since x cant taken on multiple values at once
#help-43
hexed ferry
i mean i could do d/dx both sides
wait no
$\frac_{\del f}^{\del x }+\frac_{\del f}^{\del y }dy/dx$
boreal girderBOT
Julian
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sacred lagoon
f(x) is defined for all x ?
sacred lagoon
ye
hexed ferry
lets hope gpt fixed it
\frac{\partial f}{\partial x} + \frac{\partial f}{\partial y} \frac{dy}{dx}
$\frac{\partial f}{\partial x} + \frac{\partial f}{\partial y} \frac{dy}{dx}$
boreal girderBOT
Julian
hexed ferry
yea idk what im doing lol
sacred lagoon
ye but you can write without x too
hexed ferry
oh yeah
sacred lagoon
it's just an equality between functions
hexed ferry
but f' denotes sets of ordered pairs
sacred lagoon
also I suppose f differentiable
hexed ferry
f'(x) denotes the range value
sacred lagoon
what ?
hexed ferry
ok yea that makes sense bruh
sacred lagoon
this question is act so weird
sacred lagoon
it's rly easy if you know how to do a linear diff equation of order 1
hexed ferry
they made 6 different projects
hexed ferry
the diff equation satisfied is g' = f(y) f'
we need to show this tho
sacred lagoon
f(-x)f(x)=f(-x)f'(x)
we don't know that yet
sacred lagoon
that f(x) = exp (x)
hexed ferry
f'(x)=f(x)
sacred lagoon
ok so then
hexed ferry
yea this is building the exponential function not assuming it
sacred lagoon
so you didn't define exp as the reciprecal of ln ?
hexed ferry
sacred lagoon
yea this is building the exponential function not assuming it
ok ye that's why I was so confused
hexed ferry
lol yea i should have specificed
so
i figured out yesterday
how to prove f equals its taylor series
the taylor for exp
but i had trouble multiplying series so i abandonded the idea
sacred lagoon
so g' = f(y) f'
hexed ferry
so g' = f(y) f'
we dont know this
sacred lagoon
trying to get an idea of what to do
sacred lagoon
and y is fixed
hexed ferry
and y is fixed
yea
holy shit im actually slow 💀
it legit gave the hint but i forgot to try it
ok
thanks
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✅ Original question: #help-43 message
hexed ferry
my idea is
we can let y=-x
and then
g(x+y)=g(0)=g(x)g(-x)
which proves g is differentiable for all x?
compact pewterBOT
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lofty mountain
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paper tendon
hi, can someone explain why this works?
compact pewterBOT
hi, can someone explain why this works?
paper tendon
how do we get from the equation on second row to the third row
mild sky
Factoring
compact pewterBOT
@paper tendon Has your question been resolved?
paper tendon
oh i see i just have to expand the factorials with higher values (k+1)! and (n-k)!, after that its obvious
thx
lofty mountain
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haughty smelt
I don’t know how to k ow which one is right on a test
compact pewterBOT
I don’t know how to k ow which one is right on a test
quiet sentinel
You mean whether its correct to use long division or quadratic formula?
dull moon
I don’t know how to k ow which one is right on a test
Could you please rephrase the question? Its a little vague.
haughty smelt
You mean whether its correct to use long division or quadratic formula?
I know I need to use long devision but when I tried both the factor options X+1/2 and 2x+1 I got two different answers and the bottom one is not on the answer
Key
dull moon
If you're using 2x + 1, then you maybe need to divide some other expression by two later on. Just a guess.
haughty smelt
Is the second image an amswer you were given?
Yea
haughty smelt
If you're using 2x + 1, then you maybe need to divide some other expression by t...
I thought so also but no way I tried would get the answer above
quiet sentinel
You didnt put in the a b and c correctly in the quadratic
haughty smelt
Wait where?
quiet sentinel
A is 4, b is -29 and c is -14
haughty smelt
A is 4, b is -29 and c is -14
For which one?
quiet sentinel
Oh im still messing it up lol
So basically you were supposed to long divide to get the quadratic and then use the formula
Right
haughty smelt
Yea
dull moon
I get the answer at the bottom of the second pic but I think you have an extra '2' factor there.
haughty smelt
I get the answer at the bottom of the second pic but I think you have an extra '...
Wdym?
dull moon
I mean, I see you got the correct factorization (2x+1)(2x^2-x-14), but why multiply a factor of 2?
quiet sentinel
I know where you went wrong now
Firstly you multiplied by another 2 in the denominator of the quadratic
haughty smelt
I mean, I see you got the correct factorization (2x+1)(2x^2-x-14), but why multi...
Oh yea I messed that up but I fixed it I just did not rewrite it all
quiet sentinel
You also accidentally left it as -448 in the surd
Instead of +
Since it multiplies 2 negatives together
haughty smelt
Oh wait now I see it works better now since 113 •4=452
compact pewterBOT
@haughty smelt Has your question been resolved?
sonic terrace
i dont understand ts bs
compact pewterBOT
i dont understand ts bs
chilly basalt
civic matrix
What's this?
sonic terrace
so 2 * 5 is 0 5 * 2 is 10
dull moon
okay common numerator is 10
I think you're supposed to find the value of x. Multiply the whole equation by 40 = lcm(2,5,8)
sonic terrace
obivously
strange ermine
obivously
Yeah sorry I only saw the photo
You might be surprised though that there are a lot of people who do linear (= 1st degree) equations who can't do calculations with fractions
sonic terrace
Why can’t i multiply (5(x-6))/10?
dull moon
Sorry, I read the question wrong. Multiply by 10 to remove the denominators.
strange ermine
Why can’t i multiply (5(x-6))/10?
Wdym? Maybe you want to do the correct thing but as written it doesn't sound such
A photo of your solution attempt would be much appreciated
sonic terrace
Multiply by 5 and then by 2 on both sides to remove the denominators
I did that…
dull moon
Yes, so now remove the brackets and solve for x.
compact pewterBOT
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solemn matrix
I'm trying to find all pairs of $(x, y) \in \mathbb Z^2$ in which $2x - 3xy + 5y = 1$. I tried factoring but I can't seem to find a way. I was taught a trick that if I substitute $x = k$ for some constant $k$, then I get a value of $y$. For example: if I put $x = 100$ and I got $y = 102$, then I probably have $y - x - 2$ as a factor because $y = 100 + 2 = x + 2$. But in this case, I tried this trick with $x = 10^6$ but got $y = 0$ on the LHS, whilst factoring $2x - 3xy + 5y - 1$ gives me $y = 0.2$, which is not really useful for me. What should I do?
compact pewterBOT
I'm trying to find all pairs of $(x, y) \in \mathbb Z^2$ in which $2x - 3xy + 5y...
boreal girderBOT
1 divided by 0 equals Infinity
solemn matrix
u can write x in terms of y right
yes
but i only have 1 equation, not 2 to form a system
boreal girderBOT
Cooly
open girder
but i only have 1 equation, not 2 to form a system
We need x in Z meaning 2-3y | 1-5y. We can parametrise this by setting x = k (an integer) as you mentioned. Then
1 - 5y = k(2-3y)... can you go from here?
open girder
What?
solemn matrix
this question is in the exercise of solving for integers by factoring
if there was a way to factor my equations
i have to turn it into this form
open plaza
hmm let's do this step by step
so we can try factoring out y first to get 2x + y(5-3x) = 1
now clearly the problem is we do need a 3 factor on the x here
open girder
i have to turn it into this form
Are you familiar with Simon's Favourite Factoring Trick?
solemn matrix
Are you familiar with Simon's Favourite Factoring Trick?
what's that?
it'd be useful if i know
open plaza
now clearly the problem is we do need a 3 factor on the x here
so multiply both sides by 3 to get 6x + 3y(5-3x) = 3
solemn matrix
ohhhhhh
open girder
what's that?
It'd be more efficient for you to look it up on Google
open plaza
Are you familiar with Simon's Favourite Factoring Trick?
shit name ngl
solemn matrix
*shit name ngl*
lol
solemn matrix
It'd be more efficient for you to look it up on Google
lemme try that
open plaza
so multiply both sides by 3 to get 6x + 3y(5-3x) = 3
I think you know what to do from this part
solemn matrix
It'd be more efficient for you to look it up on Google
but for this one, there's -3xy which did not really suit the environment
open girder
It doesn't really change anything
solemn matrix
It doesn't really change anything
i mean, it would be something like (x + 5/3)(y + 2/3) = 4/3
open girder
(-3x+5)y + (2x-1) = 0 and then we want the second term to contain the same linear factor (-3x+5), so we add and subtract a suitable multiple
solemn matrix
(-3x+5)y + (2x-1) = 0 and then we want the second term to contain the same linea...
im confused on how this is related to SFFT
open girder
Should be (-3x+5)(y-2/3)+7/3=0 which is just (5-3x)(3y-2)=-7
SFFT says:
Whenever you have an expression like Axy + Bx + Cy + D, you can rearragen and factor it as (Ax + C)(y + something) = constant.
solemn matrix
dang
open girder
Axy + Bx + Cy + D = (Ax + C)(y + B/A) + (D-(BC)/A)
open girder
Yes, and D = -1
open girder
Yeah, that's correct.
solemn matrix
i can ace this question using the same technique?
open girder
i can ace this question using the same technique?
Yeah
open girder
how about this?
You can use SFFT after some algebra
solemn matrix
alr
solemn matrix
how about this?
so something like:
$\frac{8}{x + 7} = \frac{2y - 1}{4}$
boreal girderBOT
1 divided by 0 equals Infinity
solemn matrix
and then $(x + 7)(2y - 1) = 32$?
boreal girderBOT
1 divided by 0 equals Infinity
solemn matrix
You can use SFFT after some algebra
this is not even needed lol
solemn matrix
**1 divided by 0 equals Infinity**
and then i can safely say that $2y - 1 = \pm 1$ since it's odd
boreal girderBOT
1 divided by 0 equals Infinity
open girder
Well, you can definitely use it on a lot of diophantine equations, but I cannot guarantee its effectiveness 
solemn matrix
Well, you can definitely use it on a lot of diophantine equations, but I cannot ...
long time seeing someone using :kek:
this can also be converted to SFFT right?
solemn matrix
Well, you can definitely use it on a lot of diophantine equations, but I cannot ...
how about this?
open plaza
sfft is shit lmao
solemn matrix
lmao
open plaza
imo it's just an excuse to fuck up your flexibility in problem-solving
solemn matrix
lol
like how i didn't realize there was a y factor there 💀
wth
how the hell are there THESE many questions
||left||
open plaza
industrial chickens
solemn matrix
SFFT ain't solving this lol
open plaza
b) should be simple
solemn matrix
dùng thủ thuật máy tính ông thầy dạy =)))
open plaza
trong phòng thi có wolframalpha à
solemn matrix
solve for $x, y \in \mathbb Z$
boreal girderBOT
1 divided by 0 equals Infinity
toxic cipher
**1 divided by 0 equals Infinity**
ah okay Z makes it easier
solemn matrix
Part c you can try completing the square for x and for y on the LHS
i've done c alr
toxic cipher
i've done c alr
part d?
solemn matrix
yea
solemn matrix
part d?
no
but all of these exercises are just converting into form of f(x, y)g(x, y) = k
where k is a constant and f(x, y), g(x, y) are factors
im going to close for now
solemn matrix
use the identity
$$a^3 \pm b^3 = (a \pm b)(a^2 \mp ab + b^2)$$
yeah i saw it
boreal girderBOT
solemn matrix
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✅ Original question: #help-43 message
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sage lynx
Prove that out of any n + 1 numbers chosen from the numbers 1, 2, 3, ...., 2n, there exist two numbers one of which divides the other.
compact pewterBOT
Prove that out of any n + 1 numbers chosen from the numbers 1, 2, 3, ...., 2n, t...
open girder
Prove that out of any n + 1 numbers chosen from the numbers 1, 2, 3, ...., 2n, t...
What have you tried?
sage lynx
!status
compact pewterBOT
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
Do you know of the pigeonhole principle
sage lynx
Yes.
kind viper
Do you know of the pigeonhole principle
question is, what n collections do we divvy up {1, ..., 2n} into
boreal girderBOT
Cooly
viscid mortar
Anjali
rotund wave
can someone tell me the answer
open girder
can someone tell me the answer
The answer to this question?
rotund wave
yeahhhh
rotund wave
question is, what n collections do we divvy up {1, ..., 2n} into
what the size of these collections? 2?
kind viper
beats me
mild sky
Every number in $1,2,...,2n$ can be written as $2^k m$ where $m$ is odd.
This is pretty much it
compact pewterBOT
@sage lynx Has your question been resolved?
open girder
yeahhhh
||The possible odd parts are precisely the odd numbers in 1,2,…,2n, of which there are n. By pigeonhole, amongst the n+1 numbers, two of them must share the same odd part. Say, a = 2^i m and b = 2^j m. WLOG, assume i < j. Then, b = 2^j m = 2^(j-i)a. So a divides b.||
rotund wave
||The possible odd parts are precisely the odd numbers in 1,2,…,2n, of which the...
nicee
safe raptor
Hello
What is the difference between a linearisation of a function and the differential of the function
They are both approximations but how can I conceptually understand the differences between them
they way I understand it is
Linearisation is an approximation of a function at a given point
compact pewterBOT
Hello
What is the difference between a linearisation of a function and the dif...
weak cobalt
Any context?
weak cobalt
Given a function differentiable around a point x=p, its linearization at that point would just be the equation of the tangent line at that point, which uses the derivative f'(p)
weak cobalt
Right so you can see it's using f'(x_0)
safe raptor
differental
weak cobalt
That tangent is an approximation of the function around x_0
safe raptor
are they the same??
weak cobalt
The derivative is the slope of the tangent
safe raptor
yes that I know but what I dont understand is the difference between the linearisation of a function and the differential of a function
like with linearisation I approximate the function at a given point
its the tangeant at point x0
but so is the differential?
weak cobalt
I don't think I see the word differential in there, but my german is poor
safe raptor
thats the whole context
yh sorry i dont how to translate it e
but yh essential df is what exactly?
barren dragon
df is the difference in f that we get from a difference dx in x (or as written here: delta x)
weak cobalt
Ok so Delta f depends on Delta x, it's just the difference in y given the difference in x
safe raptor
and linearisation doesnt account for the chnage in Delta f?
weak cobalt
When Delta x becomes "infinitesimal" (small), it becomes dx (that's not formal but intuitively it should work)
barren dragon
is there a definition for linearisierung and DIfferential in your script?
weak cobalt
df is the difference in y given a difference dx in x
safe raptor
In other words what is the relationship between p(x) = f(x) + f'(x)* (x-x0) and df
weak cobalt
Once you have df and dx at a point x_0, you have the derivative, df/dx = f'(x_0)
safe raptor
i thougth df was to approximate a complex function lets say
but linearisation also does that no??
weak cobalt
p(x) = f(x_0) + df/dx (x_0) (x - x_0)
safe raptor
its an approximation of the deritive of f(x) at point x0?
olive rivet
What grade is this
barren dragon
ok so:
the way i understand it, the Differential is not your Derivative, it is just what we call a small change in a variable
Given that x is your independent, dy is the Differential (Differenz in y) that you get from a difference dx in x
safe raptor
Uni first year
barren dragon
so it is just what you call the dy (or df in this case)
barren dragon
What grade is this
Analysis I i would say
safe raptor
yes its analysis for life sciences
safe raptor
ok so:
the way i understand it, the Differential is not your Derivative, it is j...
so the small change in df is roughly Delta f of f(x)
we approximate df and say because the change is small we estimate it to be Delta f?
weak cobalt
Do you understand the concept of limits?
barren dragon
So to sum it up:
Differential: Änderung in einer Variable, man meint idR eine kleine Änderung dy, df, dx etc.
Linearisierung: Gleichung einer Tangente, die an einem Punkt von f anliegt. Das ist eine Funktion, Das Differential betrachtet man nicht wirklich als Funktion, eher als Werkzeug
barren dragon
so the small change in df is roughly Delta f of f(x)
we approximate df and say b...
Delta X is a change in X. When that change goes to 0, we call it dX
safe raptor
und was will man dann mit dem Differential machen?
what do you do witht he differential then?
weak cobalt
You find the small change in y, and from there, the slope of the tangent
barren dragon
it is the fundamental building block for derivatives (Ableitung) and integrals
weak cobalt
I think it's more clear when we introduce another point, say x_1, and define Delta x = x_1 - x_0
Then Delta f = f(x_1) - f(x_0)
The limit as x_1 goes to x_0 of Delta f is df, and that of Delta x is dx
Delta f / Delta x is the slope of the "chord" between those two points
When the two points come together, that chord becomes the tangent
(and so the slope is df/dx = f'(x_0))
weak cobalt
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bleak flower
hello, can anyone explain how this works? 
compact pewterBOT
hello, can anyone explain how this works? <a:catscream:1350607930050871306>
bleak flower
how they know whether is 1 or 0
questions are basically
Write the un-simplified Sum-of-products Boolean Expression for the truth table above.
then
Complete the 4-variable Karnaugh map. Draw the appropriate loops around groups of 1’s on the map.
hii chartbit 
kind viper
what's "it"
quartz yoke
truth value probably
bleak flower
ye
kind viper
i have no idea what you're talking about tho like
can you highlight which part you don't understand in that slide
open girder
You mean the 0s and 1s in the truth table (on the left hand side)?
quartz yoke
how they know whether is 1 or 0
questions are basically
> Write the un-simplifi...
what does the * operation do
bleak flower
You mean the 0s and 1s in the truth table (on the left hand side)?
I mean how they make the expression from the truth table
bleak flower
what does the * operation do
amplifiers?
open girder
I mean how they make the expression from the truth table
Each K-map cell represents one row from the truth table. You fill a 1 in a cell if the output for that combination is 1 in the truth table, otherwise 0.
bleak flower
Each K-map cell represents one row from the truth table. You fill a 1 in a cell ...
how do u know though? shouldn't u start writing boolean expression first?
pine osprey
how do u know though? shouldn't u start writing boolean expression first?
Its written on the sides of the table
bleak flower
how they got it tho
strange ermine
It's given!! 🙈
pine osprey
Yeah i mean its given
open girder
how do u know though? shouldn't u start writing boolean expression first?
What do you mean? Are data are in the truth table.
pine osprey
If you want to know how they drew the table on both sides, its following the grey's code
i.e 1 bit change between rows or column
bleak flower
It's given!! 🙈
I mean the first image i sent is a solution of this original question and im still curious how they getting 1's from it
I wanna know how they get 1's from the truth table bc u need write the boolean expression first
pine osprey
If theres a bar above its 0 else its 1
open girder
I mean the first image i sent is a solution of this original question and im sti...
A bar means Input A = 0. A means Input A = 1.
strange ermine
I wanna know how they get 1's from the truth table bc u need write the boolean e...
Look at the last (rightmost) column of the left table
That's where the outputs are written
bleak flower
A bar means Input A = 0. A means Input A = 1.
ik this
bleak flower
That's where the outputs are written
wouldn't it be a'*b'*c'*d'?
strange ermine
wouldn't it be `a'*b'*c'*d'`?
What's "it"?
pine osprey
wouldn't it be `a'*b'*c'*d'`?
And this has output 0
bleak flower
boolean expression
bleak flower
And this has output 0
yep but its still incomplete expression
pine osprey
But you actually look at it as (a*b) (c*d) for the table
pine osprey
yep but its still incomplete expression
You don't need it
Well for filling the table
bleak flower
You don't need it
why?
pine osprey
After you'll need it
bleak flower
questions are basically
Write the un-simplified Sum-of-products Boolean Expression for the truth table above.
then
Complete the 4-variable Karnaugh map. Draw the appropriate loops around groups of 1’s on the map.
bleak flower
indeed
open girder
questions are basically
> Write the un-simplified Sum-of-products Boolean Expres...
It would have been nice if you'd stated this from the beginning.
bleak flower
ill just edit it
pine osprey
I wasn't asked to do things like this, rather write the simplified expression from the karnaugh map
bleak flower
idk how they figure out the boolean expression from this truth table
bleak flower
unsimplified
open girder
unsimplified
It's just all the minterms added together
bleak flower
can u elaborate how this works?
weak cobalt
It's a sum of products
Each row in the truth table is a product
To find the boolean expression, you just find which of those products yield 1, and you add them together
(by products here I mean specifically minterms)
bleak flower
like this? no idea
weak cobalt
Yes
~A.~B.~C.~D is one product where the output is 1, so it's like one part of the boolean expression. It's this product, or another product, or another, and so on. So all those parts are added ("or'ed") together to build the full expression.
bleak flower
its not +?
weak cobalt
The first half with the first three products and three + signs is good
Not sure what happened to the rest, it reads like a product of 12 variables
bleak flower
oh right i forgot to add plus 
weak cobalt
The minterms on the right of the table are correct, so if just place the plus signs correctly it should be good
weak cobalt
The Karnaugh map?
open girder
Okay, so if `Output Y` is 1 in that row, you put 1 in the cell.
Basically this^
weak cobalt
Take each minterm of your expression and put a 1 in the corresponding cell in this table
You take the row expression times the column expression
So the third row second column is A.B.~C.D
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nocturne kettle
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weak cobalt
Make a diagram
nocturne kettle
and that red line is actually perpendicular to PO
actually perpendicular bisector
$3.5^2+3.5^2=x^2$
boreal girderBOT
UCYT5040
nocturne kettle
$\frac{7\sqrt{2}}{2}$
boreal girderBOT
UCYT5040
nocturne kettle
This is what I got as my final answer
it's wrong though, the correct answer is 6
weak cobalt
I don't see why the perpendicular bisector of OP would go through S
nocturne kettle
Maybe I drew this not to scale
I don't think its perpendicular
but it is a bisector
since the radius of the smaller circle is 3.5, and S lies on that circle
and the center of PO is the center of that circle
weak cobalt
Ok but then your attempt at Pythagoras doesn't work
weak cobalt
You should see that OPS is a right triangle
nocturne kettle
how?
weak cobalt
OP is the diameter of a circle on which S lies
nocturne kettle
ok that kinda make sense
well
not really
how does that make S a right angle?
weak cobalt
That's an elementary theorem
You can prove it for example by saying that, if C is the center of that smaller circle, the angle OCP is 180º and so the angle OSP, which subtends the same arc, is half of that, so 90º
nocturne kettle
ah yeah
i could use the property of cylic quadrilaterals too for that
which probably seems more complex but, idk it works for me lol
weak cobalt
That property of cyclic quadrilaterals directly comes from the inscribed angle theorem that I just explained
nocturne kettle
yeah, that makes sense
and now its super easy to see that the answer is 6, since its an isoceles triangle with a perpendicular altitude, just split the base of 12 in half
weak cobalt
This particular right triangle theorem is known as Thales's theorem
(and it's a special case of the inscribed angle theorem)
weak cobalt
and now its super easy to see that the answer is 6, since its an isoceles triang...
Yes
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silent inlet
how do u do this by inspection without a substition i forgot
compact pewterBOT
how do u do this by inspection without a substition i forgot
native shard
how do u do this by inspection without a substition i forgot
i mean you just notice the derivative of sinx is cosx
native shard
so its like 1/u^3
silent inlet
fair enough but i feel like im missing some logic i had before idk
native shard
"by inspection" really just is the substitution being easy enough to not bother writing it
silent inlet
wait one sec
i had a different thought process behind thinking abt it before for some general cos x/sin^n x by thinking about the integral of f'(x)/f(x) just being f(x) because i dont like substitution yes ik its the same thing but nvm
native shard
i had a different thought process behind thinking abt it before for some general...
i mean this would be ln|f(x)|?
novel hull
Can someone help me with this problem? I am kind of confused. We are supposed to do it without eigenvalues I believe, and I just dont know how to. It is very obvious if you use eigenvalues but otherwise I feel that it is impossible. We also aren't dealing with complex numbers or spaces so we don't need to worry about that. The problem is: Prove that the determinant of an nxn matrix squared plus I will always be greater than or equal to 0. With eigenvalues the idea is simple, just say that the determinant is the product of all the eigen values, and since the eigenvalues of A squared will always be positive and the eigenvalues of I is just 1, we know that the determinant will always be greater than or equal to 0. Could someonw help me with the proof without using eigenvalues?
compact pewterBOT
Can someone help me with this problem? I am kind of confused. We are supposed to...
@novel hull Has your question been resolved?
signal willow
Can someone help me with this problem? I am kind of confused. We are supposed to...
i think there's a way to do this
let A be this n by n matrix
novel hull
Okay
novel hull
Wait why x?
signal willow
we can use intermediate value theorem to find a root if the determinant when x=1 is negative
still it's somewhat eigenvalue-flavoured
novel hull
I think I might just use eigenvalues and see where that goes
If he marks me down so be it
signal willow
i'll have to think about it since I just showed up
novel hull
Hmm
signal willow
i remember someone else did this problem before, and on mathstackexchange there's a similar question
it uses complex numbers though, so I'm not sure if there's an easier answer
novel hull
Yeah no we don't need to prove it using complex numbers
signal willow
Can someone help me with this problem? I am kind of confused. We are supposed to...
a problem is saying "the eigenvalues of A^2 are always positive" which is incorrect because A may have non-real complex eigenvalues
novel hull
a problem is saying "the eigenvalues of A^2 are always positive" which is incorr...
Well yes, but this class doesn't deal with complex values
So we ignore those
signal willow
Well yes, but this class doesn't deal with complex values
i.e. the idea of "A^2 having positive eigenvalues" is wrong
novel hull
Okay but the teacher does not care bro
Since this class does not deal with complex numbers
We are just taking into account non-complex values
signal willow
doesn't mean you shouldn't care
signal willow
Its not like I can just tell him no
you aren't disagreeing with the teacher, the "obvious solution" you proposed is incorrect
even if the concept of eigenvalues is allowed
novel hull
you aren't disagreeing with the teacher, the "obvious solution" you proposed is ...
Bro, this class does not deal with complex numbers, we are assuming that if the number is squared it is positive
Like pretend the problem just says only in the real space
Its not that deep
signal willow
Not all real-matrices have n real eigenvalues
novel hull
Holy bro
We are ignoring complex numbers and everything completely
Just take that out of your mind
signal willow
Then it is false that the determinant is the product of (the real) eigenvalues
novel hull
Imagine there are only real numbers
Bro isn't there a literal theorem stating that the determinant is the product of the eigenvalues
signal willow
Bro isn't there a literal theorem stating that the determinant is the product of...
only if there are n real eigenvalues
it is (usually) not the product if there are less than n real eigenvalues
you say you want to ignore non-real eigenvalues, so the theorem has to be restated
$\begin{pmatrix}0&2\-2&0\end{pmatrix}$ has no real eigenvalues but the determinant is 4
boreal girderBOT
Element118
novel hull
Okay, think about this: Since we are not dealing with complex numbers, we are also not dealing with complex eigenvalues and hence not including them or pretending they dont exist. In this case the determinant is always the product of (the real) eigenvalues as even if there are less than n eigenvalues that just means an eigenvalue is repeated, as we have not learned about complex eigenvalues and pretend they dont exist
signal willow
does the matrix $\begin{pmatrix}0&2\-2&0\end{pmatrix}$ exist
boreal girderBOT
Element118
novel hull
Yes
signal willow
but it has no real eigenvalues
so how is 4 equal to an empty product
you are resolving the problem in the wrong way
novel hull
We are adding I to it though
Which makes that impossible
To have zeros on the diagonal
Wait nvm
I am dumb
signal willow
okay, I'll show you a matrix where adding I would still make it have no real eigenvalues
novel hull
**Element118**
To get this matrix we would have to have complex numbers in the original matrix, as we are squaring it, and since we are not dealing with complex numbers this is impossible
signal willow
oh really
novel hull
The matrix A^2 in our class cant have any negative numbers bro
We are not dealing with complex values
The only case where it has negative numbers is where it has complex values
signal willow
try calculating $\begin{pmatrix}2&2\-2&2\end{pmatrix}^2+I$ and tell me what the eigenvalues are
boreal girderBOT
Element118
signal willow
it should have no real eigenvalues
novel hull
Okay give me one second
signal willow
no hurry (I have found that this is ambiguous too, urgh, is there a genuine "take your time" statement we can say in English)
signal willow
Wait, do you know what the square of a matrix is
because it seems to me you are squaring it entrywise
which is incorrect
novel hull
Oh my god
signal willow
do you remember matrix multiplication
novel hull
I am dumb as hell bro
holy shit
You do realize that that squared gives the matrix 0 8 -8 0
Which just row reduces to I
So the eigenvalues are just 1
signal willow
$\begin{pmatrix}0&4\-4&0\end{pmatrix}$?
signal willow
So the eigenvalues are just 1
eigenvalues are not preserved on row reduction
oh wait
i computed wrongly
signal willow
You do realize that that squared gives the matrix 0 8 -8 0
yeah
novel hull
I thought if you can row reduce to an upper triangular matrix then the eigenvalues are just the diagonal
If you compute it regularly then yes it is not preserved
signal willow
I thought if you can row reduce to an upper triangular matrix then the eigenvalu...
if it is similar to an upper triangular matrix
i.e. you want to write $\begin{pmatrix}1&8\-8&1\end{pmatrix}=PTP^{-1}$ where $T$ is upper triangular.
boreal girderBOT
Element118
novel hull
Yeah these are still real eigenvalues
The characteristic polynomial is just lambda squared minus two times lambda minus 63
signal willow
The characteristic polynomial is just lambda squared minus two times lambda minu...
simplify that?
wait what
novel hull
Oh yeah wait
signal willow
can you show me how you got that characteristic polynomial
novel hull
No that is right, but it does give complex eigenvalues
novel hull
But I am saying that I think we are just ignoring these cases in this class
Like for now we are assuming this does not exist
signal willow
$\begin{pmatrix}1&8\-8&1\end{pmatrix}$ is still a real matrix, and you need to be able to prove the statement when the original matrix is $\begin{pmatrix}2&2\-2&2\end{pmatrix}$
boreal girderBOT
Element118
novel hull
I think I will just focus on other stuff, I am spending too long on this
It doesn't matter this much
.close
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exotic hound
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exotic hound
I got 1/4 but the answers given pi/8
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exotic hound
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safe veldt
I got 1/4 but the answers given pi/8
The answer is pi/8. Can you share your answer? That way I can point where you went wrong.
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upper bane
verbal kiln
👀
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keen flume
Hii I need help with doing this geometry homework (again)
compact pewterBOT
Hii I need help with doing this geometry homework (again)
upper bane
so you are to work through this sheet in a specific order defined by the answers
so start from #1
upper bane
so find the box with an answer of 27
keen flume
but idk what to do forthe thing for 27,
upper bane
so find the box with an answer of 27
^
unkempt flower
but idk what to do forthe thing for 27,
have you identified which problem will be #2?
keen flume
yeah the question that’s asking “two angles are supplementary. One of them is 24 more than twice the other. How large is the larger angle”
since that’s the box for 27
unkempt flower
right. so mark that as #2 if you have not, then consider the language in the question
if you label one of them, namely the smaller one, as x, can you express the other angle in terms of x?
keen flume
can they be different
unkempt flower
wdym?
unkempt flower
oh, you can let the other angle be y, but the relationship between the larger angle and x is given.
so you can express y in terms of x.
it's not labelling them both as x, but rather the larger angle in a way that involves x.
keen flume
Ohhh
unkempt flower
example: suppose you tell me that you're 3 years older than me. I don't know your age, but I know you're 3 years older than I am.
so if my age is x, then your age is x + 3, but they both refer to different ages.
unkempt flower
do you now get the answer to #2?
unkempt flower
indeed. but first, it would be prudent to express the larger angle in terms of the smaller one first.
keen flume
okay okay
sly patio
Is this grade 9 math
upper bane
there's already a helper here
silent cedar
Do any of you have IXL
native shard
,av nzv4
boreal girderBOT
upper bane
Do any of you have IXL
!redir please
compact pewterBOT
Do any of you have IXL
This channel is only for on-topic discussion. Please take casual conversation to #discussion or #chill.
unkempt flower
X= something +24
just noting that you seem to be calling the larger angle x.
silent cedar
IXL is so hard I don’t understand anything my teacher says
keen flume
I want my x to be the bigger one and the smaller one to be y instead
silent cedar
I have 31 missing assignments
upper bane
I have 31 missing assignments
hi, please open a new channel to ask your question if you do have one!
unkempt flower
I want my x to be the bigger one and the smaller one to be y instead
that works. in the end, you'll be expressing the equation in terms of the smaller angle before using the value of the smaller angle to find the bigger one.
sly patio
Can someone motivate me to study rn
upper bane
Can someone motivate me to study rn
!redir please.
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Can someone motivate me to study rn
This channel is only for on-topic discussion. Please take casual conversation to #discussion or #chill.
unkempt flower
I don’t know where to go from here, do I add the 180 now?
so you know x = {something involving y} + 24. that something is in the question itself.
keen flume
the twice the other?
unkempt flower
correct!
keen flume
wow
unkempt flower
so how would you translate that into math terms?
(use the fact that you labelled the smaller angle as y.)
keen flume
X=2y+24
unkempt flower
precisely so.
now, for two angles to be supplementary, you know they have to add up to 180.
form an equation that says so.
unkempt flower
hang on there.
unkempt flower
you have 2y + 24 = 180. but x = 2y + 24, so what you're really saying is x = 180, which is not what we want.
I highly recommend not jumping the crucial step of writing the sum of the two angles as 180 first.
after that, you can do your substitution of x = 2y + 24.
keen flume
I’m sorry I don’t understand 😭
unkempt flower
so you know x and y are supplementary. write out that relationship first.
unkempt flower
good. now substitute your expression for x into the variable.
unkempt flower
yes! now solve this equation for y.
keen flume
I got that y=52
unkempt flower
,calc 156/3
boreal girderBOT
Result:
52
correct. but you're asked for the larger angle (x).
keen flume
Wait I know
unkempt flower
one last step here to solve for x, and you're done.
unkempt flower
that works too.
keen flume
Oh it’s 128
unkempt flower
you can then take that answer and look for the box that has it. call that #3, and continue.
wait, hm.
unkempt flower
oh there are more questions behind, okay.
keen flume
unkempt flower
I see. I'm just glad there aren't any mistaken answers or something.
do you still need help with anything else?
keen flume
do you still need help with anything else?
I wanna try this question ojt to see if I understand it
out*
unkempt flower
sure thing.
keen flume
yeah I need help, I started off with writing 7x+6 is equal to 90 and then got 12 for x
wait actually I think that’s all I had to do
unkempt flower
that is exactly the answer.
keen flume
wowow
I need help with number 4 tho it’s like number 2 but this time they angles are vertical
unkempt flower
which is number 4?
keen flume
answer 12
unkempt flower
this one, I presume?
keen flume
yeah
unkempt flower
there's not much of a difference here compared to the first question you did.
keen flume
oh wait
unkempt flower
you know that A and B are vertically opposite. what can you conclude about A and B?
keen flume
idk about their relationship are they congruent
unkempt flower
from there, what can you do to find A given the relationship B has with both A and C?
unkempt flower
idk about their relationship are they congruent
correct! vertically-opposite angles are congruent.
unkempt flower
glad to help!
keen flume
is it okay if I stay on incase I need help with another problem
unkempt flower
sure thing!
upper bane
note: the bot will periodically ping you to ask if you're done after some time of inactivity. don't forget to respond to the bot to prevent your channel from being closed
upper bane
unkempt flower
it's the same situation as the first question you did.
except that the angles are complementary this time around, and the numbers are slightly different.
unkempt flower
nice work! do remember to close the channel when done.
compact pewterBOT
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keen flume
.close
sage lynx
Can someone verify if this formula is correct? This is supposed to give the number of zeroes at the last of any natural number n.
compact pewterBOT
Can someone verify if this formula is correct? This is supposed to give the numb...
sage lynx
|x| represents the absolute value of x,
{x} the fractional part and
[x] the integral part
compact pewterBOT
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winged dock
Can someone verify if this formula is correct? This is supposed to give the numb...
Uh why not just min(a1,a2) in n= 2^a1 5^a2 ...
I guess that is what this formula is trying to do not sure though
sage lynx
Uh why not just min(a1,a2) in n= 2^a1 5^a2 ...
I want an expression purely in terms of n using elementary functions. I want this formula to be verified.
sage lynx
Another formula that is needed to be verified.
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compact pewterBOT
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glossy steppe
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silver canopy
I am trying to compute this integral [
\int_0^t \6{\op{mod}}{\4{A\tau}{T_0}, A}\dd\tau
]
compact pewterBOT
I am trying to compute this integral \[
\int_0^t \6{\op{mod}}{\4{A\tau}{T_0}, A}...
boreal girderBOT
covert spear
By mod you mean the absolute value?
silver canopy
@bold turret this is my channel. Could you open your own
Delete your messages as well
bold turret
silver canopy
By mod you mean the absolute value?
no. its the remainder function
covert spear
Remainder of what
silver canopy
mod(x,y) is x mod y
silver canopy
the function mod(At/T_0, A) should be reprseenting the sawtooth function if that helps
silver canopy
mod(y,x)?
rotund wave
yeah
silver canopy
i think you got the variables flipped
rotund wave
mod(y,x) = y - floor(y/x)*x
rotund wave
yeah write the integral as a sum or smth
silver canopy
sum?
sum of what
,,
\int_0^t \6{\op{mod}}{\4{A\tau}{T_0}, A}\dd\tau
= \int_0^t \4{A\tau}{T_0}\dd \tau - A\int_0^t \floor{\4\tau{T_0}}\dd \tau
boreal girderBOT
lilac parcel
you break it into intervals $\tau\in[nT_0,(n+1)T_0)$
boreal girderBOT
Donkey
lilac parcel
or you could rewrite your integrand as $A\left{\frac{\frac{A\tau}{T_0}}{A}\right}$
boreal girderBOT
Donkey
Compile Error! Click the reaction for more information.
(You may edit your message to recompile.)
lilac parcel
there's supposed to be a fractional part
silver canopy
you break it into intervals $\tau\in[nT_0,(n+1)T_0)$
why must tau be in that range
lilac parcel
why must tau be in that range
because for that range $\left[\frac{\tau}{T_0}\right]=n$
boreal girderBOT
Donkey
lilac parcel
so its a constant
silver canopy
because for that range $\left[\frac{\tau}{T_0}\right]=n$
right
but we would need to account for outside of that range too
lilac parcel
yeah that method takes care of it
boreal girderBOT
Donkey
$\int_0^{10.xyz}\floor{x}\dd{x} = \sum_{n=0}^9\int_n^{n+1}\floor{x}\dd{x}$ + \int_{10}^{10.xyz}\floor{x}\dd{x}
```Compilation error:```! Missing $ inserted.
<inserted text>
$
l.49 ...{n=0}^9\int_n^{n+1}\floor{x}\dd{x}$ + \int
_{10}^{10.xyz}\floor{x}\dd{x}
I've inserted a begin-math/end-math symbol since I think
you left one out. Proceed, with fingers crossed.```
lilac parcel
see, once you split it, the floor x is just n, so integration is no longer an issue
silver canopy
see, once you split it, the floor x is just n, so integration is no longer an is...
ohh i see
compact pewterBOT
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static stag
help
static stag
is this correct
mild sky
!nopdf
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slate sand
Open it in preview
visual ether
hahaha buttcheeks
static stag
pls speed my exam is in 3 hours
i see its wrong
and im supposed to break it into 3 parts
but i dont know how its from negative pie /2 over pie/6
timid lodge
from -pi/2 to pi/6, the only curve you're looking at is the cardioid, but as soon as you hit pi/6, you're suddenly looking at the circle instead since you pass the intersection
you're not looking at the area between the curves at any point, you're just switching which curve you're finding the area of
static stag
ahh
so we combine the bottom 2?
and then the top one becomes its own
so it goes from pie/6 to pie/2
timid lodge
it goes from pi/6 to 5pi/6 (using the circle) and then 2 of the area from -pi/2 to pi/6 (using the cardioid)
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gilded kayak
How do I complete the jacobian?
compact pewterBOT
How do I complete the jacobian?
trail cave
How do I complete the jacobian?
write the jacobian for (u, v, w)/(x, y, z) instead then find the reciprocal to find the jacobian for (x, y, z)/(u, v, w)
gilded kayak
how do i relate y and v
trail cave
you dont (need to)
currently you are finding ∂(x, y, z)/∂(u, v, w)
just find ∂(u, v, w)/∂(x, y, z) and flip it
trail cave
well, differentiating y with respect to v is a lot more challenging than differentiating v with respect to y is it not?
gilded kayak
wait what
trail cave
d y w.r.t v you have to do more algebra to find the relationship
d v w.r.t y it's written plainly as x - z = v so it's 0
gilded kayak
well, differentiating y with respect to v is a lot more challenging than differe...
if i dont see y and v in the same equation can i outright deduce that vy and yv are both 0
trail cave
if i dont see y and v in the same equation can i outright deduce that vy and yv ...
not necessarily, what if x was a function of y or something? then you would need chain rule
lowk when they write it as a = ..., b = ..., c = ...
you should just take up their offer and try to do ∂(a, b, c)/∂(...)
way easier than always doing ∂(x, y, z)/...
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hardy marsh
Is that correct ??
kind crane
should translate if you want someone to read your work
hardy marsh
should translate if you want someone to read your work
To english?
kind crane
indeed that's the most common language spoken in this server
hardy marsh
indeed that's the most common language spoken in this server
Ok ty (idk how to translate it😭😂😂)
pine osprey
Oui ça me paraît bien
Même si je dirais simplement la continuité en x = 2 plutôt que la continuité à gauche
enfaite c'est la limite à gauche que tu regarde pas la continuité à gauche
Peut être aussi dire que la deuxième fonction est continue comme quotient de deux fonctions continues avec dénominateur non nul
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visual kayak
Hello, I’m currently stuck on triangle congruency for a geometry test tomorrow and I’m not sure how to exactly do this besides from the work I attempted to do which I want to check if I did it right but with no way to, I’m stuck if I’m doing the work correctly to know if it’s congruent or not congruent
compact pewterBOT
Hello, I’m currently stuck on triangle congruency for a geometry test tomorrow a...
@visual kayak Has your question been resolved?
dusky agate
The congruency is either 2 sides and an angle or 2 angles and a side
You just need to look for these situations
All three side work as well
Question 3 and six are also congruent
#3 has 2 sides and an angle
They both share side MK
JM=LK
Angle K = Angle M
Therefore JMK=LMK
For #6 you did the angle part right
And you also have 2 sides
Also with Z angle, (interior angle) because DJ and ME are parallel, Angle AME = Angle DJA
There fore Triangles DJA=AMe
@visual kayak
visual kayak
Thank you for the help, I’m checking those two atm, I do have one question that comes part of the problem, is it possible if it is either SSS or SAS for #3 & #6?
dusky agate
I don’t know if you’ve learned it yet, but it is possible to prove SSS for both questions
I gtg if you don’t have any more questions
visual kayak
I appreciate the help, thank you
dusky agate
Your welcome
visual kayak
.close
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kind hollow
Hello. Suppose we have the set
A= {a_n | n element of naturals}
and a_n is bounded, so sup(A) exists. Now, if we want the term of A right before the supremum or if the sup is in A, equal to the supremum, can we let
δ>= 0 be the sufficient number to deduct from sup(A) and get the member of A right at/before it? does that make any sense or am i tripping?
compact pewterBOT
Hello. Suppose we have the set
A= {a_n | n element of naturals}
and a_n is boun...
hushed magnet
if you have A={1-1/n} then clearly sup(A)=1 but there is no number right before the supremum
kind hollow
i understand that, but can we not make a slight convention and define it using delta?
hushed magnet
what should even the result of that be here
what number do you want to get in this example
kind hollow
not any number in particular
i wanna prove that s_n = sup{a_k | k>= n} is descending
hushed magnet
ok so you want something completely different
kind hollow
i mean if i could do what i explained earlier that would become obvious
hushed magnet
consider the set A_n = {a_k: k \geq n}
show that those A_n are subsets of each other
then consider if X,Y are sets with X subset Y, what about sup X vs sup Y?
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