How does it work?
#help-39
latent quail
north talon
?
placid zealot
advice id like to give on combinations and permutations is that because they are so confusing, i always only use combinations
placid zealot
basically
latent quail
So that we can ensure that you knows how it works
north talon
sometimes I mix up when to use the choose function an the factorial for some reason
placid zealot
we can justify this
because if we imagine we have a slot for each person
like - - - - -
for the first slot
there are five different options
because we have 5 people
for the second, there are now four options
north talon
So that we can ensure that you knows how it works
so for every way to change the position of someone in the line we’re choosing 1 person so the total would be nC1 which is n!
placid zealot
as we used one option in our first slot
latent quail
so for every way to change the position of someone in the line we’re choosing 1 ...
Correct
placid zealot
this keeps on going all the way until there is only one option left in the last slot
therefore, the number of ways is $54321$
jolly parrotBOT
Amby
midnight haven
n!
placid zealot
which is also 5!
latent quail
bro its n!
We know it’s n!
placid zealot
yep we got there already
midnight haven
We know it’s n!
no no im asking the guy who asked
placid zealot
they dont understand why
latent quail
We’ve already done that part
midnight haven
i see
latent quail
no no im asking the guy who asked
He knows how it works basically
latent quail
so for every way to change the position of someone in the line we’re choosing 1 ...
Here, he did explain how it works
midnight haven
fundamental principal of counting
north talon
no no im asking the guy who asked
first i thought when you choose one person and place him into another it would change the position of another person so it’d be nC2 for some reason
I’m so bad at this lmao
dusky scaffold
👁️
north talon
.close
pearl pondBOT
pearl pondBOT
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near delta
How do i solve what 1 - 2sin^2 is without calculator?
pearl pondBOT
How do i solve what 1 - 2sin^2 is without calculator?
midnight haven
How do i solve what 1 - 2sin^2 is without calculator?
1 - cos2x = 2sin^x
midnight haven
How do i solve what 1 - 2sin^2 is without calculator?
get it to a known angle
and solve
otherwise you have to use the book
or calc
you can convert angle to multiple of 2 or 3 or etc. or 1/2 or 1/3
pseudo oxide
1 - cos2x = 2sin^x
sin^x ???
midnight haven
sin^2 x
near delta
oh forgot to mention sinx is 0,6
midnight haven
sorry
pseudo oxide
How do i solve what 1 - 2sin^2 is without calculator?
also sin^2 x not sin^2
pseudo oxide
oh forgot to mention sinx is 0,6
that's easy then?
jolly parrotBOT
ren
midnight haven
i see
near delta
ye
midnight haven
x = arcsin(0.6)
midnight haven
37 degree 53 degree 90 degree
triangle
sin 37 = 3/5
remember this
cos 37 = 4/5
this is learnt usually
near delta
is that just a known thing?'
midnight haven
Yes
near delta
oh
near delta
oh i have never heard of it
near delta
i know 0 30 45 60 90 120 135 150 180
pseudo oxide
the question is what is cos2x if sinx=0,6
okay first off
you don't need to know any of this to find that
midnight haven
okay first off
cleared
its sin 37 = 3/5
pseudo oxide
alr
midnight haven
the question is what is cos2x if sinx=0,6
@near delta @pseudo oxide
as per his given data
no need to remember the angle
near delta
ye
pseudo oxide
please do not ping me
midnight haven
please do not ping me
tf man
pseudo oxide
??
near delta
and the part where i turn 1-2(0,6)^2 to 0,28
pseudo oxide
what?
near delta
without calc
midnight haven
without calc
yes buddy
its 0.28
done there
if you want the angle
its 37 degree
near delta
i know the formula but hooow do i count that without a calc, like i cant count exponesials in my head
midnight haven
i know the formula but hooow do i count that without a calc, like i cant count e...
midnight haven
i know the formula but hooow do i count that without a calc, like i cant count e...
0.6 = 6/10
square both sides
0.6^2 = 36/100 = 0.36
prime dome
Can anyone tell me what I'm doing wrong? I tried entering infinity and that wasn't the correct answer. Neither is zero.
prime dome
I'm thinking 1/infinity, but would that make sense?
pseudo oxide
you differentiate with respect to x
not a
you can't differentiate with respect to two things at the same time, firstly
secondly it says that a is a constant
prime dome
right which would mean that if a is a constant, then d/dx of a would be 0, making the derivative of the numerator (x-a) = 1, correct?
pearl pondBOT
If you are done with this channel, please mark your problem as solved by typing .close
@prime dome Has your question been resolved?
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rich jolt
The 2011-th term of the sequence where each integer 𝑛 appears 𝑛 times
rich jolt
stuck
autumn fossil
The 2011-th term of the sequence where each integer 𝑛 appears 𝑛 times
so like 1, 2, 2, 3, 3, 3, 4, 4, 4, 4...?
pseudo oxide
hang on no-
autumn fossil
What's the position of last 1?
What's the position of last 2?
What's the position of last 3?
hmm
idt thats correct
pseudo oxide
What's the position of last 1?
What's the position of last 2?
What's the positio...
yeah, this.
rich jolt
hang on no-
1,3,6
autumn fossil
note that you could also express that as
1
1 + 2
1 + 2 + 3
pseudo oxide
and 1 + 2 + 3 ... n = n(n+1)/2
autumn fossil
because you are skipping 2 2's when getting to the last 2
then 3 3's when getting to the last 3
and so on
are you following till now?
rich jolt
i kinda get it but not getting the intuition
autumn fossil
I see... Maybe it would help if I arranged the sequence like this:
1
2 2
3 3 3
4 4 4 4
5 5 5 5 5
6 6 6 6 6 6
...
autumn fossil
1
2 2
3 3 3
4 4 4 4
5 5 5 5 5
6 6 6 6 6 6
...
now if you are familiar with triangular numbers, you should already see them coming
rich jolt
yeah i see that i need to find the last integer before 2011th term
not sure about triangular numbers tho
autumn fossil
not sure about triangular numbers tho
they are basically sums of natural numbers up to n, e.g.
1, 1 + 2, 1 + 2 + 3, 1 + 2 + 3 + 4... this is the sequence of triangular numbers
and they also denote the number of blocks used to build an n-layer pyramid / triangle
2D ofc
the formula for 1 + 2 + 3 + ... + n is n(n+1)/2, have you seen it before?
autumn fossil
cool, so n(n+1) / 2 will actually give us the position of last integer n in the sequence
e.g. 4(4+1)/2 = 10 will give us the position of last 4 in the sequence
and we need to know which integer is at position 2011.
so it would be good to find what integers appear as last ones close to 2011
for that, we can try solving n(n+1) / 2 = 2011
or just find for which n we get really close to 2011, which might be computationally less expensive than solving the quadratic above
,calc 15(15+1)/2
jolly parrotBOT
Result:
120
,calc 65(65+1)/2
jolly parrotBOT
Result:
2145
,calc 63(63+1)/2
jolly parrotBOT
Result:
2016
yeah i get that we need to solve for n(n+1)/2 <2011
i was trynna get the intuition lol
autumn fossil
hmm, what kind of intuition?
this question is about recognizing that it contains triangular numbers
once you get that, it's all about algebra
pearl pondBOT
pearl pondBOT
Available help channel!
tribal lark
Hi i need to do question 8
Its a simultaneous equation but i dont know how to even start it because i cant multiply the top or or bottem by a number to be able to cancel out a number if you know what i mean i know this is simple but its year 9 math
dusky scaffold
Hi i need to do question 8
Its a simultaneous equation but i dont know how to e...
look at the coefficients of x
4 and 3
what is the LCM of 4 and 3?
12
so multiply the top expression by 4
and the bottom one by 3
midnight haven
Hi i need to do question 8
Its a simultaneous equation but i dont know how to e...
an easier approach that would yield "nicer" numbers would be to consider the coefficients of y (which are also opposites, so you can add the two equations together)
which are -2 and 3
the least common multiple of which is -6 (but we consider the modulus of this)
so you multiply eq 1 by 3 and eq 2 by 2
tribal lark
Ill try that
midnight haven
it's okay you alreadyt did the work
midnight haven
,rotate
jolly parrotBOT
dusky scaffold
subtract the equations
midnight haven
you can subtract equation 1 from 2
tribal lark
Okay but what now the signs are the same
dusky scaffold
Okay but what now the signs are the same
yes
midnight haven
Okay but what now the signs are the same
the set of solutions doesn't change when you add one equation to another, or subtract one from another
dusky scaffold
so 12x will get cancelled out if you subtract the bottom equation from the top one
tribal lark
Oh so dont add the equations instead subtract
dusky scaffold
tribal lark
Okay
Wait this sounds really wrong. Y equals 68
Do i pick an equation to sub 68 in for y now? Im so confused
Uuuuhhhh
dusky scaffold
Wait this sounds really wrong. Y equals 68
no
you have to distribute the negative sign
so
it will become
-8y - 9 y
which is -17y
tribal lark
Oh wait sorry my brain just stopped working
Okay ill continue that hang on
Ah i did it finally so x is 3 and y is -4
Thanks alot
.close
pearl pondBOT
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Available help channel!
bold mulch
hi guys ive been stuck here for a while and i dont know how to solve with two varibles in a term
glass meadow
Try manipulating one of the two equations so that it better matches the other
pearl pondBOT
@bold mulch Has your question been resolved?
bold mulch
Try manipulating one of the two equations so that it better matches the other
i tried dividing (b) by 2 to cancel out the first and second term but it leaves me with z^2 over 2
bold mulch
Can you show exactly what you did?
glass meadow
,rccw
jolly parrotBOT
bold mulch
okok wait
bold mulch
Now add both equations together
glass meadow
Right, now you should be able to get w in terms of z, or z in terms of w, and substitute
glass meadow
Try isolating z
bold mulch
Try isolating z
i really dont know what to do from this point
glass meadow
Multiply each side by -2
bold mulch
so i can substitute this to get w in terms of z?
glass meadow
Well, kind of
It will be easier to go in another direction from here to get a second relation
You previously added both equations
What else could you have done?
Actually it'll be even easier from the start
Spoilers: you can add these two equations directly, and manipulate the resulting equation to get another expression for z in terms of w
glass meadow
so i can substitute this to get w in terms of z?
However, keep in mind that what you found here is not the only solution. If y^2 = x, then either y = sqrt(x) or y = -sqrt(x)
pearl pondBOT
@bold mulch Has your question been resolved?
simple siren
For question 5(c):
This is my graph from geogebra. is it correct?
pearl pondBOT
For question 5(c):
This is my graph from geogebra. is it correct?
@simple siren Has your question been resolved?
pearl pondBOT
pearl pondBOT
Available help channel!
silver nest
pearl pondBOT
silver nest
I know that the magnitude of PQ is root 24 and PR is root 66
Then I tried the cosine formula for this and I got undefined in my calc
woeful stump
show your work
silver nest
give me a sec
versed mica
show your work
what’s your pfp say
woeful stump
LOL i forgot i have the text uncropped
silver nest
PQ = -2i + 2j + 4k (magnitude = root 24) PR = i + 4j -7k (magnitude = root 66). Then I used the formula to get the angle which is: inverse cos (PQ * PR)/( | PQ | * | PR |)
That formula (with values) is: -22 / root 24 * root 66
I got 123° but its not an option
terse tide
PQ = -2i + 2j + 4k (magnitude = root 24) PR = i + 4j -7k (magnitude = ro...
your PQ and PR are wrong, double check them!
silver nest
Isn't V + W = (a1+a2)i + (b1+b2)j + (c1+c2)k
terse tide
remember: $\overrightarrow{AB} =\begin{pmatrix} x_B - x_A \ y_B - y_A \z_B - z_A \end{pmatrix}$
jolly parrotBOT
Emily
silver nest
Isn't V + W = (a1+a2)i + (b1+b2)j + (c1+c2)k
Is this formula incorrect?
terse tide
Isn't V + W = (a1+a2)i + (b1+b2)j + (c1+c2)k
we are not adding vectors here
if you want to find the components of the vector PQ, the vector going from P to Q
silver nest
Oh I got it, I confused the formula for adding vectors and the formula for doing a vector from 2 points in the plane
terse tide
mhm yeah
you mean 2 points in space :p
its actually subtraction of vectors here, in order to find the vector connecting 2 points in space
$\overrightarrow{AB}=\overrightarrow{OB}-\overrightarrow{OA}$
jolly parrotBOT
Emily
terse tide
which yields
$\overrightarrow{AB} =\begin{pmatrix} x_B - x_A \ y_B - y_A \z_B - z_A \end{pmatrix}$
jolly parrotBOT
Emily
silver nest
thanks
terse tide
tell me what you find
silver nest
btw where is the documentation or instructions for that notation
silver nest
thanks
silver nest
tell me what you find
thx got it
terse tide
thx got it
yep, thats what i found, noice
pearl pondBOT
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Available help channel!
wind oar
pearl pondBOT
wind oar
is this allowed? sorry for the orientation and my handwriting
heady finch
,rotate
unborn abyss
,rccw
jolly parrotBOT
terse tide
im okay up to $\ f(x,y) = g(, (ln , x)^2 - sin(x) , ) - g(x^2 -2y)$
jolly parrotBOT
Emily
terse tide
but then
terse tide
but then i dont like what you did
not because youre asked to find fxy, you differentiate wrt x then y
be smart and do fyx
you do know that fxy = fyx
mixed partials are equal, right
wind oar
some clauiauts theorem
wind oar
lemme see
wind oar
It's pretty much build up intuition, no?
terse tide
i didnt understand
wind oar
there's no way of knowing how to solve a problem in this nature without some exposure to it already
terse tide
mhm yeah, i did this kind of problems
so ig thats why it rang a bell
im gonna AFK for like 10 mins idk
ping me and show me what you find
wind oar
$\ f_y(x,y) = - g_y(x^2 -2y)*(-2)$
jolly parrotBOT
Potat
wind oar
i have a hunch that it shouldn't look like that
jolly parrotBOT
Emily
terse tide
so it becomes $2 , e^{(x^2-2y)^2}$
jolly parrotBOT
Emily
wind oar
$2 * 2(x^2-2y)*2(e^{x^2-2y}^2)$
terse tide
fyx ?
wind oar
latex is wack
terse tide
lol
wind oar
with a 2x somewhere
jolly parrotBOT
Potat
$2 * 2(x^2-2y)*2(e^{x^2-2y}^2)$
```Compilation error:```! Double superscript.
l.49 $2 * 2(x^2-2y)*2(e^{x^2-2y}^
2)$
I treat `x^1^2' essentially like `x^1{}^2'.
Preview: Tightpage -1310720 -1310720 1310720 1310720
[1{/usr/local/texlive/2023/texmf-var/fonts/map/pdftex/updmap/pdftex.map}{/usr/l
ocal/texlive/2023/texmf-dist/fonts/enc/dvips/lm/lm-rm.enc}{/usr/local/texlive/2
023/texmf-dist/fonts/enc/dvips/lm/lm-mathsy.enc}{/usr/local/texlive/2023/texmf-
dist/fonts/enc/dvips/lm/lm-mathit.enc}] (./565251253651898368.aux)
***********```
terse tide
its okay, dont worry about it
its correct, just missing an x
as you mentioned
resulting from the derivative of x²-2y wrt x
wind oar
that looks a lot neater
terse tide
$f_{xy}=f_{yx}=8 , x , (x^2 - 2y) , e^{(x^2 - 2y)^2}$
jolly parrotBOT
Emily
terse tide
its pretty :d
terse tide
that looks a lot neater
so yeah, you see
just a trick
wrt y before wrt x
changed the course
pearl pondBOT
terse tide
mhmm
terse tide
thank you for helping out
yw gl (:
pearl pondBOT
Available help channel!
random oriole
Someone help me with this cancerous IB SL question
pearl pondBOT
Someone help me with this cancerous IB SL question
random oriole
Thanks
shell totem
np
north talon
do you know how to translate a function with a vector?
random oriole
do you know how to translate a function with a vector?
erm wtf
oh nvm
yes I do
For one moment I was thinking about smth else
north talon
ive never actually learnt this before but using my intuition im guessing h(x) = g(x - sqrt(3)) - sqrt(2)
north talon
I did h(x-root3)
that would be translating h(x), not g(x) no?
random oriole
sorry g(x-root3)
random oriole
I did the algebra part but I'm kinda stuck at a dead end
north talon
why?
north talon
uhh let me do it rq
random oriole
alr
north talon
im supposed to be studying for history finals lmao
north talon
im just gonna send the ans cause i have to go sorry
north talon
north talon
real
pearl pondBOT
@random oriole Has your question been resolved?
worldly rampart
solve inequality
pearl pondBOT
solve inequality
worthy hare
it's easy
start by squaring the entire equation
and solve it just the way you'd solve an equation
worldly rampart
oh sorry, wrong equation
worthy hare
😭
worldly rampart
-x^2 + 4x
worldly rampart
yup
worthy hare
bruh moment
worldly rampart
hahahaha
worthy hare
loll
worthy hare
wtf was i doing though lol
sanest math student
worthy hare
i did -(x^2 - 4x)
😭
worldly rampart
wait, if i went that route though, is it possible to get the right answer?
cause i got it way too wrong
cause it went to => -(x+2)(x-2)
spare lark
pearl pondBOT
worldly rampart
.reopen
pearl pondBOT
✅
worthy hare
oh wait, if function is 1 / sqrt(-x^2 + 4x)
assume the thing is equal to y
rearrange such that it's likr x=g(y)
and now find the domain
worldly rampart
is there a more efficient way of evaluating the highest point given the domain without drawing a graph?
pearl pondBOT
@worldly rampart Has your question been resolved?
worldly rampart
.close
pearl pondBOT
inland lantern
is there a more efficient way of evaluating the highest point given the domain w...
ya theres calculus
pearl pondBOT
Available help channel!
keen panther
normal distribution
pearl pondBOT
normal distribution
pearl pondBOT
@keen panther Has your question been resolved?
cosmic charm
mossy folio
pearl pondBOT
mossy folio
i got this for question b
GPT says that (b) is meant to be a minimizer but i think i found a point y which contradicts the constraint
is this correct?
pearl pondBOT
@mossy folio Has your question been resolved?
sharp smelt
pearl pondBOT
sharp smelt
This is logically equivalent to proving that if $x$ is odd and $y$ is even then $x^2(y+3)$ is odd. as $x$ is odd, it's of the form $2k+1 , k \in \Z$, y is of the form $2l, l\in \Z$. As $x$ is odd, $x^2$ is odd too. We attempt to prove this now
\
x= 2k+1
\
$x^2= 2(2k^2+2k)+1$.
\
from this we can conclude that $x^2$ is odd.
\
As $y$ is even, $y+3$ is odd.
\
We attempt to prove this now.
y=2l
\
y+3 = 2(l+1)+1.
which is an odd number.
\
The product of two odd numbers is odd, thus $x^2(y+3)$ is odd.
We attempt to prove this now.
As previously proven,$x^2$ and $(y+3)$, are odd.
\
so they are of the from $2r+1, 2f+1 ; r,f \in \Z$.
\
so $(2r+1)(2f+1) = 2(2rf+r+f)+1$.
\
Which ressembles the form of an odd number
, concluding our proof.
gleaming musk
seems good but i just thought it was odd to not go through with the algebra
gleaming musk
off <=> odd
autumn fossil
x is off should be x is odd
jolly parrotBOT
Veni, vidi, perii is not f(wai)
autumn fossil
is this necessary if you dont use it?
it seems like you are using some already proven theorems, such as x odd -> x^2 odd
and even + odd = odd
which is fine, but if you use that, you dont need to go through that 2k+1 and 2l stuff
sharp smelt
Just stating it to be safe.
sharp smelt
In my mid -semester exam I forgot to justify most stuff, overcompensating for that while I'm practicing so that I don't make the same mistaken in my end of semester exam,
autumn fossil
Well, the thing is that the line you wrote doesnt justify anything
"As x is odd, x^2 is odd too" is either unjustified or justified by some previously proven theorem. I don't see how it would be justified just by saying "x is of the form 2k+1"
sharp smelt
I know, but if I write down everything I'm thinking, I'll probably write down the necessary steps in the exam.
sharp smelt
"As x is odd, x^2 is odd too" is either unjustified or justified by some previou...
Fair.
gleaming musk
seems good but i just thought it was odd to not go through with the algebra
also thought it was strange
like u set up everything to crunch it down and prove with algebra but then just didn't
sharp smelt
Is this better?
This is logically equivalent to proving that if $x$ is odd and $y$ is even then $x^2(y+3)$ is odd. as $x$ is odd, it's of the form $2k+1 , k \in \Z$, y is of the form $2l, l\in \Z$. As $x$ is odd, $x^2$ is odd too. We attempt to prove this now
\
x= 2k+1
\
$x^2= 2(2k^2+2k)+1$.
\
from this we can conclude that $x^2$ is odd.
\
As $y$ is even, $y+3$ is odd.
\
We attempt to prove this now.
y=2l
\
y+3 = 2(l+1)+1.
which is an odd number.
\
The product of two odd numbers is odd, thus $x^2(y+3)$ is odd.
We attempt to prove this now.
As previously proven,$x^2$ and $(y+3)$, are odd.
\
so they are of the from $2r+1, 2f+1 ; r,f \in \Z$.
\
so $(2r+1)(2f+1) = 2(2rf+r+f)+1$.
\
Which ressembles the form of an odd number
, concluding our proof.
jolly parrotBOT
Veni, vidi, perii is not f(wai)
gleaming musk
yesh thats good
sharp smelt
Thanks!
Have a couple of more problems 
This is logically equivalent to proving that if $a$ is not odd, then $a^2$ is divisible by $4$.
As $a$ is not odd, by definition, it is of the form $2k, k\in \Z$. so $a^2= 4(k^2), 4k^2 = 4l , l \in \Z$. Thus $4 \mid a^2$. Thus concluding our proof,
jolly parrotBOT
Veni, vidi, perii is not f(wai)
autumn fossil
Looks good, I'd just replace "not odd" with even, as talking about the definition of "not odd" doesnt really make much sense
and I think you can also leave out introducing l as a new variable
a^2 = 4(k^2) looks sufficient to conclude that 4 | a^2
pearl pondBOT
@sharp smelt Has your question been resolved?
sharp smelt
Looks good, I'd just replace "not odd" with even, as talking about the definitio...
Thanks!
No. I just used translate ( English to Latin). Sorry.
This is logically equivalent to proving that if $x<0$, then $x^5+7x^4+5x< x^4+x^2+8$.
This is the same as proving
\
$x^5-x^4+7x^3-x^2+6x-8<0$, if $x<0$
\
$x^4(x-1) + x^2(7x-1) +6x-8$.
\
We first attempt to prove $x^4>0,$ if $x<0$
\
We Have already proven earlier that $x^2>0$.
\
We now assume $x^2= q, q>0$. From which it follows $q^2>0$, os $x^4>0$
\
As x<0, it follows $x-1<0$. so $x^4(x-1)<0$.
\
It similarly follows that $x^2(7x-1)<0$
\
and that $6x-8<0$.
whose proof can be found below.
\
x<0
\
6x<0
\
6x-8<-8<0
\
6x-8<0
\
We can thus conclude that $x^5+7x^3+5x<x^4+x^2+8$
jolly parrotBOT
Veni, vidi, perii is not f(wai)
sharp smelt
<@&286206848099549185>
torn shell
its not very concise but it does the job
like its not necessary to direct us to read the next line for the proof thats kind of just implied
actually didnt notice you did a typo on the degree
7x^4 instead of 7x^3
so reordering terms just seems like a very unnecessary step
pearl pondBOT
@sharp smelt Has your question been resolved?
sharp smelt
so reordering terms just seems like a very unnecessary step
Got it, thanks
pearl pondBOT
pearl pondBOT
Available help channel!
civic ledge
can anyone explain this
pearl pondBOT
can anyone explain this
nimble osprey
given a function y=f(x), if it differentiable, you can compute its derivative f' that is another function
now if you compute the derivative of f' you will get another function, that is f'' namely the second derivative of f
civic ledge
yea
but what about the one at the bottom
it seems like stuffs magically appeared there
i can't seem to figure out how did this happen
torn shell
they used dy/dx=(dy/dt)/(dx/dt) but in this case y is the (dy/dx) part
dy/dx you differentiate y and in d/dx (dy/dx) you differentiate (dy/dx) so they just substistute it
pearl pondBOT
@civic ledge Has your question been resolved?
torn shell
how y changes with respect to x for a parametrized function is to take how y changes w.r.t. t (usually time) and divide by how x changes w.r.t. time
torn shell
where was the formula from
this is the intuition of dy/dx = (dy/dt)/(dx/dt)
torn shell
so now you are not looking at how y changes with respect to x
you are looking at how dy/dx changes with respect to x
so you take how dy/dx changes w.r.t time and divide it by how x changes w.r.t. time
make sense?
civic ledge
no
torn shell
(d/dt)(dy/dx) intuitively represents how dy/dx changes w.r.t time
and you divide that with dx/dt
torn shell
do you understand dy/dx=(dy/dt)/(dx/dt)
if so this just comes down to substitution
instead of y its dy/dx
$\frac{d(\frac{dy}{dx})}{dx}$
civic ledge
do you understand dy/dx=(dy/dt)/(dx/dt)
yes
jolly parrotBOT
Hugaxi
torn shell
this is the 2nd derivative so instead of using that formula for y you use it for dy/dx
thats how it makes sense symbolically
civic ledge
$\frac{d(\frac{dy/dt}{dx/dt})}{dx}$
jolly parrotBOT
Chronny
torn shell
$\widetilde{y}=\frac{dy}{dx}$
jolly parrotBOT
Hugaxi
torn shell
now use the formula provided for $\widetilde{y}$
jolly parrotBOT
Hugaxi
torn shell
you have $\frac{d\widetilde{y}}{dx}$
jolly parrotBOT
Hugaxi
torn shell
by using dy/dx=(dy/dt)/(dx/dt)
why would it change just because we call the function something else
this is just a substituion
torn shell
$\frac{d\widetilde{y}}{dx}=\frac{d\widetilde{y}/dt}{dx/dt}$
jolly parrotBOT
Hugaxi
jolly parrotBOT
Hugaxi
civic ledge
by using dy/dx=(dy/dt)/(dx/dt)
isn't it more reasonable to use quotient rule from this?
torn shell
no its not related
you are not taking the derivative of the quotient
but do you see how they get it now?
civic ledge
no
somehow it just seems like
it was just decided to only take the derivative of numerator
that's all
torn shell
idk how else to explain it ive already tried renaming symbols
you are having trouble udnerstanding replacing a variable in a formula
dy/dx=(dy/dt)/(dx/dt) holds for any differentiable function y
it doesn tmatter what y is
it dfoesnt have to be called y
it can be a function f
g
or dy/dx
torn shell
ok call f=dy/dx
and pretend like we are taking the first derivative
that was what i was trying to explain
df/dx=(df/dt)(dx/dt)
and just sub f back in
df/dx=(df/dt)(dx/dt)=(d(dy/dx)/dt)/(dx/dt)
torn shell
nws i think its because of the double use of the symbol y
civic ledge
is that the same for 3rd 4th 5th derivative etc?
torn shell
when they said 'we replace y with dy/dx' the y's are actually different
torn shell
is that the same for 3rd 4th 5th derivative etc?
ye
pearl pondBOT
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zinc oasis
if i have this, i can just multiply everything by ^2 to get rid of the square root right?
zinc oasis
it wont go to y^2 = 3x^2 - 9x or am i wrong
lost sigil
correct
viscid sierra
if i have this, i can just multiply everything by ^2 to get rid of the square ro...
there is nothing as such multiplying by ^2
light helm
your description is problematic though
viscid sierra
that's squaring both sides
zinc oasis
alright well that's what i meant
english isnt my native language, that's just what we say where i'm from
sorry if it came across wrong
viscid sierra
yeah, the end equation is correct
glad crater
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sterile turtle
pearl pondBOT
sterile turtle
ignoring the fact that i didn’t put bounds, what have i done wrong?
ans is 2^n/n
ah
nvm
im dumb
.clsoe
.close
pearl pondBOT
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elder wasp
pearl pondBOT
wispy meadow
hi, i'm stuck on this question, idek where to begin with this
livid goblet
hi, i'm stuck on this question, idek where to begin with this
find out what AX is and use some trigs
do you know the inverse pythagorean theorem
wispy meadow
do you know the inverse pythagorean theorem
what's that?
livid goblet
,w inverse pythagorean theorem
livid goblet
what's that?
wispy meadow
no, I never saw that before
livid goblet
you can use this in this case
wispy meadow
is there another way?
cuz I learnt only normal pythagoras, trigonometry basics and trigonometry sine/ cosine rules
.close
jolly kernel
pearl pondBOT
jolly kernel
I'm confused what I should put in
If it's speeding up the rate of change is increasing
Like just before 1 it slows down and just after one it speeds up, but how do I put that as interval notation
V is constant + from 0-1
then becomes rest
then constant - from 1-3
then rest
constant + from 3-4
or is it slightly changing between those points?
idk
slows down right before one, speeds up right after one, slows down right before , speeds up right after
i am confusion
<@&286206848099549185>
i have a postion graph, but idk how to tell when it's speeding up or slowing down
if velocity and acceleration are the same sign then it's speeding up
so if rate of change is positive and increasing or if rate of change is negative and decreasing
pearl pondBOT
@jolly kernel Has your question been resolved?
oak quiver
Share it
jolly kernel
(two different graphs, but same concept)
jolly kernel
constant until 0.9ish
oak quiver
You may think that it is a straight line
But it's not
It's a curve
Even 0 to 0.9 is a curve
jolly kernel
it is?
jolly kernel
ooh i got it
So like the halfway (2) is when velocity is 0 switching point
ty!!
oak quiver
I think so
I'm too sleepy
Gn
Dm me if you still have a doubt
I'll look into it tomorrow
pearl pondBOT
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dense sedge
wo can help
pearl pondBOT
wo can help
dense sedge
.close
ocean lily
pearl pondBOT
ocean lily
Option c is the Answer
How to simplify after this
Anyone there? ;-;
Ahh
Sad
.close
pearl pondBOT
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midnight haven
How do you solve for pi to the power of -- (30-50) pick a number
jolly parrotBOT
riemann
pearl pondBOT
@midnight haven Has your question been resolved?
balmy plank
Is there a mistake here? I do not get the "t" in f(x-yt) in the last line...
hallow cedar
i wouldnt think there is a mistake since if there wasnt that 't', then the integral would just be a function u(x), and we know that the solution is of the form u(x,t)
merry carbon
How do you define your Fourier (and inverse Fourier) transforms?
pearl pondBOT
@balmy plank Has your question been resolved?
balmy plank
i wouldnt think there is a mistake since if there wasnt that 't', then the integ...
yeah true
balmy plank
How do you define your Fourier (and inverse Fourier) transforms?
lol we've talked about this before
but I do the negative exponent in the normal Fourier, and positive exponent in the inverse Fourier if that makes sense
merry carbon
(found it 
)
balmy plank
i wouldnt think there is a mistake since if there wasnt that 't', then the integ...
But okay, do you have any idea how I can get to that? I ended up with getting a convolution $$\frac{1}{\sqrt{2\pi}}(f*g)(x,t)$$ where $f$is the one in the text, and $g = \frac 1t e^{-\frac{x^2}{2t^2}}$
jolly parrotBOT
Michael
pearl pondBOT
@balmy plank Has your question been resolved?
balmy plank
<@&286206848099549185>
balmy plank
<@&286206848099549185> some help would be greatly appreciated
pearl pondBOT
@balmy plank Has your question been resolved?
pearl pondBOT
@balmy plank Has your question been resolved?
pearl pondBOT
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midnight haven
I don't understand how to awnser algebraic word problems can someone explain how to?
light helm
show a specific question you're having issue with
midnight haven
K 1 sec
light helm
keywords are associated with mathematical operations/symbols
light helm
replacing them will give you your equations
and proceed from there
midnight haven
Wdym
versed ledge
Where it asks for the algebraic Equation
5 times number of students is 295
As you are not aware of actual number of students, just assume it's x
So 5 times x is 295
5x=295```
versed ledge
But for smt like iii)
That's even better
+
5 tickets at $4 each , that means 5 multiply by 4
=
$44```2x+20=44
versed ledge
= 44 that is given
midnight haven
So 2×+5×4=44
midnight haven
Oh okay that makes a ton of sense
I have a test in 30 mins
So like I am super nervous
midnight haven
Thx
pearl pondBOT
midnight haven
.reopen
pearl pondBOT
✅
pearl pondBOT
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dusty flame
What types of questions are these
pearl pondBOT
What types of questions are these
dusty flame
light helm
algebra
dusty flame
can u be more specific
light helm
not really
plush bramble
can u be more specific
,tex .alg lesson
jolly parrotBOT
riemann
light helm
other than what's already being stated in the question
custorm algebraic operators
divisibility
dusty flame
so theres really no trick
i thought there'd a trick for operations cuz its a speed test
😭
light helm
these take around 2 to 3 steps to solve
pearl pondBOT
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calm wing
$\sum_{k=1}^\infty \left(1-\frac{1}{k}\right)^{k^2}$
pearl pondBOT
$\sum_{k=1}^\infty \left(1-\frac{1}{k}\right)^{k^2}$
jolly parrotBOT
artemetra
calm wing
converges/diverges
gonna type my solution
$\lim_{k\to\infty} \left(1-\frac{1}{k}\right)^k = e^{-1}.$\
Since the limit expression increases monotonically, we get that $\forall k\geq 1: \left(1-\frac{1}{k}\right)^k < e^{-1}.$\
$\implies \sum_{k=1}^\infty \left(1-\frac{1}{k}\right)^{k^2} = \sum_{k=1}^\infty \left(\left(1-\frac{1}{k}\right)^k\right)^k < \sum_{k=1}^\infty \left(e^{-1}\right)^{k}$ which converges because it's a geometric series and $|e^{-1}| < 1$
jolly parrotBOT
artemetra
calm wing
is this correct-ish?
summer gorge
looks correct
summer gorge
sharp smelt
pearl pondBOT
sharp smelt
A direct proof would be advisible here, me thinks
summer gorge
wai in his NT arc?
sharp smelt
Not yet, it's offered only in odd semesters at my uni
Proof writing class
Also , nice to see you convergence, how's life been?
summer gorge
ah i see
sharp smelt
Got into a uni?
summer gorge
it has been good, I'm in uni now 
.how have you been?
summer gorge
am taking same degree once i start uni next yr
its good so far
sharp smelt
mainly applied math but yeah
Good luck!
gleaming musk
A direct proof would be advisible here, me thinks
yeah i think so too
wild fable
convergence
summer gorge
waterbeeammmm! hello fren
wild fable
hi 
sharp smelt
So we want to prove if $a \equiv b ( mod n)$, then $gcd(a,n)=gcd(b,n)$. Now we know that $n|(a-b)$, or in other words, $a-b =n(k); k \in \Z$. We first start off by assuming that $gcd(a,n)= t, t \in \Z$. so we know there exists a number say $r$ such that $rt =a$.From this we can conclude that $rt-b=nk$. We niow assume $gcd(b,n)=m$, from which we can conclude that $zm=b$.
jolly parrotBOT
Veni, vidi, perii is not f(wai)
tropic saddle
wai if you dont want to listen to my advice then you can tell me that and I wont bother in the future
sharp smelt
wai if you dont want to listen to my advice then you can tell me that and I wont...
I do
I just was hoping for a differnt method. I'll try what you suggested now.
Sorry
so $gcd(a,n) = gcd(a, n-a)$
jolly parrotBOT
Veni, vidi, perii is not f(wai)
sharp smelt
This is the idea I'm supposed to use, right
sharp smelt
I know that $\frac{a-b_}{n}=k$ so $n =\frac{a-b}{k}$. Like so?
jolly parrotBOT
Veni, vidi, perii is not f(wai)
Compile Error! Click the 
reaction for more information.
(You may edit your message to recompile.)
autumn fossil
**Veni, vidi, perii is not f(wai)**
Compile Error! Click the <:errors:5042665560...
Those are fractions, im pretty confident that wont lead anywhere
summer gorge
I know that $\frac{a-b_}{n}=k$ so $n =\frac{a-b}{k}$. Like so?
its best to avoid fractions
sharp smelt
Hmm. I have to figure out how this all ties in with $a \equiv b \mod{n}$
jolly parrotBOT
Veni, vidi, perii is not f(wai)
sharp smelt
Can I have a hint?
tropic saddle
b=a-nk
jolly parrotBOT
Veni, vidi, perii is not f(wai)
sharp smelt
From this I can conclude that $gcd(a, (n-1)k) = gcd(a,n)$ and repeating this , we eventually arrive at $gcd(a,n)= gcd(a,n)$, thus completelting the proof
jolly parrotBOT
Veni, vidi, perii is not f(wai)
tropic saddle
no
autumn trellis
How did you conclude that gcd(a,(n-1)k)=gcd (a,n)?
sharp smelt
I mean $gcd(a-nk,n)= gcd(a-nk+n,n) = gcd(a-nk+2n, n) \dots = gcd(a-nk+nk,n)$
tropic saddle
why are you adding k in each step
jolly parrotBOT
Veni, vidi, perii is not f(wai)
tropic saddle
why are you allowed to add n. the identity gcd(a,b)=gcd(a-b,b) only subtracts b
jolly parrotBOT
Veni, vidi, perii is not f(wai)
tropic saddle
k is an integer. nothing says its positive
sharp smelt
okay, this is assuming it's positive
tropic saddle
lets take a second and look at gcd(a,b)=gcd(a-b,b) again
if we look at it from left to right, we subtract b
what happens if we look at it from right to left
sharp smelt
we multiply b by -1 and add a to it
tropic saddle
uhh
what I wanted to hear is that we add b
so therefore it is indeed true that you can add/subtract any integer multiple of one side of the gcd to the other one
sharp smelt
I see, okay
I think I got how to do it now. I'll write a proof in that case. Thanks!
We want to prove that if $a \equiv b \mod{n}$ then $gcd(a,n) = gcd(b,n)$. We first note that $b=a-nk$.So we essentially want to prove that $gcd(a,n)= gcd(a-nk,n).$. We start by noting that $gcd(a,n) = gcd(a-n,n) = gcd(a-2n,n) \dots$. From which we can conclude that $gcd(a,n) = gcd(a-nk).$ This stems from the fact that $gcd(a,b) = gcd(a \pm b, b)$, thus completeting our proof.
jolly parrotBOT
Veni, vidi, perii is not f(wai)
tropic saddle
good enough I suppose
autumn fossil
,ti physicsrocks
jolly parrotBOT
The current time for physicsrocks is 10:55 PM (IST) on Thu, 03/10/2024.
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sharp kite
pearl pondBOT
sharp kite
Can someone pls explain why this is a cosine
In fact can smn please explain why all my answers were wrong 🥲
plush bramble
cos(bx) attains its max or min at x = 0
sharp kite
cos(bx) attains its max or min at x = 0
But what about the amplitude?
Does it not matter where it would be if the amplitude were 1?
plush bramble
sine and cosine have the same amplitudes so it doesn't matter
plush bramble
cos(bx) attains its max or min at x = 0
this immediately implies a * cos(bx) also attains its max or min at x=0
sharp kite
this immediately implies a * cos(bx) also attains its max or min at x=0
Ohhh so it has a min at 0 because a is negative?
plush bramble
yea that's why i said max or min
sharp kite
yea that's why i said max or min
I see
And can you show me why b is 4
I thought i understood how to calculate b, but i did it wrong
plush bramble
And can you show me why b is 4
specifically
find the period of your wave, then solve for B using period = 2pi / B
pearl pondBOT
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midnight haven
if x ∈ S, then what would we say?
if S ⊆ T, what would we say
prime bramble
if x ∈ S, then what would we say?
if S ⊆ T, what would we say
x is an element of S, S is a subset of T
midnight haven
can we say S contains x for part (i)?
prime bramble
sure
note that we also sometimes say that “S is contained in T” though
so the word contain is used for both elements and subsets; usually you need context to distinguish which one we mean
cosmic garnet
x is an element of S, S is a subset of T
so there's no distinction while speaking for $\subset$ and $\subseteq$ ?
jolly parrotBOT
Astar777
prime bramble
so there's no distinction while speaking for $\subset$ and $\subseteq$ ?
people usually call the former a proper or strict subset
cosmic garnet
so if it was $S\subset T$ then "S is a proper subset of T"?
jolly parrotBOT
Astar777
prime bramble
yeah, that’s what I would say
though I don’t like the ambiguous notation
I only use $\subsetneq$ and $\subseteq$ because $\subset$ can mean subset with or without equality depending on the author
jolly parrotBOT
higher!
cosmic garnet
hmm
pearl pondBOT
@midnight haven Has your question been resolved?