in is the whole circle and the radius is diameter/ 2
cold sage
'the whole circle'? But yeah, the radius is diameter/2
sly timber
okay sry man i have weird questions cuz its been long thx alot 🙂 can i keep it open till i solve it and send u the anw plz
to check it
cold sage
sure
sly timber
i got 8pi/3 is it correct ?
cold sage
,w 680(pi/180)
clever fjordBOT
cold sage
seems good
sly timber
thx man alot i still have alot of questions to train if i need anything i will ask again if u dont mind thx alot man
cold sage
no worries
sly timber
no worries
If you roll a 1-m-diameter wheel forward 30 cm over level
ground, through what angle will the wheel turn? Answer in radians (to the nearest tenth) and degrees (to the nearest degree).
can u tell me how should i think to solve this
i am trying to solve it but i dont understand it
cold sage
in terms of an arc length
if it rolls forward 30cm
thats essentially an arc length of 30cm thats been rolled through
sly timber
so i should imagine the wheel drew arc 30 cm right ?
cold sage
thats basically what happened yeah
cold sage
o got radian = 15
how did you get this also if i may ask
cold sage
that would get x=60
however, you need to check your units
ones in cm, the others in m
need to be the same
cold sage
that works
cold sage
,calc 0.6 * (180/pi)
clever fjordBOT
Result:
34.377467707849
seems so
sly timber
thx alot man
cold sage
no problemo
cold sage
its alright
cold sage
the ones for 0, pi/2, pi, 3pi/2 and 2pi should be easy if you know what the graphs of cos and sin look like
pi/6, pi/4 and pi/3 can be found using the trig triangles if you know those
the rest can also be found from the above and knowledge of the graphs of sin and cos
for example for -2pi/3,
if cos(pi/3)=x then cos(pi-pi/3)=x=cos(2pi/3) and so cos(-2pi/3)=x since cos is even which can be seen above
youll just have to get comfortable, drawing the graphs can be helpful
some people use the unit circle if you know of that
sly timber
yes unit circle i know it but should i memorize it so i can be able to solve it ?
cold sage
the ones for 0, pi/2, pi, 3pi/2 and 2pi should be easy if you know what the grap...
this is always an option, just depends how comfortable you are with it
find what works for you
sly timber
okay lets me check which will be esier i need a bit of time to solve it abot 10 min
sin is x and cos is y right ?
the x y axis i mean
sly timber
yes
cold sage
cos is the x, sin is the y
sly timber
okay deal
--- pi -2pi/2 0 pi/ 2 3pi/4
sin 0 -rad3 / 2 0 1 rad2 /2
cos -1 -1/2 1 0 -rad2 /2
tan 0 rad3 0 - 1
cot - 1/rad3 0 - 1
sec -1 -2 1 0 -2/rad2
csc 0 -2/rad3 0 1 2/rad2
first part
cold sage
cos of 0 isnt 0
sly timber
how i followed the circle its in middle so y and x is 0
cold sage
,w graph cos(x)
clever fjordBOT
cold sage
cos is 1 at 0
sly timber
ohh okay so alawys cos 0 = 1
cold sage
angle of 0 corresponds to (1,0) cos is 1 sin is 0
sly timber
okay so i need to fix a bit in the table
now is it correct ?
--- pi -2pi/2 0 pi/ 2 3pi/4
sin 0 -rad3 / 2 0 1 rad2 /2
cos -1 -1/2 1 0 -rad2 /2
tan 0 rad3 0 - -1
cot - 1/rad3 - - -1
sec -1 -2 1 0 -2/rad2
csc - -2/rad3 - 1 2/rad2
cold sage
tan of 3pi/4 isnt 1
sly timber
ohh - 1 sryy
sly timber
cosec of 0 isnt 0
its 0/0 isnt that 0 ??
clever fjordBOT
cold sage
ANYTHING divided by 0 is undefined
sly timber
0/0 is not 0
oh i thought beause its divided by it self okay sry
sly timber
--- pi -2pi/3 0 pi/ 2 3pi/4
sin 0 -rad3 / 2 0 1 rad2 /2
cos -1 -1/2 1 0 -rad2 /2
tan 0 rad3 0 - -1
cot - 1/rad3 - - -1
sec -1 -2 1 0 -2/rad2
csc - -2/rad3 - 1 2/rad2
i fix them
cold sage
seems okay beyond the title of -2pi/2
sly timber
that would seem like a line of thought youd get 0/0 =1 from
i took it as 0 over any number is 0 so it should be 0 i know we cant have number over 0 but did not know 0/0 is und sry man iam an idiot in math
sly timber
seems okay beyond the title of -2pi/2
is it all wrong -2pi/2 ?
sly timber
ohh okay will fix it now
thx man alot i take only 5 min break i come to continue solving i will close this chat if somebody needs to ask quesion once i come back i open new chat thx man alot 🙂
.close
mystic saffron
why is it positive 6
odd edgeBOT
why is it positive 6
sly timber
odd edgeBOT
sly timber
in number 7 x E [pi/2,pi] what does that mean ?
odd edgeBOT
sly timber
.reopen
odd edgeBOT
✅
fair prism
how should i solve 7 ?
Try drawing sin3/5 [pi/2, pi] on unit circle
sly timber
i am bit new sine 3/5 i know how to draw but what about [pi/2 , pi ]?
sly timber
Try drawing sin3/5 [pi/2, pi] on unit circle
?
sly timber
ok betwwn 90 and 180 right ?
fair prism
yeah
mystic saffron
Is calculator allowed?
sly timber
sinx=opp/hyp
yes that i know
fair prism
So we can think of a triangle with hyp length 5 and opp length 3
sly timber
Is calculator allowed?
dr said better without calculator
fair prism
Any triangle similar to this will have same sin value, so since we are drawing on a unit circle, we want to scale it so hyp is 1
mystic saffron
Ok
fair prism
So we have 3/5=x/1
This means the opposite length should be 3/5
so find the point on the circle in the 2nd quadrant where the y value is 3/5
and draw a triangle (with the base as the x axis)
sly timber
Any triangle similar to this will have same sin value, so since we are drawing o...
sry i got a question
3/5 = x/1 we did that so that hyp is = 1 ?
fair prism
Yeah
So we can draw it on the unit circle
We could have done it on a circle with radius 5 too
But usually stuff is done on the unit circle
sly timber
but isnt x/1 = x ?
sly timber
sry for asking alot of questions but i stoped university for 5 years now i returned and calculus 3 is breaking my mind realy sry for asking alot
sly timber
Ill try and make a visualization
okay man thx
fair prism
@sly timber
The big circle is radius 5. We were given sinx=3/5 in quadrant 2, so that means the height of the triangle is 3
Maybe it would be easier to ignore the unit circle and just work with the circle of radius 5
You can find the triangles base length right?
I mislabled the 3/5 ignore that
fair prism
Yeah if you do that you will get it is 4
Now we can just apply the rule of cos=adj/hyp and tan=opp/adj to find them
sly timber
thx alot man for the explanation 🙂
i will do them all can i send the anw here and u check if correct plz
fair prism
rescaled now that we know values
fair prism
thx alot man for the explanation 🙂
np
sly timber
rescaled now that we know values
i continue with this or do it in unit circle where hyp = 1 ?
fair prism
Continue with this
It doesnt matter either way though
But this is quicker since we dont need to rescale everything. Since the triangles would be similar though, that means all angles are equivalent, and therefore all sin/cos/tan values are equivalent
sly timber
okay i will start the exercises thx alot man
so for nb 7 cos = 4/5 and tan = 3/4 right ?
fair prism
yeah
sly timber
rescaled now that we know values
i will solve 8 alone will try to draw the circle and triangle on paper.
okay so i got sin = 2/rad 5 and cos = 1 / rad 5
rad = radical
for nb 8
fair prism
yeah thats good
sly timber
in 9 what does it mean - pi /2 how is there negative ?
fair prism
so for nb 7 cos = 4/5 and tan = 3/4 right ?
BTW this should actually be cos=-4/5
since the x part goes to the negative side
sly timber
BTW this should actually be cos=-4/5
ohh i forgot about that
fair prism
in 9 what does it mean - pi /2 how is there negative ?
-pi/2 is -90 degrees. negative for angles means clockwise instead of counter clockwise
sly timber
so i am in fourth part of the circle right ?
fair prism
Yeah
sly timber
okay so y will be negative and x postive
fair prism
yeah
sly timber
okay i did 9 i got sin x = rad 8 / -3 and tan x = rad 8
i was not sure which are negative how do i know whcih ones are negative ?
arctic solstice
rescaled now that we know values
that's basically the same thing if the circle was twice the size and the triangle
sly timber
i was not sure which are negative how do i know whcih ones are negative ?
@fair prism
sry sry now i know cuz in 4th quadrant all negative except cos
fair prism
sry sry now i know cuz in 4th quadrant all negative except cos
yeah theres a phrase to remember this "All students take calculus".
you seem to know how to figure it out without memorization though
fair prism
yeah
fair prism
this may help
so first one is normal sin function but compressed so the period is pi instead of 2pi
sly timber
u mean sin (k(x+c) right ?
fair prism
?
sly timber
u mean sin (k(x+c) right ?
normal sin function but compressed so the period is pi instead of 2pi
u mean this function are all of it ?
fair prism
so first one is normal sin function but compressed so the period is pi instead o...
This was for first function (sin2x)
fair prism
this may help
this is for any problem
fair prism
yeah
fair prism
k=2 but period is defined as 2pi/k so period = 2pi/2 = pi
fair prism
for 14 yeah
fair prism
cuz 2pi/1/2 is = 2pi/2 = pi
2pi/(1/2)=2pi*(2/1)=4pi
sly timber
ohhh
sly timber
this may help
now for cos can i use the same formula
fair prism
yeah. Infact its pretty much the same for all functions
Atleast the graphing part. Words like "amplitude" are for trig functions only but the math is the same
sly timber
okay thx alot man for the help i will close for today will go to sleep tomorow i come back online continue trigonometry thx alot man 🙂
its 4 am here in my country been doing trigs for about 7 hours i am slow in exercies 😦
gn man thx alot
.close
frosty ingot
odd edgeBOT
frosty ingot
<@&286206848099549185>
Can you please explain how we would solve this question
Actually wait
Nope nevermind I solve it
Easy
As
1+1
.close
frosty ingot
1+1
frosty ingot
.close
plain pendant
odd edgeBOT
odd edgeBOT
@plain pendant Has your question been resolved?
onyx atlas
odd edgeBOT
Available help channel!
brazen thorn
help me'
fair prism
Start with j and h
fair prism
Fir which and how did u get that
brazen thorn
cuz j and h look alike to 75 degrees
fair prism
Ok well j and h look completely different so I do not follow that. Either way you need to use actual mathematical properties. To find j, remember the angle of one side of a line is 180 degrees. For h use properties of transversals and parallel lines
fair prism
j is 105 and h is 75
Explain how you got these
Do not try and eyeball the values
j+60=180 by what I said earlier
odd edgeBOT
@brazen thorn Has your question been resolved?
sly timber
hey plz i only need right anw and only one helps me not alot cuz last time i did not understand anything
mystic saffron
dude
im tryingto help you
i cant give you the answer
if I do you wont learn anything
sly timber
.close
mystic saffron
so the graph is restricted on a domain
odd edgeBOT
mystic saffron
lmao
sterile blaze
good bye
odd edgeBOT
Available help channel!
tulip plover
Im having a hard time showing that the distance function is a Lipschitz function in a metric space
tulip plover
I gather it probably has something to do with the triangular inegality, but since i only have a distance and bot a norm im not sure what to do
Im trying to show
|d(x,y) - d(a,b)| ≤ 2 max(d(x,a),d(y,b) ) = δ((x,y)(a,b))
low locust
what do you get from triangle inequality?
tulip plover
So |d(x,y)-d(a,b)|≤d(x,y)+d(a,b)
Wait im not sure of the definition of δ, maybe that's why im lost
low locust
well technically just |d(x,y)| + |d(a,b)| at first but they are both positive so thats fine
and now you have to show that d(x,y) + d(a,b) <= 2 max(...)
low locust
thats the definition of delta
tulip plover
max(d(x,y),d(a,b) ) = δ((x,y)(a,b))
low locust
ah no you switched the y and a for no reason
low locust
can you show number1+number2 <= 2 max(number1, number2) ?
tulip plover
and now you have to show that d(x,y) + d(a,b) <= 2 max(...)
@low locust its not gonna work if we have this scenario
tulip plover
can you show number1+number2 <= 2 max(number1, number2) ?
I think i messed up in the definition somewhere
tulip plover
can you show number1+number2 <= 2 max(number1, number2) ?
Yeah that's obvious, but i think i lessed up the definition of one of the distances somewhere
low locust
yeah you messed up some stuff somewhere definitely. cause you also wrote d(x,y) for x,y in E^2 in the last sentence which doesnt make sense from the type of stuff that x,y are
odd edgeBOT
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.
tulip plover
It's in french
low locust
If the original problem is not in English, then post it anyway!
low locust
ok where did you get delta from?
tulip plover
Delta is the d function above
Which is the "distance produit" for E=E1 × E2× ... En
low locust
why did you have to cut out all the other bits from that
and why is that relevant for prop 15
low locust
a distance function d is always from ExE to R. that has nothing to do with product distance functions
tulip plover
Yeah but if we're looking at the distance function as a Lipschitz function we have to compare two sets of two points so we're working with (E²)²
low locust
but not as a distance on (E^2)^2
tulip plover
Well from what i see you want to compare the difference of the distances of E² to the distance in (E²)²
low locust
for a lipschitz function you have two metric spaces
in our case those are ExE and R
wait
ok sry
tulip plover
Sure
low locust
we have to show |d(x)-d(y)| <= 2 delta(x,y)
tulip plover
Ye
low locust
|d(a,b) - d(c,d)| <= 2 delta( (a,b) , (c,d))
tulip plover
Ye
low locust
2 max (d(a,c), d(b,d))
tulip plover
Exactly
tulip plover
Now how the hell do i do that
Its probably sime business with the second triangular inequality
low locust
ok
we want d(a,c) somewhere
maybe its good if we add it
and subtract again
d(a,b)+d(a,c)-d(a,c) - d(c,d)
something like this?
low locust
ah thats better
so we can instead add d(c,b)
|d(a,b)-d(b,c)+d(b,c)-d(c,d) |
<= |d(a,b)-d(b,c)| + |d(b,c)-d(c,d)|
tulip plover
Oh yeha that's it
tulip plover
Well the exercise is fairly classicak ince you get into it but the several distance less with the head
Ive done plenty of stuff like that before i feeo stupid too
low locust
yeah
its really about properly unpacking the definitions. which you did and I failed at first
tulip plover
Yeah but i was so confused i could figure with inequality to use
low locust
and then always about adding the smart zero. classic analysis
low locust
yes
neat quiver
need some help on this, not entirely sure 🙏
odd edgeBOT
need some help on this, not entirely sure 🙏
neat quiver
protractor not allowed here
i got that the outside of C is 300 degrees
<@&286206848099549185>
<@&286206848099549185>
upper onyx
i got that the outside of C is 300 degrees
can you show in the sketch what you mean with outside of c is 300?
upper onyx
ok. are you allowed to use cosine-rule and sin rule?
neat quiver
no
plain crow
upper onyx
@plain crow take your own channel.
upper onyx
take any free one. this one is occupied.
neat quiver
@upper onyxhow is it going?
upper onyx
i have no idea without sine and/or cosine-rule
neat quiver
alright
could you help me with some others?
one I have no idea where to start even.
upper onyx
you know the volume of a hemisphere and the volume of a cylinder? so you can express the volume of S as a function of x. then you can determine for which x the volume is max.
neat quiver
are you talking calculus?
if so it can't be since I haven't done calculus in my course yet
upper onyx
you can discuss my hint, or you can just try it.
-> so you can express the volume of S as a function of x.
neat quiver
I don't know anything about calculus I haven't covered it yet
upper onyx
what do volume formulas have to do with calculus? but anyway, its your question, ....
neat quiver
I think i misunderstood you, what do you mean by function of x?
upper onyx
you know the volume of a hemisphere and the volume of a cylinder? s
odd edgeBOT
@neat quiver Has your question been resolved?
odd edgeBOT
@neat quiver Has your question been resolved?
neat quiver
.close
quaint pewter
how do you do d/dx y ?
odd edgeBOT
how do you do d/dx y ?
quaint pewter
would it just be dy/dx?
quaint pewter
what about d/dx ln(y)?
clever fjordBOT
everg
quaint pewter
why have you got "x" in ln(y(x))?
pastel dew
because i think here y is a function in variable x
maybe is not ...but you have to tell before
quaint pewter
because i think here y is a function in variable x
is that not why we do d/dx instead of just saying differenciate ln(y)?
tardy lagoon
you can omit that if you want
quaint pewter
i've been doing, for d/dx ln(x): $\$ $\$ let y = $ln(x)$ $\$ $\$ $\frac{dy}{dx}$ = $1/x \cdot \frac{d}{dx} x$ $\$ $\$ $\therefore \frac{d}{dx} ln(x) = \frac{1}{x}$
clever fjordBOT
yomiko
quaint pewter
i don't know how i would incorporate y into that
tardy lagoon
do you know chain rule?
tardy lagoon
I like this form better: (f(g(x))'=f'(g(x))g'(x)
tardy lagoon
no
when doing multivariable chain rule the intuition goes away
anyways
ley f(x) = ln(y(x))
df/dx=df/dy dy/dx
what's df/dy?
tardy lagoon
?
tardy lagoon
dy/du *du/dx
ok let's use this
then y = ln(u(x))
dy/dx=dy/du du/dx
what's dy/du?
odd edgeBOT
@quaint pewter Has your question been resolved?
quaint pewter
bro left me in the dust
quaint pewter
@tardy lagoon
tardy lagoon
yo
quaint pewter
yh
tardy lagoon
what's dy/du?
quaint pewter
dx/du *dy/dx
quaint pewter
mhm
quaint pewter
1/u
signal oar
Prove that gcd(m_1, m_2) = 1, m_1 | n and m_2 | n ==> m_1m_2 | n
signal oar
This is similar to what we've done before, @spiral basalt and @pastel dew
I tried messing around with it
Though it seems we can't really construct it like before
spiral basalt
Prove that gcd(m_1, m_2) = 1, m_1 | n and m_2 | n ==> m_1m_2 | n
since you clearly reject "everything is trivial by the FTA"
I suggest looking at individual primes
glass trench
since you clearly reject "everything is trivial by the FTA"
I suggest looking at...
how is this the fta though
they arent necessarily primes
just coprime
spiral basalt
the fundamental theorem of arithmetic solves every elementary problem like that in an instant
pastel dew
its more elegant without FTA though
spiral basalt
it does
spiral basalt
its more elegant without FTA though
yeah but it's harder
idk how you do it yet
glass trench
i think theyre looking for an algebraic proof though
spiral basalt
it's algebraic dammit
pastel dew
yeah but it's harder
idk how you do it yet
yes definitely
spiral basalt
since you clearly reject "everything is trivial by the FTA"
I suggest looking at...
besides taking inspiration that is
signal oar
the fundamental theorem of arithmetic solves every elementary problem like that ...
It's not stated that I'm not allowed to use FTA or anything, perhaps we should use it then?
spiral basalt
I suggest doing both
pastel dew
kepe what was the previous question ?
signal oar
It was the one we could solve by construction
pastel dew
i don t remember the statement
signal oar
i don t remember the statement
gcd(m1,n) = 1 and gcd(m2,n) = 1 ==> gcd(m1m2,n) = 1.
This here is without any gcds
Well, one, the first statement
glass trench
i think your first step is good (the am1+bm2=1) but you should multiply by something other than m1m2 in the second step
signal oar
i think your first step is good (the am1+bm2=1) but you should multiply by somet...
Well, we want to show n * k = m_1m_2
So we need to get m_1m_2 in there somehow
signal oar
multiply by something with m1m2 in tehre
what do you mean
pastel dew
you would have said n = m_1m_2*k
signal oar
yes thats true as well
So just multiply by km_1m_2?
glass trench
So just multiply by km_1m_2?
well
we dont know what k is
we're trying to show it exists
signal oar
ah
glass trench
also, we want to get an expression with n's in it
what can we multiply am1+bm2=1 by to get some ns in the equation
signal oar
n
pastel dew
ooh i see
glass trench
so now what do we have
pastel dew
gcd(m1,m2)=1 implies exists x,y such that $xm_1+ym_2=1$
clever fjordBOT
everg
pastel dew
now multiply by n
pastel dew
$nxm_1+nym_2=n$
clever fjordBOT
everg
signal oar
pastel dew
now sub n=m1k1 in the first one
signal oar
yeah
glass trench
now like you put at the top of the page
pastel dew
and n=m2k2 in the second one
glass trench
what equations do we have for n
glass trench
and n=m2k2 in the second one
wel don't just tell him the answer 💀
pastel dew
then finally group m1m2
signal oar
n = cm_1
glass trench
yes
signal oar
n = qm_2
glass trench
yup
signal oar
.close
solar burrow
in here i use 'hopital because as h->0 the bounds of the integral are the same, so i don't even need to integrate, it's going to be zero.
is this the reasoning?
glass trench
!1c
odd edgeBOT
Please stick to your channel.
pastel dew
in here i use 'hopital because as h->0 the bounds of the integral are the same,...
or you can direcly use foundamental theorem of calculus
solar burrow
i think it's what i did... what do you mean?
what i meant is that i would end up with F(2) - F(2)
and i don't need to compute F
i already know that it will be zero
low locust
well the limit you are doing is the derivative of the numerator basically
so using lhopital is circular
if you had an antiderivative of $sin(x^2)$ and called it F, then you would have $\lim_{h\to 0} \frac{F(2+h)-F(2)}{h}$
which is just straight up the limit for the derivative
clever fjordBOT
Denascite
low locust
at the point 2
solar burrow
but that's what i wrote....
low locust
well it is what you meant maybe. but not what you wrote
this is not an application of lhopital
solar burrow
that limit is 0/0 right?
then i have to use l'hopital
i wanted to know how the numerator becomes zero.
low locust
every limit for the derivative of a function is 0/0
that does not mean that you can throw lhopital at it
because for that you need to differentiate for which you need to solve the same limit again
solar burrow
every limit for the derivative of a function is 0/0
can you explain this? i dont understand what you mean
low locust
the limit for the derivative of a function is $\lim_{h\to 0} \frac{f(x+h)-f(x)}{h}$
clever fjordBOT
Denascite
low locust
that is always of the form 0/0
but you cant throw lhopital at it
because then you end up with $\frac{f'(x+h)}{1}$ and to calculate $f'$ you need to solve the same limit as before
clever fjordBOT
Denascite
low locust
wait they used lhopital? this isnt your write up?
solar burrow
no it's the solution from an exercise from mit
low locust
ugh
well ok then they just dont care about this technicality
properly you should use the fundamental theorem of calculus, like everg said
solar burrow
i really feel doing this
(F(2+h)-F(2))/ h
i am using FTC... how should i do it instead?
low locust
well imo you should rewrite it as $\frac{d}{dx} F(x)|_{x=2}$ and then by ftc this is $f(2)=\sin(4)$
clever fjordBOT
Denascite
solar burrow
i am not sure about this rewrite you did.
what are you rewriting exactly?just the integral part? Maybe you are doing more steps in your head and i can't understand...?
what's F in this case?
if f is sin(x^2)
F is the integral of f
so f is the d/dxF
so should i write
integral ( d/dx F ) ?
should i find F? ......
prisma cairn
in here i use 'hopital because as h->0 the bounds of the integral are the same,...
Yes
solar burrow
ok thanks! but you are saying different things here. I'm lost.
solar burrow
Yes
but @low locust got a point about the limit for the derivative. Why can I use l'hopital here instead?
even if i don'tunderstand yet how should i solve it
prisma cairn
Well, what @low locust is saying is correct
Also i don't think sin (x²) is a simple integration
What you can do is say that the question is by definition the derivative of the integral
And just differentiate it
So yea fundamental theorem of calculus
solar burrow
What you can do is say that the question is by definition the derivative of the ...
whaat do you mean by this? what is the question?
oh taking the limit you mean
prisma cairn
in here i use 'hopital because as h->0 the bounds of the integral are the same,...
I meant this thing
solar burrow
ok
i'd like to understand a bit better the reasoning Denascite was suggesting if someone has some time to explain in a bit more detail
prisma cairn
In l'hospital you differentiate the thing right? For that you must know the derivative
But the question itself is asking you for the derivative
So you can't use l'hospital to solve the limit since this is the same limit you'd have to solve to find the derivative
So circular reasoning
It's like you're proving a theorem but in the proof you used the same theorem which is obviously wrong since you didn't prove it yet
odd edgeBOT
@solar burrow Has your question been resolved?
odd edgeBOT
odd edgeBOT
Available help channel!
signal oar
For which n can the fraction (6n + 17)/(4n + 8) be reduced?
signal oar
This is a gcd problem
So we need gcd(6n + 17, 4n + 8), right?
How would we find that?
pastel dew
This is a gcd problem
a new one xD
you can use euclid algorith maybe
so
$6n+17=1*(4n+8)+2n+9$
clever fjordBOT
everg
signal oar
you can use euclid algorith maybe
I found https://artofproblemsolving.com/community/c4h1351973p7387140 which seems to be very similar
pastel dew
$4n+8=2(2n+9)-10$
signal oar
And they somehow used some gcd properties there
clever fjordBOT
everg
signal oar
Oh, Merosity also agrees with the Euclidean Algorithm approach
signal oar
$4n+8=2(2n+9)-10$
Yeah
pastel dew
ok
i have been checking that discussion
now euclidian algo work because its true that gcd(6n+17,4n+8)=...=gcd (2n+9,10)
so you want that 2n+9 and 10 be coprime
spiral basalt
||so always||
spiral basalt
I failed to count modulo 5
pastel dew
ohh i see
spiral basalt
by forgetting that 2 is a generator of Z/5Z
pastel dew
so
like bezier have thought 2 enevr divides 2n+9
so we want only that 5 does no divide 2n+9
so $2n+9\not\equiv 0 \mod 5$
clever fjordBOT
everg
pastel dew
i.e. $n\not\equiv 3 \mod 5$
clever fjordBOT
everg
pastel dew
so n=h+k5 with k integers and h in {0,1,2,4}
signal oar
now euclidian algo work because its true that gcd(6n+17,4n+8)=...=gcd (2n+9,10)
Wait, why does this hold?
pastel dew
i ll prove it
if $a=bq+r$ (not necessary the euclidian division), then $gcd(a,b)=g(b,r)$
clever fjordBOT
everg
pastel dew
because if d is a common divisor of a,b then d| a-bq=r
the d is a common divisor of b,r
so gcd(a,b)|gcd(b,r)
if d is a common divisor of b,r then d divides bq
then d divides bq+r=a
slow venture
Can someone help me with geometry
celest onyx
Say d | (6n + 17) and d | (4n + 8). Then d | (68 - 48) => d | 20 => d = {1, 2, 4, 5, 10, 20}. If 2 | (6n + 17), then 2 | 17 which is a contradiction. If 4 | (6n + 17), then this would imply that 2 | (6n + 17). 5 dividing 4n + 8 implies 5 divides (n + 2) which implies that n = 3 (mod 5). Could probably check a condition on 6n + 17 as well
signal oar
we are done
Oh, alright
signal oar
Say d | (6n + 17) and d | (4n + 8). Then d | (68 - 48) => d | 20 => d = {1, 2, 4...
How does d | 68 - 48 follow
celest onyx
d divides a linear combination of 6n + 17 and 4n + 8
In particular, d divides 4 * (6n + 17) - 6 * (4n + 8) = (24n + 68) - (24n + 48) = 68 - 48
signal oar
Oh
signal oar
Say d | (6n + 17) and d | (4n + 8). Then d | (68 - 48) => d | 20 => d = {1, 2, 4...
We also need to check the cases d = 10 and 20, right?
slow venture
<@842120510661328906> i'll report you ! fu**g rabbit
What
celest onyx
10 and 20 follow from 2
Because if either of them divide 6n + 17, this implies that 2 divides 6n + 17
celest onyx
But we showed that 2 is not a divisor
So it boils down to finding a condition on when 5 does or does not divide both terms
signal oar
But we showed that 2 is not a divisor
Oh
pastel dew
nice problem indeed
signal oar
Say d | (6n + 17) and d | (4n + 8). Then d | (68 - 48) => d | 20 => d = {1, 2, 4...
For the d = 5 case, why do you not check 5 | 6n + 17
Only 5 | 4n + 8
celest onyx
You do, you just need to find a condition on when 5 | (6n + 17)
or verify that the values of n that make 5 | (4n + 8) also work for 6n + 17
signal oar
we need 5 | n + 2 for that
signal oar
Say d | (6n + 17) and d | (4n + 8). Then d | (68 - 48) => d | 20 => d = {1, 2, 4...
Ah, yeah, you got that too
But using 4n + 8
signal oar
or verify that the values of n that make 5 | (4n + 8) also work for 6n + 17
Yeah
Thank you!
.close
mystic saffron
Hello I need to know what this question is asking me I have to find the value of a in a rhombus but setting the question as a+68=a does not make sense
quick peak
The sum of angles in a rhombus is 360°
In this scenario this rhombus has angles a+68°, a, a+68° and a
With this you can calculate the a
mystic saffron
So then I would set a+68+ a=180?
mystic saffron
Thanks a lot I was just wondering what I needed to do
quick peak
No problem
odd edgeBOT
@mystic saffron Has your question been resolved?
naive agate
This looks simple enough but I haven't actually figured out how to solve it yet.
naive agate
This is sort of what I have so far, but I need to actually find an answer.
n is the amount of numbers between b and a
Please ping me if you respond, thank you!
odd edgeBOT
@naive agate Has your question been resolved?
nimble blaze
there's an issue with how you're setting up you sum,
a,b themselves aren't included in the summation
glad anchor
i think your $nb + \frac{n(n+1)}{2} = 1000$ can be arranged to be in the form ax^2 + bx + c
clever fjordBOT
jumpydino
Compile Error! Click the 
reaction for more information.
(You may edit your message to recompile.)
glad anchor
there's an issue with how you're setting up you sum,
a,b themselves aren't inclu...
b is implied with a + n
or the other way around a = b + n or whatever ig
nimble blaze
but you don't add a,b themselves
glad anchor
if b > a, n = b - a - 1
your saying they should replace all the ns in the sum with that?
cuz that makes sense they can
nimble blaze
no
naive agate
there's an issue with how you're setting up you sum,
a,b themselves aren't inclu...
The problem specifically says a and b should be.
nimble blaze
a isn't less than a
b isn't greater than b
nimble blaze
look at the wording of the question
naive agate
Huh.
My teacher told me like 4 different things on this problem so I'm very confused at this point.
glad anchor
b < a then
nimble blaze
you improperly assumed inclusive yourself
glad anchor
yeah exclusive
naive agate
My TA told me inclusive.
nimble blaze
your ta is wrong then
naive agate
Wouldn't be the first time.
naive agate
Ok, so there will be more evens than odds, or an equal amount?
glad anchor
more evens, no?
naive agate
What I was thinkin.
glad anchor
your ta is wrong then
can you elaborate on the sum thing
bc i think they just mean n = a - b - 1
(amount of integers between b and a)
nimble blaze
the way I'd set it up would be
sum to (a-1) - sum to b = 1000
with a = b + 2k