#help-17
river minnow
By contradiction, if none of the m1, …, mk were equal to o_1, you could increase the sum
torpid sequoia
Right
Sorry I left for a moment, I'm back.
So I want to see that elements in O, all belong to Max. So first I grab o_1 and to show that it belongs to Max, by contradiction, if None of the elements in m1, ... mk were equal to o_1, I could increase the sum of max, since o_1 is the biggest number in X, but this is absurd since max returns exactly the maximum possible value of the sum of the subset of X of k elements
So o_1 necessarily belongs to Max
something like this?
idk if I should @ you 😄
you suggest some form of induction ,right?
river minnow
Sorry I left for a moment, I'm back.
So I want to see that elements in O, all b...
Correct
river minnow
idk if I should @ you 😄
Feel free to
river minnow
you suggest some form of induction ,right?
Yes
Now that we can cancel o_1’s from both sums, we are left with a sum of O[2], …, O[k] and the largest sum of k elements in S minus o_1
The former can be rephrased as the sum of the first k-1 elements in the sorted list of S\{o_1}
The latter can be rephrased as the largest sum of (k-1) elements in \{o_1}
And the claim follows by inductive hypothesis
(since the size of S{o_1} is one smaller than of S)
vocal sleetBOT
@torpid sequoia Has your question been resolved?
torpid sequoia
(since the size of S\{o_1} is one smaller than of S)
yeah man amazing
that helped !
thanks a lot !
swift gorge
ABCD is right trapezoid
x=?
vocal sleetBOT
ABCD is right trapezoid
x=?
swift gorge
Helps pls😱 idk what to do
brazen wharf
Helps pls😱 idk what to do
lol
swift gorge
Xd
lone linden
ABCD is right trapezoid
x=?
Hint: Draw the perpendicular from C to AB
swift gorge
Hint: Draw the perpendicular from C to AB
yeah but i still dont see anything
🤔🤔hmmmmm
lone linden
Consider what ||the length of that perpendicular is equal to||
swift gorge
Consider what ||the length of that perpendicular is equal to||
You mean |AD| ?
swift gorge
Consider what ||the length of that perpendicular is equal to||
Well i think idk how to find it
🤯
lone linden
Normally when you want to find a segment, you look for triangles where it appears
Are there any relevant triangles that you can use
swift gorge
Are there any relevant triangles that you can use
Oooo you mean we can think this is a middle point of triangle?
Midsole?
swift gorge
You say triangle and something came to my mind
@lone linden is this true? İ dont have acces to answer soo if its true ima go sleep
lone linden
Oh this works too
My intention was to find $BC$ and $AB$ in terms of $x$ using trig
twin meteorBOT
Civil Service Pigeon
lone linden
It reduces to $\cot(35^{\circ})=\frac{1}{\sin x}+\frac{1}{\tan x}$
twin meteorBOT
Civil Service Pigeon
lone linden
$=\frac{1+\cos x}{\sin x}=\frac{2\cos^2 \frac{x}{2}}{2\sin \frac{x}{2} \cos \frac{x}{2}}=\cot \frac{x}{2}$
twin meteorBOT
Civil Service Pigeon
lone linden
So 35=x/2 -> x=70
swift gorge
How would you rate this question?
lone linden
Uhhh
Maybe 2/10? but my scale is weird
It’s more logarithmic than linear
And tops out at oly (plus it’s from the perspective of a comp math person so the entire top half is mostly reserved for that unless it’s a rlly hard school math question)
Uh scale that as you want to ig (?)
ik this is very roundabout lol
swift gorge
you 12th grade may i ask?
swift gorge
And tops out at oly (plus it’s from the perspective of a comp math person so the...
Yeah i aggre
lone linden
you 12th grade may i ask?
Yes
swift gorge
Yes
Let P(n) denote the product of the digits of the number n and a=45.45
........
What is the equivalent of the expression in terms of a?
Do you have a idea about this?
İ
lone linden
Can’t you just spam factoring on this
meager shoal
It’s not sum_n=1^n=1000 [n(45*45)]
lone linden
What
swift gorge
Yes pigeon is right i got the same
twin meteorBOT
Result:
93195
This is (very basic) Python code
I just used the first ide in the search results to run it
swift gorge
are you allowed to use this in school?
meager shoal
Could u explain to me how u got that? 
swift gorge
Could u explain to me how u got that? <:blobcry:971797574916907028>
İ think i can
meager shoal
İ think i can
Ok
swift gorge
Ok
From 100 to 999
No neee to add 1000
And How many can we write in the hundreds digit first?
Same for ten digit
same for one digit
We neee to impactvthese
Dude my english is not enought to tell to things in my mind..ik
vocal sleetBOT
vocal sleetBOT
Available help channel!
true tusk
• Ask ∫ 𝑠𝑒𝑛5(2𝑡) . 𝑐𝑜𝑠2(2𝑡) 𝑑𝑡 ( this is a Trigonometric Powers Integrals exercise )
twin meteorBOT
true tusk
yes
bitter pilot
What have you tried?
true tusk
I dont even know how to start because I couldnt assist that part of teacher class
twin meteorBOT
true tusk
ohh
bitter pilot
And then you might wanna try u-substitution.
true tusk
oh alright
bitter pilot
After distribution.
chrome raptor
sin^m(x)cos^n(x) is a common integrand and how to take the u-substitution depends on the relative parity of m and n
vocal sleetBOT
Available help channel!
buoyant elm
Can you resolve this?
vocal sleetBOT
Can you resolve this?
buoyant elm
a²+b²=2025.1/a²+1/b² ; Find the values of a and b considering a real and positive number
<@&286206848099549185>
hollow pendant
<@&286206848099549185>
!15m
vocal sleetBOT
<@&286206848099549185>
Please only use the <@&286206848099549185> ping once if your question has not been answered for 15 minutes. Please do not ping or DM individual users about your question.
buoyant elm
Sorry
hollow pendant
a²+b²=2025.1/a²+1/b² ; Find the values of a and b considering a real and posit...
$$a^2 + b^2 = 2025 \cdot \frac{1}{a^2 +b^2}$$Is this the equation?
twin meteorBOT
@hollow pendant
buoyant elm
Yes
signal pendant
a²+b²=2025.1/a²+1/b² ; Find the values of a and b considering a real and posit...
Is 2025 multiplied to both or only for the first term?
buoyant elm
Is 2025 multiplied to both or only for the first term?
The term hes connected
hollow pendant
Yes
First, set an arbitrary variable $y = a^2 + b^2$. \
Can you rewrite the equation with this?
twin meteorBOT
@hollow pendant
buoyant elm
Yes
signal pendant
The term hes connected
Ok so is this the correct equation?#help-17 message
buoyant elm
Yeah
twin meteorBOT
@hollow pendant
buoyant elm
Yes
signal pendant
.....
hollow pendant
Those are two different values.
buoyant elm
I dont know lol
hollow pendant
I dont know lol
!original
vocal sleetBOT
I dont know lol
Please show the original problem, exactly as it was stated to you, with the entire original context. A picture or screenshot is best. If the original problem is not in English, then post it anyway! The additional context might still be helpful. Do your best to provide a translation.
buoyant elm
"a²+b²=2025.1/a²+1/b² ; Descubra os valores de a e b considerando um número real e positivo"
This is how it started
signal pendant
Can you send an image?
hollow pendant
<@751533372890677398>
My instinct tells me this is what they meant.
signal pendant
........
hollow pendant
$$a^2 + b^2 = 2025 \qty(\frac{1}{a^2} + \frac{1}{b^2})$$
twin meteorBOT
@hollow pendant
buoyant elm
Yes
signal pendant
First note that this is equivalent to $$a^2b^2=2025$$
twin meteorBOT
Herbert
buoyant elm
And the enunciate
a²+b²=2025.1/a²+1/b² ; Find the values of a and b considering a real and positive number
And positive
signal pendant
Okay, so there are infinitely many solutions then
buoyant elm
Yo
signal pendant
Since you say its an olympiad i am guessing a and b are positive integers instead
lucid bane
**Herbert**
you can simplify further technically :3
signal pendant
Then there are finitely many solutions
signal pendant
you can simplify further technically :3
Indeed, ||by difference of squares||
buoyant elm
Since you say its an olympiad i am guessing a and b are positive _integers_ inst...
Yeah
signal pendant
Okay, this is like the fifth time you provided an incorrect problem
signal pendant
**Herbert**
You cant solve this one?
signal pendant
Yes so ab = 45
buoyant elm
Oh thanks
signal pendant
And a and b are integers (positive? Im guessing?)
So you can find all pairs of a and b now
buoyant elm
Tehy are positive
But i dont know if there integers
I dont have the entire question so
signal pendant
I dont have the entire answer then
buoyant elm
Yeah i k
signal pendant
The answer changes depending on if a and b are positive real or integers. But i hope you can find the answer in both situations
signal pendant
signal pendant
Yes so ab = 45
Without further information we are stuck here
glass oak
"the" answer?
vocal sleetBOT
@buoyant elm Has your question been resolved?
shadow berry
Yo how do you do estimation of peramiters?
vocal sleetBOT
Yo how do you do estimation of peramiters?
sand lodge
vocal sleetBOT
@shadow berry Has your question been resolved?
twin stone
How do I sketch this and find the zeros
vocal sleetBOT
How do I sketch this and find the zeros
meager shoal
Zeroes can be found by factorization
Max/min points can be found by first derivative test
The function is neither even or odd
meager shoal
twin stone
Does that mean I can never fix this and my life is over and I should just quit
meager shoal
Nah gang
twin stone
So what do I do now
meager shoal
Try resetting??😭
twin stone
How
meager shoal
Idk
heavy yoke
vocal sleetBOT
@twin stone Has your question been resolved?
rugged kernel
use second derivative
vocal sleetBOT
vocal sleetBOT
Available help channel!
opaque zinc
vocal sleetBOT
flat python
make cases for x
flat python
well it will become e^(lim m -> 0 1/m ln(cos(x/m)))
and as cos(x/m) will never approach anything for x != 0, ln(cos(x/m)) will never approach anything, so this will be undefined.
as 1/m of lim m-> infinity can either be + infinity or - infinity
flat python
aka the limits are different <=> undefined
opaque zinc
soo so so
for
x!=0
lim---> does not exist?
flat python
np!
flat python
opaque zinc
craxy
last hedge
Isnt it 24x + 10
vocal sleetBOT
Isnt it 24x + 10
bronze osprey
so area of triangle should be area of triangle PLUS 10
it's a rearrangement: triangle = rectangle + 10
subtracting 10 on both sides, triangle - 10 = rectangle
vocal sleetBOT
@last hedge Has your question been resolved?
bitter pilot
so area of triangle should be area of triangle PLUS 10
If you add 10 units to the area of the triangle, then you actually make the area of the rectangle greater by 10 units.
For example, if we have the numbers 5 and 3, and we say that 5 is greater than 3 by 2, then we can express that as 5 = 3 + 2. (not the other way as you would expect 5 + 2 = 3)
glossy ledge
**Number Theory Problem **
Just give a small hint, Don't give the entire solution.
https://codeforces.com/problemset/problem/2084/B
let global min = mn
glossy ledge
it is possible that there are multiple of instances of mn in the array
bronze osprey
**Number Theory Problem **
Just give a small hint, Don't give the entire soluti...
you're on the wrong server
we don't help with coding problems here
glossy ledge
but it's just math
bronze osprey
but it's just math
no, it's a coding problem that uses maths
glossy ledge
no, it's a coding problem that uses maths
im not askign about the implementation though
bronze osprey
ah hmm okay
bronze osprey
it is possible that there are multiple of instances of mn in the array
okay to answer your question, yes
glossy ledge
okay to answer your question, yes
wait
bronze osprey
the global minimum can appear more than once
glossy ledge
let me first type what I tried
glossy ledge
it is possible that there are multiple of instances of mn in the array
if there are more than one instances of the global minimum
then I can just have one mn in first part and only the mn in second part.
now, there is only one instance of mn then this mn is present in first part
gcd of all the numbers in second part is mn
each number in second part | mn
so, If I extract out the numbers from the array which divides mn then place the number/mn value in a new array
Now, I just have to choose a bunch of numbers from this array whose gcd is mn
I need a small hint
bronze osprey
im not askign about the implementation though
you said you weren't asking about the implementation though
sage cipher
if there are more than one instances of the global minimum
then I can just have...
you can just place all multiples of mn on gcd side and see if it works
also there are very well written editorials (solutions) for codeforces questions
glossy ledge
you can just place all multiples of mn on gcd side and see if it works
oh, right because the gcd will only decrease as we take more numbers and we can hope that it become exactly mn instead of some multiple of mn.
glossy ledge
https://codeforces.com/blog/entry/141155
I don't like them because I hardly see any improvement in myself when I read those editorials
glossy ledge
you are like specifically doing gcd problems?
yeah, number theory
sage cipher
cool
vocal sleetBOT
vocal sleetBOT
Available help channel!
vast shale
vocal sleetBOT
bronze osprey
Maximum height of the ball ?
it gives you the x-coordinate of the max height on the graph
but yes nice that you know -b/(2a)
vast shale
How should I calculate brother
bronze osprey
How should I calculate brother
okay so if you want h(2.5) and h(t) = 25t - 5t^2 .............
vocal sleetBOT
vocal sleetBOT
Available help channel!
cunning yew
hello , sometimes in exercices you get the question to prove that a function $g(x)$ is the image of $f(x)$ by one of these transformations : \
-dialation \
-reflection \
-rotation \
-Translation
vocal sleetBOT
hello , sometimes in exercices you get the question to prove that a function $g(...
twin meteorBOT
<rajel />
cunning yew
how do you usually prove it
peak matrix
do you get the graph or the formula?
cunning yew
the formula
afaik translation goes by f(x + k)
but i dont remember em all , neither know how to prove
cunning yew
i mean i thought they have fixed formulas
you get the expression of both functions and then you can deduce
peak matrix
you'd usually probably try to express one as a transformed version of the second, or both as transformations of some third simple function
peak matrix
you get the expression of both functions and then you can deduce
it depends on what kind of functions you get supplied with
if they are quadratic, then it's simple, all parabolas are transformations of each other
and so are all hyperbolas and ellipses
cunning yew
if they are quadratic, then it's simple, all parabolas are transformations of ea...
transformations of each other as in ?
peak matrix
transformations of each other as in ?
by moving and scaling a parabola, you can get any other parabola
and possibly rotating, but that wouldnt result in a function
unless you can be more specifc, there isnt a simple way to answer your quesiton
we have no idea what kind of functions could f and g be
cunning yew
we have no idea what kind of functions could f and g be
what about exponential functions
peak matrix
what about exponential functions
okay, so those are just functions of form f(x) = a^x + b
cunning yew
e^x
peak matrix
**<rajel />**
one more question, are you always asked about just one of those transformations? Or about all of them at once?
cunning yew
just one
peak matrix
alright then
cunning yew
wdym
peak matrix
like we need f and g
peak matrix
f has to be provided as well
cunning yew
alr thats also fine
peak matrix
can you just write out the full question
any example of it
i dont get what you mean
cunning yew
i
i'll try to find one
,, f_n(x) = \frac{e^{-nx}}{1+e^{-x}} \text{ prove that } C_0 \text{ and } C_1 \text{ are symetrical in y-axis }
twin meteorBOT
<rajel />
peak matrix
C0 and C1 are what exactly
cunning yew
n is a natrual number
peak matrix
i didnt notice the denominator doesnt have n in it
okay, so we start by finding f0 and f1
cunning yew
yep
peak matrix
now if functions are symmetrical around the y-axis, that means that when you plug in some x in f_0(x), you should get the same thing as you'd get if you plugged in -x into f_1(x)
cunning yew
definitely
cunning yew
that means f0(-x)=f1(x)
peak matrix
That's the identity we're gonna be proving, right
so we are just gonna be proving that those 2 things are equal
which is nothing, but algebra
cunning yew
you multiply
peak matrix
do you have an idea of how it could be proven?
peak matrix
btw if we add them both we'll their sum = 1 , what does that mean
nothing in particular, but all pairs of symmetrical functions will have similar property
all such pairs should add to a constant
wait no
peak matrix
nothing in particular, but all pairs of symmetrical functions will have similar ...
ignore this
cunning yew
ok
peak matrix
oh i think that it's because f0 and f1 are almost odd
they are odd functions, translated upwards by 1/2
cunning yew
i see, i guses i'll damn go for minutes
so that was the type of questions i got
if you have something to add sure
peak matrix
All such questions should be pretty similar
you follow this process
reduce the kind of geometrical question to an algebraic identity to prove, and then use algebra to actually prove it
vocal sleetBOT
@cunning yew Has your question been resolved?
cunning yew
reduce the kind of geometrical question to an algebraic identity to prove, and t...
what about rotation and dialation ?
,, f_n(x)=xe^{\frac{n}{x}} \text{ prove that } C_n \text{ is the image of } C_1 \text{ by a dialation of center } O
twin meteorBOT
<rajel />
peak matrix
what about rotation and dialation ?
rotation is most likely not gonna occur
or at least not too much
perhaps sth specific such as 90° rotation
cunning yew
rotation is most likely not gonna occur
never seen rotation , so yeah
peak matrix
**<rajel />**
whats O?
peak matrix
oh okay
well, what happens when you dilate a point (x, y) by some factor
say factor f
what are the new coordinates?
cunning yew
(fx, fy)
peak matrix
right
cunning yew
depends on the center tho ?
cunning yew
if not we can try to make a new one where the center is the origin
peak matrix
y / a = f(x / a)
Is gonna be the equation dilated by factor of a
cunning yew
only in this case ?
peak matrix
it's true whenever O is the center
cunning yew
y is the image
peak matrix
no like y / a = f(x / a) has the graph of the image
but we could also say that a*f(x / a) is the image itself
a* scales it vertically by a factor of a
x/a scales it horizontally by a factor of a
peak matrix
why the graph here is y/a
y / a = f(x/a) is the same equation as y = af(x / a)
it doesnt matter how you write it
peak matrix
well, so we have these 2 things here
cunning yew
aha
peak matrix
$y=a\cdot\left(\frac{x}{a}\right)\cdot e^{\frac{1}{\left(\frac{x}{a}\right)}}$
twin meteorBOT
MathIsAlwaysRight
peak matrix
in general, we also know that the first function scaled by a will look like this
which simplifies to x * e^(a/x)
so it should be pretty clear that x*e^(n/x) scales the graph by a factor of n
peak matrix
not really tbh
prisma parcel
I'm confused at the part when we go the other way. We say that we don't vanish for some i_0 but if we're in the variety doesn't this imply that we must vanish? Or is that saying that it isn't in V but vanishes because then it must be in W?
I can mostly follow the argument but that line is just slightly confusing me a bit
vocal sleetBOT
I'm confused at the part when we go the other way. We say that we don't vanish f...
prisma parcel
Oh the original lemma here states to show that if V and W are affine varieties, then V \cap W and V \cup W are also such
silk comet
I'm confused at the part when we go the other way. We say that we don't vanish f...
if (a_1, ..., a_n) is not in V, then at least one of the f_i does not vanish at that point
otherwise it would be in V
prisma parcel
if (a_1, ..., a_n) is not in V, then at least one of the f_i does not vanish at ...
Now if it doesn't vanish at one of the f_i's but because it still must vanish at the union then it must follow that the g's must vanish to cancel out the non vanishing f's I think
Is that right?
I apologize for my poor wording in advance, I am still getting used to some of the vocabulary
silk comet
Now if it doesn't vanish at one of the f_i's but because it still must vanish at...
not sure what "still must vanish at the union" means, but your idea is correct; I would say it like this:
if (a_1, ..., a_n) doesn't vanish at one of the f_i's, say f_{i_0} in particular, then we know f_i (a_1, ..., a_n) g_j (a_1, ..., a_n) = 0 for all i, j (since we chose (a_1, ..., a_n) from V(f_i * g_j)), so in particular we can take i = i_0 to get f_{i_0} (a_1, ..., a_n) g_j (a_1, ..., a_n) = 0 for all j. since the first term is a nonzero constant, we know g_j (a_1, ..., a_n) = 0 for all j
prisma parcel
not sure what "still must vanish at the union" means, but your idea is correct; ...
That makes sense, thank you very much!
We appreciate it
silk comet
happy to help
prisma parcel
.close
vocal sleetBOT
vocal sleetBOT
Available help channel!
proud burrow
can yall help me with the questions on the second photo i dont get it at all
outer warren
first you need to fix your graph of 2^x, its incorrect
proud burrow
i’ll check for you
ok sorry i’m an idiot i subbed in the wrong numbers
i’m redoing it now
so wait since it’s to the power of x do i do like
g(-2) = 2(-2) for example
outer warren
no
outer warren
2^(-2)
you're converting exponentiation to multiplication
proud burrow
2^(-2)
you're converting exponentiation to multiplication
is that not the same as 2(-2)
ok wait
when i put 2 to the power of -2 in my calc i get a decimal
outer warren
yes
proud burrow
yes to which one
outer warren
that entering 2 to the power of -2 in the calc will give you a decimal
proud burrow
how can i graph it then if its a decimal
outer warren
2(-2) represents the product of 2 and -2, will give you an integer
if the grid isn't precise enough,
draw additional lines with a ruler yourself or estimate it
proud burrow
hm alright
ok ur right
ok i got my new and improved graph
is (b) (i)
(2,4) and (4,16)
outer warren
they only need the x values, not the whole point
outer warren
no
proud burrow
2 and 4
outer warren
yes
outer warren
you should see it on your graph
proud burrow
well idk but my estimate would be 6
outer warren
your graphs if done properly will intersect at a third point,
but the values there aren't as nice
hence why they asked for an estimated value
show your new graph and where you're looking at
where's 6 coming from
outer warren
wdym
show your new graph
they would intersect again if it went up to 6
is that an assumption?
proud burrow
just imagine the green stuff doesn’t exist
outer warren
the graphs start to drift further apart as you go further to the right
the behaviour at x=6 is not visible on the graph
and you can't make assumptions that'll it'll intersect again there
proud burrow
oh
outer warren
look carefully for the info avaiable to you on the graph that you have
you have intersection points where x=2 and x=4
your graphs if done properly will intersect at a third point,
but the values there aren't as nice
hence why they asked for an estimated value
proud burrow
ngl i haven’t a clue the closest ones are atleast 1 apart
outer warren
do you know what it means for curves to intersect?
proud burrow
no im really bad at maths
outer warren
when they cross/touch each other
proud burrow
ye
outer warren
you would've used that property to get (2,4) and (4,16)
start at the very left
follow one of the curves
and tell be when it hits the other curve
proud burrow
well they kinda collide near -1 but that could just be me writing it wrong
outer warren
-1 is rounding to much
but yes, there is an intersection point around there
try giving a better approximation/estimate
outer warren
ideally your curves should be more smooth, but its good enough for what's being asked
don't overthink this
outer warren
again, rounding too much
don't get hung up on integer values
e.g what would be a reasonable estimate for the value at the orange
would you say that's 1 or 2?
or somethign else
proud burrow
1.5
proud burrow
well would it be 0.5
outer warren
no
proud burrow
i genuinely have no clue
outer warren
they've given you additional lighter grid lines, there's one between the -1 and 0, that'll be -0.5
and try to estimate based on that
also can you highlight where exactly you're looking at?
consider how far your location is from numbers to the left/right
outer warren
you've estimated the orange would be around 1.5
now what would be a reasonable estimate for the red?
proud burrow
1.25
outer warren
yes
now don't overthink
and apply the exact same reasoning to what you have
draw a vertical line from the intersection point until it hits the x-axis
and then ignore all other information and focus on the values on that axis
and do what you did just now
outer warren
no
you said the value was around -1,
how are you getting something relatively far away
outer warren
also can you highlight where exactly you're looking at?
the intersection point near -1
proud burrow
just imagine the green stuff doesn’t exist
proud burrow
well i don’t even understand functions at all
so this is a whole other level of confusions for me
outer warren
no
what do you mean "close"
i can literally see your graphs intersecting there
i can see your graphs crossing there
there same way they're crossing at x=2 and x=4
outer warren
outer warren
why not?
outer warren
insidious?
proud burrow
ye
outer warren
i don't know what you mean by insidious point
outer warren
do you know what it means for curves to intersect?
when they cross/touch each other
nothing in that definition says they have to be on lattice points or have nice values
proud burrow
i was being dumb
outer warren
and estimate the x value of that point
proud burrow
ah alright
hidden turret
Are these two circuits equivalent?
vocal sleetBOT
Are these two circuits equivalent?
hidden turret
both the voltage sources are 60 volts
dry yacht
What do you think?
hidden turret
What do you think?
yes but im unsure
dry yacht
Equivalent it is.
hidden turret
is it though?
dry yacht
On the right hand side, if you drag the 10 ohm resistor "up" and the 5 ohm resistor "down", you'll find those circuits similar (or equivalent).
vast shale
This is a part of a slightly bigger proof. I would like to verify whether the uniform convergence part here is done correctly or not.
vocal sleetBOT
This is a part of a slightly bigger proof. I would like to verify whether the un...
vast shale
For reference, here is about I*(a,x) (amongst the other stuff, ignore them)
Please ping if reply
<@&286206848099549185>
vocal sleetBOT
@vast shale Has your question been resolved?
vocal sleetBOT
@vast shale Has your question been resolved?
flat whale
can also post in #real-complex-analysis
vocal sleetBOT
vocal sleetBOT
Available help channel!
rare solar
I'm confused on how you go from AA^-1 to (A^-1)^-1. Is something being multiplied?
vocal sleetBOT
I'm confused on how you go from AA^-1 to (A^-1)^-1. Is something being multiplie...
hard atlas
AA^-1=I means that A is the inverse of A^-1
because its the thing that when multiplied with A^-1 gives the identity
thats just the def of what an inverse means
(well technically you also need that A^-1A=I but that also holds)
rare solar
AA^-1=I means that A is the inverse of A^-1
no I got that part. But I'm confused about how you go from AA^-1 to (A^-1)^-1. What transformation is occuring to obtain this
hard atlas
there is no transformation
rare solar
i should use a different word. i dont mean matrix transformation
hard atlas
A=(A^-1)^-1 in words means "A is the inverse of A^-1"
rare solar
i mean what algebraically is turning the left side of the arrow into the right
hard atlas
there is no algebraic rewriting going on
it would follow from multiplying both sides by (A^-1)^-1 from the right
rare solar
Proof of (a): AA^-1 = I => (A^-1)^-1 = A
Somehow I is becoming A and AA^-1 is becoming (A^-1)^-1
hard atlas
but at that point you dont technically know that (A^-1)^-1 exists
hard atlas
but at that point you dont technically know that (A^-1)^-1 exists
but again, you dont technically know that (A^-1)^-1 exists yet
idle path
Yeah, there's no algebraic manipulations going on. The point is that A^-1 is defined as the matrix B such that AB = BA = I. If you have found such a B, you can write A^-1 = B
rare solar
ugh i wish my instructor actually went over this instead of just leaving us to this ourselves
idle path
Now we want to find a matrix B such that A^-1B = BA^-1 = I. What matrix is that?
idle path
Now we want to find a matrix B such that A^-1B = BA^-1 = I. What matrix is that?
Can you answer this question, and also look up the definition of inverse in your notes? 
rare solar
the inverse of a matrix?
idle path
Yeah, the definition of the inv of a matrix
rare solar
that's the matrix when multiplied after A gives I
also no I can't answer that question lol
i dont know
actually I do
B is A
I don't know why it's A but I know that it's A. I know that AA^-1 = I, like how for real numbers aa^-1 = 1.
idle path
that's the matrix when multiplied after A gives I
Yep! 👍 You can write it in symbols, the inv of A is the matrix B such that AB = I (for square matrices atleast), and we write B as A^-1
idle path
I don't know why it's A but I know that it's A. I know that AA^-1 = I, like how ...
AA^-1 is true because it's the literal definition of the inverse. But now we're looking for the inverse of the inverse, so we're kind of reusing the definition
AA^-1 = I*
idle path
B is A
So if you know that A is the inverse of A^-1, how would you write that in symbols?
rare solar
sorry had to zoom different instructor quickly
no amount of multiplying will take me from one side to the other... so there has to be some other way
rare solar
So if you know that A is the inverse of A^-1, how would you write that in symbol...
AA^-1 = I. Right, I understand that. The firs tpart is just stating the definition of a matrix's inverse. But I'm confused on how we use this to find that it can be applied again to get the oriignal matrix
im sorry if im stupid or missing something obvious
idle path
AA^-1 = I. Right, I understand that. The firs tpart is just stating the definiti...
You're too hung up on the idea that the implication arrow means that we're "doing something" to the equation, to get from one side to the other. The implication arrow is kind of a red herring tbh, it's better spelled out in words: since AA^-1 = I, the inverse of A^-1 is A
rare solar
so... there's no proof that this is true. We're just stating that this is the case
idle path
AA^-1 = I. Right, I understand that. The firs tpart is just stating the definiti...
AA^-1 = I wasn't the symbolic representation I was thinking of. That's the definition of what an inverse is, but how do you write the inverse of A^-1, in a single expression (not an equation!)
idle path
so... there's no proof that this is true. We're just stating that this is the ca...
no, it is a proof, but it may be too simple, so that you're overthinking it
or, a simpler question: how do you write the inverse of A in symbols?
rare solar
the inverse of a matrix is the matrix such that when it is multiplied by the original matrix the identity matrix is produced
idle path
yeah, that's words again, in symbols we would write the inverse of A as A^-1
that's just notation, and we define A^-1 as the matrix B such that AB = I
rare solar
yes
idle path
now, how do we write the inverse of, say, B?
rare solar
B^-1
rare solar
stackexchange says:
By definition, B is the inverse of A if and only if AB=I and BA=I
Observe that A^−1A=I and AA^−1=I.
So the inverse of A^−1 is A.
But I fail to understand why the second line implies the third line
rare solar
and how do we write the inverse of A^-1?
(A^-1)^-1
idle path
(A^-1)^-1
Yep! So saying that the inverse of A^-1 is A is the same as saying (A^-1)^-1 = A. But maybe that's not where your confusion is?
rare solar
but we never proved that the inverse of A^-1 is A
idle path
what do we need to do to show that B is the inverse of A?
rare solar
i dont know
oh
if AB = I then B is the inverse of A
OH
so
AA^-1 = I
A^-1 is the inverse of A
becasue it produces I
and hten
we can flip it
(not sure why but I know we can)
A^-1 A = I
so A is the inverse of A^-1
but
idle path
Yep, we just use the definition. When in doubt, always use the definition
rare solar
why can we do AB = BA
idle path
So this is somewhat of a tricky point. By definition, an inverse must be two-sided, ie. we must have both AA^-1 = I and A^-1A = I. This is the property you used in your proof. But earlier I just stated that an inverse must satisfy AA^-1 = I, and this is sufficient only for square matrices. For non-square matrices you can have AB = I without B being an inverse of A
but it's correct of you to point that out, because in general AB is not equal to BA, but AA^-1 is always equal to A^-1A
uh, maybe the point about non-square matrices was confusing, but anyways, AA^-1 = A^-1A must be true by definition, otherwise it's not a true inverse
rare solar
I know it isn't exactly rigorous but
the comment is just for myself because I know for a fact that I will forget all of this
@idle path sorry for being so stubborn. My instructor is putting a tonne of emphasis on proofs, and he gets mad when you dont prove even the simplest things
idle path
yeah, I think that's fine 👍 the idea is that (1): AA^-1 = A^-1A = I by definition of the inverse of A. Secondly, the inverse of A^-1 must satisfy A^-1B = BA^-1 = I. From (1) we see that A satisfies this, so A is the inverse of A^-1
idle path
<@541326887277035570> sorry for being so stubborn. My instructor is putting a to...
yeah, it's good to be meticulous about proofs at this point 👍 this is maybe the steepest learning curve, later you can handwave much more
rare solar
Yep, we just use the definition. When in doubt, always use the definition
that's the thing. I didn't realize that since both equal I, I could equate the two
vocal sleetBOT
@rare solar Has your question been resolved?
frank flame
Question 4b is asking me,
How many unique arrangements can I make with the letters in “FORTUNES” but the vowels must be together but not in a group of 3.
I genuinely don’t know where to go from here 😭💔
gaunt finch
Question 4b is asking me,
How many unique arrangements can I make with the let...
since 2 of the vowels are together but not the 3rd, you can first imagine combining 2 of the vowels and counting them as 1 unit
there are 6 ways to do this
- OU
- UO
- EO
- OE
- EU
- UE
that means you are now ordering 7 "letters" instead of the original 8 since youre counting the 2 vowels as a single unit
since all the letters are unique, there are 7! ways to order them multiplied by the 6 ways to pick the 2 vowels in the pair
now all thats left is to subtract the situations where the 3rd vowel is adjacent to our "unit" of 2 vowels
i think you can take it from here :)
vast shale
hi I don’t understand how to interpret this question and how they got that result
vocal sleetBOT
hi I don’t understand how to interpret this question and how they got that resul...
lone linden
hi I don’t understand how to interpret this question and how they got that resul...
|z|: modulus of z (distance from z to the origin)
arg(z): argument of z (angle between z and the positive real axis)
tldr you're basically graphing the polar curve r = theta. If you're not familiar, just pick a few values of z and arg(z), plot them based on what I said above, and draw a curve
vocal sleetBOT
@vast shale Has your question been resolved?
vast shale
|z|: modulus of z (distance from z to the origin)
arg(z): argument of z (angle b...
thank you so much!
vocal sleetBOT
If you are done with this channel, please mark your problem as solved by typing .close
vast shale
would anyone be able to help me, im confused on how you know which side to shade in inequality kind of questions for argand/loci diagrams
eg these two, how do you know to shade lhs or rhs?
vocal sleetBOT
@vast shale Has your question been resolved?
vocal sleetBOT
@vast shale Has your question been resolved?
weak edge
buoyant jay
you can do a test
by taking points from one of the sides of the line you have drwan
if the answer lies is true acc to the equality
it lies that side , so you shade that portion
to check where to shade
lhs or rhs
vocal sleetBOT
@vast shale Has your question been resolved?
fringe lynx
Not sure whether this counts as Mathematics, but can someone help with my engineering work? I just started and only really know the basics
vocal sleetBOT
Not sure whether this counts as Mathematics, but can someone help with my engine...
gentle nimbus
erm the first one seems like a pain to do but the second one seems to be an application of pascal's law
Ø20? strange unit...
gusty patrol
Can anyone slove this
fringe lynx
Ø20? strange unit...
Diameter
Also I've done the pascal's law one, but the truck one is stumping me, I've asked AI a few times, and it gave me 2-3 different answers
<@&286206848099549185>
gentle nimbus
Can anyone slove this
!occupied
vocal sleetBOT
Can anyone slove this
Someone else is already using this help channel. If you need help with a question, please open your own help channel/thread (see #❓how-to-get-help for instructions).
prisma fjord
or, lets square it too
prisma fjord
x^2+1/x^2+2=a^2
gentle nimbus
Also I've done the pascal's law one, but the truck one is stumping me, I've aske...
!nogpt
vocal sleetBOT
Also I've done the pascal's law one, but the truck one is stumping me, I've aske...
Please do not trust ChatGPT or similar AI tools for mathematical tasks, as they often generate output which "sounds correct" but has numerous factual or logical errors. Use of these AI tools to answer other people's help questions is strictly against server rules (see #rules).
gentle nimbus
just use $P=\frac{F}{A}=Const.$
twin meteorBOT
parabolicinsanity
gentle nimbus
erm the first one?
prisma fjord
erm good luck on that one 💀
what?
gentle nimbus
the pivot is
prisma fjord
ok, easy equation
prisma fjord
m=fa
gentle nimbus
m=fa
never cook again 💀
anyways
theres the centre of mass right?
just use that to calculate the torques 🤔
or moments or whatever
fringe lynx
just use that to calculate the torques 🤔
Its not an actual pivot point tho no? because its not touching the ground
gentle nimbus
its the centre of mass its efficetively where all the forces act
and by extension, all the moments
fringe lynx
Its not an actual pivot point tho no? because its not touching the ground
Nvm thats wrong
prisma fjord
Nvm thats wrong
it isnt directly,but its one of many points where force is applied.
fringe lynx
But if we do use Cm for the pivot point, what is the force at the front tire?
prisma fjord
But if we do use Cm for the pivot point, what is the force at the front tire?
are you told anything about the position of the tire?
prisma fjord
So do I use that to calculate the force?
no
its later
first solve the force on the rod
from the center of mass rightwards
fringe lynx
3m to the right of c.m
But the force is unknown? because we need to find the weight?
prisma fjord
But the force is unknown? because we need to find the weight?
no way
f=ma, i remember now
force, mass, distance
fringe lynx
?? f=ma is acceleration no?
And there isn't really any acceleration nor force
2 Unknowns
vocal sleetBOT
@fringe lynx Has your question been resolved?
prisma fjord
?? f=ma is acceleration no?
y=mx+b is linear formula?
its production equation
it appears everywhere yes
fringe lynx
What????
fringe lynx
Enlighten me
fringe lynx
.close
vocal sleetBOT
vocal sleetBOT
Available help channel!
proud oar
in rates of change, i saw sal(khan academy guy) doing a video and i realized something, does answer = differentiated equation * rate of change?
vocal sleetBOT
in rates of change, i saw sal(khan academy guy) doing a video and i realized som...
proud oar
this differentiation?
vocal sleetBOT
@proud oar Has your question been resolved?
outer warren
in rates of change, i saw sal(khan academy guy) doing a video and i realized som...
answer to what? that's too vague
proud oar
oh what i mean is uhh
if they ask me what the rate in which area is increasing
dA/dt'
that is what i mean
graceful abyss
f(x)=(2x-3)^2+1, x>=1.5, f^-1(x)=?.
!occupied
vocal sleetBOT
f(x)=(2x-3)^2+1, x>=1.5, f^-1(x)=?.
Someone else is already using this help channel. If you need help with a question, please open your own help channel/thread (see #❓how-to-get-help for instructions).
outer warren
differentiate both sides wrt the the appropriate variable
and sub in your known values
proud oar
so dA/dt = differentiated formula multiplied by the rate of change?
hmm ok
wait if it is a right angle do i always do the pytahgoreas theorem
outer warren
Pythagoras is a simple way to obtain a relation between sides if you have a right triangle
it depends on what you're given
proud oar
my trouble is finding what to do
outer warren
use whatever works based on what you have
right triangle with enough info about sides → probably pythag
right triangle with some info on sides / angles → probably some trig
proud oar
oh ok
outer warren
differentiate both sides wrt the the appropriate variable
and sub in your known values
if you have
A = pq
differentiate both sides wrt t will give you
dA/dt = p * dq/dt + q * dp/dt
and sub in the info given to you in the question to get whatever they ask for
questions may vary slightly
proud oar
wait
outer warren
and you may do things in a slightly different order
proud oar
why are we writing dq/dt and dp/st
vocal sleetBOT
outer warren
.reopen
vocal sleetBOT
✅
proud oar
wait but if u differentiate both sides wont it becoem
dA/dt = dq/dt * dp/dt because the p and q got differentiated
outer warren
no
derivative of a product isnot the same as the product of the derivatives of the multiplicands
proud oar
sorry for asking dumb questions i dont get rate of change and optimizations
proud oar
x6 so 6x^5?
outer warren
you can't just differerntiate each expression being multiplied individually and multiply all those together
yes, that's what the derivative should be
the above is just an example of how logic that doesn't work
proud oar
hmm ok
vocal sleetBOT
Available help channel!
void ermine
Hi! I was wondering if I could get help on these two questions, I’m a bit stuck on them and a lot of resources seem to be giving me different answers so I’m quite confused 😭 thank you!!
vocal sleetBOT
Hi! I was wondering if I could get help on these two questions, I’m a bit stuck ...
cunning plaza
What's the definition of a parallel plane and perpendicular plane
void ermine
What's the definition of a parallel plane and perpendicular plane
two planes are parallel if they never intersect no matter how far they are extended I think and they are always the same distance apart and lie in the same three dimensional space?
Two planes are perpendicular if they intersect at a right angle. Their intersection forms a line and the angle between the planes at any point along that line is 90 degrees?
void ermine
Yes. Now try to colour in/visualize planes AEG and DHF
So DHF is the vertical right side of the box?
cunning plaza
Yup. a plane is defined by three points that lie on it. So it would be other square face of the box
void ermine
Yup. a plane is defined by three points that lie on it. So it would be other squ...
Ohh okay, I see. So the answer I chose, which was parallel is correct?
cunning plaza
Yep
void ermine
Okay, thanks!
cunning plaza
Second is also correct fyi, but make sure you understand why
void ermine
Second is also correct fyi, but make sure you understand why
Alright, I think I got it. Thanks again!
.close
vocal sleetBOT
vocal sleetBOT
Available help channel!
grand verge
Not to sure what to do after this or how inverse trig functions work...
edgy gulch
Not to sure what to do after this or how inverse trig functions work...
i think you shouldnt sub theta = arccos into the area formula
just solve it
grand verge
is it just not posssible if u do it the way i tried?
grand verge
yeah i got that
and then how do I solve for k
arccos and then anything inside = that number
like how do i get rid of the arccos
edgy gulch
you got theta = arcsin(sqrt(455)/24) yea?
grand verge
yup
edgy gulch
thats a number
about 62.72
then from (a), cos(theta) = (100 - k²) / 96
so cos(theta) = cos(62.72)
cos(62.72) is just a number
then solve away
heady vine
Is there a more optimal way to do this questions
none that I can think of