ƒ(why am I here)= MATHS
#help-38
solid kilnBOT
marsh forum
So Spivak has broken it down casewise
I was thinking of proving this is true for $\R^n; n \in\N$ instead
solid kilnBOT
ƒ(why am I here)= MATHS
nova spire
|x1 + x2 + ... + xn| <= |x1| + ...?
marsh forum
I think I've done this before, but want to do it again anyway
Let $a$ and $b$ be vectors in $\R^n$
solid kilnBOT
ƒ(why am I here)= MATHS
nova spire
what's even |...| in that case
marsh forum
so $a=(a_1,a_2, \dots, a_n);b=(b_1,b_2 \dots ,b_n)$$
solid kilnBOT
ƒ(why am I here)= MATHS
Compile Error! Click the 
reaction for more information.
(You may edit your message to recompile.)
marsh forum
what's even |...| in that case
Length of the vector
nova spire
the 2-norm then
marsh forum
wait, wrong word
magnitude of the vector
so the magnitude of the vector is $\sqrt{(a_1+b_1)^2+(a_2+b_2)^2 + \dots + (a_n+b_n)^2}$
solid kilnBOT
ƒ(why am I here)= MATHS
marsh forum
Right so far?
ionic pendant
note there are many different norms for vectors (in fact, this is one of the axioms something must satisfy to be a norm). but the norm you are using is the 2-norm or euclidian norm
nova spire
ok
well just something to say, scalar products are REALLY nice
and the triangular inequality of this norm, relies a lot on the fact that this is the norm of a scalar product
marsh forum
now $|a|= \sqrt{a_1^2+a_2^2 \dots + a_n^2}$
solid kilnBOT
ƒ(why am I here)= MATHS
marsh forum
$|b| = \sqrt{b_1^2+b_2^2 + \dots + b_n^2}$
solid kilnBOT
ƒ(why am I here)= MATHS
marsh forum
squaring both sides we get
$(a_1+b_1)^2+(a_2+b_2)^2 \dots +(a_n+b_n)^2 \zeta a_1^2+b_1^2+a_2^2+b_2^2 \dots+ \sqrt{(a_1)^2+(a_2)^2 \dots} \sqrt{(b_1)^2+(b_2)^2 \dots$
where $\zeta$ is either ><=
solid kilnBOT
ƒ(why am I here)= MATHS
supple copper
Length of the vector
You should be careful not to use the word length or magnitude
The moment you use it, you’re already claiming it satisfies the triangle inequality
Then there’s nothing to do past that
solid kilnBOT
ƒ(why am I here)= MATHS
Compile Error! Click the reaction for more information.
(You may edit your message to recompile.)
marsh forum
You should be careful not to use the word length or magnitude
Got it
supple copper
**ƒ(why am I here)= MATHS**
Instead say let a be a vector and a₁, …, aₙ be its components, define |a| to be…whatever you have here
marsh forum
I see
got it
so now I have $2a_1b_1 +2a_2b_2 + \dots+ 2a_nb_n \zeta \sqrt{a_1^2+a_2^2 + \dots + a_n^2} \sqrt{b_1^2+b_2^2 + \dots +b_n^2}$
solid kilnBOT
ƒ(why am I here)= MATHS
upper leaf
Recently I cleared PHD entrance examination, I have to appear for interview for this 1000 word research proposal needed reqired . but i have no idea how to write it. my interest is in functional analysis can anyone guide me, what can be topic and its relevent content.
how i can download books from Z library
marsh forum
hmm, I can divide both sides by n
To obtain$2 \frac{\sum_{i=1}^{n} a_ib_i}{n} \zeta \sqrt{\frac{a_1^2+a_2^2 + \dots + a_n^2}{n}} \sqrt{\frac{b_1^2+b_2^2 + \dots +b_n^2}{n}}$
solid kilnBOT
ƒ(why am I here)= MATHS
marsh forum
now what
marsh forum
Recently I cleared PHD entrance examination, I have to appear for interview for ...
!occupied
trim joltBOT
Recently I cleared PHD entrance examination, I have to appear for interview for ...
Someone else is already using this help channel. If you need help with a question, please open your own help channel/thread (see #❓how-to-get-help for instructions).
marsh forum
I think I have to use AM<QM?
nova spire
zeta?
marsh forum
stands for an inequality/equality between the two sides
nova spire
this is cauchy schwarz tho
marsh forum
I mean I can't just say using cauchy-schwarz, can I
Now that I think of it. I can prove it on $\R^2$
solid kilnBOT
ƒ(why am I here)= MATHS
nova spire
Now that I think of it. I can prove it on $\R^2$
trust me induction is not worth it
marsh forum
Not using induction
nova spire
ok but then
marsh forum
$\sqrt{(a_2-a_1)^2+(b_2-b_1)^2} \zeta \sqrt{(a_2-a_1)+ (b_2-b_1)$
nova spire
I don't know how you simplified it
marsh forum
I don't know how you simplified it
This is a fresh start
solid kilnBOT
ƒ(why am I here)= MATHS
Compile Error! Click the reaction for more information.
(You may edit your message to recompile.)
nova spire
**ƒ(why am I here)= MATHS**
Compile Error! Click the <:errors:504266556054962196...
but basically it boils down to prove, if you successfully start from this :
marsh forum
I mean maybe the $\R^2$ case will be easier to prove
solid kilnBOT
ƒ(why am I here)= MATHS
nova spire
$a_1b_1 + ... + a_nb_n \leq \sqrt{a_1^2+...+a_n^2}\sqrt{b_1^2+...+b_n^2}$
solid kilnBOT
rafilou2003
nova spire
this is what you'll have to prove later down the line
marsh forum
$\sqrt{(a_1+b_1)^2+(a_2+b_2)^2 +2a_1b_1+2a_2b_2} \zeta \leq \sqrt{a_1^2+a_2^2} \sqrt{b_1^2+b_2^2}$
marsh forum
I now square both sides
$(a_1^2+b_1^2+a_2^2+b_2^2+2a_1\cdot b_1+2a_2\cdookayt b_2) \zeta (a_1^2+a_2^2)(b_1^2+b_2^2)$
nova spire
what happened to the left hand side
solid kilnBOT
ƒ(why am I here)= MATHS
nova spire
?
marsh forum
ah, yeah, will have to take their magnitudes here
solid kilnBOT
ƒ(why am I here)= MATHS
$(a_1^2+b_1^2+a_2^2+b_2^2+2a_1\cdot b_1+2a_2\cdookayt b_2) \zeta (a_1^2+a_2^2)(b_1^2+b_2^2)$
```Compilation error:```! Undefined control sequence.
l.1421 ...+a_2^2+b_2^2+2a_1\cdot b_1+2a_2\cdookayt
b_2) \zeta (a_1^2+a_2^2...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.
Preview: Tightpage -1310720 -1310720 1310720 1310720
[1{/usr/local/texlive/2023/texmf-var/fonts/map/pdftex/updmap/pdftex.map}]```
nova spire
**ƒ(why am I here)= MATHS**
originally the right hand side is sqrt(...) + sqrt(...)
marsh forum
$(a_1^2+b_1^2+a_2^2+b_2^2+2a_1b_1+2a_2b_2)\zeta (a_1^2+a_2^2)(b_1^2+b_2^2)$
this better?
nova spire
nono
nova spire
**ƒ(why am I here)= MATHS**
```latex
$(a_1^2+b_1^2+a_2^2+b_2^2+2a_1\cdot b_1+2a_...
the left hand side here was doing great
solid kilnBOT
ƒ(why am I here)= MATHS
nova spire
and you didn't change the right hand side
marsh forum
and you didn't change the right hand side
ah, okay
nova spire
$\sqrt{(a_1+b_1)^2+(a_2+b_2)^2} \lessgtr \sqrt{a_1^2+a_2^2} +\sqrt{b_1^2+b_2^2}$
marsh forum
OK. I;ll start from here
solid kilnBOT
rafilou2003
nova spire
**rafilou2003**
just remember you're supposed to get this at the end
marsh forum
ok so do that
gn
$(a_1+b_1)^2+(a_2+b_2)^2 \lessgtr a_1^2+a_2^2+b_1^2+b_2^2 + 2\sqrt{(a_1^2+a_2^2)(b_1^2+b_2^2)}$
solid kilnBOT
ƒ(why am I here)= MATHS
marsh forum
so $[(a_1b_1)+(a_2b_2)] \lessgtr \sqrt{(a_1^2+a_2^2)(b_1^2+b_2^2)}$
solid kilnBOT
ƒ(why am I here)= MATHS
marsh forum
now again squaring both sides
$(a_1b_1)^2+(a_2b_2)^2 +2(a_1)(a_2)(b_1)(b_2) \lessgtr (a_1b_1)^2+(a_1b_2)^2+(a_2b_1)^2+(a_2b_2)^2$
solid kilnBOT
ƒ(why am I here)= MATHS
marsh forum
so $2(a_1)(a_2)(b_1)(b_2) \lessgtr (a_1b_2)^2+(a_2b_1)^2$
solid kilnBOT
ƒ(why am I here)= MATHS
solid kilnBOT
ƒ(why am I here)= MATHS
marsh forum
$0 \leq (a_1 \cdot b_2 -a_2 \cdot b_1)^2$
thus proving the triangle inequality in $\R^2$
solid kilnBOT
ƒ(why am I here)= MATHS
marsh forum
upper leaf
Recently I cleared PHD entrance examination, I have to appear for interview for this 1000 word research proposal needed reqired . but i have no idea how to write it. my interest is in functional analysis can anyone guide me, what can be topic and its relevent content.
how i can download books from Z library
solid kilnBOT
ƒ(why am I here)= MATHS
meager summit
Hello, I have a more complicated math problem that involves a bit of physics. Long story short, I'm trying to determine the shape of a spool(which I've chosen to define as a function of the radius over [0, 2pi]), such that the torque generated by a spring attached to that spool is constant over the entire rotation(or as much of the rotation as the spool allows without intersecting the string).
meager summit
I've currently determined the equations for the torque and am looking into figuring out how to determine deltaL(theta) (the extension of the spring which is deltaL0 + the amount of string currently on the spool)
trim joltBOT
@meager summit Has your question been resolved?
trim joltBOT
@meager summit Has your question been resolved?
meager summit
this is what I've been thinking of, also tried it in Desmos but I couldn't get cos(dbeta)
meager summit
<@&286206848099549185>
meager summit
I was able to solve this discretely but I would also like to solve it mathematically direct
trim joltBOT
@meager summit Has your question been resolved?
trim joltBOT
@meager summit Has your question been resolved?
trim joltBOT
trim joltBOT
Available help channel!
feral hazel
The square ADBE is given.
The point G is on the diagonal.
It is given that:
𝐺𝐹∥𝐴𝐷
𝐺𝐼∥𝐷𝐵
Prove:
a. The FDIE is a Dalton
b. The FGIE is rhombus
feral hazel
<@&286206848099549185>
feral hazel
np
wooden remnant
Well for b I would just use the properties of parralel lines to determine which corners are 90 degrees and thereby proving that the asked for shape is a square
feral hazel
im not sure what u mean let me translate that
feral hazel
yes
wooden remnant
Now use these properties
(I dont know what theyre called in english so thats why these are the dutch names haha)
feral hazel
i need to prove FD=DI and FE=IE then its dalton
wooden remnant
Whats a dalton
feral hazel
feral hazel
.close
scarlet brook
.close
trim joltBOT
Available help channel!
tired brook
How do I do this?
trim joltBOT
How do I do this?
tired brook
does anyone know?
undone sierra
Find length DB in right triangle DBC
prime lynx
Find cos C
urban isle
Find cos C
For what
urban isle
You don't need calculator here
urban isle
How do I do this?
Ok, I'll try to explain
prime lynx
You don't need calculator here
Yes
urban isle
Let's take a look to the right triangle BCD
You know 2 sides and this is right triangle
So how do you think what can you do?
sleek canopy
prime lynx
Isn't cos A = sinC
sleek canopy
also you method is quite neat
sleek canopy
Isn't cos A = sinC
ohhh yes
urban isle
Isn't cos A = sinC
It is
prime lynx
That's what I was trying to do 😅
sleek canopy
Isn't cos A = sinC
so you don t have to use the pythagora theorme
prime lynx
Yes I think
urban isle
also you method is quite neat
And I wanted him to understand all steps, not give him solution
sleek canopy
And I wanted him to understand all steps, not give him solution
I think he could ask me every step he wants 🙂
urban isle
Sorry 😟
What? I'm about everg's almost full solution and all good
Don't worry my friend
urban isle
I think he could ask me every step he wants 🙂
You're right
trim joltBOT
@tired brook Has your question been resolved?
gleaming wave
what did i do wrong
trim joltBOT
what did i do wrong
gleaming wave
wait how tho
urban isle
But you ignored it
urban isle
Wait few mins, I'll get to computer
urban isle
What set (or idk how it called) of ibp do you use?
urban isle
ill just do another problem
Cool
urban isle
Like u = ..., dv =..., du =... and v=...?
gleaming wave
here is my solution
ohhh i realize where i messed up now, i forgot to factor the 2t in the integration and hten i would have to do IBP again
thanks for the help
gleaming wave
thanks man, god bless
urban isle
<3
mighty brook
I have a quick conceptual question
trim joltBOT
I have a quick conceptual question
mighty brook
(y^a) / (z^b) = 3
How do I simplify in terms of y/z in this case? I feel as if I had similar cases in calc 2, but just can't recall after a long summer.
simple haven
if they have different exponents you cannot get everything purely in terms of y/z and a constant
at least I don't think you can
you can get y^b/z^b * y^(a-b) = 3, and then y/z = (3/y^(a-b))^(1/b)
but you have that pesky y
mighty brook
Hmmm
ancient edge
trim joltBOT
ancient edge
How do i graph this function?
worthy venture
trim joltBOT
@ancient edge Has your question been resolved?
ancient edge
<@&286206848099549185>
trim joltBOT
@ancient edge Has your question been resolved?
queen radish
How do i graph this function?
You graph this function using your calculators polar graphing setting. Or an online graphing calculator.
trim joltBOT
You graph this function using your calculators polar graphing setting. Or an onl...
dull temple
ahhh it's a bug
queen radish
Sorry, I keep responding to questions I see active in the "help" section, but it keeps saying its timed out so when I respond I take over the thread
Edit: Nvm, i messed up and was looking through unasnwered questions in the "available" section. Totally my fault/bad
gleaming wave
can someone help find where i went wrong
trim joltBOT
can someone help find where i went wrong
ancient edge
Do u have ur work
raw magnet
^
trim joltBOT
@gleaming wave Has your question been resolved?
tribal fractal
can someone help me? i cant figure out how to do this?
trim joltBOT
can someone help me? i cant figure out how to do this?
trim joltBOT
trim joltBOT
Available help channel!
brazen gazelle
trim joltBOT
mellow relic
∫
0
∞
x
n
e
−x
2
cos(2πx)dx
1
2
Γ
(
𝑛
+
1
2
)
cos
(
𝑛
𝜋
2
)
2
1
Γ(
2
n+1
)cos(
2
nπ
)
ionic pendant
use ampere's law
river willow
∫
0
∞
x
n
e
−x
2
cos(2πx)dx
=
1
2
Γ
(
𝑛
+
1
2
)
cos
(
𝑛
𝜋
2
)
= ...
ouch
brazen gazelle
∫
0
∞
x
n
e
−x
2
cos(2πx)dx
=
1
2
Γ
(
𝑛
+
1
2
)
cos
(
𝑛
𝜋
2
)
= ...
what the fuck
brazen gazelle
use ampere's law
which one was that again
brazen gazelle
how do u know what formula to use instantly
i got some troubles finding out what formulas to ue
trim joltBOT
@brazen gazelle Has your question been resolved?
ionic pendant
how do u know what formula to use instantly
practice helps. but it may help to have a list of relevant equations and an explanation of what it represents when you do these problems
twilit flume
Compare this with 2/5
trim joltBOT
Compare this with 2/5
twilit flume
uh
me and i friend have a challenge
i got the equation inside
to
idk what to do next lol
stray drum
Wdym by compare with 2/5, as in which one is larger?
stray drum
Hmm
The sum is convergent
Maybe try finding the infinite sum of that and comparing it with 2/5
wind cloak
multiplying and dividing by 2 might make it more clear to find furhur steps
twilit flume
hmmm
trim joltBOT
@twilit flume Has your question been resolved?
twilit flume
.close
vague walrus
i tried to understand what my sir said and maybe this is what he said ?? i still dont understand
Consider n=3
Prime Factorization:36=2^2×3^2
Here, p1=2 and p2==3, with a1=2 and a2=2, both of which are even.
Number of Divisors: d(36)=(2+1)×(2+1)=3×3=9
The factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, and 36, totaling 9 divisors, which is odd.
Conclusion:
Since 36 is a perfect square, all exponents in its prime factorization are even.
Adding 1 to each even exponent gives us odd numbers.
The product of these odd numbers is itself odd, meaning the number of divisors is odd.
Therefore, perfect squares have an odd number of divisors.
Number of divisors of a perfect square is always odd.
i kinda get it
but not all
this was in permutation and combination chapter lol
oh fuck i get it
.close
coarse meadow
exotic pine
alr named them some points
@coarse meadow notice how DECP is a quadilateral, so the insides sum to 360, so x+36+ECP+EDP=360
try this
coarse meadow
<@1115153680010846229> notice how DECP is a quadilateral, so the insides sum to ...
how would we find ecp or edp?
exotic pine
yes
exotic pine
alr, find ACB
i removed unnescicary stuff
im sure you know how to find ACB now
@coarse meadow
exotic pine
yup
marsh forum
could someone explain this please
trim joltBOT
could someone explain this please
marsh forum
Like isn't each column of EA equal to the corresponding row of E times A
runic dew
-corresponding row of E times A
If you multiply a row of matrix and a column you will get a single element
trim joltBOT
trim joltBOT
Available help channel!
sterile loom
trim joltBOT
sterile loom
SOLOUTIONS
I understand all working, just thought doing this method wouldn't work because the area under the curve would skew the answer.
Plz help
jagged phoenix
I understand all working, just thought doing this method wouldn't work because t...
what did you think exactly
i'm not sure what the issue is
sterile loom
I thought because there is negative area on the right hand side (only taking the right side of the y axis into account as it is odd and will times it by 2 at the end), when you find the area between the two lines your answer will be off because of the area under the axis is negative
and im trying to find the area not evaluate it
the graph is above btw
@jagged phoenix
Or are the positives and negatives not relevant when dealing with area between two curves
jagged phoenix
I thought because there is negative area on the right hand side (only taking the...
You want to take absolute value when you find the area
it's a geometrical thing
trim joltBOT
@sterile loom Has your question been resolved?
copper mantle
trim joltBOT
copper mantle
dont really understand how to approach the problem without the known dimensions
fiery perch
You can form a triangle, from the middle to Q and PQ
naive otter
Height is given 12 cm !
Base is 10 cm
Right angled triangle !
With the red line be 5√2cm
naive otter
exotic pine
sorry discord bugging hard rn
trim joltBOT
@copper mantle Has your question been resolved?
trim joltBOT
trim joltBOT
Available help channel!
graceful zenith
need help w this qu
trim joltBOT
need help w this qu
ebon plover
i think that i and j are unit vector
graceful zenith
yea i kinda forgot what projections are again
ebon plover
since it says drawn to scale
I assume that $i = \dbinom{1}{0}$ and $j = \dbinom{0}{1}$
solid kilnBOT
e_waste
ebon plover
you gotta construct a triangle where $i+j$ is the hypotenuse
solid kilnBOT
e_waste
ebon plover
and scale the other two sides
now you get the length of i and j
do the same thing to u
and div the length by i and j
you may get the result
graceful zenith
ok
graceful zenith
you may get the result
wait what does projection mean again
graceful zenith
uhhh
graceful zenith
Did I use peojection?
i dont get it lol
ebon plover
Well projection is this
u' is the projection of u in v
but how we get u'?
d is the modulus of u
which is $|u| \cos \theta$
solid kilnBOT
e_waste
ebon plover
now we need a value to represent the direction of u
as we know u and v are in the same direction
so $u' = d \frac{v}{|v|}$
solid kilnBOT
e_waste
ebon plover
as for $\cos \theta$ is $\frac{u \cdot v}{|u||v|}$
solid kilnBOT
e_waste
ebon plover
combine these together we get $u' = \frac{u \cdot v}{|v|^2}v$
solid kilnBOT
e_waste
trim joltBOT
Available help channel!
torn cave
is this the same because of the "arg(z1*z2) = arg(z1) + arg(z2)" where in this case arg(z1) = 0 because its a real number?
thorn heart
is this the same because of the "arg(z1*z2) = arg(z1) + arg(z2)" where in this c...
Yeah, let a be a real number, then arg(z)=arg(az) for all a
The only thing a will modify is the modulus of z, multiplying it exactly by a factor of a
graceful nova
is this valid sigma notation or does it always need to start from 1?
nimble stone
its valid
graceful nova
I ask because these are the correct answers for a question I did
and I determined all of these and they have the same sum
nimble stone
if theres a standard formula for the sum, they generally start from 1
nimble stone
is this valid sigma notation or does it always need to start from 1?
in which case youd have to do 1 to 33 - 1 to 2
trim joltBOT
trim joltBOT
Available help channel!
plush igloo
is there a way i can get better at thinking with probabilities and expected values without solving more until i recognize a pattern
plush igloo
it's mainly the expected value questions that take a toll on me
I can't quite wrap my head around why i got to those solutions when solving expected values
it leans towards formulaic instead of logical
this question for example
wraith hinge
expected value is just an integral, nothing too mysterious about it
plush igloo
good thing i suck at integrals then 
plush igloo
this question for example
i can only think about this in terms of probability
can't really apply expected value to it
trim joltBOT
@plush igloo Has your question been resolved?
rotund grove
rey lo
rotund grove
l
Draw the graph of x = - (b-a) for (i) b<a,
(ii) b>a where a and b are constants.
bro help me
Hey
.close
trim joltBOT
rotund grove
.reopen
trim joltBOT
✅
rotund grove
If the point A(-3, -4); B(a, -6); C(2,-2); D(-3,0) are taken in order they form a parallelogram, then find the height of the parallelogram if the base is AB.
trim joltBOT
@rotund grove Has your question been resolved?
rotund grove
,w arctan (-0.4) * 180/pi
solid kilnBOT
rotund grove
,w tan 68 deg
solid kilnBOT
rotund grove
<@&286206848099549185>
huh
<@&286206848099549185>
no one able to solve?
c'mon
zinc ginkgo
If the point A(-3, -4); B(a, -6); C(2,-2); D(-3,0) are taken in order they form ...
did you draw a picture
rotund grove
yrs
can you solve this question?
.
this question is not very hard
BRO
:
I guess I'll jsut do my work
why doesn't anyone help me?
am I the most hated person on this server
zinc ginkgo
probably because you use too many memes and make it hard to look for your relevant messages
rotund grove
If the point A(-3, -4); B(a, -6); C(2,-2); D(-3,0) are taken in order they form a parallelogram, then find the height of the parallelogram if the base is AB.
pearl lotus
am I the most hated person on this server
aww I m here for u 
rotund grove
LOl
Ok so
I have calculated the are of the parallelogram and subsequently the height but that seems too unprofessional.
Is there any better way to do it?
rustic dock
first we need to find a
rustic dock
good
rustic dock
write equation of ab
rustic dock
is it coorect?
rotund grove
how do i know
rustic dock
tell me the defination of height of parallelogram
rustic dock
rotund grove
yeah
rustic dock
this is known as height of parallelogram
rotund grove
IKR
rotund grove
no
rustic dock
have you heard of perpendicular distance formula
rotund grove
I made my own
if y = mx+b and y = mx + b1 are two parallel lines then the distance between them is given by:
(b1-b)/(root(m^2+1))
is this correct?
rustic dock
yes
rustic dock
you can use this method also
rustic dock
yes question is done write the 2 equations and its done
rustic dock
it asked height right?
rotund grove
I think it's incorrect
rustic dock
yes let me check
rotund grove
My friends also supported me
rustic dock
am getting 20/root29
rotund grove
eha
rustic dock
wrong ans given
rotund grove
,calc 20/root29
solid kilnBOT
The following error occured while calculating:
Error: Undefined symbol root29
rotund grove
,w 20/root29
solid kilnBOT
rustic dock
not at all approx its so far off from 4 ans is 3.71
rotund grove
what is the formula for the orthocenter
of a triangle given its coordinates of vertices?
rustic dock
formula is very big use the property of orthocenter to find orthocenter
rotund grove
can you calculalte the orthocenter with vertices (0,0), (3,4) and (4,0)?
rustic dock
yes
rustic dock
i will tell you process you calculate
rotund grove
Wait
rustic dock
but why you want to find orthocenter the ans is 3.71
rotund grove
ok
so
That was another question I'm done with that one
I was correct the orthocenter is (3,0.75)
😎
rustic dock
nice
rustic dock
never use the orthocenter formula
rustic dock
find altitude then intersection
rotund grove
Draw the graph of x = - (b-a) for (i) b<a,
(ii) b>a where a and b are constants.
Here
rustic dock
i think you have to shade 1st and 4th quadrant for first one
rustic dock
if b and a are arbitrary constant
rotund grove
because it can't be a line because it will create reference
rustic dock
rotund grove
or could it be
ok
so
wait
for the (i) graph you have marked b-a
the graph if a-b
= -(b-a)
rustic dock
ok i think i have to go i need to study chem
trim joltBOT
rotund grove
thanks btw
The center of a circle is (2a, a-7). Find the sum of all possible values of a, if the circle passes through the point (11,-9) and has diameter 10root2 units.
.reopn
.reopn
.reopen
trim joltBOT
✅
rotund grove
The center of a circle is (2a, a-7). Find the sum of all possible values of a, if the circle passes through the point (11,-9) and has diameter 10root2 units.
!status quo
trim joltBOT
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
3
8 is apparently the answer
butt, wouldn't there be infinite number of circles satisfying that condition?
.reopen
rotund grove
the radius is 5root2
and the point is (11, - 9)
therefore there is only one condition that the distance between this (11, -9) and the center which is (2a, a-7) is 5root2
rotund grove
ok but there are infinite right?
rustic dock
but there will only be 2 whcih satisfy all the condition
rustic dock
(11-2a)^2+(2+a)^2=50?
solve this equation you will get all possible values of a
trim joltBOT
rotund grove
I get it now the values of x and y coordinates will be in the form of a line which is fixed
trim joltBOT
Available help channel!
trim joltBOT
orchid drift
a
loud cosmos
trim joltBOT
Available help channel!
quaint pivot
the function 2^x +1 contains a horizontal asymptote of 1
quaint pivot
is there a vertical asymptote on this graph?
alpine delta
no i dont think so
quaint pivot
no i dont think so
I thought so
alpine delta
yeah cause vertical asymptotes occur when the denominator is equal to zero
so there isnt going to be one
quaint pivot
yeah cause vertical asymptotes occur when the denominator is equal to zero
and the domain of this is -infinity to + infinity right?
and the range is
alpine delta
yeah
alpine delta
yeah
quaint pivot
cool
trim joltBOT
trim joltBOT
Available help channel!
dim lodge
Does anyone have an idea on how to do this? I've tried isolating in numerous ways but it ends up looking messy and incorrect each time. I think I've been doing math for a bit too long, so my head is a bit fried
trim joltBOT
Does anyone have an idea on how to do this? I've tried isolating in numerous way...
Show your work, and if possible, explain where you are stuck.
atomic rain
Does anyone have an idea on how to do this? I've tried isolating in numerous way...
isnt asking for to find all x
it*
trim joltBOT
@dim lodge Has your question been resolved?
dim lodge
isnt asking for to find all x
Yes, it's asking to find all x. But each time I've isolated it, it just has more radicals when you raise each side to the power of two to remove the radicals. I'm unable to take a picture of my work right now because I've left my phone at home and I'm studying at a cafe (I only have my laptop).
I've tried to raise it to the power and isolate various ways, but it continues to have additional radicals
atomic rain
2x-7 = x + 2sqrt(x) + 1
atomic rain
Yes, it's asking to find all x. But each time I've isolated it, it just has more...
you suppose to muiplty both sides by power of 2
dim lodge
Yes, I've done that
atomic rain
2x-7 = x + 2sqrt(x) + 1
did you get this?
plush igloo
you can then move the right hand side to the left hand side
dim lodge
2x-7 = x + 2sqrt(x) + 1
I got this previously
dim lodge
you can then move the right hand side to the left hand side
Yes, I've done this but it still has a sqrt
plush igloo
Yes, I've done this but it still has a sqrt
you can factorize it with sqrt still in there
plush igloo
make a variable u = sqrt(x)
dim lodge
-2sqrt(x) = -x+8
Yes, I've also gotten that
atomic rain
then 4x = 64 - 16x + x^2
plush igloo
x - 2sqrt(x) - 8 = 0 si then u^2 - 2u - 8 = 0
atomic rain
or you can try u = sqrt(x)
plush igloo
which can be factorized into (u-4)(u+2) = 0
and then you can put sqrt(x) back into its place
dim lodge
Ah, ok. I probably should not have assumed that the answer would be a nice number answer then (it had been thus far). Thanks for your assistance guys
plush igloo
$(\sqrt{x}-{4})(\sqrt{x}+{2})={0}$
solid kilnBOT
sudpad
plush igloo
let's goo
and then you try to find sqrt(x) = 4 and sqrt(x) = -2
the latter one isn't possible so only sqrt(x) = 4
dim lodge
Ok, thank you!
trim joltBOT
trim joltBOT
Available help channel!
grand oriole
trim joltBOT
grand oriole
how do I solve this
trim joltBOT
@grand oriole Has your question been resolved?
grand oriole
<@&286206848099549185>
trim joltBOT
@grand oriole Has your question been resolved?
trim joltBOT
@grand oriole Has your question been resolved?
trim joltBOT
@grand oriole Has your question been resolved?
granite light
This question makes no sense
trim joltBOT
@grand oriole Has your question been resolved?
trim joltBOT
trim joltBOT
Available help channel!
unreal lark
how does that work
trim joltBOT
how does that work
thorn heart
how does that work
Alright, I think I get what they're saying
Though I have to say, θ, isn't properly defined or bounded basically anywhere, which is quite the hassle, really
thorn heart
pls explain
Are you familiar with inverse functions and injective functions?
unreal lark
im learning it right now
whole coral
i think they use the domain for arccos
(but incorrectly 
)
unreal lark
?
thorn heart
im learning it right now
Well, first of all: the cos function isn't injective, do you know why?
amber python
where did the theta even come from 
unreal lark
Well, first of all: the cos function isn't injective, do you know why?
what does injective mean
thorn heart
what does injective mean
Alright: suppose a function f(x)=y, f(x) is injective if and only if for each value of y there exists a single value of x
thorn heart
In other words, if only one x exists for each y
unreal lark
Well, first of all: the cos function isn't injective, do you know why?
yeah it fails horizontal line test
thats why they put a domain for an inverse to exist
thorn heart
thats why they put a domain for an inverse to exist
Alright, that's the gist of it
Essentially, since on the interval [-π/2, π/2] the function is actually injective, the range of the inverse function is defined as such
unreal lark
uh huh
whole coral
Essentially, since on the interval [-π/2, π/2] the function is actually injectiv...
(it's not)
unreal lark
(it's not)
why?
thorn heart
(it's not)
Oh, did I get it messed up? Let's see
whole coral
Oh, did I get it messed up? Let's see
(arcsin rather than arccos, arccos is 0 to pi)
thorn heart
Yeah, nvm, cos is symmetric, then its interval has to be [0,π]
whole coral
(but incorrectly <:SadCat:481661086399987713>)
Hence 
thorn heart
Yeah, nvm, cos is symmetric, then its interval has to be [0,π]
Hm, yeah, i'm finding it fairly confusing as well, I think I'll wait for someone more knowledgeable to answer
unreal lark
help alright
thorn heart
Though I can tell you for sure it's much more ambiguous than it should be
unreal lark
someone pls explain how this works
unreal lark
yeah 0<=y<=pi
unreal lark
0<=y<=8pi
whole coral
someone pls explain how this works
It doesn't: they messed up and got inverse cos mixed with inverse sin
knotty quest
Ok so do u know what range of a function is?
unreal lark
It doesn't: they messed up and got inverse cos mixed with inverse sin
but its a proof question
knotty quest
It doesn't: they messed up and got inverse cos mixed with inverse sin
Nop they didn't???
whole coral
Nop they didn't???
They did-
whole coral
(arcsin rather than arccos, arccos is 0 to pi)
knotty quest
Where 😶
Yes
That's what they did
Thetha between 0 to pi
When we solve that inequality
whole coral
Thetha between 0 to pi
unreal lark
oh wait
oops
i think this is a misunderstanding
they say
that let theta=arcsinx
knotty quest
Mhm
unreal lark
knotty quest
Yes
So if thetha is
Arcsinx
Then
So look
x is sina
I'll just use a instead of thetha if that's ok
unreal lark
ye
knotty quest
Ok so x is sina
And from the properties
U know that
cos(pi/2-a) is sina
Right?
unreal lark
yea
knotty quest
help thanks for explaining
Np
unreal lark
.close
steady tulip
trim joltBOT
steady tulip
Question 3
I already got the 11 obvious ones
I can't find the rest 88
Like there should be some pattern
wraith hinge
someone pls explain how this works
this is incorrect
trim joltBOT
@steady tulip Has your question been resolved?
steady tulip
<@&286206848099549185>
bleak quail
Which ones u found already?
steady tulip
The ones where the whole term is 0 and the ones where either of the terms is 1001 or 2002
steady tulip
Alr ping me if you do 🙏
bleak quail
@steady tulip
I found something but it's not like that's a whole strategy, just some thoughts
If I spend more time with this I mb will find more, but rn I want to show how smth works here
I tried placing (11 * 13) in one bracket and (7 * ?) in another bracket the way that multiples would be different by 1 or 2 or 5 or 6 or 7, cause that's all numbers brackets could differ by (actually, there are two 1s but I just didn't write another)
And the same thing with (7 * 11) and (13 * ?)
And with (7 * 13) and (11 * ?)
So that way I could find some solutions, which were 79 and 78. Check if you want
Now if we want to find more answers we might as well need to multiple the first bracket by something too
But that would be kinda tuff. Not saying that it's impossible to solve that way, though
Oh, btw it is also could be a good thing to place 11 and 7 and 13 in different bracket each and try to multiple them by something. This way could also have other solutions (although u can't multiple by 7, 11 or 13 cause it would give the first case.
And the last case u already did - where 7, 11 and 13 are all in the same bracket.
@steady tulip
lusty delta
you are not supposed to enumerate all possible values of n
read the solution here the solution explains the problem pretty well
steady tulip
you are not supposed to enumerate all possible values of n
I know
bleak quail
Well I mean if you find all solutions then you can count them and that is the answer
steady tulip
Oh, btw it is also could be a good thing to place 11 and 7 and 13 in different b...
Yea
bleak quail
It's just a diff way to go, maybe more tuff
steady tulip
Well I mean if you find all solutions then you can count them and that is the an...
There are too many to count, there should be some pattern
steady tulip
read the solution here the solution explains the problem pretty well
I don't have it , this question was sent to me by a friend
bleak quail
There are too many to count, there should be some pattern
Well I wrote some thoughts about it above
lusty delta
I don't have it , this question was sent to me by a friend
ok i see
steady tulip
Well I wrote some thoughts about it above
Yea I saw those
lusty delta
the idea is you know the number must be divisible by 7, 11, and 13, yes?
steady tulip
Yea
steady tulip
are you familiar with something called "chinese remainder theorem" ?
I know congruent modulo
lusty delta
ok close enough
steady tulip
Idk this tho
lusty delta
considering each of the terms (n - 0), (n - 1), (n - 2), and (n - 7), think about how many possible values of n (mod 7), n (mod 11), and n (mod 13) there are
bleak quail
are you familiar with something called "chinese remainder theorem" ?
Oh okay now u can go without me lol. I didn't understand this thing at school
steady tulip
considering each of the terms (n - 0), (n - 1), (n - 2), and (n - 7), think abou...
For mod 7 we can have 0, 1 , 2 right?
lusty delta
yes
lusty delta
good that is correct
steady tulip
Oh okay now u can go without me lol. I didn't understand this thing at school
Oh ty for trying tho
steady tulip
good that is correct
Now what tho?
lusty delta
Now what tho?
how many combinations of these modulo are there?
steady tulip
how many combinations of these modulo are there?
Unique combinations?
lusty delta
yes
for example number of ways you can construct a system that looks like this
n congruent 1 (mod 7)
n congruent 7 (mod 11)
n congruent 2 (mod 13)
steady tulip
Oh i see
steady tulip
how many combinations of these modulo are there?
12!/(12-3)!x3! ?
lusty delta
no it's not that complicated
steady tulip
My permutation and combination is weak ;-;
lusty delta
For mod 7 we can have 0, 1 , 2 right?
well, you pick one of these values for (mod 7)
steady tulip
no it's not that complicated
3 x 4 x 4?
lusty delta
yes that's correct
so this is where the chinese remainder theorem is used
because in your system the moduli are relatively prime to each other (7, 11, and 13), each of these systems has one unique solution for n (mod 7 * 11 * 13)
so if the question just asked you to find the number of solutions with n between 1 and 2002, how many solutions would there be?
steady tulip
so if the question just asked you to find the number of solutions with n between...
Lemme think
I don't get it ;-;
lusty delta
because in your system the moduli are relatively prime to each other (7, 11, and...
do you understand this sentence?
steady tulip
I understand till the relatively prime part
"as one unique solution" is the part I don't get
lusty delta
the "as" should be a "has"
lusty delta
yes
lusty delta
welcome
steady tulip
.close
trim joltBOT
trim joltBOT
Available help channel!
trim joltBOT
brave narwhal
.close
trim joltBOT
trim joltBOT
Available help channel!
acoustic snow
let $a_n$ be a positive sequence such that $\limsup\limits_{n\rightarrow\infty} \frac{a_{n+2024}}{a_n} < 1$. prove that $\lim\limits_{n\rightarrow\infty} a_n = 0$
solid kilnBOT
sibber
lean prawn
I think you can probably argue that if a subsequence converges to 0 then the full sequence converges to 0
acoustic snow
how
lean prawn
well let's say the limit you're given is c
if you form a new sequence that is every 2024th term of the original sequence
acoustic snow
every or shifted by 2024?
lean prawn
well anyway
there exists some $N$ such that for terms $a_n$ s.t. $n\ge N$
$\frac{a_{n+2024}}{a_n}<c<1$
for some $c$
acoustic snow
right
solid kilnBOT
Stipendi
acoustic snow
c would be the limsup no?
lean prawn
then we can choose c=(1+c')/2 for example
acoustic snow
why not choose the limsup itself?
lean prawn
well just because the limit is c' doesn't mean the quotient will stay strictly below c' all the time after some N
that's not what the limit guarantees for us
lean prawn
then we can choose c=(1+c')/2 for example
And I think this c is obtained with epsilon=|c-c'|/2
acoustic snow
wdym?
lean prawn
**Stipendi**
so with that epsilon, you will find an N such that this inequality holds for n>=N
a picture would help a lot
I'm trying to do the picture in my head
which sometimes doesn't end well
acoustic snow
**Stipendi**
but doesnt this hold for any epsilon > 0?
lean prawn
well there is no epsilon in that statement so I don't know what you mean
solid kilnBOT
sibber
solid kilnBOT
sibber
