this is also equivalent
#help-17
shy otter
Also this
peak matrix
anyway the way to start is just trying to simplify it through algebra
especially exponent rules are extremely cruicial here
shy otter
so its 1 over root 2
wait i will see what you have done
can you tell me what exactly u did to get here
peak matrix
can you tell me what exactly u did to get here
try factoring something from the inside
shy otter
and then
peak matrix
then you'll see what you get
shy otter
@scenic ravine any tips for uga preparation
peak matrix
the inside looks kinda factorable
(cos(x) - 1)(cos(1/x) - 1) = cos(x)cos(1/x) - cos(x) - cos(1/x) + 1
which is just the inside negated
shy otter
try factoring something from the inside
why is it incorrect to see what the thing inside is approaching
and then the exponent
peak matrix
why is it incorrect to see what the thing inside is approaching
you'll get 0^0 I think
peak matrix
which is indeterminate form
shy otter
1/2^ 0?
peak matrix
the inside isnt approaching 1/2, is it?
shy otter
why not?
peak matrix
2^(-something large) + 2^-(something large)
scenic ravine
<@796939110815629322> any tips for uga preparation
I just solved questions, didn't clear UGA tbh, found UGB much easier than UGA
peak matrix
that's 0 + 0
peak matrix
oh and btw once you do the algebra to get it to this $\frac{1}{\sqrt{2}}\left(\left(2^{-2^{x-1}}\right)^{3}+1\right)^{2^{-x}}$ there is a "standard" approach to follow
shy otter
uga is much easier
twin meteorBOT
MæthIsAlwaysRight
scenic ravine
uga is much easier
Yeah, but I make really silly mistakes
shy otter
can you please say what the "standard" approach is
i want to know if what i did is the fastest
scenic ravine
so many answers were off by one or two digits
shy otter
so many answers were off by one or two digits
ah, also dont mind me asking, did u get into isi?
scenic ravine
ah, also dont mind me asking, did u get into isi?
nope, I didn;t clear UGA, right
so even if I had a 100 in UGB , wouldn't have made a difference
shy otter
ah
peak matrix
to solve limit of $\left(a+1\right)^{b}$, we can take the logarithm and get $\ln\left(\left(a+1\right)^{b}\right)=b\ln\left(a+1\right)=ba\frac{\ln\left(a+1\right)}{a}=ba$
twin meteorBOT
MæthIsAlwaysRight
shy otter
wait let me read
shy otter
**MæthIsAlwaysRight**
but is that the form above?
peak matrix
**MæthIsAlwaysRight**
yeah, with a bit of algebra you can convert it to that form
that was also the motivation behind factoring out 2^(-2^(n-1))
scenic ravine
I'd honestly re-write this as $\frac{1}{x} ( sin^2(x/2) + cos(1/x)(1-cos(x))$
twin meteorBOT
math_rocks
peak matrix
**math\_rocks**
it can be also rewritten as -(cos(x)-1)(cos(1/x) - 1)
scenic ravine
it can be also rewritten as -(cos(x)-1)(cos(1/x) - 1)
yup
peak matrix
and then doing 1/x, the first factor cancels to 0 and the second one is almost like 1/x which tends to 0 as well
peak matrix
and then doing 1/x, the first factor cancels to 0 and the second one is almost l...
oh wait x goes to 0
nvm then
but still, (cos(x) -1) / x goes to 0 and cos(1/x) - 1 is bounded
so bounded * 0 = 0
shy otter
i feel im weak at functions, what should i do @peak matrix
scenic ravine
but still, (cos(x) -1) / x goes to 0 and cos(1/x) - 1 is bounded
You could use the squeeze theorm I suppose
peak matrix
yeah, sure, either squeeze or argue that cos(1/x) - 1 is bounded
peak matrix
i feel im weak at functions, what should i do <@951190447185874994>
with which limit?
shy otter
You could use the squeeze theorm I suppose
do you mean squeeze theorem
shy otter
with which limit?
in general
scenic ravine
do you mean squeeze theorem
yes
shy otter
wait let me try everything suggested here
peak matrix
I used to (and often still do) plot every single function i encountered in desmos, and that really strengthened my intuition
- it helps you avoid stupid algebraic mistakes
but like there is no single best advice
shy otter
ok. what topics do u suggest revisting
in functions,
to get a clearer understadning of limits, derivatives and integrals
i want to feel them
peak matrix
Practice ig
shy otter
okay
vocal sleetBOT
@shy otter Has your question been resolved?
twilit magnet
vocal sleetBOT
twilit magnet
I have got to (x+y)(x^2+xy+y^2) = 2024
dense cedar
x⁴-y⁴ = -2024(x-y)
(x²+y²)(x+y) = -2024
dense cedar
I have got to (x+y)(x^2+xy+y^2) = 2024
How did you get this ?
vocal sleetBOT
@twilit magnet Has your question been resolved?
twilit magnet
How did you get this ?
Like this
x^4-y^4=2024(x-y)
(x-y)(x^3+x^2y+xy^2+y^3)= 2024(x-y)
x^3+x^2y+xy^2+y^3=2024
(x+y)(x^2+xy+y^2) = 2024
<@&286206848099549185>
cursive turret
show that f(t) = t^4-2024t is increasing in the range where f(t) > 0 and t > 0.
@twilit magnet
vocal sleetBOT
@twilit magnet Has your question been resolved?
twilit magnet
show that f(t) = t^4-2024t is increasing in the range where f(t) > 0 and t > 0.
So a graph>]b
vocal sleetBOT
dense cedar
Woooooooo
strange helm
dense cedar
Eurekaaaaa
dense cedar
This is amazing !
dense cedar
Geniuses like you are seldom born on planet earth, you're one in a billion.
glad trail
Bro thinks he's Newton
dense cedar
You should publish a paper on this all
strange helm
dense cedar
Bro thinks he's Newton
Bro is Newton's father 😭
cedar burrow
What would 1 divided by -0 be?
glad trail
That sucks. I wanted the last chocolate bar
cedar burrow
1/-0.001 is -1000
So it goes to -infinity
Dividing by -0 should then yield -infinity shouldn’t it
twin meteorBOT
glad trail
It's a wrong concept not a nice theory
Idk man. I think it has legs
onyx schooner
Wow
glad trail
He might be onto something
onyx schooner
That's what the positive x asymptote of this graph tells you
dense cedar
1/∞ = 0
cedar burrow
It’s all a matter of perspective. Infinity isn’t a number - you would want 1/0 to be a number wouldn’t you?
onyx schooner
For example:
6 ÷ -0.1 = -60
6 ÷ -0.01 = -600
6 ÷ -0.0001 = -60,000
And so on...
So 6/-∞=0?
cedar burrow
Not very useful to have a result for 1/0 if you can’t keep using it mathematically
strange helm
infinity is clearly a number rotate your damn page 🔥
dense cedar
infinity is clearly a number rotate your damn page 🔥
I support this
glad trail
onyx schooner
Actually no what you're thinking is the concept of asymptote
What you're thinking is
For example:
6 ÷ 0.1 = 60
6 ÷ 0.01 = 600
6 ÷ 0.0001 = 60,000
And so on...
And I know you would think when
The denominator is 0
It will blow up to infinity
But let me tell you you have ignored a big part
Negative numbers
Let me do the same thing @sweet spindle
With negative numbers
For example:
6 ÷ -0.1 = -60
6 ÷ -0.01 = -600
6 ÷ -0.0001 = -60,000
And so on...
onyx schooner
For example:
6 ÷ -0.1 = -60
6 ÷ -0.01 = -600
6 ÷ -0.0001 = -60,000
And so on...
This will hit to -∞
Means you're saying is
6/0=∞ and -∞ at the same time
Youre saying a constant has two values at the same time
Yes -∞ and ∞ ate completely different
Are
You were dividing by 0.0000001 whatever but you're lowering your digits by positive number
The simple answer is we can't
There is no defination
Dividing 0 is undefined
By 0*
In mathematics
You can prove that using limits
But you would learn that in high school
But?
AI would agree ,chat gpt only has specific knowledge it shouldn't be your primary source of knowledge
cedar burrow
Idk if there’s much you couldn’t get ChatGPT to agree on
onyx schooner
In negative numbers yeah
vocal sleetBOT
@sweet spindle Has your question been resolved?
half imp
https://tenor.com/view/yap-yapping-yapping-alert-d1-yapper-gif-29662849594480405...
Ah, spamming dozens of messages about someone else yapping
Pending postgraduate
I didn't know grad programs accepted 12 year olds
vocal sleetBOT
vocal sleetBOT
Available help channel!
oak magnet
All good to me
vocal sleetBOT
@fervent dagger Has your question been resolved?
fervent dagger
.close
vocal sleetBOT
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iron parrot
Is the converse of the inscribed angle theorem also true?
vocal sleetBOT
Is the converse of the inscribed angle theorem also true?
iron parrot
That is, if you have an angle on the circumference with a measure of half of an angle inside the circle, and both angles subtend the same arc, does the angle inside the circle coincide with the center of the circle?
loud walrus
Sounds correct
iron parrot
.close
vocal sleetBOT
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mossy goblet
Find n>=2 such that the following statement is true: A,B nxn matrices with complex entries and A^n = On and det(A+B)=det(B)=0 implies BAB=On.
vocal sleetBOT
Find n>=2 such that the following statement is true: A,B nxn matrices with compl...
@mossy goblet Has your question been resolved?
shy sleet
Can anyone help? Idk what to do next
vocal sleetBOT
Can anyone help? Idk what to do next
kind light
But I dont see how that does anything
you can find all the angles of it
although im not sure whrre the XY=4x comes in?
shy sleet
Somethng like this?
It means chord YX is 4x im pretty sure
Or rather the angle
Idk tbh
kind light
try to find all the angles in triangle XYZ
kind light
It means chord YX is 4x im pretty sure
yeah but my solution doesent use it so maybe i went wrong somewhere
kind light
choord XW=/=XY
shy sleet
Oh
shy sleet
try to find all the angles in triangle XYZ
Gimme a hint here
kind light
inscribed angle
kind light
the inscribed angle is half of the center angle
kind light
yea
kind light
find angle YXZ interms of x
shy sleet
find angle YXZ interms of x
Its 12.5 degrees
kind light
interms of x
kind light
do you understand what "interms of x" means?
shy sleet
No
kind light
basically it has x in it, like interms of x, for example like x+10, 2x, x^2, ect
shy sleet
Oh you mean in terms of the 4x and 2x?
kind light
hint: ||draw OV (assuming O is the center||
kind light
Oh you mean in terms of the 4x and 2x?
yeah basically
kind light
whats YOV?
kind light
no
kind light
the inscribed angle is half of the center angle
remember
vocal sleetBOT
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Available help channel!
sinful nacelle
could someone help me with how to get to the integral in the middle? the first rearrangement was fine, just sub t=x-1 but i wonder if ive used the wrong substitution to get me to the middle integral
vocal sleetBOT
could someone help me with how to get to the integral in the middle? the first r...
sinful nacelle
oh hang on ill try arctan
this looks better but still not there D:
<@&286206848099549185> help pls :,)
blazing jasper
this looks better but still not there D:
is the 2nd-last integral with tan^2 (theta) + 1?
blazing jasper
nvm, I thought it was difference of two squares (it isn't)
sinful nacelle
no but it simplifies to sec^2
blazing jasper
this looks better but still not there D:
maybe you can expand the top to get sec^2(k+1)
blazing jasper
then use the definition of sec as 1/cos to get cos^-2(k+1)
sinful nacelle
i think what i eventually need to end up with is a sin2theta in the denominator
so i can use double angle and then convert to cos?
blazing jasper
then you get 1/(cos^2(k+1))(tan + 1)^k+1)
you move all terms to denominator
then as the powers are the same, you can get 1/(cos^2 (tan + 1))^(k+1)
then you should get 1/(cos (sin + cos))^(k+1)
sinful nacelle
from where?
blazing jasper
from where?
tan = sin/cos
blazing jasper
then as the powers are the same, you can get `1/(cos^2 (tan + 1))^(k+1)`
so you can get 1/(cos^2 ( sin/cos + 1))^(k+1)
sinful nacelle
oh did you pull out a factor of 1/cos
sinful nacelle
ah ok
blazing jasper
then you should get `1/(cos (sin + cos))^(k+1)`
I don't know how to simplify this into the required form, but I'm sure there's a method you can use to get it
I hope that'll be enough for ya
sinful nacelle
hmm i'll try, thanks for some tips anyway tho
i did something a bit diferent and managed to get here instead but even though its a lot simpler, now the numerator isnt 1
vocal sleetBOT
@sinful nacelle Has your question been resolved?
vocal sleetBOT
@sinful nacelle Has your question been resolved?
spiral turtle
this looks better but still not there D:
Instead of transforming to sqrt(sec^2 - 1) instead leave as 1+tan and let sec^2 become 1/cos^2 multiply through to get cos( tan cos + cos) = cos( sin + cos) and then use complex amplitudes to combine the sin + cos.
This will get you sin + cos = sqrt(2) cos(theta - pi/4) which is the same as the example
spiral turtle
(because cos is even)
vocal sleetBOT
@sinful nacelle Has your question been resolved?
sinful nacelle
Instead of transforming to sqrt(sec^2 - 1) instead leave as 1+tan and let sec^2 ...
ahhh ok i will try that thank you
vocal sleetBOT
@sinful nacelle Has your question been resolved?
fluid sandal
nehelp
rough patrol
which one
fluid sandal
but b and c are apparently wrong
rough patrol
um ok
fluid sandal
I got part a correct
silk osprey
part a
fluid sandal
i dont understand what i did wrong in b and c
peak matrix
be careful about the area below x axis
rough patrol
oh um
peak matrix
x axis, sorry
rough patrol
it isn't 2
peak matrix
Yeah, but it should be negative
since it's under x axis
areas which are under the x axis are counted negatively in the integral
fluid sandal
but area can’t be negative shouldnt you do the absolute value?
peak matrix
but area can’t be negative shouldnt you do the absolute value?
Yeah, the area can't be negative. But the integral can
integral != area
there are slight differences
fluid sandal
i see i see
peak matrix
mainly the one with sign
peak matrix
you'd have to integrate |g(x)| to get the right area
this isnt exactly true
the area is integral of |g(x)|, not g(x)
integral of g(x) is signed area
you basically did 2 mistakes
fluid sandal
how does that change the result?
peak matrix
how does that change the result?
it doesnt
because your second mistake was evaluating the integral incorrectly
fluid sandal
just a matter of writing it?
peak matrix
just a matter of writing it?
not really, its quite cruicial difference
this integral evaluates to -2
if you wrote |g(x)| instead, it would evaluate to 2 as you wrote
fluid sandal
ahhh
peak matrix
so you basically did 2 mistakes and they luckily cancelled out
peak matrix
this integral doesnt equal that though
fluid sandal
i should put those lines around it too?
peak matrix
around g(x)
fluid sandal
okay
peak matrix
$\int_{-1}^{2}\left|g\left(x\right)\right|dx=\int_{-1}^{0}\left|g\left(x\right)\right|dx+\int_{0}^{2}\left|g\left(x\right)\right|dx=2+4=6$
twin meteorBOT
MæthIsAlwaysRight
peak matrix
the trick is that |g(x)| reflects the curve that's below x-axis above it
and then we can integrate it without worrying about area getting counted negatively
peak matrix
for calculating area, yes
fluid sandal
if i’m finding area
got it
for part b
the book says 8
i got 12
from b onwards we’re not finding area i’m pretty sure
they’re just normal calculations
fluid sandal
ahh
so in that case the 2 would be -2?
c is correct now
thanks for that tip
and your help of course
peak matrix
yw :)
vocal sleetBOT
peak matrix
bye
vocal sleetBOT
Available help channel!
vast shale
How is the axiom of pairing not contradictory? Doesn't it let create the set C = {C} with A=B=C? And if A=B=D=C, what about ∃C(C∈C <--> C=C)? Or maybe the one stated in wikipedia is incomplete because in the other website it says x,z,w must be different, and y,z,w must also be different. Or maybe I forgot how to use first order logic
vocal sleetBOT
How is the axiom of pairing not contradictory? Doesn't it let create the set C =...
heavy yoke
the axiom of regularity implies no set can contain itself
dark kiln
it's not supposed to prevent you from C = {C}
peak matrix
it's not supposed to prevent you from C = {C}
It would still be contradictory if it implied that
or inconsitent with axiom of regularity
the issue is with the proof
What rule did you use here?
I suppose that you used this one, right?
Universal instatiation
it only replaces the free occurences though
vast shale
The thing that doesn't make sense to me is that axioms shouldn't interfere with each other. Technically pairing is a theorem of axiom of replacement and some other weaker axioms. But that would mean that there exists a set that belongs to itself, but also that it doesn't with foundation
vast shale
it only replaces the free occurences though
x and y are free variables in the scope of the quantifier, for example ∀x(x∧y) ⊢ t∧y. But ∀x(x∧∀x(x∧y)) ⊢ t∧∀x(x∧y)), so I don't see the problem in there
dark kiln
meth is right, it doesn't let you do it
peak matrix
oh wait i might not be right
it definitely shouldnt
but i cant figure out the reason why rn
hmm
dark kiln
if you says A is C, then the C doesn't mean the same C
peak matrix
it doesnt let it pass
The issue is that the variable that got substituted gets tied back to the existential quantifier
which is an issue
i feel like wikipedia is missing that
let me check enderton
here
t is substitutable for x in Ay a only if y doesnt occur in t
in our case, that means w isnt substitutable for x since it then gets tied by the Ew quantifier
peak matrix
x and y are free variables in the scope of the quantifier, for example ∀x(x∧y) ⊢...
to put it shortly, when eliminating forall quantifier always make the substititution only for fixed constants
dark kiln
as soon as you say let A = C, the axiom changes, the letters have to avoid C
that's the new axiom you have to use
i'm not sure that's the issue you;re talking about, must be that
peak matrix
wait what?
The axiom is fine
what isnt fine is the way in which it got manipulated in the proof
dark kiln
well yeah, that's also part of implied question, "where's the mistake in the proof"
but it's like secondary
peak matrix
The issue is that in this step, OP eliminated the quantifiers by making a substitution
x -> w
y -> w
However, this newly substituted w's got captured by the ∃w quantifier which made the proof faulty and the conclusion false
substituted variables cant get captured by any quantifier
vast shale
The issue is that in this step, OP eliminated the quantifiers by making a substi...
ah yes
w is bound
well *that was the problem
@dark kiln I feel you also have the doubt why the substitution if the variable is bound is not valid. The reason I just came with is that, intuitively, bound variables, might not refer to the same object in all cases
like this
the first one is true but the second one isn't
anyways, ty yall
.close
vocal sleetBOT
dark kiln
<@141791176540946432> I feel you also have the doubt why the substitution if the...
i wouldn't write a proof in the first place, the issue happened because you tried to write it down
"use first order logic"
peak matrix
i wouldn't write a proof in the first place, the issue happened because you trie...
It didn't?
dark kiln
oh maybe
vocal sleetBOT
Available help channel!
dark surge
I need help with question 7c, this is a new topic for me so if someone can show me how to do it slowly that would be appreciated
vocal sleetBOT
I need help with question 7c, this is a new topic for me so if someone can show ...
scenic ravine
in question (7) , do you know how to do the first two parts
dark surge
Yes I have to give one side a value so I pick 0
I got those bits it’s just that 1/2 that I don’t know
scenic ravine
Well, just proceed like you did for the other problems
dark surge
Yes
scenic ravine
can you find x for me in that case
dark surge
With the 1/2?
scenic ravine
yes
scenic ravine
okay, so $4= \frac{1}{2} x$
twin meteorBOT
math rocks
scenic ravine
yes?
dark surge
Yes
scenic ravine
can you multiply across by 2 now
twin meteorBOT
math rocks
dark surge
$4y - 6x = 24$
vocal sleetBOT
$4y - 6x = 24$
twin meteorBOT
𝐌𝐞𝐲𝐬𝐚𝐦
dark surge
I need to find the coordinates for this to sketch a graph I just don’t know how to
astral pilot
assume any value for x, then find the corresponding value of y to get a point
for, e.g. let x = 0, then find y for it, suppose you get y = 6, then one point is (0,6)
dark surge
Bro what
Could you say it a little simpler this is a new topic for me so I’m confused
astral pilot
Ok, in this question we have to find two points, so that we can sketch the graph of the line
now, this line passes through all points (x,y) such that 4y - 6x = 24
dark surge
Ok…
astral pilot
so, if we have to find two values of (x,y) such that it satisfies the line
now, we can let x = any value
say, we let x = 4, alright?
dark surge
Ok
astral pilot
now the equation of line is 4y - 6x = 24
and we assumed that x =4
so, 4y - 6(4) = 24
or, 4y -24 = 24
dark surge
Ah I see
astral pilot
yeah (4,9)
dark surge
now the equation of line is 4y - 6x = 24
In this bit why did the 6x turn into 3x?
astral pilot
seems like a mistake from my side
astral pilot
Seems like I made a mistake, lets restart from beginning
dark surge
Alright
astral pilot
4y - 6x = 24
let x = 4
now, we substitute this value in the line
4y - 6(4) = 24
4y - 24 = 24
now adding 24 to both sides
4y -24 + 24 = 24+ 24
4y = 48
y = 12
so, one of the point is (4, 12)
dark surge
Ok.
astral pilot
Now, when solving this type of question, the simplest assumption to find a point is to let x=0 and y =0
so, when x = 0
4y - 6(0) = 24
4y = 24
y = 6
so one point is (0,6)
and let y =0
dense cedar
Now, when solving this type of question, the simplest assumption to find a point...
😭 bruh !
dense cedar
You could have told him about intercept form
astral pilot
x = -4
dark surge
Yeah I dident understand a single thing until the x=0 y=0
astral pilot
You could have told him about intercept form
I don't think they've learned slope yet
anyways drawing graphs is overrated, use a graphing calculator instead 
dark surge
I have been trying to get one
It’s so expensive to get the one I need for my school
But I understand the question now so I appreciate it
astral pilot
I don't know why we need to purchase graphing calculators, using smartphones with family link or something should be allowed in school imo
vocal sleetBOT
vocal sleetBOT
Available help channel!
visual marsh
whats the answer?
vocal sleetBOT
Available help channel!
feral gorge
can someone please dumbify the solution for me 
i dont understand the first line on the second page
vocal sleetBOT
can someone please dumbify the solution for me <:sobs:1318550720051413042> i don...
bronze osprey
how are these equal?
by the chain rule, the dx on the numerator and denominator 'cancels'
also you're allowed to swap the order of the differentials
$\frac{d}{dx} \frac{dy}{d \theta} = \frac{d}{d \theta} \frac{dy}{dx}$
twin meteorBOT
south
bronze osprey
you're fine if you treat them as fractions basically
feral gorge
i see but what i dont get is in this
d/d0 (dy/dx) = d/d0 (-cot 0) for simplicity im using 0 as thetha
and diff of -cot 0 is cosec^2 0
so d/d0 (dy/dx) = cosec^2 x but then where did the 2nd d/dx spawn in from?
in here
to make it easier
bronze osprey
ah so it's just using the fact that dy/dx = -cot theta
feral gorge
ignore the bas handwriting i cant write on laptop
bronze osprey
yeah exactly
bronze osprey
to make it easier
that's correct
feral gorge
so where did the third d/dx come from?
bronze osprey
it has to do with treating the derivatives like fractions as I said
$\frac{d}{dx} \frac{dy}{\cancel{dx}} \frac{\cancel {dx}}{d \theta} = \frac{d}{dx} \frac{dy}{d \theta} = \frac{d}{d \theta} \frac{dy}{dx}$
twin meteorBOT
south
feral gorge
ohhhhhh
so essentially we're splitting dy/d0 by multiplying and dividing by dx
ohhhhh
how tf do i even think of this in the exam
bronze osprey
yeah like formally speaking that's not the correct reasoning
but yeah chain rule magically makes this work
also you might have used 1/(dx/dy) = dy/dx before
same principle
feral gorge
its so confusing sometimes
like how do i even manage to think this way during the exam
also thank you
is there any particular situation where i can treat them as fractions?
or can i just do stuff like this whenever cuz chain rule
i spent like 20 minutes tryna figure this out
thank you!
.close
vocal sleetBOT
bronze osprey
is there any particular situation where i can treat them as fractions?
literally all the time
yeah now you know haha, nw!
feral gorge
literally all the time
oh i never thought about it that way lol thanks
bronze osprey
if you want pathological examples where this doesn't hold, come learn real analysis and you'll be horrified at where your intuition can go wrong
basically all the functions in your questions will be neat enough for you not to worry
feral gorge
~~if you want pathological examples where this doesn't hold, come learn real ana...
im struggling enough as it is
bronze osprey
continuous and differentiable
also boundedness is a strong property, depends on the exact statement
anyways
feral gorge
im scared
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unkempt gate
how to prove that if a is an m digit number of all ones, and b is an n digit number of all ones, where a<= b, then a|b only when m|n?
vocal sleetBOT
how to prove that if a is an m digit number of all ones, and b is an n digit num...
unkempt gate
im stuck at $10^m-1 \mid 10^n-1$
feral stream
the vertical bar means is a divisor of
unkempt gate
**VincentBH**
should be m digit | n digit btw?
how does this help with my question?
feral stream
no the idea is suppose $\underbrace{11\dots1}{n\text{ digits}}|\underbrace{11\dots1}{m\text{ digits}}$
twin meteorBOT
VincentBH
feral stream
then you can keep chopping of n 1's from the left until you are left with either zero or $\underbrace{11\dots1}_{m'\text{digits}}$ with $0<m'<n$.
twin meteorBOT
VincentBH
feral stream
in the second case you get a contradiction since $\underbrace{11\dots1}{n\text{ digits}}$ does not divide $\underbrace{11\dots1}{m'\text{ digits}}$
twin meteorBOT
VincentBH
feral stream
since it is bigger
ye maybe you can phrase the solution a bit cleaner but that is how you solve it
unkempt gate
in the second case you get a contradiction since $\underbrace{11\dots1}_{n\text{...
second case??
feral stream
the case where you don't get zero
unkempt gate
ignore $\underbrace{11\ldots1}{m \text{ digits}}\mid\underbrace{11\ldots1}{n \text{ digits}} \implies m\mid n$
twin meteorBOT
babario
feral stream
the other direction is the easy direction
unkempt gate
**babario**
so like we r just restating the obvious and really theres not much to prove?
just think of it as recursively chopping sequences of m digits from the n digit number?
feral stream
yes basically
feral stream
you can prove the following lemma: $\underbrace{11\dots1}{n\text{ digits}}|\underbrace{11\dots1}{m\text{ digits}}$ if and only if $\underbrace{11\dots1}{n\text{ digits}}|\underbrace{11\dots1}{r\text{ digits}}$ where $r$ is the remainder of the division $m/n$
twin meteorBOT
VincentBH
unkempt gate
oh i see
feral stream
from this, your result follows immediately
unkempt gate
just think of it as recursively chopping sequences of m digits from the n digit ...
btw im kinda overthinking n having trouble trying to write it as a formal proof
@feral stream how?
feral stream
<@501097611256922113> how?
since $0\leq r<n$, the second part of the equivalence is true if and only if r=0 and that is true if and only if $n|m$
twin meteorBOT
VincentBH
feral stream
which is excessive no?
ye the way I wrote it is not very good, you might try to prove the lemma inductively on $q$ with $m=q\cdot n + r$
twin meteorBOT
VincentBH
feral stream
which is excessive no?
you can also prove it directly but you are basically proving the same
unkempt gate
you can also prove it directly but you are basically proving the same
nahh i mean wording it is literally so easy, but i cant translate it to the proof is what i mean
unkempt gate
just think of it as recursively chopping sequences of m digits from the n digit ...
as in this
feral stream
**VincentBH**
if you do this then the induction step can be proven by chopping once
unkempt gate
induction step?
twin meteorBOT
VincentBH
feral stream
unkempt gate
**VincentBH**
this feels so complicated than necessary, but its the best way to ensure rigor?
@feral stream
nahh literally why am i showing for the general case when i already have the idea on how to directly prove it?
im just asking how to write in proper math terms the idea
<@&286206848099549185>
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uneven garden
how to solve this ?
vocal sleetBOT
how to solve this ?
uneven garden
???
astral pilot
The order of a differential equation is defined as the highest derivative of the unknown function that appears in the equation.
so you tell me what do you think the order is?
The degree of the differential equation is represented by the power of the highest order derivative in the given differential equation.
I think this is pretty straightforward now
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uneven garden
The order of a differential equation is defined as the highest derivative of the...
The order is 3
uneven garden
The degree of the differential equation is represented by the power of the highe...
maybe degree 1 ?
astral pilot
yes
vocal sleetBOT
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sudden forge
Help me question 1 pls… 4. I have got an answer for angle c
but I don't get the angle of d
how to take out angel d
astral pilot
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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
- I have got an answer for angle c
but I don't get the angle of d
how to take out angel d
vocal sleetBOT
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vernal oracle
in (f) i use a^2 - b^2 result is 3/8 which is correct
in (g) i put 27n^3 in front and end up with 3n -3n =0 which is corect
Why couldn't i solve (f) same way i solved (g)
bring 16n^2 in front of the square root with result of 0
vocal sleetBOT
in (f) i use a^2 - b^2 result is 3/8 which is correct
in (g) i put 27n^3 in fron...
flat whale
What even is the question asking
vernal oracle
what happens when n goes to infinity
in (g) i just pulled 27n out of root
If i do the same in (f) i get 0 but that's not correct solution
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fluid sandal
i need help pls
vocal sleetBOT
i need help pls
atomic jasper
whats the question?
fluid sandal
how do i solve question 14 e?
i dont want the answer i just want to understand how to work it out
atomic jasper
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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
what do you have so far
fluid sandal
this is what i did, but i’m certain it’s getting me nowhere
i think i have to look at the graph
the 12 i got from working out the area between 0 and 6 on the graph
atomic jasper
whats your answer for b?
fluid sandal
-8
oh wait
a is a little before 10
since the definite integral is negative from 6 to a
i feel like there’s some simple math im missing
hello?
atomic jasper
hello?
ping helpers, cuz im confused aswell. idk why we cant just solve it analytically
fluid sandal
do i find the equation for the curve between 6 and 10?
and then just work it out from there?
atomic jasper
do i find the equation for the curve between 6 and 10?
thats what i would do yeah, get the parabolic equation and then solve it such that the area is -6
fluid sandal
let me try that
very strange
solving it on the calculator gave me x = 6 and x = 12
atomic jasper
what equation did you get for the parabola?
fluid sandal
0.75(x-8)^2 - 3
atomic jasper
what is the integral of it from 6 to a
fluid sandal
integrating that function gave me (x-8)^3 / 4 -3x
and then i plugged in a
and then 6
then i used the calculator to solve for a
oh wait
i got it
i got my answer
i forgot to substitute in -6 for the definite integral of f(x)
a is 8.695
atomic jasper
sounds about right
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cobalt ocean
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cobalt ocean
it's so intuitive, but is there any nice way to prove it?
wraith venture
I'd recommend cross-asking to #advanced-probability
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atomic jasper
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calm ice
shouldnt this be infinite solutions because of the non leading row?
calm ice
there are three variables and 4 rows
where the last row is just 0 =0
calm ice
so any real numbers work
ye
scenic ravine
but there are constraints imposed by the other equations
try row reducing it further
calm ice
but there are constraints imposed by the other equations
ahhh so we cant blindly say infinite solutions when we see 0 = 0?
calm ice
I mean try eliminating the 5000 and 20,000
you mean reduce it to row reduced echelon?
scenic ravine
yup
shell grotto
<@&286206848099549185>
bronze osprey
what have you tried so far?
shell grotto
im counting but it does not appear in the alternatives
bronze osprey
im counting but it does not appear in the alternatives
you're probably counting too few then
there's also the 2 long rectangles, and the 1 big rectangle that covers the entire thing
then you have to figure out how many more you can make with the diagonal line
vocal sleetBOT
bronze osprey
thx . close
if you label the areas 1, 2, 3 and then 4, 5
hopefully you see it's (1, 2), (2, 3), and (1, 4)
so 3 more ways
yeah convince yourself that we've counted everything systematically
for instance, (1, 2, 3) would be the long rectangle so we already counted that
shell grotto
if you label the areas 1, 2, 3 and then 4, 5
hopefully you see it's (1, 2), (2,...
That's what I'm doing, but I only counted 8. I must have missed one.
bronze osprey
That's what I'm doing, but I only counted 8. I must have missed one.
oh did you forget the original 5
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frosty moth
I have no idea how to do this
vocal sleetBOT
I have no idea how to do this
frosty moth
I need to find antiderivative of e^-x/x first but idk how
bronze osprey
I need to find antiderivative of e^-x/x first but idk how
you don't need to!
you can just compare the integral with the integral of e^(-x), which is larger than the original
frosty moth
you can just compare the integral with the integral of e^(-x), which is larger t...
Oh mah god
I didnt think of it
Thx u
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jagged cargo
hello i'd like part a checked please
vocal sleetBOT
hello i'd like part a checked please
jagged cargo
lemma 1 says list of polynomials in F with distinct degrees is LI
2.38 says LI list in U with length dimU is also a basis
spiral turtle
(1-1)+(1+1) = 0?
spiral turtle
You did it twice for the record
jagged cargo
yes i copypasted, thanks for pointing that out
signal pendant
a) looks fine other than the typos
jagged cargo
nice, thank you
signal pendant
also i think instead of strict subspace you mean proper subspace
to denote a subspace that is not the whole space
jagged cargo
i think the two terms are synonymous?
signal pendant
oh ive never heard the former
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tiny trail
how to prove dim(kerf^2)<=2dim(kerf)
vocal sleetBOT
how to prove dim(kerf^2)<=2dim(kerf)
tiny trail
i remeber a thing that if dim(a)=n , dim(a*a) does not make the dim bigger
but the question is 2dim (kerf)
can i still use this definition
no its not definition thats what i remember
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tiny trail
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vagrant oak
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vagrant oak
stuck at the right hand limit part
scenic ravine
Hint write cot in terms of tan
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vagrant oak
Hint write cot in terms of tan
my answer still comes e^0 which is 1 but the answer key has e^(2/3)
upon using the limit it also does not seem to be an inderterminant limit
<@&286206848099549185>
dense cedar
my answer still comes e^0 which is 1 but the answer key has e^(2/3)
The answer key wants you to find the value of the expression in lambda and mu.
And not of e^(cot6x/cot4x)
vagrant oak
yes but to find it we need value of the e^(cot6x/cot4x) its a continuous limit so it will be equal to mu and e^lambda
vagrant oak
how can you explain this step im stuck at this step
i cant use l hospitals rule if its not indeterminant
dense cedar
Yes
vagrant oak
it doesnt apply right??
dense cedar
But
vagrant oak
then
dense cedar
cot6x/cot4x = ∞/∞ form
Isn't it so ?
Indeterminate
And we can raise the limit to the power of e
vagrant oak
yeah give me some time ill be right back ill try this first
arent cot(3pi) and cot(4pi )= 0
then shouldnt it be 0/0 form
or rather just written 0 and e^0
thats where i cant fgure out why should we use infinite/infinite
@dense cedar sir/maam pls help im sorry im hurrying this
dense cedar
<@1255754610912530483> sir/maam pls help im sorry im hurrying this
Cot(n•pi) = ∞
Tan n(pi) = 0
And so
1/0 = ∞
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shrewd iris
hi
dense cedar
Hi
dense cedar
Simplify
shrewd iris
simplify?
dense cedar
x (1-1.15×23/27) = 5.5 + 1.15(-110/27)
dense cedar
,calc 23/27
twin meteorBOT
Result:
0.85185185185185
Result:
0.81481481481481
-27.1 is wrong
vital loom
vocal sleetBOT
outer warren
they're all given in the diagram
vital loom
they're all given in the diagram
Ik but idk which is which
like is 9 a?
is 5 the height?
outer warren
no
look at formula for area of a trapezium
that should come with a diagram telling you what
a,b,h
represent in that formula
vocal sleetBOT
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solid star
Proving that if $f$ is locally growing in $a$, then $f'(a)\geq 0$
Here is what I have so far:
We know that $\exists \delta$ so that $\forall x,y : a-\delta < x < a < y < a+\delta$ we have $f(x) \leq f(a) \leq f(y)$. (As per the definition of local growth.)
From here I wrote $\frac{f(y)-f(x)}{y-x} \geq 0$ and used the mean value theorem to see that $\exists \xi \in ]x;y[ : \frac{f(y)-f(x)}{y-x} = f'(\xi) \geq 0$.
Now, I have the intuition that $\forall \xi \in ]a-\delta;a+\delta[$ this holds, since the $]x;y[$ interval "runs through" the $]a-\delta;a+\delta[$ interval and $\xi \in ]x;y[$, but I don't know whether it is correct, and how to formalize it. Can someone please help me out?
twin meteorBOT
max
vocal sleetBOT
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feral gorge
while solving this question i dindt get it so i looked at the answers and see this (pic 2)
vocal sleetBOT
while solving this question i dindt get it so i looked at the answers and see th...
feral gorge
how do theyknow if PQ and AP are perpendicular
if its never mentioned
and then they just let a bunch of random points equal to the common point of those two lines??
where are they getting this from??
mental egret
To get image of P wrt AB, you're looking for a Q st AB perpendicularly bisects PQ
feral gorge
how can we say that the equation given is perpendicular to the line tho
i dont understand that part
mental egret
while solving this question i dindt get it so i looked at the answers and see th...
Which part of this figure is confusing to you
We are given equation for a line AB. We are given a point P. And the foot of perpendicular frm P to AB is L. Now, we're asked to find the image of P wrt AB.
Which is Q, i.e., a point with LQ = LP and PL perpendicular to AB makes QL being perpendicular to AB a necessity
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feral gorge
We are given equation for a line AB. We are given a point P. And the foot of per...
they never said the equation given was for line AB tho
thats my doubt
AB is assumed to be perpendicular to PQ
when it is never mentioned that is it
mental egret
... They named the line as AB. And PL perpendicular to AB is given.
feral gorge
... They named the line as AB. And PL perpendicular to AB is given.
this is the question
vocal sleetBOT
this is the question
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sleek moss
You are provided with a table showing pain scores before and after a treatment. The scores are based on a pain scale where higher numbers indicate more pain. The difference is calculated as: pain before - pain after
The goal is to determine whether there is a significant reduction in pain due to the physiotherapy treatment.
You may assume that the differences are normally distributed. The significance level is 0.1% (
𝛼
0.001
α=0.001).
vocal sleetBOT
You are provided with a table showing pain scores before and after a treatment. ...
sleek moss
So is it possible that's it's a t-test
With p-value=0,01?
Can someone verifiy if I'm right please
<@&286206848099549185>
Any helpers please?
<@&286206848099549185>
<@&286206848099549185>
brisk vigil
● d_1 = 5 - 5=0
● d_2 = 5 - 4 = 1
● d_3=3 - 2 = 1
● d_4 = 5 - 2=3
● d_5 = 5 - 4 = 1
● d_6 = 5 - 3 = 2
● d_7 = 4 - 4 = 0
● d_8 = 3 - 3 = 0
○ \sum_{i = 1}^{n}d_i=0 + 1+1 + 3+1 + 2+0 + 0 = 8
○ n = 8
○ \bar{d}=\frac{\sum_{i = 1}^{n}d_i}{n}=\frac{8}{8}=1
5. Calculation of standard deviation of differences s_d:
○ First, calculate the variance of the differences:
● (d_1-\bar{d})^2=(0 - 1)^2 = 1
● (d_2-\bar{d})^2=(1 - 1)^2 = 0
● (d_3-\bar{d})^2=(1 - 1)^2 = 0
● (d_4-\bar{d})^2=(3 - 1)^2 = 4
● (d_5-\bar{d})^2=(1 - 1)^2 = 0
● (d_6-\bar{d})^2=(2 - 1)^2 = 1
● (d_7-\bar{d})^2=(0 - 1)^2 = 1
● (d_8-\bar{d})^2=(0 - 1)^2 = 1
○ \sum_{i = 1}^{n}(d_i-\bar{d})^2=1+0+0+4+0+1+1+1 = 8
○ s_d^2=\frac{\sum_{i = 1}^{n}(d_i-\bar{d})^2}{n - 1}=\frac{8}{7}\approx1.143
○ s_d=\sqrt{1.143}\approx1.07
○ t=\frac{\bar{d}}{s_d/\sqrt{n}}=\frac{1}{1.07/\sqrt{8}}\approx2.67
why it cannot transform to the normal formula
mild grove
why it cannot transform to the normal formula
you need to use the bot i guess
to get the image with it inside