ok
#help-10
modest ingot
the whole idea of inverse function is just rewinding
nah
let’s say you start at your school
i tell you to walk straight for 1 miles
let’s say a function told you to do that
warped zinc
bruih ok
modest ingot
now the inverse function tells you to rewind that action
which means you walk straight back for 1 mile
so you’re back at your school
warped zinc
okay cool
modest ingot
this is the idea of that
x is your current position
f inverse is the set of instruction that told you to go straight for 1 mile
f is the set of instruction that tells you to go back 1 mile
which means you had to return to your starting position
which is why it’s x on the right side of your equation
so whenever you do a certain instruction and then do the opposite of that instruction
clearly you just go back to how u were before following the instructions
modest ingot
yes the inverse of an inverse is just normal
lol okay
anyway this idea is what they wanted you to use
i’m pretty sure
so you should go and review it or smth
modest ingot
modest ingot
well whatever we talked about for the last 30 mins
should’ve amounted to something lol
modest ingot
also it says “math contest”
so it’s trying to get you to use “tricks” lol
like for 26) you’d use simon’s favorite factoring trick
to factorize x+ xy + y = 19
and then test those values on that other thing
which again i’m going to guess u haven’t learnt
modest ingot
just don’t do contest problems
warped zinc
only way to help your understanding tho
well at least for me
it helps alot
thanks for thehkep
modest ingot
sure if u say so lol
obtuse pebbleBOT
odd mantle
Am I doing something wrong? I'm left at -8costsint + 10sintcost
tawny copper
LMAO
obtuse pebbleBOT
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Available help channel!
warped zinc
obtuse pebbleBOT
marsh geyser
what characterizes a linear function in terms of its graph and equation?
marsh geyser
So can u determine the slope given the points?
marsh geyser
Do you think it matters in terms of still being a linear equation or do u think it is important?
warped zinc
no i think its still linear
just the question is asking f(0)
im guessing the answer is C
but how could i solve this with algebra
marsh geyser
So f(3) = a
That makes f(a)=2
Right?
Same way for the other one
Now write the equations of f(a) and f(c)
Better use c instead of b
Not to get confused with b of the equation
warped zinc
ssorry im back
warped zinc
That makes f(a)=2
why does that make f(a) = 2?
marsh geyser
f(f(3))=2
warped zinc
a being just a variable?
warped zinc
how do you know to put a inside of the brackets tho
so f(3)=a
then why is input a into f(x)=2
marsh geyser
Lets do this in a more striagjt way then
So f(f(3))=2
We also know that f(3)= 3m + b
So f(3m+b) = m(3m+b)+b=2
And from the other one
m(2m+b)+b=1
Now i have a system with 2 unknowns
U solve it and after getting the value of m and b
U can calculate f(0)
Is that clearer for u?
@warped zinc
obtuse pebbleBOT
@warped zinc Has your question been resolved?
warped zinc
okay thanks
yea i understand nnow a bit more
when you have lets just say
f(f(x))
you never include x in the outermost function right>?
polar cargo
I simplified the equation to n/8, is there a reason why it wouldn't be infinity?
worn yoke
that simplification does not look correct
polar cargo
that simplification does not look correct
It isn't?
worn yoke
can you show how you got that?
polar cargo
can you show how you got that?
give me a second to type it out, I lost my phone
polar cargo
can you show how you got that?
here
worn yoke
where did n^4 come from?
worn yoke
you should substitute (n+1) for n
polar cargo
lemme rewrite it rq
obtuse pebbleBOT
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shell oar
obtuse pebbleBOT
pseudo swift
if e^(k) = M e^(k-1) doesn't shock you, there shouldn't be any problem repeating that thing
that equation works for every iteration
so you have e^(k) = M e^(k-1) = M (M e^(k-2)) = ..etc
@shell oar
shell oar
if e^(k) = M e^(k-1) doesn't shock you, there shouldn't be any problem repeating...
ok acc how did that get e^(k) = M e^(k-1)?
pseudo swift
that's what they're telling you just above
pseudo swift
as I said, it works whatever the iteration
shell oar
oh ok
pseudo swift
shifting by 1 doesn't matter at all
shell oar
shifting by 1 doesn't matter at all
secondly i dont understand this summation here
whats going on here
i understand the above line
but why e^0??
pseudo swift
well e^(0) is a vector
you can write it as a lin combo of the eigenvectors
as to why they're doing that
well read what's next
shell oar
well e^(0) is a vector
but why e^0 why not e^1 or e^2 etc?
pseudo swift
I guess you could start from somewhere else yea
we're only interested to what happens in the limit anyway
it's just a natural choice to start with the error at iteration 0, cause that's what we're starting with
but you could proceed similarly with e^1942 if you wanted
it's just a bit weird
shell oar
it's just a natural choice to start with the error at iteration 0, cause that's ...
yh kl makes sense
shell oar
it's just a bit weird
why do we need the modulus of lamda to be less than 1?
for the lhs to be true
pseudo swift
well imagine you have a lambda_i >= 1, what happens to e^(k) ?
shell oar
well imagine you have a lambda_i >= 1, what happens to e^(k) ?
oh yh gets bigger
less than 1 makes it smaller as k gets bigger
pseudo swift
yep
shell oar
so the p(M) < 1 condition basically bound it so that all e-values r less than 1
obtuse pebbleBOT
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weak wasp
I need help solving a question on applications of definite integrals, it asks to
weak wasp
Number 33. a)
I’m having trouble figuring out if I should solve wrt to y or x and why
bold bane
I’m having trouble figuring out if I should solve wrt to y or x and why
What methods of calculating the volume of revolution do you know?
weak wasp
Oh in that third picture
This is what my teacher recommends for the cylindrical shell method
But it goes differently if you are integrating wrt to x vs y so I need to understand when to do which
bold bane
In both cases, you will need to know the radius.
Whether you use the Washer Method or Cylindrical Shell Method.
weak wasp
Okay
bold bane
The first is the Cylindrical Shell method and the second is the Disk Method.
As you can see, in both cases, you will need to know the radius.
How would you measure the length of the radius, r?
Would you measure it in terms of x or in terms of y?
weak wasp
In terms of y?
bold bane
If you were to use the Distance formula to measure the length, what variable would you end up with?
$D = \sqrt{ (x_2 - x_1)^2 + (y_2 - y_1)^2}$
warm shaleBOT
Kookiemon
weak wasp
Wait you need the distance formula here??
I haven’t been using that at all for these questions
bold bane
I'm teaching you an intuitive way to determine which variable to integrate with.
weak wasp
Okay
bold bane
Give these two points, A and B, if you were to measure the distance, aka radius, using the above Distance formula, which variable in the Distance formula wold be eliminated?
weak wasp
Oh y
bold bane
Correct.
weak wasp
So it’s x then
bold bane
Yes.
weak wasp
But don’t you know that because it’s a vertical line
Sorry I mean integrating wrt x
That’s what I said x for
weak wasp
Oh okay
bold bane
Does that clarify how you should setup the integrla?
weak wasp
I’ll try what I think you mean one second
I feel like this is wrong
And I still don’t know if I’m solving wrt to x or y
bold bane
Oops, I gave you incorrect information about the Cylindrical Shell method.
bold bane
The CS method is based on the height, not the radius.
bold bane
It will be x for the CS method.
weak wasp
Always?
bold bane
Not always.
The same logic applies when evaluating the length, aka distance, of h above.
weak wasp
Oh I saw something where he said that we use y for the washer method and x for the CS method
For a question he did
Obviously specific to that one
But I didn’t understand it though because I came late to class
I just know it’s a thing I guess
bold bane
In the CS method above, the distance would be x_2 - x_1, however those x values would be calculated in terms of y. --> x(y) = f(y)
weak wasp
Uh you mean this?
bold bane
x2(y) - x1(y)
bold bane
Yes, however you need to use the plus/minus square root.
weak wasp
So that’s what I’m supposed to do??
bold bane
It would be wrt y.
bold bane
Yes.
bold bane
That's my bad.
weak wasp
Still not working I don’t know
I tried this way where you split the integral in half because you are using -sqrt with one side and +sqrt for the other side
bold bane
Because you are solving wrt y, the interval will be from y=0 to y=1.
weak wasp
Oh right
But then how do you use the - and +
If you can split
Cant*
Oh actually if I’m only able to reach one half anyways with the + then I’m not actually needing to split an integral cause it’s already split for me due to the sqrt issue
I just need to write out the other half that’s -
bold bane
You should only need one integral for this.
weak wasp
How?
bold bane
Ok, let's focus on one part of the integral at a time because you did set it up incorrectly.
Let's focus on the length of the radius.
You have already determined the lengths need to be in terms of y.
weak wasp
How do I know it’s the primary value?
bold bane
The radius is the secondary value.
weak wasp
Is it because it’s longer
bold bane
Because that's how the Cylindrical Shell method works.
We calculate a Shell, or rectangular area, along the segment h.
weak wasp
Okay so how do I know if height is a vertical or horizontal line
bold bane
For the CS method, the height will be parallel to the axis of rotation.
weak wasp
Okay
bold bane
For any given value of y on the interval from [0,1], you are calculating the area of the shell that is made by an arbitrarily small width, dy.
bold bane
bold bane
Yes, because the axis of rotation is about y=1.
weak wasp
Okay so I knew that already then 😭
Ig that’s not what I’m stuck on
So how do you make it only one integral
Because this way still works right?
bold bane
No.
So you have this integral. The basis for the CS method wrt y.
A more proper way to write it would be
Because you need to know the length of the radius and the height which will both be in terms of y because you are solving the integral wrt y.
The radius, r(y), isn't as simple as r(y) = y in this problem.
weak wasp
Why not?
bold bane
If you look at the Geogebra file I posted, you will see that the radius of the shell changes as y is integrated from [0, 1].
The radius of these two shells are not the same.
weak wasp
Well actually is it y-1?
bold bane
Close.
bold bane
You always subtract the smaller value from the larger value to ensure you get a positive length.
weak wasp
So 1-y
bold bane
In this case, for y on the interval [0,1], it will always be less than or equal to 1.
Yes, so r(y) = 1 - y.
weak wasp
Because the line is above it
But if it was at y = -1
For the rotation axis
Would it be y - (-1)?
bold bane
Correct. You are getting it now.
bold bane
Understanding the order in which you subtract is a pretty big part of understanding how to evaluate these types of integrals.
weak wasp
Yeah I think I get that part
bold bane
So now let's focus on the height.
weak wasp
Alright
bold bane
So to find the length of h, you essentially apply to the Distance formula to get x_2 - x_1, or x_2(y) - x_1(y).
The one key thing to recognize here is that because x = sqrt(y), you need to use the positive and negative results of the square root to get the x values in Q1 and Q2.
Does that make sense?
weak wasp
Yeah I understood that part when you explained it a while ago
I just don’t know how to implement that into only one integral
bold bane
Give that h = x_2 - x_1, what would h(y) be equal to?
weak wasp
2sqrt(y)
bold bane
Correct.
bold bane
Looks good.
weak wasp
Why am I getting a negative number?
Oh wait
It’s that 1-y thing
I never wrote it in my book
bold bane
Yeah, y-1 should be 1-y.
weak wasp
Okay can I ask something else?
bold bane
Sure.
bold bane
So using the CS method, you have the radius in the image above.
As I mentioned earlier, you ALWAYS want the length to be a positive value.
weak wasp
Also side question , is the rule I made for the reason of when to use integration wrt to y for all 3 types of methods or only CS?
weak wasp
Hey before we move on, check this shit out
bold bane
Well, the Disk Method and the Washer Method are basically the same thing with r(x) = 0 iin the Disk Method.
bold bane
I prefer cats.
weak wasp
Bro is tired from doing nothing all day
I’m here doin first year uni math in grade 12
Anyways
Ok so the side question thing is right?
bold bane
What was the reason you made?
weak wasp
If the axis of rotation is a vertical line (y axis or x = #) integrate wrt to x
And if it’s a horizontal line (x - axis or y = #) integrate wrt to y
bold bane
For the CS method, that would be correct.
weak wasp
Okay but not for the other two?
bold bane
No, not for the Disc nor the Washer method.
bold bane
Yes.
bold bane
When in doubt, refer to the Distance formula.
weak wasp
That made it more confusing
Also is that distance formula idea for all 3 types
Of solving
bold bane
Yes.
weak wasp
I probably should understand it
bold bane
It is how I learned to understand when to solve wrt x or y.
weak wasp
Because you just cancel out one piece because it’s already either a horizontal or vertical line
So one half never gets used
bold bane
Correct.
weak wasp
But I don’t know how you know which is vertical or horizontal
bold bane
Just look at the endpoints of the radius or height.
weak wasp
Okay and height is the one parallel to the axis of rotation
But only for the case of CS
bold bane
If you have (x, y_2) and (x, y_1), then you know it is y_2 - y_1 or y2(x) - y1(x).
weak wasp
Okay yea
Shit I didn’t read ur bio mb
Clearly 😭
Even tho the distance thing isn’t that hard ig
bold bane
It helps to understand how some of these formulas are essentially derived.
weak wasp
Yeah true
Actually for when to split integrals
Is it meant for stuff like number 2 here
bold bane
You split integrals when the boundaries change.
weak wasp
Because in that graph to the left I did it because when solving you would end up with two different equations coming to make a corner
So then I split it to only have the one equation
There’s sort of a line I drew from that centre right corner to the left side
That’s where I cut the integral
bold bane
If you were to calculate the volume of the bounded region above about the purple axis, you would find that the upper and lower boundaries change regardless if you used the Washer or CS Method.
The height changes from [0,2] and [2,3] so you would need split integrals for the CS method.
The width changes from [0,2] and [2,4] so you would need to split the integrals for the Washer method.
weak wasp
Right cause it’s being done with a different equation now
bold bane
Correct.
bold bane
yw
weak wasp
I’ll know in a bout 2 or so weeks if I get into Waterloo university that’s why I’m going all the way to a discord server for math help lmao
bold bane
What degree program are you interested in?
weak wasp
Oh Mechatronics Engineering
I got this dope project as well I’m working in
On*
A flying car
Problem is my current avg is terrible
Like 90% on the dot
I have this project tho and a part time job and Im a design and CAD co lead on my schools robotics team
So hopefully I get in
bold bane
Nice.
Ok, well I'm out for the evening. Best of luck to you in getting into Waterloo. 👍
red ice
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elder jasper
w
daring salmon
how do i do this
obtuse pebbleBOT
how do i do this
brave bramble
Have you reduced the matrix yet?
daring salmon
i will do that now
daring salmon
Have you reduced the matrix yet?
brave bramble
You did find a reduction for the matrix?
,w matrix row reduce {{1,2,3},{4,5,7},{-5,-1,0},{2,7,11}}
daring salmon
yes
warm shaleBOT
brave bramble
Okay hopefully it looks like that
Writing those equations out, these two facts would have to be true:
x1 - 1/3 x3 = 0
x2 + 5/3 x3 = 0
daring salmon
yes
brave bramble
Heck I'll even clear denominators here:
daring salmon
shouldnt the signs be opposite?
brave bramble
3x1 = x3
3x2 = -5x3
brave bramble
My reduction doesn't read that way, no
brave bramble
3x1 = x3
3x2 = -5x3
We just want ANY vector that obeys these two equations
daring salmon
wouldnt it be x_1 + 0x_2 = -1/3x_3
from the first row
then u move x_3 to the other side and the sign changes
or am i doing it wrong
i think you are right
daring salmon
We just want ANY vector that obeys these two equations
how do we get the vector
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obtuse pebbleBOT
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wheat pebble
Anyone help me to unpack a few of the things in this example? Book is from Khalil on nonlinear control and I dont understand what he means by the set and both equal to and smaller than a, if that just a box in R^2 with the same inverval between -a to a on both x1 and x2 axis?
obtuse pebbleBOT
Anyone help me to unpack a few of the things in this example? Book is from Khali...
wheat pebble
Also in the the parts I have highlighted, where have they come from ? $x_{1} \cdot x_{2} - y_{1} \cdot y_{2}$
warm shaleBOT
Stebsy
gilded needle
yes, W is the box you described
obtuse pebbleBOT
@wheat pebble Has your question been resolved?
wheat pebble
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obtuse pebbleBOT
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graceful marten
obtuse pebbleBOT
graceful marten
So, if V is constant
It will be p ∝ T
How can it be written also as
p1/ p2 = T1/ T2
graceful marten
yep
graceful marten
So p = kT
Are you allowed to assume that?
When V is a constant
ok, i see
If V wasnt constant you cant assume that, is that right?
brave bramble
Pick two points, we get these two equations:
p1 = kT1
p2 = kT2
graceful marten
right
brave bramble
Divide one by the other
graceful marten
ok yeah haha
got it :3
I wasnt sure if your allowed to assume P= kt
tytytytyytyt
:>
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obtuse pebbleBOT
graceful marten
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agile star
Hey guys is this considered as upper triangular matrix?
obtuse pebbleBOT
Hey guys is this considered as upper triangular matrix?
safe haven
yes
obtuse pebbleBOT
agile star
is this considered triangular matrix?
nvm, found out there's a thing called diagonal matrix
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stiff stump
Question on math behind transformers, saw 3blue1browns last video on it and he describes that only the final tokens hidden state vector is used to generate the next token, why is that? Is it true? Why would you ignore all the valuable info in the other vectors? https://youtu.be/wjZofJX0v4M?si=xjG1aMGzizelL5B9 21:15 in the video for this specific question.
obtuse pebbleBOT
Question on math behind transformers, saw 3blue1browns last video on it and he d...
@stiff stump Has your question been resolved?
obtuse pebbleBOT
@stiff stump Has your question been resolved?
blissful bane
Suppose R and R' be two rings, let 1 and 1' be unity of respective rings
blissful bane
is it true that any homomorphism sends 1 to 1'?
blissful bane
my definition says both operations are preserved
f(a+b)=f(a)+f(b)
f(ab)=f(a)f(b)
my attempt: $f(1 \cdot 1)= f(1) \cdot f(1)$
warm shaleBOT
Dubs
blissful bane
$1'f(1) =1' f(1) f(1)$
warm shaleBOT
Dubs
blissful bane
oh cancellation isn't necessarily possible with in rings right?
kind hawk
yes
blissful bane
i thought i can cancel it an say f(1)=1'
I'm asked to find the homorphism between Z_20 to Z_30
i know this is of the form f(x)=ax
to determine all of them, can i choose each generator of Z_30 for a and conclude ?
kind hawk
why should a be a generator
blissful bane
i thought a homorphism maps generator to generator
kind hawk
the important observation is that whatever f(1)=a is, you need that f(0)=f(20)=20*a=0
kind hawk
i thought a homorphism maps generator to generator
no not necessarily
blissful bane
atleast to become group homomorphism ?
blissful bane
the important observation is that whatever f(1)=a is, you need that f(0)=f(20)=2...
but 20 isn't in Z_20
kind hawk
it is
blissful bane
congruent to 0
kind hawk
so?
blissful bane
so find a in Z20 such that 20a=0?
kind hawk
a is in Z30
blissful bane
right, find a in Z_30 such that this happens
20(3)=0
f(x)=3x
f(20)=3(20) = 60 =0 mod 30
is it the order of image should divide the order of pre image?
kind hawk
yes
obtuse pebbleBOT
@blissful bane Has your question been resolved?
obtuse pebbleBOT
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meager ember
obtuse pebbleBOT
meager ember
answer is supposed to involve Euler's identity but ive struggled with trying to apply it. answer key || lists the limit as 0 which I find confusing||
merry peak
Why does this have to be done with eulers identity
Its pretty intuitive that in the brackets, it converges to 1/2 and the exponent goes to infinity
So 0 is the answer
solar otter
isn't the exponent infinity
meager ember
Why does this have to be done with eulers identity
because this is a question set to practice euler's identity
gilded needle
isn't the exponent infinity
that isn't sufficient on its own
meager ember
it's possible this is some kind of error
merry peak
Oh
versed stratus
L= this, take the log of both sides
$L=\left(\frac{\left(x+4\right)}{2x-5}\right)^{\frac{\left(2x^2-1\right)}{x-6}}$
warm shaleBOT
ƒ(Why am. I here)=I don't know
versed stratus
$\ln\left(L\right)=\frac{\left(2x^2-1\right)}{x-6}\ln\left(\left(\frac{\left(x+4\right)}{2x-5}\right)\right)$
warm shaleBOT
ƒ(Why am. I here)=I don't know
versed stratus
l'hopital's rule now
gilded needle
or just use continuity of log
the fraction inside the log goes to 1/2, so the log factor goes to log(1/2), which is a negative constant
the other factor goes to infinity
result: -infinity
obtuse pebbleBOT
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empty salmon
how do I do this question
obtuse pebbleBOT
how do I do this question
red ice
but how do I apply it to a cubic equation
Cause you know 1 is a root, you could just divide the cubic by (x - 1)
To get a quadratic
empty salmon
oh right on
red ice
You could also use the sum and product of roots for a cubic
Yeah and also consider $(\alpha - \beta)^2$ instead
warm shaleBOT
south
red ice
You can write this in terms of alpha + beta and alpha beta
obtuse pebbleBOT
red ice
No worries!
obtuse pebbleBOT
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random olive
"Parameterize the space curve"
obtuse pebbleBOT
"Parameterize the space curve"
random olive
can someone help me solve this?
fossil crag
what have you tried?
random olive
i mean nothing really i have never parameterizised anything before
obtuse pebbleBOT
@random olive Has your question been resolved?
fossil crag
i mean nothing really i have never parameterizised anything before
you know how to find the coordinates of points on a circle?
see if you can give me the link between:
- equation of a circle
- expression of the points on a circle
then find the expression of x and y by carefully swapping in the second equation
and you'll deduce expression of z
fossil crag
x^2 + y^2 = r^2?
that's a circle centered around (0,0)
what about a circle centered around any point (a,b)?
random olive
(x−a)^2 + (y−b) ^2 = r ^2?
fossil crag
ok now we're almost there
if I wanted an ellipse centered around (a,b), what would it look like?
random olive
(x-a)^2 / a^2 + (y-b)^2 / b^2 = 1?
spare field
3x^2 + 6y^2 ???
fossil crag
(x-a)^2 / a^2 + (y-b)^2 / b^2 = 1?
so not a good idea to use a and b again
we'll say (x-x0)^2 / a^2 + (y-y0)^2 / b^2 = 1
fossil crag
(x-a)^2 / a^2 + (y-b)^2 / b^2 = 1?
How do we extract x = ... and y = ...
fossil crag
x=r⋅cos(θ) y=r⋅sin(θ)
like you did here similarly?
fossil crag
x=x0 +a * cos(θ) y = y0+b * sin(θ)?
now we know (x-x0)^2 / a^2 + (y-y0)^2 / b^2 = 1 leads to the parametrisation x=x0 +a * cos(θ) y = y0+b * sin(θ)
fossil crag
"Parameterize the space curve"
you can now apply to our problem
random olive
okay how?
random olive
we'll say (x-x0)^2 / a^2 + (y-y0)^2 / b^2 = 1
do we need to get our equations to look like this?
fossil crag
do we need to get our equations to look like this?
yes
random olive
okay do we put them equal to each other and then remove z?
fossil crag
okay do we put them equal to each other and then remove z?
replace z by ...
random olive
ohh i thought both z were positive becuse then i could have moved one over and they would be eliminated
but dont we need to remove z in some way
fossil crag
yeah just directly plug in z = 3x^2 + 6y^2 into second equation
random olive
oh so for -3x^2 + 30x we can keep the 3 outside?
fossil crag
yes sure
btw every term is divisible by 3
so you could divide everything by 3 to simplify
random olive
ohh okay
so we get (x^2 + 10x + 25)?
and 2(y^2 + 12y + 36)?
and then (x+5)^2 - 2(y+6)^2?
fossil crag
No there's a problem
if you divide by 3 you get :
x^2 + 2y^2 - 10x - 12y = -31
and so when you factor the terms in y by 2 you forgot that 12 becomes 6
(also the sign problems)
fossil crag
and then (x+5)^2 - 2(y+6)^2?
so try again
random olive
why do we get -31 we have = 93
fossil crag
why do we get -31 we have = 93
we divided by 3
and we have to swap signs
in order for the coefficients in front of x^2 and y^2 be positive
otherwise at the beginning we have
30x + 36y **-**3x^2 **-**6y^2 = 93
so try to get the same thing yourself if you're not convinced
random olive
okay so we have 30x + 36y -3x^2 -6y^2 =-93 from the beginning?
or 20x + 36y +3x^2 + 6y^2 = -93?
fossil crag
"Parameterize the space curve"
we just took equation 2 and plugged in z = 3x^2 + 6y^2
fossil crag
why do we get -31 we have = 93
So start from 30x + 36y -3x^2 -6y^2 =+93
fossil crag
if you divide by 3 you get :
x^2 + 2y^2 - 10x - 12y = -31
and try to get to this
random olive
(x-5)^2 + 2(y-3)^2?
fossil crag
(x-5)^2 + 2(y-3)^2?
that looks better
random olive
but = what do i take -31^2?
fossil crag
not at all
just write this x^2 + 2y^2 - 10x - 12y = -31
and if you made terms appear to get (x-5)^2 + 2(y-3)^2, you need to subtract them
perhaps expand (x-5)^2 + 2(y-3)^2 again to see what needs to be done
random olive
= -12?
random olive
x^2−10x+2y ^2−12y+43.
fossil crag
yes
random olive
but i dont understand what i am supposed to move over 43?
random olive
-31
random olive
12?
random olive
so is that it?
random olive
do i divide by 12 now?
fossil crag
yes now's a good time to do so
random olive
okay but there is nothing times itself that becomes 12
fossil crag
not good for x or y
random olive
why not
fossil crag
because
fossil crag
we'll say (x-x0)^2 / a^2 + (y-y0)^2 / b^2 = 1
what is b in our case?
random olive
sqrt(12)
fossil crag
no
random olive
why nit
fossil crag
yes b = sqrt 6
random olive
okey and what is the degree?
fossil crag
okey and what is the degree?
any theta works
random olive
okay so that is the answer
random olive
how?
fossil crag
x = 5 + sqrt(12) * cos
.
random olive
okay but i dont understand
fossil crag
which part you didn't get?
random olive
how to find f
fossil crag
how to find f
you already expressed x and y in terms of theta
all you need to find now is z
and then (x,y,z) = (...,...,...) = f(theta)
the (...,...,...) will be your definition for f
random olive
3(5 + sqrt(12) *cos)^2 + 6(3 + sqrt(6) *sin)^2
random olive
f = that ?
random olive
yes okay
fossil crag
= (5 + sqrt(12) * cos(theta), 3 + sqrt(12) * sin(theta), 3(5 + sqrt(12)*cos(theta))^2 + 6(3 + sqrt(6)*sin(theta))^2)
random olive
okay
fossil crag
and that's the parametrization
obtuse pebbleBOT
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timid silo
Transform in product tan x + tan 2x- 3tan3x
obtuse pebbleBOT
Transform in product tan x + tan 2x- 3tan3x
river forge
Sup
timid silo
hi
timid silo
yes
river forge
tan(2x)= cos(2x) / sin(2x)
here $\tan(3x) = \frac{3\sin(x) - 4\sin^3(x)}{4\cos^3(x) - 3\cos(x)}$
$\tan(3x) = \frac{3\sin(x) - 4\sin^3(x)}{4\cos^3(x) - 3\cos(x)}$
warm shaleBOT
Cha0s
river forge
we can substitute these into the original equation and solve
timid silo
could you solve it please? i don't understand
<@&286206848099549185>
Transform in product tan x + tan 2x- 3tan3x
lapis barn
tan ( x+2x) = tan x + tan2x/ 1- tanx tan2x
tan3x - tanx tan 2x tan 3x = tanx + tan2x
tan3x = tanx + tanx 2x + tanx tan2xtan3x
lapis barn
Transform in product tan x + tan 2x- 3tan3x
u wanted this way?? ..is it okay @timid silo ?
timid silo
u wanted this way?? ..is it okay <@456226577798135808> ?
I don't understand Could you explain please?
lapis barn
I don't understand Could you explain please?
i just applied tan(A+B)= tan A + tanB/ 1 - tanA* tan B
timid silo
yes
timid silo
tan x + tan 2x- 3tan3x I need to write this whole thing as a product
how do I do that
lapis barn
tan x + tan 2x- 3tan3x I need to write this whole thing as a product
are u sure that coefficient of tan3x is minus 3 ??
lapis barn
yeahh maybee
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lunar rover
$$y'-2y=3$$ Im not sure where im doing wrong if anyone can check my work
warm shaleBOT
Totalani
lunar rover
I got my integrating factor to be $I(x)=e^{2x}$
warm shaleBOT
Totalani
lunar rover
$y'e^{2x}-2ye^{2x}=3e^{2x}$
warm shaleBOT
Totalani
lunar rover
am I good so far?
nvm I think I see my mistake
$$ye^{2x}=\int 3e^{2x} , dx$$ how do I think here to solve for y?
warm shaleBOT
Totalani
lunar rover
integrate first?
alpine verge
your integration factor should be $e^{-2x}$
warm shaleBOT
y0shi
alpine verge
since using the product rule on $ye^{2x}$ would not give what you have up there
warm shaleBOT
y0shi
lunar rover
I see
Oh right
its on homogenious diff equations that yo switch right?
Since you move it to the other side
think I confused with that
$$ye^{-2x}=\int 3e^{-2x} , dx$$ so how do I go on with this?
warm shaleBOT
Totalani
lunar rover
Im a bit rusty when it comes to intragrating Ce anything but $\frac{3}{2}e^{-2x}+C$ this should be it?
warm shaleBOT
Totalani
lunar rover
oh yea
lunar rover
then you move over the e^{-2x}
alpine verge
then we divide by the derivative of the exponent
lunar rover
right
warm shaleBOT
Totalani
alpine verge
yw!
lunar rover
.close
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timid silo
It is well known that the axiom of choice implies LEM. What about weaker forms of axiom of choice? Is there some LEM-flavoured statement that is provable from them?
obtuse pebbleBOT
It is well known that the axiom of choice implies LEM. What about weaker forms o...
@timid silo Has your question been resolved?
obtuse pebbleBOT
@timid silo Has your question been resolved?
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plucky lintel
he NASA wants to transport the Space Shuttle from Cape Canaveral, which is at sea level, to the orbit of geostationary satellites, at a distance h = 36 • 10^3 km from the Earth's surface. Knowing that the Shuttle has a mass m = 2.03 • 10^6 kg, determine the work done by its engines to overcome only gravitational interaction.
Shouldn’t it be the integral of (m g x)?
obtuse pebbleBOT
he NASA wants to transport the Space Shuttle from Cape Canaveral, which is at se...
plucky lintel
shouldn't it be the integral of mgx?
worn yoke
g is only approximately constant at low altitudes
for a question involving going to space you'll need newton's law of gravitation
plucky lintel
ok clear, but if the exercise doesn't give any information about the mass, are the previous passages correct in order to estimate the value of the work done?
worn yoke
you will still integrate the force of gravity but you need to use newton's law:
[ F = G\frac{mM}{r^2} ]
warm shaleBOT
pnoןɔ
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deep ravine
Why is the answer I and III instead of II?
obtuse pebbleBOT
Why is the answer I and III instead of II?
worn yoke
we want the overall power to be less than -1
deep ravine
we want the overall power to be less than -1
Wouldn’t that n^p diverge if p was less than -1?
It would have to be between -1 and 1 to converge right?
worn yoke
you're thinking of geometric series
deep ravine
you're thinking of geometric series
Yeah i rewrote it in geometric series
worn yoke
$\sum_{n=1}^\infty r^n$ converges for $|r|<1$ \
$\sum_{n=1}^\infty n^p$ converges for $p<-1$
warm shaleBOT
pnoןɔ
worn yoke
they look similar but are very different
deep ravine
they look similar but are very different
Wait what is this series called?
worn yoke
this is called a p-series
deep ravine
this is called a p-series
I thought that was only 1/n^p?
worn yoke
1/n^p is the same as n^(-p)
indigo pendant
did I do something wrong here
obtuse pebbleBOT
did I do something wrong here
frank monolith
it was sin(theta), how did the theta become x 
indigo pendant
Ignore that @frank monolith
indigo pendant
Ok wait
frank monolith
why do you ask though
indigo pendant
frank monolith
they changed the limits
frank monolith
sin(x) has the same behaviour between pi/6 - pi/2 and pi/2 - 5pi/6
so you can just integrate over one interval and multiply by two
and yeah you can solve the integral without all this too
frank monolith
show your working
frank monolith
if you are finding the area between the two curves
why are you squaring them
isnt it just integral (f(x) - g(x)) dx?
indigo pendant
Wait
frank monolith
like the whole thing
frank monolith
ah i'm not familiar with polar so wont comment on it.
indigo pendant
ok its ok
frank monolith
your mistake in the integral was that the integral of cosx is sinx and not -sinx
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wide oracle
obtuse pebbleBOT
wide oracle
what did i do wrong here? this should be the answer
i didnt finish but you can tell its not the same
latent walrus
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quartz apex
obtuse pebbleBOT
quartz apex
I've also solved the wronskian as 1/x
But when I plug it into, I'm not getting the correct answer
obtuse pebbleBOT
@quartz apex Has your question been resolved?
obtuse pebbleBOT
@quartz apex Has your question been resolved?
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lime scaffold
How do you do c
obtuse pebbleBOT
How do you do c
obtuse pebbleBOT
@lime scaffold Has your question been resolved?
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obtuse pebbleBOT
https://www.coursehero.com/file/104429044/1619564229-OVS-test2-3UPDF/
stiff leaf
https://www.coursehero.com/file/104429044/1619564229-OVS-test2-3UPDF/
We can't view that
sly mica
.close
obtuse pebbleBOT
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timid silo
Hey! i am a bit stuck on what to do here
obtuse pebbleBOT
Hey! i am a bit stuck on what to do here
timid silo
The last part is what the answer should be
But I have no idea how to get it since there is a plus sign
timid silo
Hey! i am a bit stuck on what to do here
well what is ax + bx equal to, as an example to guide you?
Aaa im not sure, I havent gotten there yet
well you should if you are dealing with exponentials
do you know like-terms
I know ax equals 3(4)^x
Yes
alright
so 3x + 2x is what for example
5x
ok
you have [
3\22^{2x} + 2^{2x}
]
If you consider $2^{2x}$ to be your '$x$' from the example I gave above, what does the sum equal to?
warm shaleBOT
