get ur own channel
#help-28
dusk drum
twilit agate
dusk drum
sturdy spruce
What’s the constant of integration
sturdy spruce
Are you sure
sturdy spruce
Does it make sense when t=0
sturdy spruce
That’s why there’s an mg at the bottom
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sturdy spruce
i just set c=0
Idk what you mean by c
Physically you want the time elapsed at rest to be 0
Those are the initial conditions
dusk drum
.reopen
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✅
glossy valveBOT
>_<
dusk drum
oh nvm this is equal to -m/b ln(gm-bv)
or
is it
yea differentiating both yields the same result
but thats since -mln(m)/g is a constant
oh yeah constant of integration my bad
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gusty carbon
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robust topaz
u get same eq in denominator if you combine fractions
thats what it looks like
just check if thats true
gusty carbon
so do i need to multiply x by x-5 and 2 by x+8?
steel cove
yes
robust topaz
yes u get same fraction in denominator
just combine the fractions cancel denomicator and then solve for the quadratic
should be simple
astral lagoon
torn jolt
hello
robust topaz
could be
im not doing it
make sure ur calculations are correct
and then solve for the quadratic
gusty carbon
-20?
vestal rampart
msyeve
gusty carbon
i got 4,-5
gusty carbon
it was right :)
narrow helm
nice
gusty carbon
how should i start this oen?
narrow helm
like what you did with the last question
gusty carbon
i got x^2 +0x-57=0
narrow helm
should be correct
gusty carbon
- or - sqrt 228
narrow helm
how did you go from x^2 - 57 = 0 to +- 228?
gusty carbon
yes
- or -
Oh yeah oops
I think so
I always forget if i can
+or- sqrt 57
Yay thank you
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little fractal
how can I deduce the identities of the half angle?
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how can I deduce the identities of the half angle?
fathom saddle
cos(2x) = 1 - 2sin²(x)
Solving that for sin(x):
sin(x) = √[(1 - cos(2x)) / 2]
That's basically the half angle formula right there. We can make it "look right" by subbing in x/2 in for x.
fathom saddle
That's a picture! Did you have any thoughts on what I put above?
little fractal
That's a picture! Did you have any thoughts on what I put above?
ah? I did what you told me
little fractal
cos(2x) = 1 - 2sin²(x)
Solving that for sin(x):
sin(x) = √[(1 - cos(2x)) / 2]
and for the tangent?
How can I obtain the last identities?
narrow helm
rationalise the numerator of the last expression and you'll get those 'last' identities
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@little fractal Has your question been resolved?
bleak wadi
Simplify:
$\frac{1}{2!}((a+b)(a+b-1)-a(a-1)-b(b-1)$
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Simplify:
$\frac{1}{2!}((a+b)(a+b-1)-a(a-1)-b(b-1)$
glossy valveBOT
ChocolateFudge
bleak wadi
$\frac{1}{2!}((a+b)(a+b-1)-a(a-1)-b(b-1))$
glossy valveBOT
ChocolateFudge
bleak wadi
Halp pls
bleak wadi
fix your parentheses
..
minor crater
what is 2! equal to?
bleak wadi
2
minor crater
yep
bleak wadi
$\frac{1}{2}((a+b)(a+b-1)-a(a-1)-b(b-1))$
glossy valveBOT
ChocolateFudge
minor crater
**ChocolateFudge**
inside the brackets you just expand really
bleak wadi
1/2*((a+b)^2 - (a+b)) - (a^2-a) - (b^2-b)
bleak wadi
I am tryign to simplify this equation to its simpilist form
minor crater
then expand all brackets
bleak wadi
1/2*(a^2 + 2 a b + b^2 - a - b) - (aa-a) - (bb -b)
bleak wadi
then expand all brackets
I don't think I can expand it farther
minor crater
cancel some stuff
bleak wadi
halp
minor crater
$1/2*(a^2 + 2 a b + b^2 - a - b - a^2 + a - b^2 +b )$
glossy valveBOT
artemetra
bleak wadi
ok
minor crater
$=\frac{1}{2}*(\cancel{a^2} + 2 a b + \cancel{b^2} - a - b - \cancel{a^2} + a - \cancel{b^2} +b )$
glossy valveBOT
artemetra
minor crater
yes
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bleak wadi
.end
minor crater
it closed
bleak wadi
thx
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deft tulip
what's the query to get texit bot to graph some R3 equation using wolfram? like for example x^2 + y^2 = 9z
deft tulip
,w graph x^2 + y^2 = 9z
glossy valveBOT
minor crater
well that was easy lol
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deft tulip
,w graph y^2 - z^2 = 0
glossy valveBOT
deft tulip
well that was easy lol
actually, q what if the x variable isn't present but still wanna see a 3d graph?
minor crater
0*x 
deft tulip
lul
minor crater
,w graph y^2 - z^2 + 0*x = 0
glossy valveBOT
minor crater
aw
minor crater
cuz no x
deft tulip
,w graph y^2 - z^2 + x - x = 0
glossy valveBOT
deft tulip
,w 3dgraph y^2-z^2=0
glossy valveBOT
minor crater
,w 3dgraph y^2-z^2=0
minor crater
oh i didnt even know desmos does 3d plots
it's a new feature
deft tulip
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deft tulip
it's a new feature
gotcha thx, thatll serve my purposes for now
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ruby wolf
15.33
ruby wolf
I'm a little stuck
fickle shore
I think you use the formula of (a+b)²
ruby wolf
heh
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mossy palm
hello I am introducing myself to the notion of Laplace transform to solve differential equations, I do not understand the process between the lines that I have arrowed in red
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@mossy palm Has your question been resolved?
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@mossy palm Has your question been resolved?
fast peak
the missing 5 looks like a typo
for the last step, add 2 on both sides and then divide by p+5 on both sides
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vital cosmos
how would i show a function 2xsin(1/(y^4 + x^2)) - 2(x^2 + y^2)x*cos(1/(y^4 + x^2))/(y^4 + x^2)^2 is or isn't continuous (x,y) = (0,0)? How would I go about it?
mortal kraken
$2x\sin(\frac{1}{y^4+x^2})$
glossy valveBOT
The Prophet Of The Damned
mortal kraken
is that the first term you typed
if yes
x=0 y=0
1/0 undefined
which makes the whole undefined
vital cosmos
yes it is
okay thanks
I guess a step back then
my original function was f = (x^2+y^2)*sin(1/(y^4+x^2))
did i do df/dx properly?
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vital cosmos
<@&286206848099549185>
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@vital cosmos Has your question been resolved?
deft tulip
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deft tulip
@hallow walrus how would you prove that? i guess that is not obvious to me at all 😄
craggy tapir
bisector theorem?
hallow walrus
Draw a line parallel to the angle bisector from one of the bottom two vertices
crisp rover
what do you want to prove
hallow walrus
And extend the opposite side
Then bpt should give you the result
Maybe a few more steps
deft tulip
ok gotcha, sorry i did not understand this, thanks for the clarification 😄
learn something new
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peak dome
how do i integrate cos(x^2)
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how do i integrate cos(x^2)
fast peak
how do i integrate cos(x^2)
thats easy, you dont
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twin wolf
fresnel 
scarlet parrot
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scarlet parrot
I need help with the second question
I need to calculate that using the Integration by part
neon geyser
I need to calculate that using the Integration by part
yes so why don't you get started with that?
scarlet parrot
I did
neon geyser
where are you stuck then?
,, \int \ln (1+x) \cdot \frac{1}{x^2} \text{ dx} = - \frac 1x \cdot \ln (1+x) + {\color{red}\int \frac{1}{1+x} \text{ dx}}
glossy valveBOT
nyxie9151
scarlet parrot
I need to find the primitive of that
neon geyser
the red thing you already solved in question 1)
neon geyser
that does answer it lol
u already know the answer to this integral
u did it in 1)
so you have the entire thing
scarlet parrot
So how does the process of replacing x by 2 and 1 work?
glossy valveBOT
nyxie9151
scarlet parrot
Just F(2) - F(1)?
neon geyser
Just F(2) - F(1)?
yes
scarlet parrot
Okay okay
neon geyser
if the integral was x^3/3 + c
then sub in 2 then subtract when subbing in 1
so you have (2^3/3 + c) - (1^3/3 + c)
then you see that the c's canacel out so you never actually deal with it anyway
scarlet parrot
What about the steps it took to get here
glossy valveBOT
nyxie9151
neon geyser
u = ln(1+x) and dv = 1/x^2 dx
plain fox
u = ln(1+x) and dv = 1/x^2 dx
u only have 1 variable so shouldnt integral by parts be like this?
$$\int uv dx = u \int v dx - \int (\frac{d}{dx}u \cdot \int v dx) dx$$
glossy valveBOT
JustToPro
plain fox
**nyxie9151**
what u stated is integration by parts but when u are integrating with a different variable which in here is v
scarlet parrot
Idk what u said to me just now bc i'm only studying highschool math, but what i do know is that i need to learn how to use this bot
scarlet parrot
u only have 1 variable so shouldnt integral by parts be like this?
$$\int uv dx ...
dollar signs??
plain fox
dollar signs??
u can use dollar signs for this also
plain fox
Idk what u said to me just now bc i'm only studying highschool math, but what i ...
have u studied integration by parts before?
scarlet parrot
**nyxie9151**
2 days ago and its this formula
scarlet parrot
u can use dollar signs for this also
What's the programming language called?
plain fox
What's the programming language called?
im pretty sure TeXit or LaTex
plain fox
2 days ago and its this formula
ur question is $$\int^2_{1} \frac{\ln(1+x)}{x^2}dx$$
scarlet parrot
Yes i solved it
glossy valveBOT
JustToPro
plain fox
u can not use the formula u are saying for this one tho
plain fox
**nyxie9151**
this is integrating u with respect to another variable v
in the question u are integrating x with respect to x
so the formula i sent is applicable here only
this can be rewritten as $$\int^2_{1} \ln(1+x)x^{-2}dx$$
and here now $u=\ln(1+x)$ and $v=x^{-2}$
glossy valveBOT
JustToPro
plain fox
$$\int uv dx = u \int v dx - \int (\frac{d}{dx}u \cdot \int v dx) dx$$
and now use this
glossy valveBOT
JustToPro
plain fox
is this formula new to u?
scarlet parrot
**JustToPro**
Yes it is but u made a mistake here
scarlet parrot
**JustToPro**
.
plain fox
?
plain fox
is ur answer correct (checked with the key book or a guide or smthing)
im 99% sure u can not use the formula u used (the one with integral u dv) in this question because u have variable "x" and are integrating with respect to that "x" (dx)
scarlet parrot
I made a slight mistake at first but the answer should be: 3ln(2) - 3ln(3)/2
That 1% might just break through
Try calculating it using the formula u propose
plain fox
the formula i am proposing makes sense in this 100%
scarlet parrot
Use it
plain fox
and ill explain it wait
this is ur question
the question number 1 , u calculated that integral , yes?
now using my formula
scarlet parrot
Yes
plain fox
$$\int \ln(1+x)x^{-2} dx = \ln(1+x) \int x^{-2} dx - \int (\frac{d}{dx}\ln(1+x) \cdot \int x^{-2} dx) dx$$
glossy valveBOT
JustToPro
plain fox
the integral literally simplifies into this
scarlet parrot
Lemme solve that
scarlet parrot
Okay
plain fox
$$= \frac{1}{(1+x)} x^{-1} \cdot -1 dx - \int (\frac{1}{1+x} \cdot x^{-1} \cdot -1 dx$$
simplify further and we get
glossy valveBOT
JustToPro
plain fox
$$= -\frac{1}{x(1+x)} + \int \big{(}\frac{1}{x(1+x)} \big{)}dx$$
which is literally ur first question
scarlet parrot
Yes
glossy valveBOT
JustToPro
plain fox
so this just makes more sense (And is also correct)
$$ \int u \text{dv} = uv - \int v \text{du}$$
This is pretty advanced stuff for high school
glossy valveBOT
JustToPro
scarlet parrot
Oh yeah i can tell
plain fox
the formula u used it advanced
scarlet parrot
**JustToPro**
Liquify it
plain fox
**JustToPro**
so the first fraction basically becomes
$$-\frac{1}{2(1+2)} - (-\frac{1}{1(1+1)})$$
glossy valveBOT
JustToPro
plain fox
so the answer becomes uh
scarlet parrot
the formula u used it advanced
Well its the second last math lesson in the last highschool year but what do i know
Whip out the calculator
plain fox
**JustToPro**
so this becomes $$\frac{2}{6}$$
glossy valveBOT
JustToPro
scarlet parrot
2ln(2) -ln(3)
glossy valveBOT
JustToPro
plain fox
is there a key guide or smthing we can use to verify our answer
scarlet parrot
Nope
plain fox
as i said im 99% sure that to solve this question u have to use the formula i sent
but idk why u guys havent studied that yet
scarlet parrot
I think we were supposed to study it
But some issues and delays happened and removed some parts of each lesson
For example we're studying integrals for area only, when we were supposed to also study volume
Stinky political stuff affected our educational system
plain fox
sequences are also imp
scarlet parrot
Yes but the part where it's only child's play
plain fox
oh so u will be studying the harder stuffs?
plain fox
,w solve \int^2_{1} \frac{\ln(1+x)}{x^2}dx
glossy valveBOT
plain fox
,w \frac{1}{3} + 2\ln2 - \ln3
glossy valveBOT
plain fox
hm
plain fox
There never was any advances sequence lessons
u said u skipped the childs play stuff
scarlet parrot
I was right :D
scarlet parrot
u said u skipped the childs play stuff
No they kept it in
plain fox
I was right :D
shouldnt be
scarlet parrot
It took us 2 hours to finish this year's sequences
plain fox
thats rough
scarlet parrot
Yeah well what can u do
It made the exam easier thats for sure
The finals that is
scarlet parrot
But not a great start for higher education in terms of lessons
scarlet parrot
but u skipped like half the stuffs
Not half but i'd say like a good 15%
The only logarithmic function we studied this year is ln
No log
plain fox
But not a great start for higher education in terms of lessons
well good luck to u for the future
log and ln is basically the same thing
ln is log with base e
scarlet parrot
would be harder in future classes
They make us go through a thing called preparatory school
The make a whole lot more money from it than highschool
plain fox
hm
scarlet parrot
So u see where heads meet there
plain fox
so they reduced the stuffs from high school and have another school to teach that?
scarlet parrot
Yup
plain fox
thats just a business for money making
plain fox
hm
scarlet parrot
Meaning its the same thing for almost every former french colony
So
A good chunk of the world
plain fox
hm
scarlet parrot
Guess what
In those preparatory schools, u also pay before u enter the SELECTION PROGRAM
Its not abt the part where u pay
Its abt the SELECTION PROGRAM
plain fox
so u pay before even selecting what u want to study ?
scarlet parrot
Yes
plain fox
so its just a way to make money
scarlet parrot
guess what
If u dont make it through the selection program they recommend u to some institutions with no future
And if u do
Make it through
U pay to apply to a set of universities that THEY choose for u
plain fox
is selection program some sort of test / exam where they see which institution u are able to go?
plain fox
U pay to apply to a set of universities that THEY choose for u
so u are stuck in the hands of the education commitee or smthing like that?
scarlet parrot
is selection program some sort of test / exam where they see which institution u...
No they do a thing where they use a formula to calculate ur final grade, which puts u on a graph competing with other students, if u make it, you get to study at the preparatory school, then u pass ur finals, and the grades u get determine which universities they will send u to
plain fox
oh ok
plain fox
well where i live , u study , u pass exam , u choose which university to go to but u have to pass the entry exam of that university
scarlet parrot
If u apply to some of those universities (u have a limit) and u don't find a place there, they force u to study atleast a year in one of the other universities that u didnt choose
scarlet parrot
so u are stuck in the hands of the education commitee or smthing like that?
Yes
scarlet parrot
well where i live , u study , u pass exam , u choose which university to go to b...
Yes
I'm studying to get a good enough grade so that i can go to one of the countries like the one ur in, and get accepted at a uni there
plain fox
If u apply to some of those universities (u have a limit) and u don't find a pla...
ez , just apply to the universities with a low score and dont apply to the ones that are famouse , they will force you to study in the one which u didnt choose which would be a famous university xD
plain fox
I'm studying to get a good enough grade so that i can go to one of the countries...
im not in the country u are thinking of
im in Pakistan , not many good universities in here or anything
scarlet parrot
ez , just apply to the universities with a low score and dont apply to the ones ...
It is funny, but in reality they ban u from applying to unis for a year
To go redo ur last year of highschool
plain fox
so going to university is like a must ?
scarlet parrot
Not really
U can just immigrate to another country but of course they make it difficult for u
Because they need doctors and engineers
plain fox
but what if u want to stay in France
scarlet parrot
U can lol
plain fox
no but without going to a university , like what if i want to be a dropout and start a business?
scarlet parrot
Just go to a shit uni where they ask for minimum grades
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sonic thicket
hi guys
spare owl
just send the problem
sonic thicket
ok ok
spare owl
and the steps you tried to solve ti
sonic thicket
this is the problem
and
I got the first one right
but the answer that I get for the second one
is 1
but the actual answer is
0
and I dont know why
lemme send you what I have
sorry if I have some mistakes with my english btw
spare owl
no problem
sonic thicket
basically since it is a quadratic function I can calculate that as a parabolla
and I tried to found the vertex of that parabolla using the derivative
and setting it equal to zero to found the maximum value in this case
putting that answer in my equation and tried to solve it in that way
but I do not know why the answer is zero
spare owl
it's not really a parabola
sharp vortex
is it?
spare owl
in that case no
sonic thicket
Im reaaally confused with this one
I also tried to solve it with the vertex formula but it gave me the same thing
Idk if it has something to do with the ceiling function or something like that
wanton shore
@spare owl
sonic thicket
i honestly have no idea
dw
Im in the same situation hahaha
I´ll wait til someone comes up with something
spare owl
what have you done in the 1st part
spare owl
the only thing i can think of is vieta's formulas but i'm unsure how that would work
i guess i dont get what the condition for 2 real roots would be
sonic thicket
and then I switched the b in the equation of the problem with the b that I got here
And did the same thing that I did in the last one
Found the minimum value
@spare owl
spare owl
fair i missed that substitution
i don't see why it wouldn't work then
looks correct to me
sonic thicket
yeah for me too
I dont understand why the answer is zero in the last one
<@&286206848099549185>
can somebody help me please
saddest man on earth rn
bc of this thing
😦
If u help me you will have Megan Fox at your house tomorrow
Its the prize
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vocal coyote
Can you give me a single example of an equation $x^4+ax^2+b$ that has exactly two real roots when $b \geq 0$
am I reading the question wrong?
glossy valveBOT
AkishBrabus
vocal coyote
right, I'm asking for a single example
just one graph that I can plot
like x^4? Does that count as two solutions
they are including multiplicity?
Either way, I think it is 1
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@sonic thicket Has your question been resolved?
sonic thicket
right, I'm asking for a single example
ohh ok lemme think
sonic thicket
Either way, I think it is 1
yeah me too
vocal coyote
You should use desmos and just play around with it
sonic thicket
I just saw a video about it but it says that bc a cannot be greater than 0 the solution cannot be 1
vocal coyote
The minimum is achieving when b=o
sonic thicket
but I didnt understand hahaha
vocal coyote
Hmmm I'll have to play around with it more
sonic thicket
The minimum is achieving when b=o
why?
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thorn comet
whats in the boxes labelled as 1 and 2?
sonic thicket
the answers
thorn comet
oh ok
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blazing dome
Help! in need of a cool trig problem to solve. of any kind
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Help! in need of a cool trig problem to solve. of any kind
torn gust
o7
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red stone
Anyone know how to do 33,34,35
red stone
I thought mine got deleted sorry
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dense kestrel
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dense kestrel
I have no clue how to start this
hot herald
identify the numbers between 1 and 47
that have a factor of 17
determine their prime factorisation
and count the number of times 17 appears in that
dense kestrel
Yes yes thank you i got it
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plain depot
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plain depot
Basically they are saying "verify that the function z satisfies that equation"
spiral vigil
can you calculate δz/δx ?
spiral vigil
i think that's intended to be multiplication
plain depot
yeah it is ( at least thats my percepcion too ) , but how should i do it ?
how can i differenciate that function if i dont know what is that function ?
do i need to use the "Implicit function theorem" or something like that
spiral vigil
well you can just work with φ' I guess?
spiral vigil
but idk, is that not just supposed to be x • φ • x/y?
kindred grove
well it's a PDE, you often end up with arbitrary functions when solving them
it's prolly not multiplication
spiral vigil
guess it would be written differently then
plain depot
but idk, is that not just supposed to be x • φ • x/y?
ahh i think not , but if thats the case it would be much easier i think
plain depot
well it's a PDE, you often end up with arbitrary functions when solving them
doest that mean Parcial differencial equation ? Cause i didnt talk about that yet in my Calc 2 class
kindred grove
yes
spiral vigil
i think platypus is right, it should work with any φ? can you show your work
plain depot
well i am studying for my test , and i am just doing some exercice sheets maybe i am doing something i am not supposed to
But for example i did this exercice
spiral vigil
de classe C¹ -> you can differentiate it
plain depot
the are saying consider z to be f((x+y)/(x-y)) , and f a function of class C1 prove the following:
spiral vigil
those are neat exercises
plain depot
and i did this exercice , as aPlaytypus said i worked with arbitrary functions , but in the previous exercise i dont know how to do it
little mica
There is formula I don't remember the name but spacific for these type of questions
Where we have to calculate x(dz/dx)+y(dz/dy)
plain depot
idk for this one i didnt use any specific formula
spiral vigil
can you show work for the φ one?
kindred grove
Where we have to calculate x(dz/dx)+y(dz/dy)
yea I saw something like that once, but forgot
¯_(ツ)_/¯
little mica
it is like if
f(Lemda x,Lemda y)=(Lemda)^n f(x,y)
Then x(df/dx)+y(df/dy)=nf
little mica
For this question n=0
kindred grove
that's pretty much it yes
the only problem is when you talk about d(phi)/dx, d(phi)/dy, that doesn't make sense, phi is a 1d function
plain depot
Ohhh yeah that makes sense
kindred grove
picking up this part of your workings, what you have when you apply the chain rule is
x dφ/du du/dx, where u=x/y
and du/dx is just 1/y
so in the end you indeed have the nice cancelling out
plain depot
and du/dx is just 1/y
I got this part
plain depot
so in the end you indeed have the nice cancelling out
But not this one sorry 😭😭😭
Also i am not that good at English
kindred grove
i meant this part is essentially correct
plain depot
Ohh ok, but it cancels out?
kindred grove
it's just that you're talking about dphi/du in both terms, there's no dphi/dx, dphi/dy as we mentioned earlier
plain depot
In other woeds why are they equal
plain depot
it's just that you're talking about dphi/du in both terms, there's no dphi/dx, d...
So you are saying dphi/dx , dphi/dy are both 0?
kindred grove
I'm saying your work wasn't totally correct but you had the right idea
plain depot
I got that it doesent make sense to talk about parcial derivatives since phi its 1d function
kindred grove
So you are saying dphi/dx , dphi/dy are both 0?
I just said these don't even make sense
that's what I was talking about the whole time
plain depot
Ohhh i see
kindred grove
and also your x^2 * 1/y became x^2 y at some point
plain depot
Tyy so much!!!
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lavish whale
How would I be able to take a derivative (not numerical) on a TI-nspire CX-II?
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runic spruce
If the determinant of a matrix is zero does that mean its singular
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warm harbor
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warm harbor
problem
here is where I am so far
why I need help: the fact that this is a degenerate random variable is what throws me off, so in my mind, the sequence converging to single point such as "a" feels like its literally the same thing as something converging in probability, but idk how to show this graphically
this video does a great job explaining how this concept works if you are interested https://www.youtube.com/watch?v=Ajar_6MAOLw
torn gust
yea it's just that once you're past n0, the "distribution" has P(outside range)=0, so P(outside range) tends to 0 as well
the "tail" outside is 0 and converges to 0
warm harbor
but they are degenerate random varialbes
torn gust
yea that's just like a discrete variable right
warm harbor
sort of but the probability is 1 at a single point and zero eslewehre
at least thats how I understand it
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sterile swift
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sterile swift
this might not really be the place
but I am simulating two boxes labeled 1 and 2
and basically if there are no empty boxes, then we would have 1 + 2 in our distrubtion or 3
and if there is one empty box, we either have 1 + 1 or 2+2 which is 4
so clearly the probability of one empty box is twice that of zero empty boxes
however when I run this simulation I get them as equivalent
stiff musk
what output is printed
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stiff musk
what result do you expect
0.5, 0.5, 0 sounds right to me
you have four possible outcomes if you draw two numbers with replacement from {1,2}:
1,1 : one empty
1,2: zero empty
2,1: zero empty
2,2: one empty
all four have equal probability
sterile swift
you have four possible outcomes if you draw two numbers with replacement from {1...
I see, I think I need to adjust my r code then
because when you distribute two balls into two boxes, the probabiity to have 1 empty box is double that of two empty boxes
but I'm not sure how to simulate that then...
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sterile swift
yeah so it confuses me
because doing 1+2 and also 2 + 1 as two distinct results implies the balls are distinguishable which they are not
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scarlet parrot
Balls hehe
sterile swift
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round sluice
hello can you help me with a question involving integral and limit?
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hello can you help me with a question involving integral and limit?
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round sluice
craggy tapir
what have you tried
craggy tapir
round sluice
I forgot the minus part of the integral at the denominator
at the last line
Look to be honest I didn't try to answer the question I just don't really know how I should answer this type of question
craggy tapir
have you learned about one sided limit?
round sluice
yes, you mean when x approches zero from the right for example?
craggy tapir
yeah
round sluice
It is interesting that they specifically chose to use x approaching from the right
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round sluice
I'll try answering it somehow and if I'll need help I will come back here thank you for your time!
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torn jolt
im trying to solve the ODE [
2y'' + (x+1)y' + 3y = 0, \q x_0 = 2
]
I set up my series and get [
2\sum_{n=2}^\infty a_nn(n-1)(x-2)^n + 3\sum_{n=1}^\infty a_nn(x-2)^n + \sum_{n=1}^\infty a_nn(x-2)^{n+1} + 3\sum_{n=0}^\infty a_n (x-2)^n = 0
]
but im really unsure on how to proceed with cleaning this up lmao
glossy valveBOT
torn jolt
oh no i messed up
,align 2\sum_{n=2}^\infty a_nn(n-1)(x-2)^{n-2} + 3\sum_{n=1}^\infty a_nn(x-2)^{n-1} \+\sum_{n=1}^\infty a_nn(x-2)^{n} + 3\sum_{n=0}^\infty a_n (x-2)^n = 0
i think this is fine now
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torn jolt
,, 2(n+2)(n+1)a_{n+2} + (2n +3)a_n = 0, \q a_2 = 0, \q a_0 = 0
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full marsh
About the second term
Why do you have n-1 in the exponent now
And third term has n now
torn jolt
well
so what i did is like
we have [
y = \sum_{n=0}^\infty a_n(x-2)^n, \
y' = \sum_{n=1}^\infty na_n(x-2)^{n-1} \
y'' = \sum_{n=2}^\infty n(n-1)a_n(x-2)^{n-2}
]
so substituting accordingly gets us [
2\9{\sum_{n=2}^\infty n(n-1)a_n(x-2)^{n-2}} + (x+1)\9{\sum_{n=1}^\infty na_n(x-2)^{n-1}} + 3\9{\sum_{n=0}^\infty a_n(x-2)^n} = 0
]
As for the middle term, the coefficient can be rewritten as $x+1 = x -2 +3$. So: [
2\9{\sum_{n=2}^\infty n(n-1)a_n(x-2)^{n-2}} + \9{\sum_{n=1}^\infty na_n(x-2)^n} \[1.5ex]+ 3\9{\sum_{n=1}^\infty na_n(x-2)^{n-1}} + 3\9{\sum_{n=0}^\infty a_n(x-2)^n} = 0
]
torn jolt
yeah messed up my copypasting lmao
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full marsh
ok I guess u can rewrite it that way yes
full marsh
why does that help
You just want to have all summations with the same (x-2)^something
and start from the same index
torn jolt
because we want to represent (x+1) some way in terms of (x-2). We want to have a recurrence relation independent of x after all
full marsh
oh true true
full marsh
not really fixed
you need same exponents, and make the summations start from the same index
torn jolt
yeah i meant imma fix it now
full marsh
Make every summation have (x - 2)^n
full marsh
lol
torn jolt
we have [
y = \sum_{n=0}^\infty a_n(x-2)^n, \
y' = \sum_{n=1}^\infty na_n(x-2)^{n-1} \
y'' = \sum_{n=2}^\infty n(n-1)a_n(x-2)^{n-2}
]
so substituting accordingly gets us [
2\9{\sum_{n=2}^\infty n(n-1)a_n(x-2)^{n-2}} + (x+1)\9{\sum_{n=1}^\infty na_n(x-2)^{n-1}} + 3\9{\sum_{n=0}^\infty a_n(x-2)^n} = 0
]
As for the middle term, the coefficient can be rewritten as $x+1 = x -2 +3$. So: [
2\9{\sum_{n=2}^\infty n(n-1)a_n(x-2)^{n-2}} + \9{\sum_{n=1}^\infty na_n(x-2)^n} \[1.5ex]+ 3\9{\sum_{n=1}^\infty na_n(x-2)^{n-1}} + 3\9{\sum_{n=0}^\infty a_n(x-2)^n} = 0
]
We have this after fixing the powers: [
2\9{\sum_{n=0}^\infty (n+2)(n+1)a_{n+2}(x-2)^n} + \9{\sum_{n=1}^\infty na_n(x-2)^n} \[1.5ex]+ 3\9{\sum_{n=0}^\infty (n+1)a_{n+1}(x-2)^n} + 3\9{\sum_{n=0}^\infty a_n(x-2)^n} = 0
]
so all our powers are the same, so we fix the indices to become:
[
4a_2 + 3a_1+ 3a_0 + 2\9{\sum_{n=1}^\infty (n+2)(n+1)a_{n+2}(x-2)^n} + \9{\sum_{n=1}^\infty na_n(x-2)^n} \[1.5ex]+ 3\9{\sum_{n=1}^\infty (n+1)a_{n+1}(x-2)^n} + 3\9{\sum_{n=1}^\infty a_n(x-2)^n} = 0
]
oh wait actually
i messed up
ok i think its fine
wait
this is incomplete still one sec
@full marsh what do u think 
glossy valveBOT
torn jolt
so our recurrence relation has conditions $a_2 = 0, a_1 = 0, a_0 = 0$ and then the recurrence relation itself is [
2(n+2)(n+1)a_{n+2} + 3(n+1)a_{n+1} +(3+n)a_n = 0
]
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naive stone
Guys , not a numeric question but
See we have been using squares to measure Area , Volume etc etc cm squares , meter squares ect ect
ik why are only using squares... i am just wondering suppose for a movement if we choose say circle for measuring these things now (area , volume ect ect) then
All the formula's we have created so far or of no use now , right ? or we can still use them and just say ... okay this shape is 45 circle units lol
by formulas i mean those simple 2D , 3D formulas like l*b blah blah
spiral vigil
haha good question, yes in principle we could define a "circle meter" to be, say, the area of a circle with diameter 1m; or we could define a "triangle meter" to be the area of a triangle with side length 1m. But squares are simpler because they're just meter times meter
robust topaz
id suppose its because you can form tessellations with squares
and also ig you can never truely know the area of a circle
because pi is irrational
slate violet
?
wide sundial
A bit hard to cover a shape with circles, they don’t cover a flat surface without overlapping very well
slate violet
Basically all that matters for the area is the units
It's pretty hard to know that the area of a circle has units (length)^2 unless you know the pi r^2 formula
robust topaz
you can know the area of a circle
not to 100% precision
slate violet
It's pretty easy to know the area of a rectangle has units (length)^2 cause base * height is pretty obvious
robust topaz
when your working on like a rocket
if it were circles
it would become a competition of who can aprox pi to the highest degree
wide sundial
I mean…you can very much measure things in polar coordinates
naive stone
haha good question, yes in principle we could define a "circle meter" to be, say...
haha yeah
so all the formulas we have created so far are suddenly junk right ? lol
i was just wondering what would be the formula of simple 2d square shape if are using 'unit circles' lol
spiral vigil
i think they'd just be off by a scale factor
slate violet
haha yeah
so all the formulas we have created so far are suddenly junk right ? ...
Ah you might actually be interested in polar coordinates then
robust topaz
I mean…you can very much measure things in polar coordinates
yea but
naive stone
Ah you might actually be interested in polar coordinates then
what are they ?
robust topaz
imagine trying to teach area in grade 4 then
wide sundial
what are they ?
We can measure things by a distance and an angle
slate violet
wide sundial
Like that
naive stone
i think they'd just be off by a scale factor
uh yeah smth like that , thanks
wide sundial
Instead of having grid lines that you normally see you have circles
slate violet
Yep
naive stone
We can measure things by a distance and an angle
sounds like trigno , calculus to me
wide sundial
It’s very cool
slate violet
sounds like trigno , calculus to me
Yeah exactly
It's a mix of both trig and calculus
slate violet
This only works when theta is in radians
It's not too hard to show from A_sector = pi r^2 * theta/360
But 2pi radians = 360 deg
So if you make $\theta$ be tiny, $d \theta$ and sum all of these wedges up
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south
naive stone
It's a mix of both trig and calculus
woah interesting
do u know some books about it ?
spiral vigil
(radians is the correct way to measure an angle)
slate violet
You get the formula for polar integration $A = \int \frac{1}{2} r^2 \ d \theta$
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naive stone
https://tutorial.math.lamar.edu/Classes/CalcII/PolarCoordinates.aspx
Thanks mate i will check it out
any good youtubers who teaches that lol if i may ask ? maybe not just that but for maths as a whole
wide sundial
haha yeah
so all the formulas we have created so far are suddenly junk right ? ...
It’s quite an interesting thing to think about, it goes to show that our x y coordinate system may be a bit challenging to work with when measuring circles or circular things, which is why we sometimes try polar coords to see if the problem might be easier from this perspective
slate violet
Thanks mate i will check it out
any good youtubers who teaches that lol if i ma...
organic chemistry tutor, lmao
wide sundial
It’s some sort of same thinking as “why don’t we try thinking in terms of circles rather than squares”
slate violet
It’s some sort of same thinking as “why don’t we try thinking in terms of circle...
The issue with a circle is that the circles will overlap if you want the area of a square
naive stone
It’s quite an interesting thing to think about, it goes to show that our x y coo...
Yeah ig formulas would be much more difficult if we stared to measure things in unit circle (If we want to measure precisely ) all that trigno calculus would come in
slate violet
But you can approximate a circle with rectangles / squares just fine
Or even a blob
naive stone
organic chemistry tutor, lmao
ah that guy
naive stone
But you can approximate a circle with rectangles / squares just fine
really ? interesting
slate violet
wide sundial
Yeah ig formulas would be much more difficult if we stared to measure things in ...
I think the problem here is that circles don’t tessellate 2D
slate violet
You can estimate the area of a blob by grid counting
1 if completely inside, 0 if completely outside
1/2 if partially inside
slate violet
I think the problem here is that circles don’t tessellate 2D
Yeah
wide sundial
You can definitely try measuring things with shapes that do tessellate 2D
naive stone
oh i just realize i am talking to two different ppl lol
slate violet
Only triangles, squares, and hexagons do if you're talking regular polygons
wide sundial
Shapes like equilateral triangles and hexagons
slate violet
Basically you're asking to measure area using triangles
Cause a hexagon can be broken up into triangles
wide sundial
Hexagons are the bestagons
wide sundial
But even better is the hexaflexagon
slate violet
Too much maths YT
naive stone
Ah you might actually be interested in polar coordinates then
ah fk i just realized i know this already but they haven't taught me well
wet hollow
hello, so my math problem is to find the derivative of the function in the first picture and my answer is;
5x^2 - 20
(x^2 + 4)^2
but math . com insists that the numerator is -5x^2 + 20 (refer to the second picture) , can i ask why is it that? and also can i further simplify the my derivative?
wide sundial
,w d/dx (-5x)/(x^2 + 4)
glossy valveBOT
wide sundial
hello, so my math problem is to find the derivative of the function in the first...
Same as your answer
wet hollow
Same as your answer
am i able to further simplify my derivative by expanding the denominator and then dividing it for example
5x^2 - 20
(x^2 + 4)^2
5x^2 - 20
(x^2 + 4)(x^2 + 4)
5 - 5
(x^2 + 4)
0
(x^2 +4)
?
wide sundial
am i able to further simplify my derivative by expanding the denominator and the...
What
If you think that you need to go revise your fractional algebra
wide sundial
Yes
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tropic wedge
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tropic wedge
I tried solving this by replacing ctg with cos / sin
applying substitution
and I reached infinity
after doing the limit -> 0
could someone let me know if my result is right?
restive sequoia
I’m getting divergent with my calculator, because of the ln term.
tropic wedge
I’m getting divergent with my calculator, because of the ln term.
so ig I got the right result then?
restive sequoia
I think so.
narrow helm
could someone let me know if my result is right?
is this a test?
tropic wedge
nope, it's an exercise
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sturdy fulcrum
(i) I get the answer as 25, even though the answer key says it's -7
My process:
Σα^2β^2
=> α^2β^2 + β^2γ^2 + α^2γ^2
=>(αβ + βγ + αγ)^2 -2αβγ(α+β+γ)
=>(Σαβ)^2 - 2αβγ(Σα)
=>(c/a)^2 - 2(-d/a)(-b/a)
=>9 + 16
=> 25
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tardy copper
For b
proven mango
they subbed it back in
so they don’t need to change the bounds
either you change the integrand and change the bounds
or you can change the integrand back to how it was originally and keep the bounds
basically they just substituted u for sin(x) again in the end
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simple badge
I'm given P(A1) = 0.22, P(A2) = 0.26, P(A3) = 0.28, P(A1 ∩ A2) = 0.11, P(A1 ∩ A3) = 0.07, P(A2 ∩ A3) = 0.05, P(A1 ∩ A2 ∩ A3) = 0.01
And asked to find P(A2 ∪ A3 | A1)
So P(A2 ∪ A3 | A1) = P(A1 ∩ P(A2 ∪ A3)) / P(A1)
How am I supposed to find P(A1 ∩ P(A2 ∪ A3)) when P(A1 ∩ P(A2 ∪ A3) = P(A1) * P(P(A2 ∪ A3)|A1) would require me to know P(A1 ∩ P(A2 ∪ A3))
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I'm given P(A1) = 0.22, P(A2) = 0.26, P(A3) = 0.28, P(A1 ∩ A2) = 0.11, P(A1 ∩ A3...
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pliant heron
Can I receive some help with this question. I think my formula for odd is incorrect. I was not sure how to do it with the alternating signs that I found in the table
leaden ermine
Can I receive some help with this question. I think my formula for odd is incorr...
I would work with this notation:
$$\text{even } n: x_{2n} \text{ and } \text{odd } n: x_{2n+1}$$
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pliant heron
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pliant heron
I would work with this notation:
$$\text{even } n: x_{2n} \text{ and } \text{od...
Okay thank you!!
pliant heron
I would work with this notation:
$$\text{even } n: x_{2n} \text{ and } \text{od...
I am just confused on what the equation for odd n would actually be?
Since it alternates between negatives and positives
leaden ermine
The first term is x_1 = -4
x_3 = 16
so it quadruple every time
changing sign every time
how would you describe the quadrupling?
without alternating sign yet
@pliant heron
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hazy onyx
i need help with factorials
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hazy onyx
I have this exercise:
list of numbers = {0, 1, 2, 3, 4}
How many possible numbers if 1 always comes before 2 and the number can't start with 0
torn gust
two parts:
how would you create such a number digit by digit, starting with the starting digit, like how many options are there each step
if you find there are say 100 of these numbers, how many of them have 1 before the 2, instead of 2 before the 1? is there something special there?
torn gust
ok I assumed that