Bcos it doesn’t matter what you get from the first deck
#help-39
naive zinc
It is a classic result of inclusion-exclusion principle
shut flicker
can you elaborate please
naive zinc
I have to go running. The term I believe should be derangement.
shut flicker
okok il have a look thanks
hot timber
do you wanna check the concept or shall I continue the simple explanation
shut flicker
sure continue
hot timber
so we do this overall 52 times
so the cumulative probability is (51/52)^52
you would get a similar answer through that concept as well
do check it out
hot timber
so the cumulative probability is (51/52)^52
‘Tis about 36 percent btw
pearl pondBOT
shut flicker
.reopen
pearl pondBOT
✅ Original question: #help-39 message
shut flicker
so the cumulative probability is (51/52)^52
wait can you explain why its /52 each time
like since the first deck doesn't matter, the second we pick theres a 51/52 chance its not matching, then the third card can be whatever, then the fourth card there's a 50/51 chance that its not matching, im pretty sure im wrong somewhere tho
pearl pondBOT
hot timber
wait reopen
but then the probs would be 1/52
no it wouldn’t
that’s incorrect
I think you should use derangements
mb I’m not thinking straight
I’ve not had good sleep for a while due to exams
pearl pondBOT
Available help channel!
dusty jungle
f: N->N, f(n+1) > f(f(n)) for all n in N. Prove that f(n) = n
dusty jungle
i can prove f(n) = n if i know that f(n) >=n. (prove f is increasing then get that f(n) < n+1)
i tried to do an induction with base case f(1) >=1, f(f(1))>=1, then assumed the same for k
f(k+1) > f(f(k)) => f(k+1) >= k+1
but i cant prove f(f(k+1)) >= k+1
please dont give me full answers
pearl pondBOT
@dusty jungle Has your question been resolved?
dusty jungle
<@&286206848099549185>
surreal verge
Hmm have you considered trying to find a contradiction?
dusty jungle
yeah i tried
didnt get far
well i know that if f(f(k+1)) <= k
then using my induction hypotheses, f(f(f(k+1))) >= f(f(k+1)), f(f(f(f(k+1)))) >= f(f(k+1))
surreal verge
One thing that might help is that for all natural numbers n,k n > k implies that k +1 >= n
dusty jungle
f(k+1) > f(f(k)) => f(k+1) >= k+1
yeah i used that here
naive zinc
Hint: Prove by induction on n that f([n,infty)) is contained in [n, infty)
dusty jungle
so prove f(n) >= n?
naive zinc
It will deduce this
dusty jungle
wait thats actually such a smart way to do it
f(n+1) > f(f(n)) , but f(n) >=n so f(f(n)) >= f(n) >= n
so f(n+1) >=n+1
by induction
thats exactly what i needed
naive zinc
Not by me. I found the post on MSE, and I can’t send you the link because once you open it you will see the whole proof. So I decided to take the gist of it.
naive zinc
Exactly
dusty jungle
f is increased
*ing
which means f(a)>f(b) <=> a>b
so f(n+1) > f(f(n)) => n+1>f(n)
so f(n)=n
naive zinc
Yeah
dusty jungle
Not by me. I found the post on MSE, and I can’t send you the link because once y...
damn but thanks for getting me the idea
ive never seen induction like that
naive zinc
It’s brilliant. I can’t come up with such solutions. Never have trained myself for competition questions
It says it’s a IMO problem
dusty jungle
yeah its IMO 1977 tho
naive zinc
I see
dusty jungle
It’s brilliant. I can’t come up with such solutions. Never have trained myself f...
off topic question tho, are competitions that important to get the opportunity to study higher math
naive zinc
I don’t think it’s mathematically directly relevant. Though indirectly it to some degree reflects you are very devoted to math.
So no in general but not 100 percent no
pearl pondBOT
restive garnet
true
honest oyster
Look for triangles with side BC
naive zinc
Given a triangle ABC, M on AB, N on AC. MN parallel to BC, triangles ABC and AMN being congruent will gives you MN/BC=AM/AB=AN/AC. this case it’s all 1/2. use it
Use this on your ABC and your ABD
naive zinc
Given a triangle ABC, M on AB, N on AC. MN parallel to BC, triangles ABC and AMN...
This is not letters in your diagram, my bad
Given a triangle XYZ, P on XY, Q on XZ. PQ parallel to YZ, triangles XYZ and XPQ being congruent will gives you PQ/YZ=XP/XY=XQ/XZ. this case it’s all 1/2. use this on your ABC and ABD
Letters changed
I gave you a statement about a pair of triangles (XPQ, XYZ). Use this on your (AME, ABC) and (BMF, BAD)
Exactly
Do again on (BMF, BAD)
Exactly
Solved
Back to my statement. XP/XY=XQ/XZ implies PQ parallel to YZ
Let me think why
pearl pondBOT
pearl pondBOT
Available help channel!
rare holly
for comparing the coeficients they choose OP and OA+AP
pearl pondBOT
for comparing the coeficients they choose OP and OA+AP
rare holly
what if i choose AO+OP would that be wrong?
tough ore
what if i choose AO+OP would that be wrong?
What do u mean exactly
rare holly
to compare the coefiecints
tough ore
Wait
tough ore
to compare the coefiecints
How will u get AP tho
rare holly
hello how do i find b
pearl pondBOT
hello how do i find b
glass meadow
there are two solutions to this
lilac ocean
what does f(bx) do to a graph, where should the intersection points be
rare holly
<:pandahmm:858834883760750602> there are two solutions to this
how do i find them
rare holly
what does f(bx) do to a graph, where should the intersection points be
whatever the angle is it is being devidid by b
rare holly
yh
glass meadow
Where is the first peak of sin(x)?
rare holly
Where is the first peak of sin(x)?
90,1
glass meadow
And where is the first peak of this graph?
rare holly
90,2
glass meadow
... ?
glass meadow
Ok so this graph goes down from (0,0), right?
rare holly
yh
glass meadow
Ok, how?
glass meadow
You should know how to reflect any function f(x) about the x-axis
rare holly
y is -2 we can time it by -1
glass meadow
Ok so then what's the value of a?
rare holly
-1
rare holly
Ok so then what's the value of a?
but is 2 aswell?
glass meadow
glass meadow
Ok, then let's keep a=2 and go back a step
This graph goes down from (0,0) instead of up like sin(x), how can you fix that? Reflecting about the x-axis is not an option
rare holly
This graph goes down from (0,0) instead of up like sin(x), how can you fix that?...
why
glass meadow
... because you arrived at the conclusion that a must be -1 but also 2 at the same time
rare holly
if i was in a exam i would not think of both ways
glass meadow
I'm not sure what you mean by that
glass meadow
Well, think of the graph of sin(x)
Yes if you follow its graph from 0, it goes up, but there is another way you can go
rare holly
Yes if you follow its graph from 0, it goes *up*, but there is another way you c...
down?
glass meadow
I mean... maybe? How?
rare holly
the only way i can think of it is by reflection in the x axis
glass meadow
Look at it closely...
I really can't say any more without solving the thing for you
toxic lichen
Look at it closely...
showing a graph with radians might be a bit unhelpful
pearl pondBOT
glass meadow
showing a graph with radians might be a bit unhelpful
Fair... the point is more about the overall shape though
pearl pondBOT
Available help channel!
random hatch
pearl pondBOT
random hatch
can somone help sovle this for me
lone mesa
Did you graph it? The washer method should work
pearl pondBOT
@random hatch Has your question been resolved?
random hatch
Did you graph it? The washer method should work
I did graph it
But i dont know if i should solve by rotating around x axis
or y axis
because it doesnt tell me to rotate around an x axis or a y axis or a line
it just says bounded
lone mesa
It says about y=6, so it should be most likely around the line y=6
random hatch
wait so would the cross sections be perpendicular to the x axis?
lone mesa
Yes, the region is being rotated around the horizontal line 𝑦=6. That means we slice the region vertically (along x), and each slice forms a washer whose area is based on vertical distances from the curves to the axis
random hatch
But how do i find the volume
can you do it and i wantto see
because
first the paabola thing shape opens up forever
so like how do we get the volume of that
and also like idk what the limits would be
humble root
Hi! Before all else, did you try drawing the region?
humble root
I did graph it
Okay well you have,
humble root
But how do i find the volume
Well, we want to rotate over the line y = 6, which is a horizontal line.
We first need to decide what formula we are going to use to find the volume. Have you decided?
random hatch
No i dont know what to use
Im new to this i just started learning this yesterday
I know theres the pi(r)^2 or pi(r^2-R^2)
lone mesa
We first need to decide what formula we are going to use to find the volume. Hav...
We have been through all of this already
humble root
I know theres the pi(r)^2 or pi(r^2-R^2)
Good! These formulas should be useful.
So, we want to rotate this region in red here using the formula you mentioned.
random hatch
1 curve 2 lines
humble root
1 curve 2 lines
Good! One curve only.
humble root
I know theres the pi(r)^2 or pi(r^2-R^2)
Which of these uses only one curve?
humble root
but what about the inside part do we only care about the shaded area?
Yep we do.
We can bound the shaded area using the limits of integration (which will find in a bit).
random hatch
so we are findnig the integral from 0 to 1? or 0 to 6
humble root
so we are findnig the integral from 0 to 1? or 0 to 6
Wait wait
random hatch
huh
humble root
Before we rotate, here's the deal
There is one caveat with the integral pi(r)^2. That caveat is that the volume of rotation has axis of rotation at y = 0 (since r measures the distance from the origin to the curve).
humble root
The problem here is that the integral is not rotated at y = 0.
So we need to fix this problem first.
How might we change it so that the axis of rotation becomes y = 0? Hint: We want to apply a transformation to this entire graph. What transformation might be useful here?
humble root
moving the lines down 6 notches
Good!
That's exactly rightm
So, we want to move the curve six units down so that the axis of rotation becomes y = 0
Would you agree that you get something like this?
Note that the curve now has equation y = x^4 - 6
random hatch
oh yeahhh
wait but like
im not allowed to use graphing calculators on my exam tommroow
how do i identify what if i need to do this
humble root
I mean
random hatch
just if the ling we are bound to isnt zero?
humble root
Draw the graphs manually :)
random hatch
wether if its the x axis or y axis
Uh i havent had enough practice to imagine the graphs in my head
humble root
I mean on paper lol
Well in any case
If you don't have enough graphing practice in general,
humble root
just if the ling we are bound to isnt zero?
Then you need to do this type of transformation whenever the axis of rotation is not y = 0 or x = 0, so yes.
humble root
If you don't have enough graphing practice in general,
(For future reference, it is very important to know how to graph!)
random hatch
yeppp i know...i just forgot everything over summer break from when i was in pre calc
now im in calc and im confused on everhtihign
humble root
Oof
lone mesa
this gonna be a one long question
humble root
It takes some time to understand the concepts so
humble root
this gonna be a one long question
Agreed, we havent even gotten to the expansion bit 😭
humble root
Anyway, now you might be asking - what are the limits of integration?
random hatch
yeah
humble root
Well, let me ask you this: Using the formula pi(r)^2, what does this formula simplify to, if you know that y = r = x^4 - 6?
random hatch
wait so the integrand is definitly pi(x^4-6)?
humble root
wait so the integrand is definitly pi(x^4-6)?
Squared, yep.
random hatch
oh yeah the squear part
humble root
So! If you know that the integral has x in it,
What do you think the bounds of the integral are?
random hatch
and since we are inegrating with to y we have to cahnge it right?
oh
i dont understand
i thoguht the intergral was from -1 to 1
humble root
i thoguht the intergral was from -1 to 1
Good! No your exactly right
The area of the region starts at point A and ends at point B
If you find the x coordinates of A and B,
You get -1 and 1 respectively.
So, the integral is between those two values.
As in the integration limits.
random hatch
so the integral from -1 to 1 of pi(x^4-6)^2 with dx?
humble root
Yep, exactly.
plush bramble
are you supposed to give an exact answer
random hatch
no teacaher said its okay if i round it too but idk what i did wrong
okay let me get it wrong on purpose to get the answer
plush bramble
explain how you got -1 and 1 in your limits
random hatch
the asnwer was 54.733525342542
see she rounds it
oh cus thats where the curve intercrpts the line
the line that we dont rotate on
random hatch
wait what how you get int from 0 to 1
humble root
@plush bramble I don't think I did anything wrong, I might be missing something incredible obvious.
random hatch
my brain hurts
I have my exam tommrow i cannot understand this im cooked guys help me 😭
humble root
random hatch
huh
humble root
When you integrate about the origin,
random hatch
why we finding the blue part
humble root
You also get this volume here as well.
Exactly so,
Its an uninteded part of the area
So actually you need to use the r^2 - R^2 formula, my mistake!
random hatch
wait huhhh
humble root
Okay let me explain
random hatch
so r1 is going to be the y=-5
humble root
We only want this tiny red portion here, right?
humble root
so r1 is going to be the y=-5
Yep!
random hatch
yes
humble root
yes
The problem is,
If we directly use pi(r)^2 here,
That will be the entire distance
From the curve y = x^4 - 6 to y = 0
We don't want this though
We want only the distance from y = x^4 - 6 to y = -5.
So we have to subtract it out.
random hatch
so doing top minus bottom or bottom minus top now?
humble root
random hatch
wait i thought x^4-6 was the bottom function
humble root
Oh yeah but its squared so
x^4 - 6 when squared becomes larger then y = (-5)^2
I meam
It doesn't really matter how they are oriented to be honest, you can just put an absolute value sign on the final integral.
random hatch
ohhhhhhh so algebraically its bigger so we use that as the top function?
oh
hahahaha
random hatch
okay so igot this one wrong and i cant redo it so im gonna generate a new question and try to solve it
humble root
Yeah that's my mistake sorry
random hatch
humble root
I should say
random hatch
can you find out the answer and then when im finished ill see if im correct
humble root
Do r^2 - R^2
humble root
can you find out the answer and then when im finished ill see if im correct
Sure!
humble root
As a check, the number of your answer that is before the decimal places should be a multiple of 9.
random hatch
i got 27.5967354668
humble root
i got 27.5967354668
Yep that looks about right
humble root
Well,
Remember that r just means radius
So
In other words, r means distance.
So,
What is the distance between the axis of rotation y = 3 and the curve y = x^8?
random hatch
uh y=x^8?
humble root
Hint: What is the distance between y = 3 and y = 0?
random hatch
-3
humble root
oh beecause i counted the space...
Oo okay well
Whats the distance from y = 100 and y = 500?
random hatch
ooooo dang im slow
humble root
What is the distance between y = 3 and y = x^8?
humble root
ooooo dang im slow
You're fine :)
random hatch
x^8-3
humble root
Just don't be slow on the exam lol
random hatch
I try 😭
humble root
So whether you remember to shift the grapb down
random hatch
ohhh so that was our equation
humble root
ohhh so that was our equation
Exactly.
random hatch
so
1 more, this looks hard beuase of cosign 😭
wait is it just pi/14?
no im slow
we know that we are integrating from 0 to pi/14 right?
I got this
humble root
I got this
Looks about right 👍
lost steppe
Maybe I don’t get the intuition behind it, please could I have some help
lost steppe
What do I do with the information
So I see that 10^k is congruent to 4mod6
I have done thus far
But I don’t rly get what this tells me
Like the general idea behind it ig
pulsar flax
if you write 10^100 a 6c + r then
10^(10^100) = 10^(6c + r) = 10^r mod 7 by fermat
lost steppe
Ohhhhh my
Tysm 😭
Nah that’s frustrating, how did I not spot it
It just felt oveeprwhelkmg w the information
.close
pearl pondBOT
lost steppe
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pearl pondBOT
✅ Original question: #help-39 message
pearl pondBOT
pearl pondBOT
Available help channel!
cedar harbor
how does line 1 become line 2
pearl pondBOT
how does line 1 become line 2
cedar harbor
just have problem foiling the polynomial
cinder flower
awful name
pearl pondBOT
@cedar harbor Has your question been resolved?
pearl pondBOT
@cedar harbor Has your question been resolved?
naive zinc
Everything inside <> is linear
pearl pondBOT
topaz pecan
Hey there! Need some help with understanding this division
pearl pondBOT
Hey there! Need some help with understanding this division
nocturne grail
Hey there! Need some help with understanding this division
which one
topaz pecan
13 and 14 mainly
topaz pecan
Mhm!
nocturne grail
hm
so
verbally, tell me what you understand from
n * k
(a bit off but just making sure you get get get what multiplication is)
nocturne grail
verbally, tell me what you understand from
n * k
and by 'tell me what you understand' i mean, describe this notation to me. what does it say?
what is it trying to denote?
topaz pecan
we can start with smth easier, what is 2 * 3 trying to denote?
2 x 3? Or the exponents
nocturne grail
oh yeah 2 x 3
but- since x is what we usually call our variables, in discord, we denote multiplication with *
and on paper too!
topaz pecan
and by 'tell me what you understand' i mean, describe this notation to me. what ...
In the best way u can describe it a way of adding but instead of going 2 + 2 + 2, 2 x 3 is much simpler
topaz pecan
but- since x is what we usually call our variables, in discord, we denote multip...
Ok, great!
topaz pecan
Thank you bro ^^
topaz pecan
Since it’s an exponent times a variable I assume it would be 5n?
nocturne grail
Since it’s an exponent times a variable I assume it would be 5n?
okay lets just get to the point of all this
basically what im trying to say is
multiplication is the process of repeated addition
as you said
instead of 2 + 2 + 2
we say 2 x 3
for the sake of simplicity!
so when we have
n x 5
we are trying to repeat addition n five times! n + n + n + n + n
which is simplifed to 5n!
nocturne grail
Since it’s an exponent times a variable I assume it would be 5n?
(correct)
topaz pecan
which is simplifed to 5n!
Right!
nocturne grail
aaaaaand if we have something like n x k
we are trying to denote
'n added to itself k times'
if you dont get it, its fine.
you'll need that like- 2 years from now!
alright
lets get onto exponents!
what do you think n^2 is denoting?
^ is how we denote 'raised to the power of' in discord
topaz pecan
what do you think n^2 is denoting?
N times itself once ?
nocturne grail
N times itself once ?
well... that's an interesting way to say it but yeah! correct.
just like multiplication being repeated addition for simplicity
there we go
instead of denoting n * n
we said its 'n multiplied by itself once'
so for the sake of simplicity
we said n^2
which means 'n multiplied by itself once'
ok
so if we have smth like
n * n * n * n
how do you think we'd do that
with exponentiation notation
hm, how about we rephrase with exponentiation is
It's repeated multiplication.
instead of saying
n * n * n * n * n
we say
'this is n, repeated 5 times with multiplication. and thus, we'll write it as n^5'
so if we have 7 * 7 * 7
we can say
'this is 7 repeated 3 times with multiplication, so 7^3'
are you with me?
topaz pecan
I think so, yes
nocturne grail
I think so, yes
ok lets start from the very beginning
we thought
2 + 2 + 2 + 2 +2 +2
looked VERY ugly and it took up TOOO much space
so instead of writing it like that
we decided to create a process, something we do to simplify this ugly expression.
and we named this process, 'multiplication'
we said, we're gonna use multiplication to represent repeated addition, just bc of how ugly it looks!!!!
topaz pecan
Right !!
nocturne grail
so there, we have
2 + 2 + 2 + 2 +2 +2
which is 2 repeated 6 times with addition.
so instead of saying
2 plus 2 plus 2 plus 2- EW TOO MUCH TALKING
we say
it is
2 TIMES 6
which means 2 added 6 times
I dont wanna use 'added to itself 6 times' bc that would be incorrect.
just know '2 added 6 times' means adding up six 2's
are you with me?
topaz pecan
Yes!
nocturne grail
perfect
so 6 added 7 times or 'adding up six 7's'
....
I didnt mean for that to happen 😭
I swear 😭
ok changing the two numbers
12 added 8 times
topaz pecan
I didnt mean for that to happen 😭
LOLLL
nocturne grail
LOLLL
how would you represent adding one hundred 8's?
you wouldnt dare to...
8+8+8+8+8... because that is toooo much
how would you denote it instead?
with our fancy x (multiplication)
nocturne grail
were you here the whole time? or are you doing something on the side?
unique ibex
x not ^
topaz pecan
were you here the whole time? or are you doing something on the side?
Just studying while reading this!
nocturne grail
Yoiud do 8^100, right?
yeah well- I'm not sure if this is a discord issue but...
it is 8 x 100
^ is used for something we didnt discuss yet!
topaz pecan
yeah well- I'm not sure if this is a discord issue but...
it is 8 x 100
Oh yeah sorry, that was on me lol
I see!!
nocturne grail
mhm mhm
so how'd you go on about adding two billion 2's?
(I'm being sarcastic... but if you want to, you can try. if you dont want to, we can move onto our next topic)
topaz pecan
2 x 2,000,000 ect
nocturne grail
hm
first of all, a billion is nine 0's. which is on me
no one deals with billions these days anyways
second of all, a billion is 1,000,000,000
not 2,000,000,000
ok just
dont bother with that example
its unrealistic
unique ibex
second of all, a billion is 1,000,000,000
you said two billion, so 2,000,000,000 would be correct
topaz pecan
adding 5 n's?
Denotino?
nocturne grail
you said two billion, so 2,000,000,000 would be correct
oh FUCK
topaz pecan
Denoting *
nocturne grail
I didnt- okay
@topaz pecan goodjob boo- the only mistake is that a billion is written with 9 zeros not 6
sorry for that
nocturne grail
It's 6am, excuse me.
topaz pecan
Oh right !!
nocturne grail
you can do this! know that n is just any other number... we just dont really know which one so we say n.
topaz pecan
representing 'adding 5 n's with multplication!
Ok, that would just be 5 x n, right?
nocturne grail
Ok, that would just be 5 x n, right?
YES!
absolutely
you've got the hang of it!
so if i say 'adding 10 n's with multiplication'
that's just 10 x n
adding 912830129381093821 n's is just
n x 912830129381093821
and so on!
ok
now
we grew up!
we became concerned with something that is just as ugly as 2 + 2 + 2 + 2 + 2 + 2
repeated... MULTIPLICATION!
topaz pecan
Right, I see!
nocturne grail
disgusting stuff that are like
2 x 2 x 2 x 2 x2
that again, takes too much space
so now we're creating a process to write ' 2 multiplied by itself 5 times'
that phrasing isn't very accurate, but we'll go with it.
alright!
so we decided to pick a fancier name for this operation
we called it, exponentiation!
so
instead of having this disgracful piece of shit that is 2 x 2 x 2 x 2 x2
we can say, 'this is 2 multiplied by itself 5 times' which translated to
2 TO THE POWER OF 5
now, idk why we used the term 'power'
but- english people did it and so will we
so 2 TO THE POWER OF 5 just means 2 x 2 x 2 x 2 x 2
also, order here matters A TON
saying 5 TO THE POWER OF 2 means 5 x 5
which is NOT equivalent to 2 TO THE POWER OF 5
but still... we didnt solve the redundancy problem of 'too much writing'
actually, before i get to the next point,
@topaz pecan what is 12 RAISED TO THE POWER OF 4 written like
in the disgusting form.
topaz pecan
actually, before i get to the next point,
<@847901979569291264> what is 12 RAISE...
Ok
Ok
Twelve to the four
In its worst possible form
Is 12 x 12 x 12 x 12
Right??
nocturne grail
YEAH
EXACTLY
so now we can get rid of the worst possible form by simplifying it!
now discord wont let me do this properly so give me a sec
wait
$12^4$
jolly parrotBOT
car
nocturne grail
ONG
topaz pecan
W
sly garden
car you are doing awesome
nocturne grail
1-
nocturne grail
W
alright! so remember, something RAISED TO THE POWER OF something
means we write the first something on the bottom-
actually
lets say we have a number called sarah
and we wanna raise it to another number called patria
when we want to write this
in the simplified form
we do
&(sarah)^(patria)&
bro
OH THE
$(sarah)^(patria)$
what the fuck?
ok lets rename our numbers... sigh
again
if we have a number, lets call p
and we wanna RAISE IT TO ANOTHER number called k
the way we'd write it is
$p^k$
jolly parrotBOT
car
nocturne grail
now notice something here
in the phrase 'p RAISED TO THE POWER OF k'
p came first, k came second
whoever comes first is the 'base'
is the number that's the other number is gonna be on top of
and the number that came second
is the 'exponent'
which is the number that we put ontop of the base
so
4 RAISED TO THE POWER OF 7 (which is 4 x 4 x 4 x 4 x 4 x 4 x 4)
4 came first, we put it as the base
$4$
unique ibex
what the fuck?
You need curly brackets
jolly parrotBOT
car
nocturne grail
7 came second, so we put it as the exponent
unique ibex
${\text{sarah}}^{\text{patria}}$
jolly parrotBOT
soup_norm
nocturne grail
${\text{sarah}}^{\text{patria}}$
I just wrote my first latex text 2 mins ago 😔 but ty!
wait a minute...
@topaz pecan are you- there...
topaz pecan
Yeah!! Just reading this out right now
nocturne grail
you might get the most benefit if you're active interactive with me! but as you wish
topaz pecan
Ah my fault then, sorry! I just didn’t know when to interject, I didn’t want to interrupt you🙏
nocturne grail
you def have the fulllll right to interrupt me if ANYTHING DOESNT make sense
nocturne grail
Ah my fault then, sorry! I just didn’t know when to interject, I didn’t want to ...
so- does any part of what i explained make sense to you?
oh my god i keep phrasing it incorrectly
did anything NOT make sense to you?
unique ibex
I just wrote my first latex text 2 mins ago 😔 but ty!
Yeah it can be a bit annoying but just use curly brackets in place of parentheses, and when you really need curly brackets, you can escape them by putting backslashes
topaz pecan
did anything NOT make sense to you?
No no it all made sense!!! Your doing great so far, all very helpful
nocturne grail
lovely!
nocturne grail
No no it all made sense!!! Your doing great so far, all very helpful
would you mind writing
7 x 7 x 7 x 7
in exponent form
for me
please 🥺
nocturne grail
would you mind writing
7 x 7 x 7 x 7
I'd like you rewrite this on paper and send me a pic
or paint if you're on pc or wtv.
nocturne grail
PERFECT!
okay now that you understand
multiplication and exponentiation
I shall redirect you to our king
our queen
our legend
our myth
the organic chem tutor!
topaz pecan
https://youtu.be/ZtTO8uUGy2Y
HOORAYY
nocturne grail
you'll watch this video and if ANY part confuses you, you come here and inform the people here!
pearl pondBOT
If you are done with this channel, please mark your problem as solved by typing .close
glacial mulch
the organic chem tutor!
mans helped me through hell
unique ibex
you'll watch this video and if ANY part confuses you, you come here and inform t...
Oh also
@topaz pecan once you ask people here, please describe what part of the video confused you
topaz pecan
<@847901979569291264> once you ask people here, please describe what part of the...
Sure!
unique ibex
Because it might be hours or days later that you come back to ask, and by then we may be sleeping or unavailable or something
So it might be a different person
topaz pecan
Thank you so much for all the help @nocturne grail, really appreciate it <: )
topaz pecan
Because it might be hours or days later that you come back to ask, and by then w...
Course !
unique ibex
And so you want to make sure that they can understand your question quickly, so give them any necessary context
nocturne grail
Thank you so much for all the help <@858810915663511553>, really appreciate it <...
no problem sir!
now...
!done
pearl pondBOT
If you are done with this channel, please mark your problem as solved by typing .close
@topaz pecan Has your question been resolved?
tawdry solar
,tex Find the sum [\frac1{\sqrt[3]{1} + \sqrt[3] {2} + \sqrt[3] {4}} +\frac1{\sqrt[3]{4} + \sqrt[3] {6}+ \sqrt[3] {9}} + \frac1{\sqrt[3] {9} + \sqrt[3]{12} + \sqrt[3]{16}}]
jolly parrotBOT
calvin
proper nova
hm
lucid anvil
difference of cubes maybe
try multiplying the numerators and denominators of each of those by something so that you can get a difference of cubes factorization on the denominator
for example $\frac{1}{x^2+xy+y^2}=\frac{x-y}{x^3-y^3}$
jolly parrotBOT
qianqian07
tawdry solar
,,\frac1{2^0 + 2^{\frac13} + 2^{\frac23}} + \frac1{2^{\frac23} + 2^{\frac 13}3^{\frac13} + 3^{\frac 23}} +\frac 1{3^{\frac 23} + 2^{\frac23} 3^{\frac23} + 2^{\frac43}}
jolly parrotBOT
calvin
tawdry solar
i was thinking something along the lines of this 😭
proper nova
i was thinking something along the lines of this 😭
?
tawdry solar
**qianqian07**
how does this help tho
proper nova
let's take the first fraction
if you rewrite the denominator, it's $\frac{1}{(\sqrt[3]{1})^2 + \sqrt[3]{1} \cdot \sqrt[3]{2} + (\sqrt[3]{2})^2}$
jolly parrotBOT
1 divided by 0 equals Infinity
proper nova
@tawdry solar
tawdry solar
Yeah
proper nova
you can do the same for each other fractions
proper nova
**qianqian07**
and then it's easier now
tawdry solar
Alr thanks
tawdry solar
and then it's easier now
So i was relatively close right
pearl pondBOT
Available help channel!
green dew
Let H = bary{(A,1),(D,-2)}
G = bary{(A,1),(D,-2),(C,5)}
Determine the set of points M of the geometric plane such that:
-3AM-2MD+MC and HM+5CM have the same norm
(all vectors btw)
naive zinc
Do you have the original question
a screenshot or something
Tons of unspecified terms appear in your question. I prefer to see the original
green dew
sure
thought that H,G,A,D and C being fixed points was enough
ABC is an equilateral triangle, I and J are the respective midpoints of [AC] and [BC], D is the image of A by translation of vector BA, and K is the point such that:
B = bary{(K,4),(D,-3)}
1)a) Show that K is the barycenter of A and D with coefficients that must be determined.
b) Construct D and K
2)a) Determine the type of triangle BCD
b) Determine the type of quadrilateral AKIJ
3) (this is what i stated in the help channel for H and G)
a) Show that G = bary{(H,1),(C,5)
b) Show that I,J and K belong to a same line.
c) (the problem that i wanted to verify the answer of)
pearl pondBOT
@green dew Has your question been resolved?
pearl pondBOT
@green dew Has your question been resolved?
naive zinc
Then what’s your answer (should be a line)
Or a point
I got a point anyway, (-6m+2u+5v, -6m+2u+5v)=(6m-(10/3)u-(25/6)v, 6m-(10/3)u-(25/6)v), m=AM, u=AB, v=AC, let m=xu+yv I got x=1655 (2(u,v)=(u,u)=(v,v) are cancelled out or assume them to be 1 in the first place)
I am very bad at calculation
naive zinc
And I probably miscalculated it
green dew
-3AM-2MD+5MC = 3AM-2MD+5MC = (3-2+5)MG =6MG cause G is barycenter of the three points A,D,C with same coeffs
same thing to HM + 5CM
its equal to 6GM
so norm of 6MG and 6GM is always the same for any point M
so the set of M is the plane?
naive zinc
5CM, I misread it as 5GM
naive zinc
so the set of M is the plane?
Yes. I found out that those two vectors are exactly the opposite of each other
naive zinc
Np, glad to confirm
green dew
.close
shut flicker
how are you meant to do the first part, think it involves complex numbers
compact ridge
how are you meant to do the first part, think it involves complex numbers
yeah, let's go with the complex number way
if you use the unit circle, hopefully you can see that cos(8pi/7) = cos(6pi/7), cos(10pi/7) = cos(4pi/7), cos(12pi/7) = cos(2pi/7)
shut flicker
yeah im just imagining cos curve
compact ridge
yeah, then you just need to find cos(2pi/7) + ... + cos(12pi/7)
but cos(2pi/7) is the real part of e^(i 2pi/7)
shut flicker
yeah, then you just need to find cos(2pi/7) + ... + cos(12pi/7)
so this would be equal to 2(cos(2pi/7) + cos(4pi/7) + cos(6pi/7))
compact ridge
so this would be equal to 2(cos(2pi/7) + cos(4pi/7) + cos(6pi/7))
yes, but that's not the idea we're going with here
compact ridge
but cos(2pi/7) is the real part of e^(i 2pi/7)
that is to say, if you find e^(i 2pi/7) + ... + e^(i 12pi/7)
then take the real part of that sum, it'll be the same
compact ridge
is it to do with like roots of unity
yep
shut flicker
ok so the sum equals 1 i think
compact ridge
that is to say, if you find e^(i 2pi/7) + ... + e^(i 12pi/7)
*then* take the re...
but we are missing one of the terms here
shut flicker
1/2?
shut flicker
14pi/7?
shut flicker
1
compact ridge
yeah
shut flicker
so are cos2pi/7 + ... + cos12pi/7 equal to -1, since this plus cos14pi/7 = 0, which means that since we only want 2pi/7 to 6pi/7, we divide by 2, to get -1/2
compact ridge
so are cos2pi/7 + ... + cos12pi/7 equal to -1, since this plus cos14pi/7 = 0, wh...
yep! that's the reasoning
pearl pondBOT
pearl pondBOT
Available help channel!
quick venture
can someone explain how we get the bit in the red brackets
pearl pondBOT
can someone explain how we get the bit in the red brackets
vital estuary
quick venture
yh okay
tawdry solar
pearl pondBOT
tawdry solar
!status
pearl pondBOT
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
1
last summit
polynomial long division
tawdry solar
I forgot to send the whole question
My apologies for the sarcasm
or
nvm
so I have to do polynomial long division
But i dunno what to do after
tawdry solar
,,\polynomlongdiv{x^4-3x^3 -5x^2 +x-7}{x^2 +5}
jolly parrotBOT
chronically.mahir
Compile Error! Click the 
reaction for more information.
(You may edit your message to recompile.)
tawdry solar
Well i did it in my Books
tawdry solar
It becomes more clear once you do it
And I got remainder x-7
last summit
so what does that make you think
last summit
And I got remainder x-7
if there was no x-7 term on the polynomial, it would be divisible by x^2+5
right
tawdry solar
yeah
last summit
so what does that make you think
tawdry solar
so it has to be 0?
tawdry solar
-5x^2 - ax-b=0?
tawdry solar
idk wym
last summit
If you did the division
and the remainder was ax+b
when would x^2+5 divide that remainder
tawdry solar
tawdry solar
when would x^2+5 divide that remainder
when its 0?
light helm
Division looks wrong
last summit
any other times?
tawdry solar
idk
light helm
you want to multiply the quotient component to the whole divisor,
Not just the leading/whatever comes first
tawdry solar
wdym
light helm
in that first iteration, you indicated subtraction by x^4
tawdry solar
light helm
But you need to subtract
x^2(x^2 + 5)
tawdry solar
why?
light helm
Or x^4 + 5x^2
light helm
Because that's how the division algorithm eorks
tawdry solar
light helm
the leading term cancels, and you repeat until the degree of the remainder is lower than the divisor
tawdry solar
the degree will be the same no?
light helm
Wdym
tawdry solar
x^2 and x^2
light helm
those have the same degree, yes
light helm
much better
tawdry solar
Now what?
tawdry solar
much better
Now what?
light helm
consider an example with numbers
88/9
would be 9 remainder 7 right?
tawdry solar
Yeah
light helm
now know that, what could you subtract from 88 so that the remainder would be 0
tawdry solar
7
tawdry solar
Subtract (16x +43) from P(x)?
light helm
Yes
light helm
Yes, subtract from the original polynomial
tawdry solar
Yes, subtract from the original polynomial
a=-15, b =-50
light helm
Yep
tawdry solar
.close
pearl pondBOT
tawdry solar
Tysmmm
pearl pondBOT
Available help channel!
digital pewter
lwk
digital pewter
how does one differentiate the sqrt of xy
implicit differentiation just makes zero sense to me
versed mica
how does one differentiate the sqrt of xy
chain rule
do you know the derivative of xy wrt x?
digital pewter
is it, um, y dy/dx?
versed mica
not quite
digital pewter
x+y?
versed mica
nope
digital pewter
😔
versed mica
so there shouldn't be any confusion for sqrt(xy) now
versed mica
yea but use parentheses
digital pewter
(1/2)(xy)^(-1/2) * (y+xy')
versed mica
yep
digital pewter
ok good
and then the derivative of x-4y would be 1-4y'?
perfect
ok
so if im asked to use implicit differentiation to find dy/dx of sqrt(xy) = x - 4y, would the answer be
dy/dx = xy/4sqrt(x)?
pearl pondBOT
@digital pewter Has your question been resolved?
digital pewter
.close
pearl pondBOT
pearl pondBOT
Available help channel!
tacit sedge
Hello i have my final tomorrow and am really struggling with learning the concept of laplace and how to do it as well as vocab like basis subspace an etc
waxen agate
Hello i have my final tomorrow and am really struggling with learning the concep...
Do you know the definition of Laplace Transform? Have you memorised the core Laplace Transforms you must know?
jolly wing
Do you know the definition of Laplace Transform? Have you memorised the core Lap...
I need help with discrete math
tacit sedge
I know like the intergal equation and we get a sheet on the test I just don’t even know how to like
what it does exactly? Or how to match terms with its laplace
waxen agate
I need help with discrete math
Please open your own channel. See #❓how-to-get-help
jolly wing
Please open your own channel. See <#488120190538743810>
ok
waxen agate
what it does exactly? Or how to match terms with its laplace
Since your exam is literally tomorrow, I would suggest that spend your time learning how to match terms and execute the steps. Understanding comes later.
tacit sedge
Okay that’s true
It’s just hard bc I’ve found no YouTube video that helps any like
Suggestion
waxen agate
I think you can search for something like Laplace transform solved examples, for example.
waxen agate
To be honest, you don't have enough time to fully understand all of this...
Walk through one exact problem step by step and identify the areas you don't understand.
bitter herald
Hello i have my final tomorrow and am really struggling with learning the concep...
If you want a tip, just try to understand the laplace transform table
Inverse laplace transform is basically always partial fractions decomposition
So work on that as well
viscid shale
Hello i have my final tomorrow and am really struggling with learning the concep...
To be honest, id treat it like learning derivatives once again. Id say you first learn the formal definition and understand why we do it, then just learn the tables and practice a lot.
I think my college has a pdf with like 30 ex.
bitter herald
But also if you can't remember most of the properties, deriving them from the definition
[
\LLL{f(t)} = \int_0^\infty f(t)e^{-st}\dd t
]
Isn't too bad
jolly parrotBOT
tacit sedge
Inverse laplace transform is basically always partial fractions decomposition
I love you. Thank you sm
For the advice
It’s cuz
My exam 3 had a question
asking us to use the intergal way vs the worksheet
and idk I just really blanked bc
i didn’t know how
and the laplace one was like a higher order ode non homogeneous
So is the sheet like a short cut of doing it from the integral basically
viscid shale
Similar to how derivatives tables are just the solved limits
bitter herald
and the laplace one was like a higher order ode non homogeneous
Yeah all IVTs should turn into algebraic equations you have to solve in the frequency domain
Where you again have to do PFD
viscid shale
Yeah, the laplace table is just solutions people find just so we dont have to do the integral 1000 times
tacit sedge
OH similar to derivatives is a great way to describe it holy wow
That’s great thank you guys I have a few more questions if that’s okay
bitter herald
I have some solved example slides if you want some