Dang i left my math book in school
#help-33
knotty trellis
I guess i will just ask my teacher
That's a good idea probably
wind stratus
I have an important exam coming up and my teacher isn't answering mails, so I need help for this. I thought that (1/√x)' became 1/2.√x but I don't end up anywhere near the solution. Any help?
marsh citrusBOT
I have an important exam coming up and my teacher isn't answering mails, so I ne...
wind stratus
it's the solution by the way
limber hearth
Well the derivative of a product isint the product of derivatives
$(uv)' = u'v + uv'$
wind stratus
oh so i should have multiplied before derivating?
elfin berryBOT
YakuBros
limber hearth
oh so i should have multiplied before derivating?
Or use the product rule
wind stratus
Or use the product rule
makes sense
limber hearth
Which is better
wind stratus
hold on let me quickly try that again
see i already had that formula but it's worded differently in my country
limber hearth
Moreover, if you have $(\sqrt{u})'$ then it would be equal to $\frac{u'}{2\sqrt{u}}$
elfin berryBOT
YakuBros
limber hearth
see i already had that formula but it's worded differently in my country
Well, word it as it is in your country, but apply it when there is a product
wind stratus
thanks
limber hearth
And by the way, for 1/sqrt(x), it is 1/u so it will become -u'/u^2, so (-1/2sqrt(x))/x, which is basically -1/(2x*sqrt(x))
And not 1/2sqrt(x)
As you said
Be careful to those
wind stratus
And by the way, for 1/sqrt(x), it is 1/u so it will become -u'/u^2, so (-1/2sqrt...
oh yeah sorry
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marsh citrusBOT
marsh citrusBOT
Available help channel!
still temple
pls help
quaint elm
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hazy dragon
why can we add e^c/(n+1)?
marsh citrusBOT
why can we add e^c/(n+1)?
hazy dragon
to RHS
considering LFS and RHS without it are f(x)=P_n(x)
also x0=0 we can see for creating the taylor poly
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marsh citrusBOT
deep lily
Finished 1, can someone help with 2?
marsh citrusBOT
Finished 1, can someone help with 2?
jovial latch
Finished 1, can someone help with 2?
ACTR is a right trapezoid, do you know the formula for the area of a trapezoid?
deep lily
ACTR is a right trapezoid, do you know the formula for the area of a trapezoid?
in our case its (CT+AR)xAT/2
jovial latch
no.. TR is the height you're looking for
TR is the side with 2 right angles
we want to find the area of the red trapezoid
deep lily
we want to find the area of the red trapezoid
Oh
jovial latch
in our case its (CT+AR)xAT/2
with that in mind, give this another try
jovial latch
yep u got it
deep lily
yep u got it
Thank you so much
jovial latch
cheers 
jovial latch
you too
lunar bolt
How do I prove that the lower limit of aka. that a_n is down limited with the number 1 using inductive generalization?
marsh citrusBOT
How do I prove that the lower limit of aka. that a_n is down limited with the nu...
lunar bolt
I got this but a collogue says its incorrect, at least the part that is crossed. And I dont know what to do with the part before the crossed part to prove thsis
hazy lion
its not clear what youre trying to do here
i may show its decreasing
and show that 1 is a fixed point
anddd
always greater than 1
placid nova
How would you show it's decreasing?
lunar bolt
induction
yes, I meant this
also I want to prove that its decreasing towards 1 but never passes lower that 1 so its lower limit is 1 aka. its down limited with the number 1 (I dont know how to translate it to English, this is my best translation)
placid nova
also I want to prove that its decreasing towards 1 but never passes lower that 1...
You want to show that the lower bound is 1
hazy lion
also I want to prove that its decreasing towards 1 but never passes lower that 1...
thats the idea
show there isnt a fixed point between a1 and 1
show that an is decreasing
and show that it is never less than 1
which part is giving u trouble
lunar bolt
both 😢
placid nova
we found the decreasing part tricky
hazy lion
okay,give me one moment
placid nova
i have a probably correct proof, but it's a bit long and uses induction twice
hazy lion
thank u for patience 
lunar bolt
reacting with cats and dogs for encouragement
placid nova
thank u for patience <:happy:908558103992225822>
are you still attempting to scribble something? the proof we have is probably not ideal as it seems like it has too many steps for an exam question
hazy lion
are you still attempting to scribble something? the proof we have is probably no...
this is an exam question?
yea i got it
we cant help with exams
lunar bolt
this is an exam question?
yes, the first out of 4 questions
placid nova
its not from an in progress exam
lunar bolt
oh no, we already finished it
placid nova
lemme bust out lappy ig then
using this line next time i get invited to a work meeting
hazy lion
lol
sorry yall im at a stupid airport
okay so the base step is easy yea?
really interested in the inductive step
so we assume that $a_n > a_{n+1}$
elfin berryBOT
jan Niku
hazy lion
and what we want to show is that $a_{n+1} > a_{n+2}$
elfin berryBOT
jan Niku
hazy lion
really you have two usual tricks up your sleeve and youll probably use one
either and show $a_{n+1} - a_{n+2} > 0$ and at some point add and subtract the same quantity
elfin berryBOT
jan Niku
hazy lion
so i started at
\begin{align*}
a_{n+1} - a_{n+2}
&= a_{n+1} - \frac{3}{4-a_{n+1}} \
&> a_{n+1} - \frac{3}{4 - a_n} \
&= a_{n+1} - a_{n+1} \
&= 0
\end{align*}
oh
elfin berryBOT
jan Niku
hazy lion
@lunar bolt
second step is the hard one
a_n is bigger than a_n+1
so 4-a_n is smaller
so 3/(4-a_n) is bigger
so a_n+1 - 3/(4-a_n) is smaller
thats the induction done
what do you think
placid nova
**jan Niku**
im not entirely convinced of the jump from line 1 to 2
elfin berryBOT
jan Niku
hazy lion
$-a_n < - a_{n+1}$
elfin berryBOT
jan Niku
hazy lion
$4 - a_n < 4 - a_{n+1}$
elfin berryBOT
jan Niku
hazy lion
$\frac{1}{4-a_n} > \frac{1}{4-a_{n+1}}$
elfin berryBOT
jan Niku
jan Niku
$\frac{3}{4-a_n} > \frac{3}{4-a_{n+1}}$
jan Niku
$-\frac{3}{4-a_n} < -\frac{3}{4-a_{n+1}}$
hazy lion
$a_{n+1}-\frac{3}{4-a_n} < a_{n+1}-\frac{3}{4-a_{n+1}}$
elfin berryBOT
jan Niku
placid nova
wrong tag
lunar bolt
@placid nova is eating ice cream, his hand is occupied
placid nova
Looks good to me
hazy lion
its a bit of a tricky step
but i feel confident in it
creating 0 is usually the hard part
lunar bolt
is there no simpler way of proving this? because this looks very hard and something that I think little students in our class would remember
hazy lion
Hmm well i was wondering
its usually easier to not work like this
and just treat it as a continuous function
but i never like doing that
,w derivative 3/(4-x)
elfin berryBOT
placid nova
you don't even need to derivate if you take properties of elementary functions as a given
hazy lion
so, you argue that as long as x<1, then 4-x > 3
err
yea i never really do it this way
so i wouldnt be certain
hazy lion
is there no simpler way of proving this? because this looks very hard and someth...
hmmmmmmmmmmm
you could use epsilon delta
but thats no fun
usually people find that more unwieldy
weildy
It really depends what you have from analysis at that point I guess
like, say, monotone convergence theorem
but youre crossing from calculus into stuff most people dont like using
unless theyre math ppl
placid nova
ok hear me out, am i stupid?
a_n > a_(n+1)
let's say
a_n = 1000
a_(n+1) = 1
then
3/(4-1000) is negative
3/(4-1) is positive
yet we said the unequal sign is the opposite way
placid nova
It really depends what you have from analysis at that point I guess
we are comp sci students so pretty bare bones
hazy lion
ah you should deal in discrete world then its good for you
placid nova
didnt even have any epsilon delta problems
hazy lion
what you said makes no sense to me sorry
hazy lion
ok hear me out, am i stupid?
a_n > a_(n+1)
let's say
a_n = 1000
a_(n+1) = 1
the...
here
i mean, a_n cant be above 3/2
for any n
right?
i mean we assumed it was decreasing
placid nova
yes, but we haven't proved it's decreasing here, right?
the derivation is based only off of the fact that a_n > a_(n+1)
hazy lion
i dont understand what your argument is im sorry
are you arguing that some a_n may be less than 0
placid nova
i picked arbitrary a_n and a_(n+1) such that a_n > a_(n+1)
placid nova
are you arguing that some a_n may be less than 0
yes
hazy lion
so show that a_n > 1 for all n
placid nova
but that shouldn't be necessary for the above derivation to be correct, no?
placid nova
where do we use it impicitly though?
placid nova
we used it implicitly when we raised both sides to -1
we would have to somehow justify that step i think
hazy lion
the argument that 1 < an for all n is straightforwand
it has basically the same structure as the argument i typed out to your last doubt
start with $1 < a_n$
elfin berryBOT
jan Niku
hazy lion
then $4-a_n < 4-1$
elfin berryBOT
jan Niku
hazy lion
so $a_{n+1} = \frac{3}{4-a_n} > \frac{3}{4-1} = 1$
elfin berryBOT
jan Niku
hazy lion
and its established
placid nova
this is very simple, yes
I never approach problems that way, constructing something that i can later use
hazy lion
its like constructive wishful thinking
youre lucky youre compsci so in theory like
you can quickly test properties of sequences
helpful to code
placid nova
this is true
hazy lion
maybe you look into cobweb diagrams
theyre pretty straightforward to code
and they make a lot of intuitive sense
especially for monotone convergence like you have here
anyways
any other doubts
placid nova
its like constructive wishful thinking <:happy:908558103992225822>
is this often useful? sounds like something i have to add to my arsenal
hazy lion
for analysis yes
discrete math
but idk you will develop your own tactics
this is just mine
it can be slow
but i would always prefer to work stuff out for myself, you know
no doubts left
lunar bolt
what is your major if I may ask?
lunar bolt
well thank you for the help 🙂
hazy lion
np good luck on your exam yall
shell plume
A beverage is dropped from a heigh of 100m. How much time will it take to fall through the bottom of 20m? I keep getting 4.04 seconds but it's saying that its wrong.
marsh citrusBOT
A beverage is dropped from a heigh of 100m. How much time will it take to fall t...
shell plume
runic temple
shell plume
So how long it would take to fall through the last 20m?
I got it thank you the answer was 0.477~ seconds :D
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marsh citrusBOT
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Available help channel!
timber imp
can somebody explain why these triangles are all equal?
marsh citrusBOT
can somebody explain why these triangles are all equal?
timber imp
its stated its a regular hexagon
and
the blue triangle is equilateral
its spoiler mb
proud basin
maybe you could go backwards, imagine a grid of triangles where you make a hexagon in it
basically a hexagon is made of these 6 congruent equilateral triangles (yellow), and each of them are divided up into 4 equal pieces
timber imp
maybe you could go backwards, imagine a grid of triangles where you make a hexag...
how do we know the grid would be similar tho?
proud basin
how do we know the grid would be similar tho?
a grid of congruent triangles :p
proud basin
basically a hexagon is made of these 6 congruent equilateral triangles (yellow),...
i think this approach is nicer
marsh citrusBOT
@timber imp Has your question been resolved?
timber imp
i think this approach is nicer
hi so sorry my dad called me to eat lunch
timber imp
basically a hexagon is made of these 6 congruent equilateral triangles (yellow),...
wait how do u tell if the pieces are equal?
timber imp
@proud basin ?
timber imp
<@&286206848099549185> ?
marsh citrusBOT
@timber imp Has your question been resolved?
marsh citrusBOT
@timber imp Has your question been resolved?
proud basin
wait how do u tell if the pieces are equal?
each piece is an equilateral triangle, which has the same sides
timber imp
each piece is an equilateral triangle, which has the same sides
so it just is equilateral and fits into it like that?
not really a reason as to why it is tho right
proud basin
basically a hexagon is made of these 6 congruent equilateral triangles (yellow),...
ok, what if i change it, if i only have the top triangle, i can add 5 more and they must be congruent since i can rotate it and they must have symettry, so the angle thats facing away from the middle is 60 degrees because its equally split between 6 triangles, and since each interior of a regular hexagon is 120, each angle of the triangle is 60 since symetry
timber imp
ok, what if i change it, if i only have the top triangle, i can add 5 more and t...
which triangle is this?
marsh citrusBOT
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Available help channel!
low mica
marsh citrusBOT
low mica
how do we sketch this
asymptote at y= 0
but no vertical asymptote
that graph interescts (0,0)
what do i ffind now
late geode
yeh
elfin berryBOT
pixel
low mica
is this righjt
late geode
no
derivative of a quotient isn't the same as the quotient of the derivatives of the numerator and denominator
low mica
ngl
late geode
quotient rule for derivatives would be relevant here
low mica
derivative of a quotient isn't the same as the quotient of the derivatives of th...
idk what ur trying to say here
low mica
quotient rule for derivatives would be relevant here
ahh
late geode
$\left[\frac{g(x)}{h(x)}\right]' \redneq \frac{g'(x)}{h'(x)}$
elfin berryBOT
ℝαμOmeganato5
low mica
isnt iit like
$\frac{g'(x) \cdot h(x) - h'(x) \cdot \g(x)}{[h(x)]^2}$
smthn like that
elfin berryBOT
pixel
Compile Error! Click the 
reaction for more information.
(You may edit your message to recompile.)
marsh citrusBOT
Available help channel!
velvet yarrow
marsh citrusBOT
velvet yarrow
,rotate
elfin berryBOT
velvet yarrow
Q2
marsh citrusBOT
@velvet yarrow Has your question been resolved?
velvet yarrow
<@&286206848099549185>
strange escarp
Hello
velvet yarrow
hi
strange escarp
Do you have more information
question b. But can you also take a screenshot of the question.
strange escarp
Okay yeah. wait.
I can't help you more. One of my weak spots is probabilities, sorry.
velvet yarrow
thats ok
strange escarp
<@&286206848099549185>
marsh citrusBOT
@velvet yarrow Has your question been resolved?
velvet yarrow
<@&286206848099549185>
marsh citrusBOT
@velvet yarrow Has your question been resolved?
tough igloo
Q2
we know that n=38 and p= 0.59 as given by the question
we should start by finding what q is and the standard deviation
we know that q=0.41 (q=1-p)
and standard deviation is sqrt(npq)
so in this case its sqrt(9.1922)
the question asks us to find P(x>0.63 games won)
we can change it into z score to find the probability where z= (x-mean)/standard devation
thorn marsh
i dont understand why it exactly is multiplication here
marsh citrusBOT
i dont understand why it exactly is multiplication here
thorn marsh
and why the example asks us to list down 2 boys 1 girl, which i did
like i cant see it after listing down, its just two possibilities
raw hawk
i dont understand why it exactly is multiplication here
Most cases multiplication is used when you see "and"
raw hawk
what are the exceptions
I'm not sure I have seen any exceptions yet
And I have not solved every question similar to this so I used most
near elbow
marsh citrusBOT
near elbow
guys, when finding an inverse function, do we need to check the answer with its domain or range?
I got sqrt(x-2) +1
But the answer on the mark scheme is -sqrt(x-2) -1
near elbow
I got sqrt(x-2) +1
But the answer on the mark scheme is -sqrt(x-2) -1
Are they the same answer?
severe owl
y = x²+2x+3
Or y-1 = (x+1)²
Or ±√(y-1) = x+1
Or x = ±√(y-1) -1
You wrote (y-1)²
It will be (y+1)²
severe owl
near elbow
ohhhhh yeaaaa
near elbow
You wrote (y-1)²
thankuuu so much
i just realised
Is that the same with -sqrt(x-2) -1?
marsh citrusBOT
@near elbow Has your question been resolved?
marsh citrusBOT
near elbow
.reopen
marsh citrusBOT
✅
near elbow
It was given in question or you found it ?
it was in the question
near elbow
this one
kind solar
When you go from (y+1)^2 = x - 2 to the next line, you should have the plusminus in there
and of course in order to define a valid function, you need to choose either the plus or the minus for each of the inputs
and the range constraint will help you there
dreamy pollen
it was in the question
Correct now see you got answer as
Y = +√x-2 -1
Y =-√x-2 -1
For y≤-1
If you consider the first answer only for x=2 it will be correct if you take x=6, y will become 3 which will false the y condition
That's why your mark scheme considered the 2nd answer
Y = -√x-2 -1
You can chose any positive value of x ≥2 and it will always be less than equal to -1
marsh citrusBOT
@near elbow Has your question been resolved?
stark field
marsh citrusBOT
pine zealot
Find the standard equation of the circle passing through (−2, 1) and tangent to the line
3x − 2y = 6 at the point (4, 3). Sketch. (Hint: The line through the center of the circle and
the point of tangency is perpendicular to the tangent line.)
marsh citrusBOT
Find the standard equation of the circle passing through (−2, 1) and tangent to ...
quaint elm
pine zealot
I did some weird shit getting the perpendicular of the ecuation, than getting the secant, bla bla but the result was wrong(I can say this since this is from a book that tells you the answer a the end of the book, so you can say I am practicing)
This is the answer: ( x + 2/7)^2 + (y − 41/7)^2 = 1300/49
But I cant get to it
marsh citrusBOT
@pine zealot Has your question been resolved?
grim hemlock
But I cant get to it
Well they tell you y=3x/2 -3 is tangent
Therefore you know that y = -2x/3 + 17/3 must pass through the center. Do you know what to do with that?
pine zealot
Mmm not really
grim hemlock
Mmm not really
Well you know the equation must be in this form.
You have 2 unknowns (x0 and r) and 2 points. Now do you have an idea?
marsh citrusBOT
marsh citrusBOT
Available help channel!
north knoll
could someone please explain how to get from the first line to the second
hazy lion
could someone please explain how to get from the first line to the second
is this not just commutativity?
I'm staring at it, and this is what it seems like
or have we switched the order
north knoll
as in you can just move the things
hazy lion
idk im trying to tell if we really just moved things, but we swapped the differentials, so maybe we did swap the order
north knoll
yea i can see that but i didnt know that was something you could do T^T
hazy lion
let me see
devout mauve
with fubinis theorem you can switch the order of integration. its true like "most of the time", the conditions arent too bad
hazy lion
with fubinis theorem you can switch the order of integration. its true like "mos...
yea this is what i was trying to verify
hazy lion
oh, are they saying, because the fourier transform exists, then swapping order is okay
north knoll
oh so is that like a general rule
this is all so confusing to me :')
anyways ty i guess my question is kind of answered
hazy lion
sorry i thought it was just algebra 
north knoll
LOL sorry
hazy lion
i wouldnt have picked a fubini question for the first problem of the day
i might be slow
maybe @devout mauve can say better
north knoll
i guess it is more of a physics thing
hazy lion
it is not
north knoll
it is not :o
hazy lion
I believe we are fine we need what that both A and phi are square integrable?
which
ugh
north knoll
hrmm im trying to look into it and im just left even more confused
im gonna move on from it and see if it comes up again
anyways ty guys for having a look :)
.clsoe
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marsh citrusBOT
hazy lion
hrmm im trying to look into it and im just left even more confused
yea, I'm sorry I couldnt help 
I'm curious now too
I dont think the conditions for a function being fourier transformable imply that its absolutely integrable
marsh citrusBOT
Available help channel!
celest folio
marsh citrusBOT
celest folio
yes im aware so i applied chain rule
cobalt widget
Yr
cobalt widget
Why did u remove minus?
celest folio
i multiplied a negative by a negative
cobalt widget
Why tho
celest folio
idk 😭
cobalt widget
Ur supposed to multiply with 1/2 no?
celest folio
(-2*1/2) times (5s) times (u)
wait
im confused
is the -2 part of the inside function
celest folio
when i get the derivative of -2(5s^2+8) do i do -2(5s)
celest folio
right
cobalt widget
Now we do first derivative
We take 1/2
We multiply with ut
And power becomes 1/2-1
So power is -1/2
So 1/2 in the denominator
Basically -1/(5s^2+8)^2
That's the first thing
Now for inside right?
So basically it's -10s
Instead of 10s
Got it?
celest folio
so for the inside we dont apply the power rule
celest folio
So basically it's -10s
cause we're doing -2*5s
cobalt widget
No
We do
2 and 2 got cancelled
We are doing -1x10
Cuz numberator there is 2 and denominator u got 2 cuz of multiplying with 1/2
celest folio
so the inside we're getting the derivative of 2(5s^2+8)
which is 2*5s so 10s
then we multiply that by the outside
so -10s/u
cobalt widget
No we r only taking the derivative of (5s^2+8) after
celest folio
lol i was confused on which step we were on
it would be so much easier to see this all on paper
i get it now
cobalt widget
it would be so much easier to see this all on paper
Ikr
marsh citrusBOT
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Available help channel!
leaden island
How can I prove these types of questions without a calculator or is it fine to just say the answer
marsh citrusBOT
How can I prove these types of questions without a calculator or is it fine to j...
spark otter
How can I prove these types of questions without a calculator or is it fine to j...
well it helps writing the range of functions used to create g(x)
since the range of cos(x) is [-1,1]
sec(x) possible values are 1 over that
so either negative and it's (-inf, -1]
or positive and [1, inf)
multiply by 10, the intervals are changed accordingly
+1, changed accordingly
lucid turret
May I receive the blessings of Prof. Alberto Z.
marsh citrusBOT
May I receive the blessings of Prof. Alberto Z.
lucid turret
What is the best method to check whether 3 points are collinear?
My method was this:
let the line connecting two of the points be ax+by+c=0
put the values of x and y and solve for a,b and c
now check whether this equation is satisfied by the third point
raw hawk
Seems good.
unborn condor
"best" is subjective. The easiest is usually the definition.
Let's say the points are A, B and C.
compute vectors AB and AC
if AB = k*AC, they are
lucid turret
vectorss?
lucid turret
Seems good.
have you a good experience of maths?
raw hawk
have you a good experience of maths?
No
lucid turret
may proffessors see my messages
lucid turret
and tell me the 'best method'
unborn condor
My method was this:
let the line connecting two of the points be ax+by+c=0
put ...
seems good
raw hawk
I love the confidence
Thank you 🙂.
lucid turret
to check whether three points
are collinear or not
kind solar
The best method is one that makes you learn something new
unborn condor
woah there
lucid turret
The best method is one that makes you learn something new
Is this a hint?
unborn condor
we're on a schedule here!
raw hawk
and tell me the 'best method'
"Best" depends on the person to be honest
lucid turret
i mean the fastest
unborn condor
fastest depends on the person
lucid turret
I mean 2+2 =4 is faster than (1+1+1+1) = 4
unborn condor
depends on how the operations are implemented
lucid turret
ofc
kind solar
By "fastest", do you mean fastest for a computer or fastest in an exam?
lucid turret
that I can use by myself
raw hawk
that I can use by myself
You can create a good method of finding out if 3 points are collinear
lucid turret
yes
i created one
suppose we're given three points
cyc - >(x1,y1) -> x3 y3 <-
and x2 y2
kind solar
a potentially fast method is computing A-B and A-C and adding a third zero component to them and then checking if their cross product is the zero vector
lucid turret
I am not familiar to that
raw hawk
Cool
kind solar
but that depends on how quickly you can compute cross products
I can compute them pretty quickly and with a little practice I think you can too
lucid turret
you mean cross multiplication?
raw hawk
you mean cross multiplication?
Vector cross product I think 🤔
lucid turret
nvm this
wait
what wiLL be the area of the triangle formed by the three points?
0 right?
no way
use the area of triangle formula then
!
lucid turret
(1/2) |x1(y2 − y3) + x2(y3 − y1) + x3(y1 − y2)|
if this is 0 then the points are collinear
then x1(y2 − y3) + x2(y3 − y1) + x3(y1 − y2) must be 0
which is very easy to compute
🧠
is my method correct bro/
<@&286206848099549185>
oops
i should have waited a minute more
,w substitute x1 = 2, x2 = 5, x3 = 16, y1 = 5, y2 = 11, y3 = 33 in x1(y2 − y3) + x2(y3 − y1) + x3(y1 − y2)
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lucid turret
,w 2(-22) + 5(28) + 16(-6)
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cya
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naive rock
hello i need help knowing what i did wrong
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hello i need help knowing what i did wrong
raw hawk
You didn't divide it properly
fervent rampart
you wrote [ y = \frac{\ln x}{x} + C ] but the original problem says [ y = \frac{\ln x+C}{x} ]
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cloud
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near elbow
Correct now see you got answer as
Y = +√x-2 -1
Y =-√x-2 -1
For y≤-1
If you cons...
Thankuu so much for answering
marsh citrusBOT
Thankuu so much for answering
near elbow
When you go from (y+1)^2 = x - 2 to the next line, you should have the plusminus...
Thankss for answering, sorry for the late reply
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elder grotto
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elder grotto
What did I get wrong
Everything is right
Is it wanting me to do the perimeter times 13??
Like what tf
There’s nothing wrong
elder grotto
That’s the same thing
You basically just said
It’s saying that P=(…) KM=(…) Not P=(…) KM(…)
You just said the same thing
Does it want me to multiply 39 and 13
@cunning fiber
cunning fiber
It’s saying that P=(…) KM=(…) Not P=(…) KM(…)
?
It’s saying that P=(some number) times KM
cunning fiber
You entered as if it was asking for P and KM separately
cunning fiber
It’s asking for P = (what number) times KM
Then asking for the perimeter after that
cunning fiber
It’s saying that P=(…)*KM=(…) units
That’s why there’s a multiplication symbol here
elder grotto
P=39 times 13
rough trellis
Could be written as (P=... ) (Km) if I'm not mistaken
elder grotto
I know what a perimeter is
elder grotto
I don’t understand what it’s saying
I know
I solved it
I just don’t understand the wording
cunning fiber
Can you please let me finish?
elder grotto
I am
cunning fiber
There’s 3 sides
So P = 3(side length)
And the side length is just KM
Hence P=3(KM)
elder grotto
The perimeter is 3?
cunning fiber
It’s asking for P = (what number) times KM
elder grotto
What is P
cunning fiber
Perimeter
elder grotto
The perimeter is 39
cunning fiber
Are you reading what I’m saying?
elder grotto
So if I input the numbers into what you said
39 = 3(13)
Basically it’s just asking what multiplied by 3 to make it 39?
cunning fiber
Yes
elder grotto
Ik but it’s weird because it makes you want to say P = 39
But it’s 3
So this should be right
cunning fiber
Yes.
elder grotto
Appreciate it
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novel juniper
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nobody
elfin berryBOT
nobody
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@novel juniper Has your question been resolved?
still temple
I would say so. How did you define reflections?
novel juniper
I would say so. How did you define reflections?
the way I would in physics
still temple
I don't know how it's done in physics
novel juniper
Uh, basically I would treat this as a plane mirror and find the virtual imahge
still temple
If you have a point (a,b,c) and reflect it along the xy plane are you able to tell me what are the coordinates of the reflection?
novel juniper
(a,b,-c)
novel juniper
I have
still temple
We need to know what happens to e1, e2 and e3
novel juniper
**nobody**
This
still temple
What makes you unsure about your answer?
still temple
Just to be sure you understand the answer
You know what each column in a matrix represents?
still temple
Yep
novel juniper
Another question, if you don't mind
still temple
So if the reflection sends (a,b,c) to (a,b,-c) the first two bases go to themselves and the last to -itself so your answer is correct
still temple
Another question, if you don't mind
Go on
novel juniper
So I'm trying to derive the rotation matrix in $\R^3$
elfin berryBOT
ƒ(why am I here)= MATHS
novel juniper
so I'm rotating the xy plane by angle theta
and depressing it downwards by phi
Like assume it's hinged about the origin
I displace it downwards by an angle theta
so first I figure out the rotation matrix with zero tilt of the z-axos
$$\begin{bmatrix}
\cos(\theta) & \sin(\theta)&0\
-\sin(\theta) & \cos(\theta)&0\
0&0&1
\end{bmatrix}$$
elfin berryBOT
ƒ(why am I here)= MATHS
novel juniper
now I depress it by angle $\phi$
elfin berryBOT
ƒ(why am I here)= MATHS
novel juniper
so I feel the rotation matrix will be
$$\begin{bmatrix}
\cos(\theta) & \sin(\theta)&\cos(\theta)\
-\sin(\theta) & \cos(\theta)&\sin(\theta)\
-\sin(\phi)&-\sin(\phi)& \cos(\phi)
\end{bmatrix}$$
is that right?
still temple
What does depressing something mean?
novel juniper
Y/N
elfin berryBOT
ƒ(why am I here)= MATHS
novel juniper
What does depressing something mean?
I mean pressing it downwards
so the x axis makes an angle with the OG x -axis
still temple
Rotating it again?
novel juniper
sort of
Maybe I'll see 3b1b first
let's say I want to do something like this
Wait a minute
$$\begin{bmatrix}
\cos(\theta) & \sin(\theta)&\cos(\theta)\
-\sin(\theta) & \cos(\theta)&\sin(\theta)\
1-2cos^2(\phi)&1-2cos^2(\phi)& \cos(\phi)
\end{bmatrix}$$
elfin berryBOT
ƒ(why am I here)= MATHS
novel juniper
is this the rotation matrix in $\R^3$
elfin berryBOT
ƒ(why am I here)= MATHS
novel juniper
oops
this isn't a composition
rather the idea is right, but I have to multiply them
don't I
Ok, what I mean to say is I'm yawing and pitching the system, by 𝜃 and 𝜙 respectively, how would I find such a rotation matrix ( say roll is 0)
still temple
If your transformation is obtained by composing two transformations together, you should use matrix multiplication. I'm not familiar with the concepts of yawing or pitching or roll though
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naive topaz
Prove that for any $n>2$, $\sqrt{n+2}+\sqrt{n-2}$ is irrational
marsh citrusBOT
Prove that for any $n>2$, $\sqrt{n+2}+\sqrt{n-2}$ is irrational
elfin berryBOT
Khw
naive topaz
I have tried by contradiction, saying that sqrt(n+2)+sqrt(n-2)=a/b, for a,b integers and b not equal to 0. I squared the equality and got sqrt(n-2)=((a/b)^2-4)/(2(a/b)). Since a/b is rational then sqrt(n-2) is also rational, which is a problem??
static quarry
from that you can conclude that sqrt(n+2) must also be rational
but n-2 and n+2 are integers, so if their square roots are rational then n-2 and n+2 must both be perfect squares
naive topaz
but n-2 and n+2 are integers, so if their square roots are rational then n-2 and...
n is for any real number greater than 2
static quarry
oh, real numbers... that makes it harder
grim hemlock
Do you know the rational roots theorem?
static quarry
who the hell writes n for a real variable haha
naive topaz
Do you know the rational roots theorem?
nope
naive topaz
who the hell writes n for a real variable haha
yeah, I should have mentioned that in the beginning
naive topaz
Is this a real analysis course?
No, this is a Olympics maths problem
For high schoolers
grim hemlock
ok well the rational roots theorem can come in handy when proving things are irrational
naive topaz
ok well the rational roots theorem can come in handy when proving things are irr...
Oh I see, the x here is n then?
But there are square roots so wouldn't be polynomial
grim hemlock
here is an example of a proof with RRT
static quarry
but wait, sqrt(x+2) + sqrt(x-2) is a continuous function for x>2, it has the intermediate value property, it can't skip rationals
naive topaz
here is an example of a proof with RRT
Ahh I see
The equation is x^2-2n-2sqrt(n^2-4)=0
So the root p/q should satify: p divides 2n+sqrt(n^2-4) and q divides 1 so q=1. Getting x=(2n+2sqrt(n^2-4))/k for k integer
And so for x to be rational, sqrt(n^2-4) must be rational as well
We know that between square numbers greater than 2, the difference between them is greater than 4 so for n integer it does not exist
But I don't know for n rational
naive topaz
but wait, sqrt(x+2) + sqrt(x-2) is a continuous function for x>2, it has the int...
ahh you are right... So it should be integer then
sorry
radiant bough
How would i approach this harder question
marsh citrusBOT
How would i approach this harder question
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brave spire
Say u had a recurrence relation, which was built of two recurrences you already know the closed form solution to (so for eg. you had to solve a_(n+1) = (a_n)^2 + a_n and uk the solution to a_(n+1) = (a_n)^2 and a_(n+1) = a_n) then could you combine those two solutions to get the solution of their combination?
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Say u had a recurrence relation, which was built of two recurrences you already ...
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stable axle
How do I calculate the limit of a function if it's given that the function has a limit value at x = 1?
(I just started studying limits so i'm a little confused)
marsh citrusBOT
How do I calculate the limit of a function if it's given that the function has a...
thorny bison
How do I calculate the limit of a function if it's given that the function has a...
What's the function?
stable axle
What's the function?
I have to find for the these three:
f(x) = (x^2-2x)/x-1
g(x) = 1/(x-1)^2
h(x) = sqrt x-1
stoic slate
And what have you tried?
stable axle
And what have you tried?
I have tried to plot the first function
But I just started reading about limits, so I don't know so much about it
marsh citrusBOT
@stable axle Has your question been resolved?
stable axle
<@&286206848099549185>
magic hare
Limits are the concept of approaching, to know what value the function is gonna take as x is approaching 1 is simply plug in 1 into the function
@stable axle
e.g. if we have a function f(x)=x²+3 to know the limit as x is approaching 3 in that function simply you first try to plug in 3 into the function so 3²+3=12 and thats value the function is getting close to as x is getting closer to 3 basically
If you try doing that in the functions you have provided what do you get?
stable axle
So you mean I have to do it like this to find the value of the function:
the first function:
(1^2-2*1)/1-1
magic hare
You try doing that first, now as you can see what do you get when you try to plug in 1 into that?
-1/0 right?
stable axle
yes
stable axle
0 or undefined
magic hare
Not 0 its undefined
And now as you can see
We have a problem
We cant plug in 1
Because we will get an undefined value
So now we go to the second way
Since we ended up with something/0 we need to try and simplify the function
Can you do that?
magic hare
Since we ended up with something/0 we need to try and simplify the function
The reason we try to simplify is to get rid of the part that makes the function undefined when we plug in 1
stable axle
Ok, I will
magic hare
You cant simplify it right?
stable axle
no, not really
stable axle
yes
magic hare
What did you see?
stable axle
I don't know if I understand right to explain what I saw but it was two lines. The first one was a little curved and then straight up. The other one was a little curved and then oblique
magic hare
Mhm
Now
You can visually see that in this particular function when you approach x=1 from the right side its going traight down to negative inf right?
stable axle
yes
magic hare
And from the left side as you get closer to x=1 the function so going up to +inf right?
stable axle
yes, I see it now
magic hare
Now visually you can tell that this function is diversing because its not taking a finite value and its taking two different values when you are getting closer to 1 now right?
stable axle
yes
magic hare
So you say the limit is divergent.
But
As mathematicians
We dont only believe in numbers
So lets try to prove it in numbers
We go back to the function
stable axle
ok
magic hare
(x²-2x)/x-1 right?
stable axle
yes
magic hare
Do you how "the limit as x approaches 1 from the left side" is written?
lim-->1^-
Like this
stable axle
yes
magic hare
Exactly
When its + that means from the right side
And when its - it means from the left side
The limit is actually when both of those values are equal to eachother
So now lets try from the left side
What is our first step?
When taking limits
stable axle
I don't know (just started it, so i know almost nothing about it)
stable axle
oh, okay
magic hare
So we will do the same thing here but we will do it with 1^-
So if we plug it in what do we get?
1^- acts as one but its a number that is just a tiny little bit smaller than 1 so you can think about 0.999999999...
magic hare
Mhm
stable axle
don't know if I did it right but this is what I got
magic hare
Now it acts as 1
Not 0.999999
Like this
Its infinitely close to 1 so its not 0.999999 to be exact
But its smaller than 1
You just pretend like its one until you substract is by itself
For example
magic hare
After this if you go on you will get
The reason you get that is because when you substract something smaller than 1 by 1 you get a negative number
So it something a little bit smaller than 0
Its a negative number but its infinitely close to 0
Now remember what we said
About -1/0?
stable axle
yes
magic hare
About it being undefined
stable axle
undefined
magic hare
Here its not the case
Because 0^- is not 0 its something a little bit smaller than 0
And now i want you to imagine this
When we devide 1 by 0.1 what do we get?
stable axle
10
magic hare
What if it was 1/(-0.1)
stable axle
-10
magic hare
Now 1/(-0.01)
stable axle
-100
magic hare
1/(-0.001)
stable axle
-1000
magic hare
As you can see
As the denominator
Is getting close to 0
Its getting a smaller value in negative numbers
So if we thing about something really really really a little bit smaller than 0 so a negative number that is close to 0
1/ by that number
Shouldnt we get -inf?
Or get closer to -inf
stable axle
yes
stable axle
-inf
stable axle
wouldn't that be written as -1/(-inf)?
magic hare
-1*(-inf) but yeah
And -1*( a negative number)= a positive value right?
So that means
Its +inf
stable axle
yes
magic hare
Now
We just proved
That from the left side
When x is approaching 1
Its going to inf
And if you look at the graph
Thats exactly what was happening
Now we need to check the right side, if the right side also gives us +inf that means the limit is +inf but lets see
We do the same thing
But with 1^+
Try it yourself this time
magic hare
Yeah dont worry
So lets take it from the beginning i kinda made it a bit complex
So
To find the limit of as x is approaching a certain point
The limit from the left side must be equal to the right side
And that will be our limit
Our first rule of taking limits is to first plug in the number and see if you can get a value
In this case we couldnt get a value it was undefined
So we have to check the limits from the left and right side
To make sure if the function even has a limit
At that certain point i mean
So we start with checking the left side
