RunaAnn
#help-26
thorny flameBOT
past cradle
You'd need to add 9/8 on the other side
craggy haven
remember to scale that 9/16 on the right because it's inside the 2() on the left
yeah
craggy haven
you almost have it
just if you add 2(9/16) on the left you need to add it on the right as well
glass crescent
so it should be plus 2(9/16) on the right too?
craggy haven
yeah instead of the raw 9/16
glass crescent
$(x-1)^2+2(y+3/4)^2=113/8$
thorny flameBOT
RunaAnn
glass crescent
correct?
past cradle
,w 1 + 12 + 9/8
thorny flameBOT
past cradle
Wonderful
glass crescent
is this already the standard form?
craggy haven
i think it's supposed to be $\frac{(x-h)²}{a²} + \frac{(y-k)²}{b²} = 1$
thorny flameBOT
hayley
glass crescent
oh so i have to divide
craggy haven
yeah
topaz sinewBOT
@glass crescent Has your question been resolved?
glass crescent
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shut obsidian
There are three boxes, in which two of them there is acid. The boxes respectively say: \begin{enumerate} \item This box is safe. \ \item There is acid in this box. \ \item There is acid in box 2. \end{enumerate} We also know that atleast one of these statements is false and atleast one of them is true. \ Which of the boxes is safe?
topaz sinewBOT
There are three boxes, in which two of them there is acid. The boxes respectivel...
craggy haven
those all sound pretty dangerous
shut obsidian
shut obsidian
those all sound pretty dangerous
Not with the previous-to-last sentence
(Also congrats on mod, Hayley)
cursive thorn
long stirrup
no
shut obsidian
i still maintain that none of them are safe
No, there is exactly two times acid
One is safe.
long stirrup
true false false: no
false true false: no
false false true: no
true true false: no
true false true: no
false true true: yes
the third one is safe
shut obsidian
the third one is safe
How did you deduce that?
shut obsidian
true false false: no
false true false: no
false false true: no
true true false: ...
Why no in the first 5 cases?
craggy haven
i just checked three cases, based on which one was safe
shut obsidian
i just checked three cases, based on which one was safe
What did you do when you e.g. assumed box 1 was safe?
long stirrup
2 and 3 are the same statement, except for "false true true" and "true false false" that's how you can tell they can't be right
craggy haven
What did you do when you e.g. assumed box 1 was safe?
just checked the truth value of each statement and then saw whether we had at least one true and at least one false
shut obsidian
just checked the truth value of each statement and then saw whether we had at le...
So if we say box 1 is safe, then statement 1 is correct.
Statement 2, 3 could both be correct too.
Oh, one must be false
Then both must be false
Statement 1 correct, 2 false, 3 false
If box 1 is safe.
craggy haven
So if we say box 1 is safe, then statement 1 is correct.
Statement 2, 3 could ...
if we say box 1 is safe then the other two have acid
shut obsidian
if we say box 1 is safe then the other two have acid
Yes
long stirrup
my way is the smart
shut obsidian
if we say box 1 is safe then the other two have acid
So three true statements, but we need atleast one false. So box 1 can't be safe.
craggy haven
exactly
shut obsidian
Ah, so that's your way, Hayley. Thanks
long stirrup
at least 1 false and and least 1 true
means it's 6 cases, out of which 2 are possible, the ones where truth values 2=3
native raptor
why is this so contentious...
shut obsidian
why is this so contentious...
It's not?
long stirrup
do we not get the same answer?
shut obsidian
are we discussing efficiency then?
Yeah
Pretty sure Frowny got the same answer as Hayley
long stirrup
i'm not sure what hayley says though
shut obsidian
i'm not sure what hayley says though
She e.g. looks what would happen if box 1 was safe. That would lead to a contradiction
Then what would happen if box 2 was safe, etc.
native raptor
since 2 and 3 are the same statement effectively
the fastest way would be to deem 1 false no?
long stirrup
yes
shut obsidian
but she said you can;t tell
Wasn't that at the very beginning? I don't know if she already thought of her way then.
Let me try writing out her way
native raptor
from what I read, hayley was basically just suggesting brute forcing
shut obsidian
1 is safe: s1 is correct, s2 or s3 needs to be false. They must both be correct though, since 1 is safe. Contradiction.
2 is safe: s1 is false, s2 or s3 must be correct. s2 is false. s3 must be correct, but s3 is false. Contradiction.
3 is safe: s1 is false, s2 is correct, s3 is correct. No contradiction.
I think that's what Hayley thought of
shut obsidian
true false false: no
false true false: no
false false true: no
true true false: ...
You are looking for contradictions here too, right?
long stirrup
no
shut obsidian
true, false, false is a no because 1 is safe, but 2. false means 2 must be safe too. Contradiction to the statement there is only one safe box and thus no
long stirrup
i just can't think of a way that wouldn't do that
shut obsidian
I wasn't asking for a way that wouldn't use contradiction?
I was just asking if you used contradiction in your argument to say "no"
long stirrup
you always reject things until one remains, you reject because a hypothesis leads to a contradiction
i didn't use the word contradiction at any point in my head
shut obsidian
i didn't use the word contradiction at any point in my head
Well what did you use then to say "1 is a no"? Just "it can't be correct"?
That would be equivalent to saying "contradiction"
long stirrup
okay
shut obsidian
true false false: no
false true false: no
false false true: no
true true false: ...
Alright, thanks Frowny
.close
topaz sinewBOT
topaz sinewBOT
Available help channel!
patent hollow
topaz sinewBOT
patent hollow
hey bit stuck on 6
idk how to use the formula and i tried drawing a tree diagram but it doesn’t work
reef harness
$rotate
I'm a bit rusty in probability
but I'll try
but I don't understand the question, 2 condition exactly 2 heads and at least 1 head?
topaz sinewBOT
@patent hollow Has your question been resolved?
patent hollow
but I don't understand the question, 2 condition exactly 2 heads and at least 1 ...
yes
topaz sinewBOT
@patent hollow Has your question been resolved?
mental basin
is this a valid matrix operation or did I make a typo in there? (it's supposed to be part of a proof on the Vandermonde-determinant)
The idea is to divide each jth row by (x_j-x_0). This is a slight variation of what our textbook does, but I'm not sure I understand where the - signs disappeared to? I could probably just copy this into the assignment because "that's what the textbook did", but I'm trying to understand why and how this is a valid operation (or how to properly do it if this isn't one, as I suspect)
odd pagoda
well multiply eg the entries in the first row by x1-x0 and simplify
what do you get
btw, \det
mental basin
oh that's just the 3rd binomial formula...
I thought I was missing something obvious, thanks!
.close
trim flame
I need notes on Euler's method to study for your differential equations class. Can you help me with that?
topaz sinewBOT
I need notes on Euler's method to study for your differential equations class. C...
@trim flame Has your question been resolved?
mental basin
neon iron
- Prove for a matrix A:
If A^TA = A, then A is symmetric and A = A^2.
This is extremely obvious in the other direction. Got 2 almost-proofs but they each had a small flaw
neon iron
Attempt 2: If A^TA = A, we know that either A is not invertible or A^T is the identity because if A were invertible, we would have A^T = I upon multiplying A^-1 at the right.
However, if A^T is the identity then A is symmetric because we can also prove that A = I by the following:
Raise both sides to T to get
AA^T = A^T
A^T is invertible if A is invertible and its inverse is (A^T)(-1)
Multiply on right to get A = I
So A^T = I = A .
Hence A^TA = A implies AA = A -> A^2 = A.
That proves it if A is invertible.
If A is not invertible then we know det(A) = 0.
det(A) = 0 implies a certain relation between the diagonal entries of A; for 2x2 we have ad = bc, for 3x3 we have aeg + … = …)
)
This one i couldnt think how it would end. Then I have a\ttempt 1 where thers just a flaw in logic
If A^TA = A, then A is symmetric and A = A^2.
This is extremely obvious in the other direction. A has to be equal to A^2; since A is symmetric, A^TA = AA. So A = AA.
Hence we will do a proof by contrapositive. Prove that
if A is not symmetric OR A =/= A^2,
then A^TA =/= A.
Part 1:
If A is not symmetric, then A =/= A^T.
Case 1: A is not symmetric and A = A^2
Hence A^tA =/= A because if A was symmetric that would be implying A^TA = AA = A^2 which would be equal to A if A was equal to A^2. Since A is not symmetric we can infer this is not true.
Case 2: A is not symmetric and A =/= A^2
A is not symmetric
So A^TA =/= AA
So A^TA =/= A^2 =/= A
Since we have proven the contrapositive, we have proven the original
topaz sinewBOT
@neon iron Has your question been resolved?
neon iron
Think about what you get when you transpose A^TA again
I already did that
Raise both sides to T to get
AA^T = A^T
But the onlyr esults I found with this was if we assumed A is invertible
eternal wing
(It's not AA^T)
.
eternal wing
Yup
neon iron
LMFAO
eternal wing
lol
eternal wing
yeah and the rest is very easy
eternal wing
the first time I read through what you wrote I also missed that
neon iron
Yeah
Like I know youre supposed to swap them
I think i just somehow didnt
Ok I have one moreq uestion
eternal wing
sure
neon iron
Use row reduction to show that the determinant of
1 1 1
a b c
a^2 b^2 c^2
is (b − a)(c − a)(c − b).
Lemme grab the work for it
We know
1 1 1
a b c
a2 b2 c2
R2 -> R2 - aR1 (so doesnt affect the determinant)
1 1 1
0 b-a c-a
a2 b2 c2
R3 -> R3 - a^2R1 (so doesnt affect)
1 1 1
0 b-a c-a
0 b^2-a^2 c2-a^2
R2 -> 1/(b-a)R2 (so multiply final det by 1/(b-a))
1 1 1
0 1 c-a
0 (a+b)(a-b) (c-a)(c+a)
R3 -> R3 - (a+b)(a-b)R2
1 1 1
0 1 c-a
0 0 (c-a)(c+a) - (a+b)(a-b)(c-a)
Maybe its turning out weird bc im doing Gauss Jordan
eternal wing
I would maybe try to do it without the division by b-a
You don't know if it's 0 or not
neon iron
😭 I only really know how to use gauss jordan
And gaussian
I think thats all that was taught
Wolframalpha does these weird methods of guess and check
Or like 'observing' stuff
But ive just been foloowing the method i was told to use
strange whale
also you screwed up the division anyway
strange whale
why is c-a still here
neon iron
(c-a)/(b-a)
Lmao
Im really not sure where the c-b term comes from in the final answer
But ill try doing like
Well idk i still dont know what to do
neon iron
What would i do if not dividing by b-a
\
Oh syhit
What if we do R2 -> R2 - R2. no nvm LMAO
Like, c-a - (b-a) is c-b
strange whale
R2 and R3 have factors in common you know
eternal wing
R2 -> R2 - R2?
neon iron
Oh.... so R3 is just R2 multiplied by (b+a) on columkn 2 aqnd (c+a) on column 3
neon iron
R2 -> R2 - R2?
Ignore that
I meant more like
column 3 -> column 3 - column 2
But that doesnte xist lmao
strange whale
R3 <- R3 - (b+a)R2 kills the coeff in (R3, C2) indeed
or maybe +(a+b)R2 idk, haven't read your thing in detail
neon iron
1 1 1
0 b-a c-a
0 b^2-a^2 c2-a^2
R3 -> R3 - (b+a)R2
That doesnt actually kill it tho....?
That's (b-a)(b+a) - (b+a)(b-a)
Which is (b-a)(b+a-1)
OH SHI8T
OH SHIT
UR RIGHT
Okay
1 1 1
0 b-a c-a
0 b^2-a^2 c2-a^2
R3 -> R3 - (b+a)R2
1 1 1
0 b-a c-a
0 0 (c-a)(c+a) - (b+a)(c-a)
So the final determinant would be (b-a)( (c-a)(c+a - (b+a)))
💀
Since they said u can just take the determinant once its in tringular form
And its equal to the product of the diagonal main
strange whale
So the final determinant would be (b-a)( (c-a)(c+a - (b+a)))
careful with your parentheses
neon iron
Oh yea
strange whale
Since they said u can just take the determinant once its in tringular form
that's what's nice about triangular ye
neon iron
This still doesnt have c-b anywhere... We would need b - c and then negate it
strange whale
This still doesnt have c-b anywhere... We would need b - c and then negate it
(c+a - (b+a))
neon iron
Ohhhhhhhhhhhhhhhhhh
So true
(b-a)( (c-a)(c+a - (b+a)))
(b-a) ( c-a)(c-b)
YES LETS GO
Ty
Knew i could count on math discord
frank narwhal
topaz sinewBOT
frank narwhal
trying to solve this using partial fraction decomposition
i can find B and C but i dont know how to find A if i could get some help
where i got it rn is
7x + 126 = A(x - 3)(x + 4) + B(x + 4) + C(x - 3)^2
B and C being 21 and 2 respectively
topaz sinewBOT
@keen zodiac Has your question been resolved?
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@keen zodiac Has your question been resolved?
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@keen zodiac Has your question been resolved?
tall trench
<@&286206848099549185>
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@keen zodiac Has your question been resolved?
topaz sinewBOT
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languid mirage
Three points P(3,4), Q(3,1) and R(8,4) are the vertices of a right angeled triangle, find the length of the perpendicular from P to QR
topaz sinewBOT
Three points P(3,4), Q(3,1) and R(8,4) are the vertices of a right angeled trian...
urban pulsar
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topaz sinewBOT
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
2
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lofty prairie
HOW TO SOLVE THIS?
topaz sinewBOT
HOW TO SOLVE THIS?
lofty prairie
i did consider two groups (ppl) and (eoe) wich can permutate among itself, also letter permutaions ie 3!/2! * 3!/2! * 2!
=18 i get is this correct way?
topaz sinewBOT
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lofty prairie
<@&286206848099549185>
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waxen arrow
How do i find the side for this equation to find the height?
“A square based pyramid has a total surface area of 224cm^2. If the length of the base is 8 cm, what is the height of each triangular face?”
livid tusk
A square base pyramid has a single base, and 4 triangles
Total surface area = Area(base) + 4Area(Triangle)
Area(base) = 88 = 64
224 = 64 + 4Area(Triangle)
Area(Triangle) = 160
Area(Triangle) = side * height/2
side = 8(the base)
height = 1602/8 = 40
clear lodge
you're not supposed to calculate the solution for him, you're supposed to guide him for him to get it himself
waxen arrow
A square base pyramid has a single base, and 4 triangles
Total surface area = Ar...
I get this however the answer sheet says the height should be 10 cm, so i’m very confused
i have the formulas to figure out the height and all, except i will need the length of the side to do so and this equation doesn’t provide one
<@&286206848099549185>
waxen arrow
you're not supposed to calculate the solution for him, you're supposed to guide ...
can u help 😭
clear lodge
you know that the pyramid has a square base of length 8cm
with that you should be able to calculate the area of the base. Compute it and tell me the answer
waxen arrow
64
waxen arrow
yeh mb forgot the units
clear lodge
After that, you know that the total area of the pyramid is the sum of the areas of the base and each side
since it's a square pyramid, you got the base, and 4 sides
waxen arrow
mhm
clear lodge
since you got the total (given) and the area of the base (just calculated) you can get the area of the other 4 sides together. Compute it and tell me
waxen arrow
40 cm^2?
clear lodge
how much is 224cm^2 minus 64cm^2?
waxen arrow
160
clear lodge
remember i said the other 4 together
since they are all 4 equal, obviously the area of only one is 1/4 of that, which is the 40cm^2 that you got
clear lodge
which is specifically where Davit made a mistake.
He said 4A = A
now, you have the area of one side triangle (40cm^2), and you got the base of the side triangle (given, 8cm). With that you should be able to compute the height
waxen arrow
hmm
still a little confused cuz to figure the height it’s just Pythagoras theorem right?
so it be square root of c^2 - a^2 to find the height yeh?
so how would i get the c
cuz it cant be 40^2 - 4^2 can it?
clear lodge
the area of a triangle is half the product of the base times the height
you got the area, and you got the base
waxen arrow
oh wait
40 divided by 8 times 2
OHHHH
tysm i was just thinking too hard
.close
vagrant field
You (Alice) and your friend (Bob) go to a homecoming party, where there are two boxes of pizza.
Box 1 contains:
3 pepperoni slices
4 cheese slices
Box 2 contains:
3 pepperoni slices
2 cheese slices
You ask Bob to get a slice for you. Bob, being Bob, chooses which box to get the slice from randomly. Let 𝐵
be the event that Bob chooses Box 1.
Bob, being Bob again, also chooses a slice from the box at random. Let 𝐴1
and 𝐴2
be the events where Bob chooses a pepperoni slice from Box 1 and Box 2, respectively.
topaz sinewBOT
You (Alice) and your friend (Bob) go to a homecoming party, where there are two ...
vagrant field
Write down the composite event 𝐶 in which Bob eventually gets a pepperoni slice.
remote cypress
how to distribute 2(x-1)
topaz sinewBOT
@vagrant field Has your question been resolved?
stiff knoll
how to distribute 2(x-1)
2*x - 1*2
topaz sinewBOT
@vagrant field Has your question been resolved?
carmine pelican
unsure if this is correct, just want someone to check my work basically. the question also asks to report any x values for which the derivative does not exist, i said 0 and -2
topaz sinewBOT
unsure if this is correct, just want someone to check my work basically. the que...
topaz sinewBOT
@carmine pelican Has your question been resolved?
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@carmine pelican Has your question been resolved?
carmine pelican
<@&286206848099549185>
topaz sinewBOT
@carmine pelican Has your question been resolved?
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@carmine pelican Has your question been resolved?
gloomy ivy
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acoustic tangle
Why is the power 3/2
\begin{align*}
L^{\frac23} &= 120 \
\left(L^{\frac23}\right)^{\frac32} &= \left(120\right)^{\frac32} \
L^{\frac23\cdot\frac32} &= 120^{\frac32} \
L^1 &= 120^{\frac32} \
L &= 120^{\frac32} \
\end{align*}
thorny flameBOT
A Lonely Bean
topaz sinewBOT
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nimble crown
am I going insane or does sin(pi/4) make 1/sqrt2 not sqrt(2)/2
topaz sinewBOT
am I going insane or does sin(pi/4) make 1/sqrt2 not sqrt(2)/2
drifting swift
sqrt(2)/2 and 1/sqrt(2) are one and the same
or do you perhaps wish to claim they aren't
junior acorn
am I going insane or does sin(pi/4) make 1/sqrt2 not sqrt(2)/2
sqrt(2)/2 = 2^{1/2 - 1} = 2^{-1/2} = 1/sqrt(2)
junior acorn
because I'm looking at this to mark my past test paper and I used the 1/sqrt2
it'll still be correct
junior acorn
sqrt(2)/2 = 2^{1/2 - 1} = 2^{-1/2} = 1/sqrt(2)
.
ionic oar
Wouldn't it be faster to simply factor x^2 out from the denominator and sub 1/x^2 as u
junior acorn
they're the same thing
nimble crown
Wouldn't it be faster to simply factor x^2 out from the denominator and sub 1/x^...
yea but these tests you can't let u = something, it has to be given
if I could i would, it would make my life 10x easier
junior acorn
Wouldn't it be faster to simply factor x^2 out from the denominator and sub 1/x^...
what
no
nimble crown
well thanks for the help
ionic oar
$- \frac 12\int_0^1 \frac{-2x^{-3}\dd x}{(1 + x^{-2} )^{\frac 52}} = \frac 12 \int_{1}^{\infty} \frac{\dd x}{x^{\frac 52}}$
thorny flameBOT
jan Nejon
ionic oar
Would I want a job where Im forced to follow procedure
nimble crown
¯_(ツ)_/¯
ionic oar
Where I'm not allowed to express creative freedom and use the beauty of math to simplify processes
🎨
nimble crown
you could go into the highest level of math in my school system
they let you do anything there
even make up stuff for u=
but anyways
thanks for the help jan
topaz sinewBOT
junior acorn
**jan Nejon**
no
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ionic oar
Can you find the error 
topaz sinewBOT
Can you find the error <:thonk:406575732563902485>
topaz sinewBOT
ionic oar
Ah got the error it was in the limits
It would be from 2 to infinity
Now it works fine
junior acorn
yeah
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warped bane
Long Polynomial Division
topaz sinewBOT
Long Polynomial Division
graceful lynx
2/3x
olive thicket
there is horner way
olive thicket
oo you are using the normal one
warped bane
yeeee
olive thicket
alright wait
warped bane
my prof havent teach it yet
graceful lynx
its divide multiply subract bringdown
2x^3-x^2-(3x^2+1)(2/3x) =0x^3-x^2-2/3x
warped bane
can u explain we arent on this topic yet so idk
graceful lynx
graceful lynx
He divides by 3x^2+1
olive thicket
this one
graceful lynx
He divides by 3x^2+1
Not by 3x^2+x+1
warped bane
1x-2 right?
olive thicket
@graceful lynx but our answer is quite different
i'll do number 2 but in other way
horner way
it is more simple
olive thicket
olive thicket
horner method
warped bane
we cant use it yet T-T teach havent teach it yet so were stuck with long division
olive thicket
yeah
warped bane
shit hurts
graceful lynx
we cant use it yet T-T teach havent teach it yet so were stuck with long divisio...
Feels bad
warped bane
how should i divide this using long division?
warped bane
no worries bro thanks for your help
olive thicket
sure man
valid nest
What's the answer to this?
topaz sinewBOT
What's the answer to this?
mild hearth
!status
topaz sinewBOT
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
@valid nest Has your question been resolved?
valid nest
e?
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@valid nest Has your question been resolved?
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neon iron
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neon iron
how does me do
stucc on finding the inverse of f(x)
cos(what) + 3(what) = x
not sure what what could be
main shale
I don't think you can find the inverse in terms of elementary functions
but the "differentiable" part of the question hints that you might have to use the inverse function theorem, or some other result like that
neon iron
what 😭
but this problem comes shortly after learning the chain rule in the ap calc ab textbook
sad
.close
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golden rain
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golden rain
Can we ??
golden rain
r u asking if u can distribute -x into the parentheses
Yes but like
steady escarp
what is the original problem?
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neon iron
Hello, I need to find FG, HG, FG , I got FH
topaz sinewBOT
Hello, I need to find FG, HG, FG , I got FH
hybrid coral
Hello, I need to find FG, HG, FG , I got FH
observe triangle FEH
neon iron
oke we got gfh 15
i cant use trigono i need 90
degree triangle
yk
@hybrid coral any ideas?
hybrid coral
triangle feg is 90
neon iron
yuh
neon iron
owh
wait
sin 30 = 8 / FG
right?
and then sin 30 = 1/2
so 1/2 = 8 / FG
and FG would be 16?
hybrid coral
use tan
obtuse remnant
permutation equation
let x be
1 2 3 4
a b c d
topaz sinewBOT
permutation equation
let x be
1 2 3 4
a b c d
obtuse remnant
this is where I got, whatever I do, it just blocks me everywhere
I observed that the solutions are x=e, x=A, x=A^2
But I need to prove them
Where e is
1 2 3 4
1 2 3 4
topaz sinewBOT
@obtuse remnant Has your question been resolved?
exotic arch
So, I'm solving this problem
topaz sinewBOT
So, I'm solving this problem
exotic arch
"Find the module and argument of the complex number z, for which: z=1-i"
everything is going good, find the module being sqrt(2)
and for me the easiest way to find the argument is to use tan
tan being a/b
so I get 1/(-1)
now I look for what it equals to in radians
it equals to 90 degrees or pi/2
but my textbook says it's 7/4*pi which is 315 degrees
and I'm confused as to why
keen venture
,calc tan(pi/2)
thorny flameBOT
Result:
1.6331239353195e+16
Oh it actually spits out a number lol.
exotic arch
truly a wonder to behold
odd jacinth
You can plot it in your head, it becomes more obvious that the angle is -pi/4 so the argument is 7/4 pi
C is the same as R^2 when you think about its geometry
1-i is (1, -1)
exotic arch
I still don't really understand
how do I get to 7/4 or 315 degrees
I get that it's (1, -1)
but idk
odd jacinth
Draw it
Draw a graph
-i is the point at a 3/2 pi angle
1-i is further
The angle is half a right angle with the x-line
-45° = -pi/4 = 7pi/4 as argument
odd jacinth
And if you draw it, you'll also see that it gives tan(argument) = -1
It becomes way more obvious on a drawing and by drawing the right triangle and everything
exotic arch
And if you draw it, you'll also see that it gives tan(argument) = -1
this
for a=1
b=-1
odd jacinth
Yeah now place (1, -1)
obtuse remnant
.reopen
odd jacinth
And draw a line from the origin to it
exotic arch
And draw a line from the origin to it
odd jacinth
Link it to the x-line at the right border of a
And to the origin
It's a right angled triangle
So tan(argument) = a/b
= -1
exotic arch
odd jacinth
And also to the right border of a
exotic arch
yeah?
exotic arch
now I see
odd jacinth
The argument is the angle of the triangle glued to the origin
So tan(argument) = a/b = -1
nova orbit
topaz sinewBOT
nova orbit
Could someone explain how this is dividing the denom and numerator by the highest power of n that occurs in the denominator?
bold nebula
the denominator is n + 1
so the highest power of n is 1
here
and this is why they divided by n both
bold nebula
n = n^1
bold nebula
yes, then you would divide by n^2
bold nebula
you are supposed to divide both the numerator and the denominator by n
the numerator is n
n divided by n is 1
the denominator is n + 1
nova orbit
ohhhhh
bold nebula
n+1 divided by n is (n+1)/n = 1 + 1/n
nova orbit
ayo ima write that down
thorny flameBOT
Modus
bold nebula
.close
topaz sinewBOT
nova orbit
.reopen
topaz sinewBOT
✅
bold nebula
yes
nova orbit
and the extra +1 doesnt matter?
bold nebula
it doesn't because as you can see
if n tends to infinity
2/n^2 tends to 0, and generally any positive number at the top won't change it
nova orbit
Gotchya
bold nebula
even 10000000/n^2, because 100000000 compared to the infinity is nothing
topaz sinewBOT
nova orbit
.reopen
topaz sinewBOT
✅
nova orbit
Hmm one sec
Ehhh I dont understand why num and denom got flipped
I know n can be sqrt of n^2
bold nebula
What I've typed is only for the denominator
bold nebula
I know n can be sqrt of n^2
simply, e.g.:
5 = sqrt(25) = sqrt(5^2)
6 = sqrt(36) = sqrt(6^2)
10 = sqrt(100) = sqrt(10^2)
etc.
nova orbit
oh gotchya
Could you also explain to me what "the numerator is constant and the denominator approaches 0" means?
I saw a graph of convergant and divergent but aye aye aye
One approaches 0 as something approaches infinity or somethin like that, one doesnt
I assume its divergent because the num is always 1, and the denom is approaching 0.
which is something like 1/0
bold nebula
ye smth like 1/0
it's indeterminate, but here you can imagine it as 1 divided by something really really small
bold nebula
limit is infinity and this is why we can call it divergent
nova orbit
If we solved the equation like above and it goes past the limit, what does that mean then?
Trying to know the difference between divergent and convergant while learning what a limit as something approaches something else actually means/looks like
Pretty sure diverge just means difference, or divide, converge obv means come together so I'm assuming convergant means as x aproaches whatever, ? is convergant if points get closer together)
.close
nova orbit
How does -1^n/n turn to 1/n?
nova orbit
Well I know its absolute value but its kind of confusing how theres a 1 above
My algebra is not so great
wary tundra
topaz sinewBOT
dense rain
Technically the fundamental theorem of algebra and conjugate root theorem implies there is only 1 valid answer out of those choices without solving anything.
dense rain
Do what they said, find the x's which cause division by zero
wary tundra
would be either c or d correct?
odd jacinth
it is one of them yes
wary tundra
im not sure how the 5 would apply so I assume its C?
odd jacinth
it's c
next sinew
sorry i didnt realise i was in a different channel
odd jacinth
when x < 0, |x| = -x
so on x < 0 side, you draw -x, on the other, you draw (x-2)²
,w plot (x-2)²
thorny flameBOT
odd jacinth
which looks like that, because it's a parabola with a minimum at x = 2
next sinew
,w plot (x-2)²
do i need a table of values as well?
odd jacinth
you can do it if it helps you, but you can just see some values from head
topaz sinewBOT
@next sinew Has your question been resolved?
odd jacinth
the parabola should go to 4 at x = 0
so arrange your y-scaling a bit
but yeah the shape is essentially it
odd jacinth
,rotate
thorny flameBOT
odd jacinth
the parabola isn't defined on the left side of x = 0
it should hit 4 precisely at x = 0
topaz sinewBOT
next sinew
.reopen
topaz sinewBOT
✅
next sinew
is this correct?
also is there anything else I need to do to it? bc I asked another person and they said I needed dots or something to fit the original question
odd jacinth
you can put a full dot at (0, 4) and an empty dot at (0, 0) to say that for 0 it's the 4
next sinew
so now it’s finally done?
odd jacinth
yeah
odd jacinth
domain and range are intervals
odd jacinth
are your answers intervals ?
odd jacinth
[1, 8] and [2.2, 5.3]
next sinew
howd you get those
odd jacinth
by looking at the drawing
and you can't even understand what to do if you don't know what's an interval
next sinew
it's like the space between them right?
next sinew
<https://en.wikipedia.org/wiki/Interval_(mathematics)>
also for c) is it [4, 2] [5, 5] [3, 7]?
odd jacinth
no, c) are the x coordinate of intersection points, so 2, 5, and 7
odd jacinth
check the definition of range of a function
when you plot a function, the y axis are the values the function takes
next sinew
so the range is all the possible y values
next sinew
i think i get it now
so it's the first far right red dot (2.2) and the far left red dot (5.3)?
next sinew
oh i am totally wrong then
odd jacinth
it gets a bit higher than 5.5 right there
2.2 is the far right y value
your left green point is at y = 3
which has nothing to do with the question
odd jacinth
more like this yeah
odd jacinth
a) [1, 8]
b) [2.2, 5.6]
again, these are intervals
odd jacinth
answer for d) is [1, 2[ and ]5, 7[
next sinew
can you show me step by step how you got there?
odd jacinth
step 1: I see on the graph that f is below g on the interval [1, 2[
step 2: I see on the graph that f is also below g on the interval ]5, 7[
step 3: f is above g on the rest so there's nothing else to add
next sinew
i understand now. thanks
topaz sinewBOT
@next sinew Has your question been resolved?
next sinew
so how would i amend it?
odd jacinth
f(x-3) = (x-3)²+(x-3)+1 = x²-6x+9+x-3+1 = x²-5x+7
odd jacinth
f(2) = 4+2+1 = 7
so g(f(2)) = g(7) = 4
next sinew
i still dont understand sorry. are there steps in between those? like what do you do with the exponents etc?
next sinew
it's complicated 
topaz sinewBOT
next sinew
.reopen
topaz sinewBOT
✅
next sinew
(ignore the f at the bottom)
odd jacinth
dude I gave the solution how is this what you got lmao you can just read what I wrote
next sinew
ye im saying i wrote down what you wrote
next sinew
ye im saying i wrote down what you wrote
im still a bit confused tho
glossy fable
what are you confused about
next sinew
what are you confused about
f(2) = 2^2 + 2 + 1
= 4 + 2 + 1 = 7 makes sense
but then im not sure how i go from there to the final answer
glossy fable
you plug in 2 in f(x) and that gives you the "x" value for g(x)
its just a matter of plugging in numbers
next sinew
so the x value for g(x) is 7?
topaz sinewBOT
@next sinew Has your question been resolved?
next sinew
so the x value for g(x) is 7?
ok ive solved it
final question before i go to bed:
@glossy fable
glossy fable
what does it mean by "invertible"
next sinew
idk this is all the context i found
thats why im asking in this channel lol
maybe it means like a horizontal line test?
topaz sinewBOT
@next sinew Has your question been resolved?
topaz sinewBOT
topaz sinewBOT
Available help channel!
jovial ingot
How to compute this integral? I tried it in polar coordinates but still garbage…
jovial ingot
Polar: $$\int_{0}^{\frac{\pi}{4}}\ln(\sec^2(\theta)+1)d\theta$$
thorny flameBOT
(ᗜ ˰ ᗜ)
jovial ingot
<@&286206848099549185>
sweet shard
Polar: $$\int_{0}^{\frac{\pi}{4}}\ln(\sec^2(\theta)+1)d\theta$$
Show how you got this
jovial ingot
Show how you got this
2r/(1+r^2), r from 0 to sec(theta)
tough cargo
his polar form seems right
I tried to do it in my head tho so I'm not ruling out a dumb mistake
jovial ingot
garbage…any other ways to do that double integral? Or…Can this anti derivative evaluate to 0.63951
craggy haven
i'm not sure on those bounds
tough cargo
Yeah cus if u do half the square u multiply by 2 from symmetry, but after integrating wrt to dr the integrand has 1/2
Anyways if there's a better way it's not by FTC
topaz sinewBOT
@jovial ingot Has your question been resolved?
sweet shard
Did you try arctan yet
topaz sinewBOT
obtuse nest
$$\forall x, y, z \in S.\ x R y \land y R z \implies x R z$$
$$\forall x, z \in S.\ (\exists y \in S.\ x R y \land y R z) \implies x R z$$
obtuse nest
S is a set, R is a relation on S
topaz sinewBOT
topaz sinewBOT
Available help channel!
sleek haven
Hello, just checking that this is correct.
topaz sinewBOT
Hello, just checking that this is correct.
sleek haven
It's taking a moment to load
pallid phoenix
Do u know what c means in y = mx + c?
whole wagon
How to find and prove what's limit of (1+1/n)^n sequence?
topaz sinewBOT
How to find and prove what's limit of `(1+1/n)^n` sequence?
whole wagon
It seems to stay close to 2.71 as it reaches inf but have no idea why 2.71 exactly
drifting swift
ever heard of euler's constant? a.k.a. e?
topaz sinewBOT
@whole wagon Has your question been resolved?
whole wagon
ever heard of euler's constant? a.k.a. *e*?
Yes, but how to prove it
drifting swift
lim (1 + 1/n)^n is one of the definitions of e
if you have a different definition on hand then perhaps you could try to prove that one is equivalent to this one
whole wagon
Well, so I can just say that e is upper limit of the sequence?
drifting swift
"upper limit"?
whole wagon
Actually question was if it had upper and lower limit
Lower limit would be a(1) 2^0.5
And after that it raise to reach e
But it can never reach e as it's unending, right?
So I can say that limit of sequence is sqrt(2) and e
Wait I got mixed up
It's not sqrt it's 2 and e
drifting swift
do you mean like limsup and liminf...?
topaz sinewBOT
topaz sinewBOT
Available help channel!
hollow hazel
Can I please get some help with this?
topaz sinewBOT
Can I please get some help with this?
hollow hazel
I already think I got one direction
Idk how to show the other
Or if this is the best way to show it
topaz sinewBOT
@hollow hazel Has your question been resolved?
topaz sinewBOT
topaz sinewBOT
Available help channel!
royal sky
topaz sinewBOT
royal sky
isnt the h at the bottom supposed to be transfered to the top top?
but in this exercise they put it to the middle
after multiplying it
drifting swift
$\frac{a/b}{c} = \frac{a}{bc}$
thorny flameBOT
AnnGhost
topaz sinewBOT
topaz sinewBOT
Available help channel!
unborn bone
topaz sinewBOT
drifting swift
,rccw
thorny flameBOT
unborn bone
Is this correct, and if so, how do I get there?
modern rock
well i dont really know but 2 sqrt(r/g) is equal to (2 sqrt r)/sqrt g
then u times both sides by sqrt g
and then u can use some exponential rules to get the rest
topaz sinewBOT
@unborn bone Has your question been resolved?
fathom sinew
Hi how would i prove the first limit
topaz sinewBOT
Hi how would i prove the first limit
fathom sinew
I mean i know what the definition of a limit is, but i don`t see how i can prove that function has a limit P
strange whale
Wdym you prove the first limit
strange whale
You don't have to prove it, it's the hypothesis for the proof
strange whale
If you know the first limit = P, then the second limit = KP
That's what you have to show
fathom sinew
but tbh i have no idea how to prove it
like
how
can i say its part of the limit law
or do i use k = any real number and p is any real number
mellow cypress
what is the definition of a limit given to you?
strange whale
can i say its part of the limit law
Nah, imagine you didn't knew that the limit law was true
mellow cypress
is it epsilon delta definition?
fathom sinew
what is the definition of a limit given to you?
yh i think
fathom sinew
is it epsilon delta definition?
yh i think
epsilon greater than zero, delta greater than zero such that so and so
mellow cypress
then just take modify your initial epsilon so it works nicely for K\phi(\theta)->KP
fathom sinew
phi is that circle with vertical line?
mellow cypress
yee
fathom sinew
alr
fathom sinew
then just take modify your initial epsilon so it works nicely for K\phi(\theta)-...
dont know how to do that, sos.
wait
|phi(theta) - P| < epsilon
modify that?
mellow cypress
[\forall \frac{\varepsilon}{|K|}>0 \exists \delta:(|\theta-\theta_0|<\delta\Rightarrow |\phi(\theta)-P|<\frac{\varepsilon}{|K|})]
agreed?
mellow cypress
thats what we are doing
if this is true for all positive real numbers e, its also true for e/K right?
fathom sinew
yh
mellow cypress
oh maybe I should replace K with |K|
fathom sinew
maybe
mellow cypress
yh
so thats it, the above statement is true, now multiply by |K| both sides of the inequality
fathom sinew
then I get |K||phi(theta) - P| < epsilon
thorny flameBOT
mellow cypress
yea
fathom sinew
hmhm so what next
mellow cypress
thats it
fathom sinew
i never heard of the edge case, is that in first year math?
i can just say K does not equal to = 0 cuz its not part of the set R
oh wait
thats N
mellow cypress
because K is any real number, K=0, the modified e/|K| will be undefined
fathom sinew
ye