It does
#help-49
delicate sage
timber arrow
so if i wasnt counting for it it wld be 5 x 4/2
dreamy lichen
so if i wasnt counting for it it wld be 5 x 4/2
if you just wanted 2 letters (without caring about order), then yes
timber arrow
if they said as a precondition, that A B is the same as B A then i would do 5 x 4 divided by 2
dreamy lichen
it would be 5 * 4 / 2
timber arrow
if you just wanted 2 letters (without caring about order), then yes
ah i see
im a bit confused how does 5 x 4 count for the double
AB and BA
dreamy lichen
im a bit confused how does 5 x 4 count for the double
There are 5 ways to choose a first letter. So we can choose A. Then there are 4 ways to choose the other letters, so we can choose B
So it counts AB
There are 5 ways to choose a first letter. So we can choose B. Then there are 4 ways to choose the other letters, so we can choose A
so it counts BA as well
timber arrow
_ _ _ _ _ so i can put A B C D E in the first _ and then in the next one i only have 4 options
timber arrow
There are 5 ways to choose a first letter. So we can choose A. Then there are 4 ...
ohh nice that was a good explanation
i can put any of the 5 letters in the first, any 4 in the second
nice thanks
dreamy lichen
i can put any of the 5 letters in the first, any 4 in the second
indeed
then any 3 in third, any 2 in fourth and the last one goes to the fifth
so it would be 5 * 4 * 3 * 2 * 1
dreamy lichen
u divide by 2
if you are choosing only 2 things, then yes
if you were choosing 3 things, you would have to divide by 6
dreamy lichen
so like npr so n!/(n-r)!
this one is with order
and ncr is without order
that's when you are additionally dividing by r!
timber arrow
i alw get confused whats the diffewrence between without and with order
if it says order doesnt matter
AB = BA
?
timber arrow
so order doesnt matter we have fewer total combinations or more
than order does matter
dreamy lichen
than order does matter
did you mean "does matter"?
dreamy lichen
you said doesnt matter twice
dreamy lichen
so order doesnt matter we have fewer total combinations or more
there would be fewer
timber arrow
Order doesn't matter = NCR so n!/(n-r)!r! - we have fewer combinations as AB = BA
Order does matter = NPR so n!(n-r)! we have more combinations because AB and BA are distinct combinations
dreamy lichen
you could say it like that
timber arrow
there would be fewer
yeah that makes sense cuz AB = BA we have to divide by the number of objects factorialed
cuz each time u add an object u add more repeats
yeah that makes sm sense
thanks
.close
midnight plankBOT
midnight plankBOT
Available help channel!
rigid turret
is A correct
midnight plankBOT
is A correct
kind herald
is A correct
well if you want a simple answer, yes.
Do you want me to explain to you, why?
midnight plankBOT
@rigid turret Has your question been resolved?
rigid turret
well if you want a simple answer, yes.
Do you want me to explain to you, why?
no thanks
thanks for the answer
timber arrow
midnight plankBOT
timber arrow
im not sure about how to begin
noble elbow
im not sure about how to begin
The key point here is getting the closest number of tickets per two day period (since tickets are saved for two days, and more tickets are worth less per ticket)
So for example, Monday and Tuesday are 12 tickets. The cheapest combination is a book of 8 tickets and a book of 4
Which costs $8.80
Wednesday and Thursday are 9, which can be 8 + 1 tickets
For a cost of 6.60
timber arrow
So for example, Monday and Tuesday are 12 tickets. The cheapest combination is a...
is it?
cuz
cant we buy 8 tickets on Monday, have 3 spares which we can use for Tuesday. Then we have 4 left to buy on tuesday. We can buy another 8 so we have 1 left on wednesday to buy. Then we can buy 4, 4 and 1 final on saturday
so B
i think its B
cursive pawn
cant we buy 8 tickets on Monday, have 3 spares which we can use for Tuesday. The...
This is unrelated but why is your text like that
Didn’t mean to reply
flat veldt
Discord bug
noble elbow
cant we buy 8 tickets on Monday, have 3 spares which we can use for Tuesday. The...
thats what im saying
since you use your tickets every two days, then it's most efficient to calculate what the cheapest pairing of days is
as in
the ones that you can most easily split up into 8 or 6 or four tickets
midnight plankBOT
midnight plankBOT
Available help channel!
timber arrow
high hill
yoo
high hill
ight we back
so the answer key is saying the domain is -infinity to 1
But how I don’t understand? does the graph not end at infinite too
blissful totem
so the answer key is saying the domain is -infinity to 1
the arrow on the end of the piece wise suggests that it extends infinitely
since the gradient is defined for the 2 piece wise that meet at x=1, it will continue extending infinitely towards the left side
hence domain of (-inf,1]
high hill
wait so since it’s two lines
you have to mark where it bounce
?
i’m so cooked
but i think im getting it no cap
midnight plankBOT
@high hill Has your question been resolved?
high hill
sure
tame quarry
How do I evaluate this integral?
midnight plankBOT
How do I evaluate this integral?
unique stream
How do I evaluate this integral?
remember a^x = e^(x * ln(a))
tame quarry
Ah i never thought about that
unique stream
wdym
deriving e^(x * ln(a)) gives you ln(a) * e^(x * ln(a))
and integrating e^(x * ln(a)) gives you 1/ln(a) * e^(x * ln(a))
tame quarry
scratch what i said
sorry lol did it wrong
I’m a little confused on the integrating e^(x*ln(a))
i thought the e would remain and i would divide it by its exponent +1
unique stream
I’m a little confused on the integrating e^(x*ln(a))
You can find it out by looking at the behavior of the derivative
deriving e^(a * x) always gives you a * e^(a * x)
which implies that integrating it instead divides by the factor a
@tame quarry mostly clear? 🙂
midnight plankBOT
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midnight plankBOT
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Available help channel!
vast pine
can someone explain/help me understand trigonometric identities i dont know what there for
midnight plankBOT
can someone explain/help me understand trigonometric identities i dont know what...
vast pine
or how to use them
midnight plankBOT
@vast pine Has your question been resolved?
vast pine
.close
twilit field
Prove that the limit of x^3 at a is a^3
midnight plankBOT
Prove that the limit of x^3 at a is a^3
grand pondBOT
Veni, vidi, perii
twilit field
so I can re-write (x^2+ax+a^2) as $(x+a)^2 -ax$
grand pondBOT
Veni, vidi, perii
Compile Error! Click the 
reaction for more information.
(You may edit your message to recompile.)
tribal temple
You could, if you wanted to 
twilit field
Would that help?
tribal temple
It might (the idea I have is different, but I shall not spoil it for now 
)
twilit field
I think I'll try variations of f(x)=x^2 first
I don't think I fully understood that in the first place
let's say x^2+x insetad
I want to prove the limit is a^2+a
$|a^2+a| < \varepsilon$
grand pondBOT
Veni, vidi, perii
tribal temple
(I shall say that doing so would be very close to the proof of showing both x^2 and x have their limits, and "the limit of a sum is the sum of the limits"...)
grand pondBOT
Veni, vidi, perii
tribal temple
**Veni, vidi, perii**
Also more that |(x^2 + x) - (a^2 + a)| < eps
sinful topaz
I’m assuming you want to prove it via this epsilon delta definition
Otherwise you could just prove that all polynomials are continuous via proving that if f and g are continuous then so is fg (probably by sequential continuity)
And then that being continuous at a is basically just equivalent to having the limit exist at a
Ie lim x^3 = f(a) = a^3
twilit field
.reopen
midnight plankBOT
✅
twilit field
**Veni, vidi, perii**
So continuing from here
Trying to find the limit of $x^2+x$ at any a \in \R
grand pondBOT
Veni, vidi, perii
Compile Error! Click the reaction for more information.
(You may edit your message to recompile.)
twilit field
so I have to find a delta such that for any $\varepsilon$, there exists a $\delta$ st $|f(x)-L| < \varepsilon$,, there exits a $\delta $ st $|x-a| < \delta$
?
grand pondBOT
Veni, vidi, perii
prime hornet
indeed
twilit field
so this gives me $|x-a||x+a+1| < \varepsilon$
grand pondBOT
Veni, vidi, perii
grand pondBOT
Veni, vidi, perii
prime hornet
do some scratch work first
twilit field
ooh
prime hornet
usually, the way these go is to "know" what L is, assume the inequality |f(x) - L| < eps, and then find a delta by doing some algebra
grand pondBOT
Veni, vidi, perii
prime hornet
then you can start writing your proof
prime hornet
I'm a little confused by what you're doing 
twilit field
I'm trying to find a < inequality involving |x+a+1|
twilit field
**Veni, vidi, perii**
I mean this gives me $|x-a| < \frac{\varepsilon}{|x+a+1|}$
grand pondBOT
Veni, vidi, perii
prime hornet
**Veni, vidi, perii**
where did this come from? I'm just a little confused haha
grand pondBOT
Veni, vidi, perii
prime hornet
correct
grand pondBOT
Veni, vidi, perii
prime hornet
right, okay
twilit field
Was thinking of writing |x+a+1| as |x-a+2a+1$
prime hornet
hm.. I can't exactly see where you're headed here 
you still need to find an explicit delta
prime hornet
may I suggest doing the following?
turn $|x^2 + x - (a^2 - a)| < \varepsilon$ to $-\varepsilon < x^2 + x - (a^2 - a) < \varepsilon$
grand pondBOT
higher!
prime hornet
you can do this by the definition of the absolute value
prime hornet
okie
twilit field
I think I have an idea
prime hornet
grand pondBOT
Veni, vidi, perii
prime hornet
why can we say that?
the product of two terms being less than a number does not mean each term is less than that number
prime hornet
indeed
twilit field
I think, on seeing what I did for x^2, I have the right idea, but have to work on justifying this
so |x+a-(-1)|
wait, I also know that $|x+a+1| < \frac{\varepsilon}{|x-a|}$
grand pondBOT
Veni, vidi, perii
grand pondBOT
Veni, vidi, perii
twilit field
Can I have a hint
or ykw
let's do something even simpler
limit of x+1 at a\in R
twilit field
prove that the limit of $x+1$ at any $a \in \R$ is $a+1$
midnight plankBOT
prove that the limit of $x+1$ at any $a \in \R$ is $a+1$
grand pondBOT
Veni, vidi, perii
grand pondBOT
Veni, vidi, perii
grand pondBOT
Veni, vidi, perii
sharp coral
that doesn't look quite right
twilit field
Now?
grand pondBOT
Veni, vidi, perii
sharp coral
yes
twilit field
OK. Now I'll try $x^2+x$
grand pondBOT
Veni, vidi, perii
twilit field
at $a \in \R$
grand pondBOT
Veni, vidi, perii
twilit field
$|x^2+x-(a^2+a)|<\varepsilon$
grand pondBOT
Veni, vidi, perii
twilit field
$|x^2-a^2+x-a|< \varepsilon$
grand pondBOT
Veni, vidi, perii
twilit field
$|x-a||x+a+1|< \varepsilon$
grand pondBOT
Veni, vidi, perii
grand pondBOT
Veni, vidi, perii
twilit field
now I want $|x+a+1|>1$say
grand pondBOT
Veni, vidi, perii
grand pondBOT
Veni, vidi, perii
quiet hinge
this makes no sense to me
Ask yourself exactly what you are trying to do here
You are trying to prove that the limit x->a x²+x = a²+a and in order to do that, it seems like you are using the epsilon delta definition of a limit
However you are not correctly using the epsilon delta definition, so write it out in its entirety on a piece of paper and keep it next to you
solid iris
However you are not correctly using the epsilon delta definition, so write it ou...
we need p!=x0 and |f(p)-L|
quiet hinge
Okay, so you are given f, L and x_0 here. Your job is to find delta (for a given epsilon) such that for all points (not x_0) in a delta width neighbourhood of x_0 satsify |f(p)-L|< epsilon
quiet hinge
Okay, so you are given f, L and x_0 here. Your job is to find delta (for a given...
@twilit field does this make sense?
twilit field
yes
quiet hinge
Okay, so for what delta will
|x²+x -(a²+a)|<epsilon for a given epsilon (keep in mind 0<|x-a|<delta)
And your delta must be in terms of epsilon , since you want that inequality to hold for all epsilon
twilit field
so I was thinking of |x^2-a^2|+|x-a|
quiet hinge
What do you plan to do next?
quiet hinge
Try finding an upper bound for |x²-a²|+|x-a| in terms of delta and a, since then you can choose delta so that the upper bound is lesser than epsilon
twilit field
|(x+a)(x-a)|+|x+a|$
quiet hinge
Try finding an upper bound for |x²-a²|+|x-a| in terms of delta and a, since then...
Does this make sense?
twilit field
so $|(x+a)||x-a+1|< \varepsilon$
quiet hinge
Hmm, please recheck
grand pondBOT
Veni, vidi, perii
quiet hinge
Okay, so you have simplified the expression
twilit field
ooh
quiet hinge
Try finding an upper bound for |x²-a²|+|x-a| in terms of delta and a, since then...
Now:
- does this make sense?
- if yes, try doing this
twilit field
I know use that $|x-(a-1)|>|x|-|a-1|$
grand pondBOT
Veni, vidi, perii
quiet hinge
Wouldnt it be better to split it as |x-a+1| \geq |x-a|+1
Because you know |x-a| < delta, you would have found an upper bound for the |x-a+1| factor (|| delta +1 is an upper bound||)
Now find an upper bound for the |x+a| factor (keep in mind the upper bound must be in terms of delta, a and other constants because only then can you control it to be as small as you like)
quiet hinge
Wouldnt it be better to split it as |x-a+1| \geq |x-a|+1
@twilit field
twilit field
Hav a class now, will come back to this in a bit
midnight plankBOT
@twilit field Has your question been resolved?
sweet spire
Proove that in any scalene triangle this is false: lenght of IH=2024 and GH=2023, where G is the center of gravity of the triangle and I is the center of the inscribed circle
midnight plankBOT
Proove that in any scalene triangle this is false: lenght of IH=2024 and GH=2023...
delicate helm
and H?
delicate helm
no
sweet spire
Proove that in any scalene triangle this is false: lenght of IH=2024 and GH=2023...
<@&286206848099549185>
sour stirrup
sweet spire
What do you mean by close )))
midnight plankBOT
@sweet spire Has your question been resolved?
sweet spire
Proove that in any scalene triangle this is false: lenght of IH=2024 and GH=2023...
<@&286206848099549185> Can you help me?
midnight plankBOT
@sweet spire Has your question been resolved?
delicate helm
After playing around with this for a bit i think this question is a red herring
And in fact IH <= GH in any triangle
delicate helm
conjecture
sweet spire
how do I write this?
delicate helm
I suspect that this can be proven using copious amounts of vector algebra
it is enough to show that GH² - IH² >= 0
given positions of vertices as vectors you can get both GH² and IH² explicitly using https://en.wikipedia.org/wiki/Barycentric_coordinate_system and in the end you will get an expression in terms of side lengths that you need to show to be non-negative
which i would guess is possible
That said i don't recommend this method unless you wanna kill entire day expanding alebraic expression
But i also don't know a better method
midnight plankBOT
@sweet spire Has your question been resolved?
midnight plankBOT
@sweet spire Has your question been resolved?
midnight plankBOT
@sweet spire Has your question been resolved?
sweet spire
<@&286206848099549185>
midnight plankBOT
@sweet spire Has your question been resolved?
sweet spire
<@&286206848099549185>
sweet spire
Proove that in any scalene triangle this is false: lenght of IH=2024 and GH=2023...
Can you help me?
deep hollow
do u at least have a drawing or a graph
deep hollow
sorry algebra isnt my strongest of works im good with calculus
sweet spire
Oh okay
midnight plankBOT
@sweet spire Has your question been resolved?
last slate
ce ai de găsit sau de dovedit
rigid dune
Proove that in any scalene triangle this is false: lenght of IH=2024 and GH=2023...
this one. or following the discussion so far, GH > IH.
radiant trout
if you know this you can just solve an inequality
its still not nice but simplify it and see if you can get somewhere
midnight plankBOT
@sweet spire Has your question been resolved?
jaunty canopy
it looks like u have an expression for IH and GH so you might be able to prove GH > IH from that
i was never really any good at olympiad geometry but a better place to ask for help might be the maths olympiad discord server
u'll be able to find ppl who can actually do oly geometry that'll be able to help u
radiant trout
its still not nice but simplify it and see if you can get somewhere
@sweet spire how far did you get
sweet spire
I tried multiple things but didnt got too far
I also tried cos theorem in IGH triangle
radiant trout
what about the inequality
pure hollow
A(x)= root of x to the fourth power simplify
sudden yacht
A(x)= root of x to the fourth power simplify
!occupied
midnight plankBOT
A(x)= root of x to the fourth power simplify
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).
midnight plankBOT
@sweet spire Has your question been resolved?
radiant trout
what is the inequality though after simplifying
@sweet spire
keen sierra
https://classproctor.com/blog https://classproctor.com/blog/the-evolution-of-online-exams-trends-and-insights-for-2024
bleak steppe
does it have something to have with euler's line?
sweet spire
I got the answer
Thank you for your help
It was something with the angle GIH which has to be less thank 90°
And euler s line
remote tangle
I got the answer
Please close the channel if you're done
radiant trout
.close
slender talon
Can someone help me w this one?
midnight plankBOT
Can someone help me w this one?
slender talon
Not sure if I even did the other questions right but I got the answer… all I did was (in previous questions) was to calculate what would make RHS turn into LHS
slender talon
Wait gimme a sec
wild fern
Not sure bc I can't do the calculation atm
But basically do the first line again
But don't move the numbers around
Just calculate the right side
slender talon
Okay thank you! Lemme try that
wild fern
Left side should be y
And then you're left with y = some fraction
Then add (or subtract) the appropriate fraction to get
y + (something) = 3/2
That something will be your horizontal translation
And of course you would represent that as a vector translation if that's what you've been taught
slender talon
I see
Well
I got 37/10 this time
I think
I’ll be able to figure this out
Thanks!!
.close
midnight plankBOT
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Available help channel!
abstract cape
midnight plankBOT
abstract cape
2f(x)f'(x)= (f(x))^2+(f'(x))^2 (after differentiating)
from the equation I can only understand that this type of function is impossiple. because if of the highest degree of lhs is x^(2n-1) then the highest degree of RHS will be x^2n which is contradictory
<@&286206848099549185>
carmine void
2f(x)f'(x)= (f(x))^2+(f'(x))^2 (after differentiating)
Maybe devide by f(x)^2 (Assuming f ≠ 0)
carmine void
<@&286206848099549185>
Read #❓how-to-get-help
abstract cape
Maybe devide by f(x)^2 (Assuming f ≠ 0)
then?
abstract cape
I get y'=y^2/(2y-1)
carmine void
I get y'=y^2/(2y-1)
Where do you get it from?
abstract cape
Where do you get it from?
y= f(x) and y'=f'(x)
abstract cape
Where do you get it from?
after simplifiyng
ok I made a mistake
is this e^x?
I get (f(x)-f'(x))^2=0
carmine void
is this e^x?
Yes, that's what I got, too
abstract cape
I didn't notice it makes a perfect square
abstract cape
from the equation I can only understand that this type of function is impossiple...
this explains this also
I only thought of polynomials
carmine void
Yes, you figured out that it can't be a polynomial. But ofc it doesn't have to be
abstract cape
I only thought of polynomials
I tried root(x), 1/x and other stuffs but didn't try exponential , logarithmic and trigonometric functions
carmine void
Well done 👍 You almost solved it immediately on your own
abstract cape
Well done 👍 You almost solved it immediately on your own
yeah I mean it wasnt so hard after all
half meteor
Well done 👍 You almost solved it immediately on your own
just wanted to ask, was there a way to obtain that e^x other than intuition?
abstract cape
I just overlooked it
carmine void
We still need to handle the initial conditions. It could be C•e^x for any C in R.
abstract cape
just wanted to ask, was there a way to obtain that e^x other than intuition?
what are you talking about?
if you simplify you get f(x)=f'(x)
carmine void
just wanted to ask, was there a way to obtain that e^x other than intuition?
Well they get x'= x as an ode. Are you a bit familiar with ode's?
half meteor
ah makes sense
abstract cape
if you simplify you get f(x)=f'(x)
there's only one function that satisfies this condition
abstract cape
np
the prove is cool you can watch it from mit ocw video
other's proves are a little vague
carmine void
i wasnt taught diff eqs yet
There is Picard iteration giving you the unique solution as the right side is lipschitz continuous
abstract cape
.close
midnight plankBOT
Have you solved for the C yet?
abstract cape
Have you solved for the C yet?
nope
I will try that later
I am working on another problem
right now
carmine void
I will try that later
Ok, this shouldn't be a problem. Just wanted to mention it as you may over read it when we went a bit of topic
.close
midnight plankBOT
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Available help channel!
static lion
how to check convergence if x is the power?
midnight plankBOT
how to check convergence if x is the power?
static lion
which is the dominant value?
last slate
,, \forall x>1
grand pondBOT
anjali
fresh garden
static lion
use uhhhh x^2 + 5^x > 5^x for all x > 1
if I do this, what would be the num power?
thats if for the denom
but what about the numerator
fallow scarab
try to find something similar
static lion
so I pick 3^x?
fallow scarab
not that similar
3^x < your numerator so proving the convergence of something smaller doesn't prove anything
static lion
so I choose x alone?
fallow scarab
you should pick something that's BIGGER
if old numerator / denominator < new numerator / denominator and the latter still converges, then so does the former
static lion
but I still need to do the P test to know if I need to use a bigger or smaller number right?
fallow scarab
yes this problem is more than just 1 step
static lion
what do you mean by picking something bigger? choose any number or only from the equation?
fallow scarab
something that's bigger than the numerator for all x
fallow scarab
use uhhhh x^2 + 5^x > 5^x for all x > 1
the "for all x > 1" is important
fallow scarab
if `old numerator / denominator` < `new numerator / denominator ` and the latter...
does your function satisfy this inequality
old numerator = x + 3^x + 2
static lion
i dont get what you mean for picking the new numerator
so you saying like pick bigger i adjust numerator?
if so how I adjust it
fallow scarab
so you saying like pick bigger i adjust numerator?
no
fallow scarab
use uhhhh x^2 + 5^x > 5^x for all x > 1
did you understand this
static lion
nope
fallow scarab
probably should have said that 30 minutes ago
midnight plankBOT
@static lion Has your question been resolved?
last slate
what would be approximately the value of x^x/(1-x!) at x = 500?
midnight plankBOT
what would be approximately the value of x^x/(1-x!) at x = 500?
sinful trout
midnight plankBOT
robust forge
what would be approximately the value of x^x/(1-x!) at x = 500?
−2.503765396E215
last slate
mhm
midnight plankBOT
Available help channel!
feral prawn
Could someone explain number 5 please I don’t get the processs used
midnight plankBOT
Could someone explain number 5 please I don’t get the processs used
fallow scarab
Could someone explain number 5 please I don’t get the processs used
What process used
feral prawn
Nvm I got it
I think
Hold up lemme try to explain it the way I think and then lmk if I’m wrong
Ok so is this correct @fallow scarab
midnight plankBOT
@feral prawn Has your question been resolved?
midnight plankBOT
@feral prawn Has your question been resolved?
midnight plankBOT
@feral prawn Has your question been resolved?
midnight plankBOT
@feral prawn Has your question been resolved?
floral apex
.close
pliant sorrel
midnight plankBOT
pliant sorrel
how does that definite integral turn into e^2x - C^2
isnt it suppose to be e^2x - 1
tribal temple
how does that definite integral turn into e^2x - C^2
It doesn't, because...
tribal temple
isnt it suppose to be e^2x - 1
...you're multiplying the whole integral by C^2
midnight plankBOT
midnight plankBOT
Available help channel!
pliant sorrel
midnight plankBOT
pliant sorrel
where did the +pi/4 in the answer come from...
twilit jetty
where did the +pi/4 in the answer come from...
which +pi/4
twilit jetty
the complettet solution
is the original question tan(x) + tan(pi/4 - x) = 1?
pliant sorrel
how did the solution set i get the pi/4
twilit jetty
how did the solution set i get the pi/4
you need to answer the question for me to help you
"original question" means you need to screenshot the book problem
right now youre showing the answer key for the original question which doesnt tell me enough information
feral sedge
how did the solution set i get the pi/4
mtt has a point in that you posted something incomplete
pliant sorrel
cerulean sapphire
Try taking Tan of equation (A)
midnight plankBOT
@pliant sorrel Has your question been resolved?
lofty chasm
how to get range fort his
midnight plankBOT
how to get range fort his
lofty chasm
no
tribal temple
(0, infinity)
novel lion
the function is $g(x) = \frac{1}{x}$ with bound $x \in \mathbb{R}$ and $x \geq 3$
grand pondBOT
Astar777
novel lion
so x can only have values ranging from 3 to infinity
so 1/3, 1/4, 1/5 .......... 1/inf
starting point of range is 1/3
and end point is 1/inf which approaches 0
lofty chasm
the range is 0 < g(x) </ 1/3
novel lion
so range is (0, 1/3]
lofty chasm
this is how the graph looks like
can i get an explaination using the graph
tribal temple
can i get an explaination using the graph
"Everything to the right of x = 3"
pure dust
"Everything to the right of x = 3"
i see the black line in the graph now
okay i understand
lofty chasm
ohhhhhhhh
midnight plankBOT
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Available help channel!
late palm
midnight plankBOT
late palm
Solving the given equation I got x=7/2y
But don't know what to do after cuz y। is canceling out afterwards
grand pondBOT
Astar777
late palm
Sorry it was a silly mistake
late palm
**Astar777**
I got this niw
novel lion
now on to $\frac{x^2 + y^2}{x^2-y^2}$
grand pondBOT
Astar777
late palm
What to do further?
novel lion
u have value of x/y
novel lion
**Astar777**
how can you make x/y here
late palm
-x ²/y² + y²/X2
novel lion
what
take y^2 common from both numerator and denominator
you'll be left with $\frac{y^2(\frac{x^2}{y^2} +1)}{y^2(\frac{x^2}{y^2} -1)}$
grand pondBOT
Astar777
novel lion
y^2 cancels out, then just substitute the value of x/y
novel lion
thats right
late palm
Thank-you so much
novel lion
late palm
.close
hard shard
im very confused as to why the value for Rb is wrong
midnight plankBOT
im very confused as to why the value for Rb is wrong
hard shard
i used wolfram alpha
feral sedge
maybe this site is weird and expects 0.001613
hard shard
didnt work lol
feral sedge
feral sedge
(without rounding any value, it comes out to 0.00161246976619…)
feral sedge
ill bring it up with the prof i guess
sounds good probably if all else fails
midnight plankBOT
feral sedge
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Available help channel!
last slate
hi
last slate
if 1<a, b, c
novel lion
hi
pearl hull
Hello
last slate
then is this true
little yoke
hi
last slate
and 1/a = 1/b + 1/c
then a<b, c
last slate
yeah
modern sapphire
if a>b, then what about 1/a and 1/b? is 1/a > 1/b? Do you need 1/a > 1/b?
keen seal
what are they trying to sya here
midnight plankBOT
what are they trying to sya here
sharp coral
matrix multiplication is not commutative in general, but some matrices "commute" in the sense that AB = BA
so in this case, they are saying that polynomials of a particular square matrix have this property, that they commute with each other
(a polynomial of a matrix has essentially the same definition as a polynomial of a real number)
midnight plankBOT
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Available help channel!
twilit field
midnight plankBOT
twilit field
Need help with the last one
so the contradiction statement would be $P \implies \neg(P \land Q)$
grand pondBOT
Veni, vidi, perii
twilit field
so $P \vee \neg(\neg(P \land Q))$
grand pondBOT
Veni, vidi, perii
twilit field
or $P \vee (P \land Q)$
grand pondBOT
Veni, vidi, perii
grand pondBOT
Veni, vidi, perii
twilit field
midnight plankBOT
twilit field
what's wrong with this
dreamy lichen
nothing? What should be wrong with it
twilit field
oh
nvm
got my mistake
I'm really confused
what columns would teh truth table have here
P,Q, \neg P , P \implies \negQ, \neg Q \implies P, P iff \neg Q, P \lor Q, P \land Q
anything else
dreamy lichen
If you want to prove this using truth table, then it should have at least P <=> -Q and (P v Q) ^ -(P ^ Q)
you can include few more columns that will help you in building the truth table
twilit field
I think the truth table will be overkill here
dreamy lichen
yeah, it might be
twilit field
so $P \iff \neg Q$ is the same as $ (P \vee \neg (\neg Q)) \land (\neg Q \vee \neg P)$
grand pondBOT
Veni, vidi, perii
twilit field
so $P \vee Q$is true
grand pondBOT
Veni, vidi, perii
dreamy lichen
Yep, if P <=> -Q is true, then P v Q is true as well
dreamy lichen
**Veni, vidi, perii**
Or in other words, this is equivalent to
(P v Q) ^ (-Q v -P)
the second part can be further simplified to exactly what you need btw, through a certain rule I wont spoil
twilit field
If A is true, \neg A must be false
dreamy lichen
what is A?
twilit field
so this is the same as $\neg(P \land Q)$
grand pondBOT
Veni, vidi, perii
dreamy lichen
Yes, (-Q v -P) is same as -(P ^ Q), by ||De Morgan's law||
grand pondBOT
Veni, vidi, perii
twilit field
Yes, (-Q v -P) is same as -(P ^ Q), by ||De Morgan's law||
I forgot that even had a name 💀
So basically here I have to show that $\implies$ and $\iff$ can be expressed in terms of $\land, \lor, \neg$
right
grand pondBOT
Veni, vidi, perii
twilit field
Like can a statement form have $\exists$?
grand pondBOT
Veni, vidi, perii
twilit field
or $\forall$
grand pondBOT
Veni, vidi, perii
twilit field
dreamy lichen
**Veni, vidi, perii**
This is first order logic symbol
that's not in propositional logic
twilit field
ah, okay
dreamy lichen
you dont need to worry about it
dreamy lichen
So basically here I have to show that $\implies$ and $\iff$ can be expressed in ...
You need to prove that any statement can be reduced to an equivalent statement containing only negation and implication
twilit field
oh
dreamy lichen
Like I can have a string of n nested implications
If that's what you have, then you are already done
twilit field
not too bad
dreamy lichen
A -> B -> C does already contain only negation and implication
the thing is, you need to deal with statements like A ^ B, A v B, A <=> B
twilit field
Not sure if we've defined well formed yet
dreamy lichen
the thing is, you need to deal with statements like A ^ B, A v B, A <=> B
if your book is not that formal, you'll only need to explicitly do these 3 things, and then say the rest can be obtained by applying those 3 rules multiple times
dreamy lichen
Not sure if we've defined well formed yet
Yeah, it's not that formal then
dreamy lichen
if your book is not that formal, you'll only need to explicitly do these 3 thing...
this will be sufficient
twilit field
I mean that's not too hard
just a minute
So $A \land B \equiv \neg(\neg(A) \vee \neg B) \equiv \neg(\neg A \implies B)$
grand pondBOT
Veni, vidi, perii
twilit field
Similarly $A \vee B \equiv (A \vee \neg(\neg B)) \equiv A \implies \neg B$
dreamy lichen
**Veni, vidi, perii**
this is wrong
grand pondBOT
Veni, vidi, perii
lunar ocean
yeah they're both somewhat wrong
try putting in that A and B are both true and you can see the equivalence doesn't quite work because the expressions you ended up with are false
midnight plankBOT
@twilit field Has your question been resolved?
midnight plankBOT
midnight plankBOT
Available help channel!
night gyro
someone help i got the answer as yes but idk bc its 4 marks
midnight plankBOT
someone help i got the answer as yes but idk bc its 4 marks
novel lion
someone help i got the answer as yes but idk bc its 4 marks
its not yes
he doesnt have enough cement to make 180kg concrete mix
night gyro
how
novel lion
1 : 3 : 5
night gyro
ye
novel lion
if weight of cement required is x
then sand and gravel would be 3x and 5x respectively
in total this all should add up to 180kg
night gyro
wait isnt cememnt 15
novel lion
so x+3x+5x = 180
x= 20
so he needs 20kg of cement
but only has 15kg
hence he doesnt have enough to make the concrete mix
night gyro
ooo ok
novel lion
wait isnt cememnt 15
yes
but thats the amount he has
its not the amount required to make 180kg of concrete mix
night gyro
tht wht he has oooooooo
novel lion
what i did was to find how much cement he would need for 180kg of concrete mix
night gyro
ok
i see
so questions like this u have to involve x
wht abt this i was savin this to ask
but none responds
idk if its correct but we do 6x1.8
night gyro
which is 10.8
novel lion
and also area of the tile (in m^2)
novel lion
which is 10.8
thats of tile
not the rectangular wall
nvm
my bad
yes
its 10.8 m^2
novel lion
and area of a tile is 0.18 m^2
now find how many total tiles will be used to cover that wall?
night gyro
yh
novel lion
so that will be?
novel lion
yes
night gyro
60
novel lion
right
but thats total tiles
u need the number of tiles of each color
see what information is given that could be used for that
novel lion
yes
novel lion
so how many tiles left after that?
night gyro
yh
novel lion
how will you do that?
check the question, do you see any information that can help here?
night gyro
green to blue is 1:3
novel lion
yes
night gyro
24/3?
night gyro
yh
novel lion
right?
novel lion
1/4 : 3/4
night gyro
24/4
novel lion
yes
novel lion
yes
novel lion
correct
novel lion
wot
night gyro
like as soon as i talked to u i understood everything
novel lion
skillz
novel lion
well, firstly you had dimensions of wall and tile given
so you can find the number of tiles
night gyro
yh
novel lion
thats it
for the rest part info was given
like 3/5 while tiles
so u just multiply 60 with 3/5
and 1:3 green to blue tiles
so u use that to find green and blue tiles
question solved
night gyro
my brain started tweakin yesterday bc i couldnt solve this
novel lion
night gyro
thanks man i have exams soon so im doin much questions as possible
night gyro
.close
robust forge
Show that if G is simple and bipartite then epsilon<= v²/4 
midnight plankBOT
Show that if G is simple and bipartite then epsilon<= v²/4 <:wut:126916740889601...
@robust forge Has your question been resolved?
frail carbon
Suppose the partitions have m and n vertices respectively (so m+n = v)
what's the most edges you can have in this graph?
for what values of m and n is that number maximised?
robust forge
what's the most edges you can have in this graph?
mn
robust forge
for what values of m and n is that number maximised?
v/2 hmm but the book derives the answer is a diff way
If shows that the answer is maximised when m = n = v/2
And no of edges would be equal to n/2C2 + (n/2+1)C2
and in the next ques they ask to generalize this
To derive a formula for k-partite graph
@frail carbon they have used this formula in the answer key any hints over this que
frail carbon
do the brackets [.] and {.} refer to floor and ceiling?
as in the two integers closest to n/m (unless n/m is an integer in which case just n/m)
robust forge
do the brackets [.] and {.} refer to floor and ceiling?
I believe [] refer to floor but the other might be referring to set I mean one part can't have less than 1 vertice
robust forge
as in the two integers closest to n/m (unless n/m is an integer in which case ju...
Yup
Any hints @frail carbon 
robust forge
<@365017527484416001> they have used this formula in the answer key any hints ov...
Anyone hints ?
robust forge
<@&286206848099549185>
midnight plankBOT
@robust forge Has your question been resolved?
frail carbon
sorry I had to leave for a while. If a vertex is in a subset with i vertices to maximise the number of edges in the graph you want an edge for each of the n-i vertices not in the same subset. You can also then explicitly find how many subsets are of what size
robust forge
sorry I had to leave for a while. If a vertex is in a subset with i vertices to ...
Ok but how does this relate to the above question
Also to maximize the no of edges every point in I should be connected to each point in n-i ?
U there ? @frail carbon
frail carbon
Each part has k = [n/m] vertices, or possibly one more
you can explicitly write n = kr + s for some r, some s < k
then in terms of r and s find the maximum number of edges, and then rearrange the result to get the formula
you are asking for hints for 2.9 now right? or have I misunderstood
robust forge
you are asking for hints for 2.9 now right? or have I misunderstood
Yes 1.2.9
robust forge
then in terms of r and s find the maximum number of edges, and then rearrange th...
Ok il try that
robust forge
@frail carbon i believe I was able to do it algebraically
But I still don't get the intuition behind it and the result looks like a combinatorial counting
Can you explain how can we arrive at the result intuitively
frail carbon
I don't have an intuition for this either I'm afraid
There probably is one
But I just don't see it
midnight plankBOT
robust forge
But I just don't see it
Do you happen to know anyone who could explain the result using a combinatorial method
frail carbon
I would not
robust forge
Hmm
frail carbon
Hmm
ask in #combinatorial-structures maybe
robust forge
ask in <#576511723574525962> maybe
I asked in math discussion cause it would be more reachable to ppl lol
I also asked a prodigy frnd of mine he couldn't find one either
midnight plankBOT
Available help channel!
last slate
Require help the b part of this question
midnight plankBOT
Require help the b part of this question
last slate
$$\frac{dT}{dr} = \frac{-q}{2 \pi k_{\text{insulation}} r_3}$$
grand pondBOT
Edmund Cloudsley
last slate
if we integrate this, we should get
$$\int dT =\int \frac{-q}{2 \pi k_{\text{insulation}} r_3} dr$$
grand pondBOT
Edmund Cloudsley
last slate
$$T = \frac{-q}{2 \pi k_{i}} \ln{r_3} + C$$
grand pondBOT
Edmund Cloudsley
last slate
we can equate temperature to 50 degs celcius, we know q, we know k_i
how do we find C?
unlike part (a) wherein we can find C by using the temperature at r_1 however here, the equation doesn't share the same k as r1 and r2
The worked solution in this book, they have simply utilised the C value derived from part (a). But I am unable to make sense of it.
since in part (a), we utilise $k_{\text{pipe}}$ instead of $k_{\text{insulation}}$
grand pondBOT
Edmund Cloudsley
midnight plankBOT
@last slate Has your question been resolved?
last slate
<@&286206848099549185>
fallow scarab
how do we find C?
Usually some initial condition or boundary value
last slate
yup
fallow scarab
The worked solution in this book, they have simply utilised the C value derived ...
Do you know how to find T(0.2)
last slate
So the intial condition they have utilised involves a different value of k
last slate
Do you know how to find T(0.2)
yeah. we would just have to substitute 0.2 for the r value
.close
midnight plankBOT
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Available help channel!
turbid nova
x^3-15x^2+68x-96=0
midnight plankBOT
x^3-15x^2+68x-96=0
turbid nova
Solve by factoring
midnight plankBOT
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
small jasper
In general, when you're given a cubic to solve by hand, there's generally going to be at least one integer/rational root
you can use the rational root theorem to narrow down your possibilities and then test them
feral sedge
use rational root theorem
small jasper
you can use the rational root theorem to narrow down your possibilities and then...
and yes, ik it says to solve by factoring, but you can fudge that once you find the root lol
glacial aurora
(x-8)(x-4)(x-3)=0
turbid nova
ye thank u but how tf u get that answer
delicate sage
ye thank u but how tf u get that answer
Wolfram
delicate sage
I MEANT WORKING OUT
Can find a root as they mentioned
turbid nova
ur kidding me
delicate sage
Or proceed by making the cube depressed
delicate sage
and see if they work
Trial nd hit 🎯
turbid nova
Or proceed by making the cube depressed
i'm depressed 😭
delicate sage
Else I said , go ahead to make it a depressed cubic
feral sedge
so i have to go through all the factors of 96
not all of them
i.e. start with smaller numbers
100% of the time you will not need to test much to find a root
delicate sage
fuck maths
🥲
turbid nova
.close
midnight plankBOT
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last slate
what is the derivative of x pentation n
last slate
x^x^x^x^x^x........ n times
delicate sage
n tends infinity?
runic hamlet
well maybe you can do something with induction but I doubt this has a nice formula
fresh sparrow
i don't really have a proof for this but i assume this is what a general formula would look like
(let me type it out)
let $^n x$ be the nth tetration of x (if you were talking about pentation it would be a different story)
grand pondBOT
artemetra
midnight plankBOT
@last slate Has your question been resolved?
fresh sparrow
$\frac{d}{dx} ,^n x = ,^n x F(x,n-1)$\
where \
$F(x,k) = \begin{cases}\log(x) + 1 \text{ if } n=1 \ \frac{^{n-1} x}{x} + ,^{n-1} x \log(x) F(x,k-1) \text{ otherwise}\end{cases}$
grand pondBOT
artemetra
