#help-41
keen pawn
amber waspBOT
keen pawn
Just the first part
as to why I'd expect it to be this
I'd expect it to be the same as we're dealing with sets
cunning birch
I'd expect it to be the same as we're dealing with sets
dimension(subspace) = card(a basis)
though the analogy is imperfect, it comes from that
wheat grove
Contracts for two construction jobs are randomly assigned to one or more of three firms, A, B, and C. Let
Y1
denote the number of contracts assigned to firm A and
Y2
the number of contracts assigned to firm B. Recall that each firm can receive 0, 1, or 2 contracts.
2 jobs are available. But I'm confused about the last sentence. It says each firm can recieve 0-2 contracts. I take that just to mean that out of the 3 firms, one firm (say A) can get all the jobs and the rest would have 0. Do I have that right?
zealous turtle
amber waspBOT
zealous turtle
How do I find the critical numbers in a closed interval trig function?
Because I know from the graph of this I have three points.
And I got where my first derivative equals 0.
Which is pi/6.
But aside from that, I'm stuck and would like help in this.
I did.
solar relic
I already saw it now
we know that cos of a varible is 1/2 when variable is 60 degres, so 2pi/6
solar relic
im looking for find the last critic points
but the cos function is par
then -pi/6 also is critic point
zealous turtle
then -pi/6 also is critic point
Because it's within the -pi/2-to-pi range?
solar relic
yes
zealous turtle
Fair.
solar relic
so how cos is a periodic function
if we sum 2pi to pi/6 we can find the last
but this is not on the domain
mmmm
zealous turtle
Not -pi/2?
Can we ping Helpers?
<@&286206848099549185> We both don't know how to solve this problem, I was wondering if anyone else might know?
deft ice
Because I know from the graph of this I have three points.
the end points must also be considered. looking at the graph, there's a global min at the beginning of the interval.
zealous turtle
the end points must also be considered. looking at the graph, there's a global m...
So those would be my CNs?
Pi/6, -Pi/2, and Pi?
deft ice
there's three points for where f’ = 0 and a fourth point for the absolute minimum.
zealous turtle
Crap.
Okay I'm lost.
How would I be able to find where my f' = 0?
On the closed interval?
deft ice
How would I be able to find where my f' = 0?
this should help:
- draw a unit circle.
- find the points on the circle that have an x value of 1/2 (because cosine is the x value of triangles on the unit circle)
zealous turtle
this should help:
1. draw a unit circle.
2. find the points on the circle that...
5pi/3.
And pi/3.
deft ice
where's the start of the interval -pi/2 on the unit circle?
solar relic
Imagine that you are travelling around the unit circle but clockwise
solar relic
yes
solar relic
zealous turtle
Huh?
solar relic
The green ones are the values of 2x to be 1/2
solar relic
It is the first
zealous turtle
Oh I mean alone.
Like I'm trying to figure out other ways to find it through using ONLY pi/6.
But I don't know if that's practical or not.
amber waspBOT
deft ice
test
zealous turtle
.reopen
amber waspBOT
✅
zealous turtle
Wassup zyx?
deft ice
Like I'm trying to figure out other ways to find it through using ONLY pi/6.
it may be easier to rewrite the interval in terms of cosθ:
θ = 2x, so the interval in theta is [-pi, 2pi]
deft ice
shoot
zealous turtle
I'll save this question for my teacher's office hours tomorrow.
So for this, I got -sinx^2 = -2cosx.
How do I use that equation so far to find the derivative?
warm burrow
Those who know how to find the derivative: 💀💀🔥🔥
zealous turtle
Oh lol.
Okay.
How do I use that equation so far to find the LOCAL MIN AND MAX lol.
tawdry lintel
Where did sin2x come from
final matrix
Where did sin2x come from
cos^2(x) you would apply the chain rule to take the derivative so it would become 2(cosx)*-sin(x)=-2sinxcosx=-sin(2x)
tawdry lintel
Oh nevermind
final matrix
How do I use that equation so far to find the LOCAL MIN AND MAX lol.
you let the derivative equal 0
and then find the value of x which makes that equation true
this gives you stationary points or where the rate of change is 0 which indicates a local max/min
you can then either graph or find the second derivative and find the concavity
zealous turtle
I see.
So far, I made it so that -Sin2x = -2Cosx.
But aside from that, I'm kind of stuck as to what to do next.
What's the next step for me?
warm burrow
I love you unit circle
zealous turtle
What's the next step for me?
What about this?
warm burrow
So far, I made it so that -Sin2x = -2Cosx.
Is this the derivation
tawdry lintel
Both shouldn’t be negative
zealous turtle
Is this the derivation
The OG derivative was -sin2x-2cosx.
tawdry lintel
But aside from that, I'm kind of stuck as to what to do next.
You will end up having two cases (as reflected in the graph with two local extrema), one where cos is divided out and one where cos (since it is in both terms) equals zero.
deft ice
cos^2(x) you would apply the chain rule to take the derivative so it would becom...
the derivative is incorrect. see the comment i'm replying to for the correct version.
warm burrow
Can anyone fact check this I got something different
deft ice
final matrix
the derivative is incorrect. see the comment i'm replying to for the correct ver...
sorry, how is it incorrect?
is it to do with the conditions of the question
deft ice
sorry, how is it incorrect?
sorry, what i meant to say is:
eldritch-y, your derivative is incorrect. see Gizmic's derivative for the correct version.
zealous turtle
Okay.
zealous turtle
Question.
Now that I found the critical points, what do I do next to find the local max and min?
light kelp
a critical point is either a local max or a min
were u taught the first derivative test?
zealous turtle
were u taught the first derivative test?
Oh yeah, find the critical points and plug them into the OG equation to find the answers right?
light kelp
umm actually in the case of trig functions i think it would be easier to take the second derivative and if f''(x) < 0 its a local max and f''(x) > 0 its a local min
deft ice
alterative: unit circle. you can look at the circle and see the sign of cos, sin, and when they're equal to or less than 1.
deft ice
Oh yeah, find the critical points and plug them into the OG equation to find the...
do a quick review of the first derivative test, my g. you have a textbook?
zealous turtle
No.
deft ice
pro tip: find a free textbook online. at the very least, even if you don't read each chapter fully, do the examples.
that is step 1 to doing well in most classes.
deft ice
oh nice!
warm burrow
Insanity
amber waspBOT
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manic sky
I understand that the equation is a hyperboloid and I know what the equation of a line is but I am lost as to how to start
manic sky
yes
i am not sure because since the set is curved isnt there a possibility that the line would go outside of the set to connect the two points
full elk
You want to write a generic line in vector form with unknown coefficients whose coordinates (for all values of the parameter) satisfy the equality
manic sky
i dont understand im sorry
full elk
So consider a line like $r(t) = (a_0, b_0, c_0) + t(u_x, u_y, u_z)$
grizzled pagodaBOT
Azyrashacorki
full elk
That gives $x = a_0 + tu_x$, etc
grizzled pagodaBOT
Azyrashacorki
manic sky
yeah, i understand that
am i supposed to substitute the parametrization of the equation line into my hyperboloid equation set?
full elk
Yes
And then you'll get some polynomial in t that has to be identically 0, which gives 3 equations for the 6 unkowns.
Since you want one line, you can try and guess a solution for the first equations and solve for the remaining variables
manic sky
okay thank you very much
amber waspBOT
@manic sky Has your question been resolved?
keen pawn
amber waspBOT
keen pawn
As we want to minimise $F$, we wish to maximise the denominator
grizzled pagodaBOT
A dense set
keen pawn
that gives us $tan(\theta) =\mu$ trivially
grizzled pagodaBOT
A dense set
keen pawn
and we're done
cedar plaza
trivially how?
amber waspBOT
@keen pawn Has your question been resolved?
keen pawn
trivially how?
$\mu cos(\theta) = sin(\theta)$
grizzled pagodaBOT
A dense set
keen pawn
on taking the the derivative
cedar plaza
ah ok
keen pawn
if $f(x)=x^3$, find $f'(a^2); f'(x^2)$
amber waspBOT
if $f(x)=x^3$, find $f'(a^2); f'(x^2)$
grizzled pagodaBOT
A dense set
keen pawn
I'm using first-principles to do these problems
The derivative of a single variable function is defined as follows : $\lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$.
\
Applying this definition, we find
\
$f'(x)= \frac{(x+h)^3-x^3}{h}$.
\
This gives us $f'(x) =\lim_{h\to 0} \frac{x^3+3x^2h+3xh^2+h^3-x^3}{h} = 3x^2$.
\
As $a^2 \in \R$, we find that $f'(x)= 3(a^2)^2= 3a^4$
grizzled pagodaBOT
A dense set
keen pawn
We know attempt to find $f'(x^2)$. We implore the reader to notice that the function is now a function of $x^2$ and not $x$. We thus have to perform a substitution in order to use our definition.$x^2=u$.
\
this gives us $\lim_{h\to 0} \frac{f(u+h)-f(u)}{h}$.
As the reader can verify, this is
$3u^2$
grizzled pagodaBOT
A dense set
amber waspBOT
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Available help channel!
stone sentinel
please help
amber waspBOT
please help
lusty pine
SAS
stoic walrus
please help
The Triangles have a common line, they also have equal lines, the mentioned angles are also equal as they are alternate to each other.
So as of the SAS theorem, they are congruent
(Hope this makes sense, English isn't my first language so pardon any mistake)
amber waspBOT
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verbal lark
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amber waspBOT
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verbal lark
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hearty sleet
Oops I'm back,
amber waspBOT
Oops I'm back,
hearty sleet
Ok first off is the grazing area a circular shape or a heart shape I can't visualize shit RN for some reason
Second how would you get grazing area
Ik half of it is legit just a semicircle but how do you get the other half
<@&286206848099549185>
Oh 84 btw
Anyone 😔
amber waspBOT
@hearty sleet Has your question been resolved?
amber waspBOT
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Available help channel!
smoky basin
Im stucked
amber waspBOT
Im stucked
amber waspBOT
@smoky basin Has your question been resolved?
austere cipher
Just a notation query, if i wanna say x = infinity, how do i do it the proper way using a limit?
grizzled pagodaBOT
Hemesfere
crisp stratus
uhhhh
lavish zenith
Just a notation query, if i wanna say x = infinity, how do i do it the proper wa...
can you give more context
crisp stratus
maybe just ``$\lim_{x\to \infty}$'' ?
grizzled pagodaBOT
artemetra
crisp stratus
**Hemesfere**
this doesn't mean anything as is
austere cipher
can you give more context
I wanna say $\dot x = 0$ at $ x = \lim_{a\to\infty} a$
Wait
grizzled pagodaBOT
Hemesfere
austere cipher
is it for a function ?
im graphing x dot against x
crisp stratus
do you want to say that the derivative of x is 0 at infinity?
$\lim_{x\to\infty} \dot x = 0$
grizzled pagodaBOT
artemetra
crisp stratus
why won't this work
lavish zenith
why won't this work
how do i get helpful role
crisp stratus
help a lot idk
austere cipher
$\lim_{x\to\infty} \dot x = 0$
Hmmmmmmmmm that works
crisp stratus
help a lot idk
i didn't ask for it lol
craggy sundial
ig $\dot x \to 0$ as $x \to \infty$ works too if $\dot x$ is approaching 0
grizzled pagodaBOT
Sepdron
crisp stratus
also good
austere cipher
Alright thank u all
crisp stratus
Alright thank u all
if you are done, type ".close"
sharp rapids
Let ( f, g : A \subset \mathbb{R} \to \mathbb{R} ) and let ( c \in A' ). Assume that ( f ) is bounded in a neighborhood of ( c ) and that ( \lim_{x \to c} g(x) = 0 ). Prove that ( \lim_{x \to c} f(x)g(x) = 0 ).
grizzled pagodaBOT
Halex
sharp rapids
Since $f$ is bounded in a neighborhood of $c$, we have that $$\exists M > 0, \exists \delta_1 > 0: |x-c| < \delta_1 \implies |f(x)| \leq M.$$ Now since $\lim_{x \to c} g(x) = 0$ we have $$\forall \epsilon > 0, \exists \delta_2 > 0: |x-c| < \delta_2 \implies |g(x)| < \epsilon.$$
Choose $\delta = \min{\delta_1, \delta_2}$
grizzled pagodaBOT
Halex
sharp rapids
Is this correct?
Since $|g(x)| < \epsilon$ for every positive epsilon, in particular we have that $|g(x)| < \frac{\epsilon}{M}$
grizzled pagodaBOT
Halex
sharp rapids
not for the same delta though
wdym?
stuck vapor
you want delta associated to e/M, not associated to e
sharp rapids
you want delta associated to e/M, not associated to e
should have stated that from the start right
So for ( |x - c| < \delta ), it follows that ( |f(x)| \leq M ) and ( |g(x)| < \frac{\epsilon}{M} ) hold, which implies that ( |f(x) g(x)| < M \cdot \frac{\epsilon}{M} = \epsilon ).
grizzled pagodaBOT
Halex
sharp rapids
Show that if ( f : (0, \infty) \to \mathbb{R} ) is such that ( \lim_{x \to \infty} x f(x) = L ), where ( L \in \mathbb{R} ), then ( \lim_{x \to \infty} f(x) = 0 ).
grizzled pagodaBOT
Halex
sharp rapids
I'm not sure what to do here
sharp rapids
use defn of limit
from $|xf(x) - L| < \epsilon$ we have $$\frac{L-\epsilon}{x} < f(x) < \frac{L+\epsilon}{x}$$ we might conclude by Squeeze Theorem that $\lim_{x \to \infty} f(x) = 0$?
hearty inlet
yeah
grizzled pagodaBOT
Halex
fierce edge
from a bag of 20 marbles with 5 orange, 5 green, and 10 blue marbles. We draw all of them one after another without replacing
fierce edge
what’s the probability that orange will be picked 5th
how to use calculations for this
i’ve used another argument to get 1/4 as the answer
curious about the full calculation though
amber waspBOT
@fierce edge Has your question been resolved?
dull ivy
what’s the probability that orange will be picked 5th
Hi once again
fierce edge
learning is also engaging with smarter people than yourself
dull ivy
learning is also engaging with smarter people than yourself
Yes become smart then asking from smart people
warm burrow
there is no winning or losing in a good conversation only things learned
amber waspBOT
@fierce edge Has your question been resolved?
mild cradle
test
mild cradle
.close
amber waspBOT
stable quest
Hiya, I messaged a mod thingy for this. The same thing happened to me. 💀
mild cradle
Hiya, I messaged a mod thingy for this. The same thing happened to me. 💀
ok thx
amber waspBOT
Available help channel!
normal dove
is it possible to assume a point p perpendicular to A in plane BCD which will mean AP.BP=0 And then find Magnitude of P?
fierce edge
normal dove
?
fierce edge
what’s the actual question?
keen topaz
is it possible to assume a point p perpendicular to A in plane BCD which will me...
Perpendicular to the base?
normal dove
Perpendicular to the base?
yea
normal dove
what’s the actual question?
finding the volume
normal dove
they took projection of BA on Perpendicular vector
i was just wondering if this is possible
normal dove
Should probably be.
what do i assume point P?
keen topaz
what do i assume point P?
Yes.
normal dove
mb is perpendicular
normal dove
yes
fierce edge
no i mean this is the standard way
but well if your method involves finding the perpendicular height
then sure
it’s just geometry but with vectors
and u know in geometry you need the area of the base * height
normal dove
im just wondering if it is possible and if there enough information to do that
normal dove
..
fierce edge
what
fierce edge
if u can show how itll be helpful
i’m not on pc 
let P be a point on BCD
it should be parallel to the normal vector of that plane
and also the other restriction where AP * BP = 0 where * is the dot product
try first and send progress
denote P as some (x,y,z)
fierce edge
there are constraints for all points on the plane BCD
amber waspBOT
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fervent oasis
amber waspBOT
fervent oasis
the eigenvalue of v is 2, so the last can't be
can someone help me to get the eigen value for MW
oh nevermind I got it
.close
amber waspBOT
split sail
Which question
Okay
So let's start off with what you already know
Okay so what's the speed of the freight train and passenger train?
Yes the speed is the rate
You ate looking for the time
Are*
Ya so you use the equation of t equals d/r
No
Its
Okay so
By combining them
Yes
YEAHH
💀 oh
Shoot
I gotta be dyslexic
Is it not asking for the time 😭
Yeah time is t
Yayaay
rustic gyro
Hello! Can someone explain why we multiply 4 twice in this problem? Thank you! This is my teachers work, and I don't understand why they multiplied the 4 twice in the second step,
grave dawn
when u do it the first time
a factor of 4 comes down
and now u do it a 2nd time
another factor of 4 comes down
amber waspBOT
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silk owl
https://mathproblems123.wordpress.com/2022/09/13/integrating-polynomials-on-polygons/ If I got this correct, for straight line, 2d, closed, and not self intersecting polygons, take x power rule of origin function, go to end where it is with x(t) and y(t) subbed in and y'(t) multiplying outside integral and with those x and y functions of t in an integral 1 to 0, power rule of t, just do for 1 t, then you get a sum of all values for a function over a polygon with Green's Theorem, right?
silk owl
Just want to know, is this a correct assessment?
Please ask if I did not phrase that correct?
So f(x,y) to its anti-derivative to in integral that anti-derivative with x(t) y(t) subbed in dt y'(t) to solving that integral to sum to solved, correct?
Can you please just use that last one and ask if you need more info?
<@&286206848099549185> exactly 15 minutes passed, is here correct place?
amber waspBOT
@silk owl Has your question been resolved?
amber waspBOT
@silk owl Has your question been resolved?
split sail
yes b_n need converge bc i tried creating counterexamples but couldnt. but idk how to prove😭
amber waspBOT
yes b_n need converge bc i tried creating counterexamples but couldnt. but idk h...
@split sail Has your question been resolved?
split sail
<@&286206848099549185> 🥹
sudden arrow
Hi
split sail
hello:D
till now i have said let b_(2n) = L_1 b_(2n+1) = L2 and b_3n = L_3
im thinking of taking subsequences common to b_(3n) and either ti show L1 = L2 = L3
ok i did that
and just wrote out |b_2n - L| < epsilon for all 3
then took my N=2 so |b_n - L| < epsilon for all n>=N is true so converges to L
.close
amber waspBOT
amber waspBOT
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amber waspBOT
split kraken
real analysis is fake
real and true 
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solemn escarp
how do i approach this?
amber waspBOT
how do i approach this?
lusty pine
what is it searching the maximum for?
solemn escarp
(a^4 + 1)(b^4 + 1)
lusty pine
ok
solemn escarp
can you do it?
lusty pine
not sure im trying it rn
solemn escarp
thanks in advance
lusty pine
this is what i got so far, if you can find the minimum of the bottom function you can do it
no idea how tho
lusty pine
hold on what if i ask my teacher about this lmao
amber waspBOT
@solemn escarp Has your question been resolved?
craggy sundial
maybe you just need to solve for a and b?
it's 2 equations and 2 unknowns, so it should work right?
solemn escarp
im doing it right now but its just a ton of roots
probably not intended but
give me a minute
can sqrt(17 - 10sqrt(22)) be cancelled out?
the first sqrt
lusty pine
who says a and b are real
solemn escarp
its probably real
its from uhh 9th grade competition
can an inequality be used
anyways heres whqt i got for a^4
craggy sundial
I got x as -2±√7 and -2±i
craggy sundial
I got something wrong
solemn escarp
oh
craggy sundial
I got x as -2±√7 and -2±i
oh +2 ± √7
craggy sundial
I did it the brute force way of using ((a+1)(b+1))^2 = 4
then expanding the (a²+1)(b²+1) = 34 and substituting stuff in that last equation
also, I did the whole thing in desmos because it's easy to check, because then the graph is wrong if I did something wrong
I don't think this is the way to do it though
solemn escarp
definetly not
solemn escarp
its probably inequalties idk
solemn escarp
that's the value of y btw
okay
does anyone know a better way of doing this?
<@&286206848099549185>
amber waspBOT
@solemn escarp Has your question been resolved?
solemn escarp
.close
amber waspBOT
amber waspBOT
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proud thorn
<@&286206848099549185>
amber waspBOT
<@&286206848099549185>
proud thorn
I have a Quiz in about 7 hours, it’s currently 2 am
I need help with simplfying radicals
How would I simplify something like this?
crisp skiff
What’s the full question?
proud thorn
Express each radical as a mixed radical in simplest form
Ahhhh thank you
Also how would you simplify the one on the bottom?
crisp skiff
full question?
proud thorn
Express each mixed radical as an entire radical
crisp skiff
including how to do it and stuff
proud thorn
Oh dang thanks
crisp skiff
ye
final quest
amber waspBOT
@proud thorn Has your question been resolved?
amber waspBOT
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keen pawn
amber waspBOT
cloud scarab
Fun fact: such function must be linear over Q
keen pawn
hmm
cloud scarab
In a sense that $\forall a, b \in \mathbb{Q}$ we have $\varphi(aw+bz) = a\varphi(w)+b\varphi(z)$
grizzled pagodaBOT
EQUENOS
cloud scarab
So it can't be continuous, because otherwise it would be linear nvm it would be linear over R but not necessarily over C
keen pawn
I was thinking of defining a function as follows
\varphi(z) = \begin{cases}
1, \text{ if z is not real }\
0, \text{ if n is real}
\end{cases }
cloud scarab
$\varphi(2i) \neq 2\varphi(i)$
grizzled pagodaBOT
EQUENOS
keen pawn
hmm
grizzled pagodaBOT
A dense set
Compile Error! Click the 
reaction for more information.
(You may edit your message to recompile.)
night jacinth
$\varphi(2i) \neq 2\varphi(i)$
that's not a problem tho
the bigger problem is that it's not even additive
cloud scarab
the bigger problem is that it's not even additive
Yes, and it demonstrates that
night jacinth
ah yeah oops
keen pawn
Is this supposed to be an easy question
night jacinth
\varphi(z) = \begin{cases}
1, \text{ if z is not real }\\
...
however having an additive function that only outputs real numbers is a good idea
cause you can be pretty sure it won't be C-linear
night jacinth
Is this supposed to be an easy question
it's not hard yeah
night jacinth
you might wanna think of something more conventional
keen pawn
firstly how do I avoid the error in the TeX output
night jacinth
what error does it say
can you click on the warning sign ?
well too late
$$\varphi(z) = \begin{cases}
1, \text{ if z is not real }\
0, \text{ if n is real}
\end{cases}$$
grizzled pagodaBOT
aPlatypus
night jacinth
yeah it was just the dollars you missed ig
cloud scarab
Think of the simpliest R-linear function from C you know
night jacinth
it's not linear but good guess
cloud scarab
It's not R-linear
keen pawn
arg(z) may though
keen pawn
Re(z_
cloud scarab
nice
keen pawn
$\varphi(z) = Re(z)$ is linear though
cloud scarab
nope
night jacinth
not C linear tho
cloud scarab
It's not C-linear but it's R-linear
grizzled pagodaBOT
A dense set
keen pawn
isn''t $\varphi(az)= aRe(z)$
grizzled pagodaBOT
A dense set
keen pawn
yes
cloud scarab
Yeay try a=i and it will fail
keen pawn
oh right, I'm working over the field of $C$
grizzled pagodaBOT
A dense set
cloud scarab
yes
keen pawn
I thought i was still working over $|$
grizzled pagodaBOT
A dense set
grizzled pagodaBOT
A dense set
keen pawn
thaanks
so
This is basically saying if T(p)= q(p(v)), then $T$ is a linear map
right
right?
robust isle
what happened to the previous question 
keen pawn
Done
robust isle
where is your solution?
oh Re(z)
okay
the x is a bit hmmm
not a fan of the notation but that's the idea
grizzled pagodaBOT
A dense set
keen pawn
okay
robust isle
no thats not exactly any better
if you want to plug in something into the function to make it look more like its defining a function, you should write something like
,, T(p)(x) = q(p(x))
grizzled pagodaBOT
robust isle
or maybe $(Tp)(x) = q(p(x))$
grizzled pagodaBOT
keen pawn
let $q(x)= 2x;p(x)=x^2$.It then follows that $q(p(x))= 2x^2$. If $q(p(x))$ were indeed a linear transformation, $q(p(2x)) = 2q(p(x)) = 4x^2$. But $q(p(2x)) = 8x^2$. Thus the transformation isn't linear
robust isle
sin(x) isnt a polynomial
grizzled pagodaBOT
A dense set
amber waspBOT
@keen pawn Has your question been resolved?
keen pawn
<@&286206848099549185>
opaque nexus
?
white yacht
Yep
tidal wadi
<@&286206848099549185>
?
opaque nexus
oh lemme see
opaque nexus
**A dense set**
is this it?
keen pawn
yes
opaque nexus
so alr
i got till about 7:00 cst
the reasoning in this example is correct, but the presentation has minor formatting issues that could be clarified.
keen pawn
the reasoning in this example is correct, but the presentation has minor formatt...
What are said issues?
opaque nexus
hold on im currently spacing the equation out so its ez to read lol
mb alr
let 𝑞 (𝑥) = 2𝑥q (x) = 2x and 𝑝 (𝑥)=𝑥2p(x)=x2.
keen pawn
Just a heads up, this server supports TeX
tidal wadi
**A dense set**
you want me to work this out?
keen pawn
I've worked it out, I would like it checked
opaque nexus
idk what tex even is
keen pawn
idk what tex even is
LaTeX
tidal wadi
I've worked it out, I would like it checked
were is the question
opaque nexus
well so this isnt really a linear
keen pawn
This is the question
tidal wadi
t is not a linear map
opaque nexus
then it would satisfy the property of homogeneity (scaling)
keen pawn
thats what i got
hmm?
keen pawn
then it would satisfy the property of homogeneity (scaling)
Yeah, it would
but it doesn't
so it isn't
right
tidal wadi
yes it isnt a linear
yh its nto a linear
opaque nexus
mb
robust isle
let $q(x)= 2x;p(x)=x^2$.It then follows that $q(p(x))= 2x^2$. If $q(p(x))$ were ...
this reasoning is flawed
can you figure out why?
opaque nexus
so
q(p(2x)) = 2 ⋅ q(p(x)) = 4x2.
however, calculating 𝑞(𝑝(2𝑥))q(p(2x)) directly:
q(p(2x)) = q((2x)2) = q(4x2) = 8x2.
keen pawn
this reasoning is flawed
It would be $q(2p(x))$ that I would have to look at?
grizzled pagodaBOT
A dense set
robust isle
It would be $q(2p(x))$ that I would have to look at?
correct, because the input into T is p
keen pawn
I see
okay
so say $q=x^2, p=(2x)$. It follows that $q(p(x)) = 4x^2$. Testing for linearity, we find that if it were linear $q(2p(x))$ would be $8x^2$. However, we find that $q(2p(x))= q(4x)= 16x^2$. Clearly $16x^2 \neq 8x^2 \forall x \in \R$. It this follows that $T$ isn't linear
grizzled pagodaBOT
A dense set
keen pawn
correct, because the input into T is p
Is this fine?
robust isle
yes
keen pawn
cool
grizzled pagodaBOT
A dense set
keen pawn
So we want q to be a linear function
robust isle
what do these q look like?
keen pawn
$ax$
grizzled pagodaBOT
A dense set
robust isle
robust isle
wdym by uniqueness
keen pawn
that only functions of this form work
robust isle
not really
robust isle
no this is a bad idea
grizzled pagodaBOT
A dense set
keen pawn
why so?
robust isle
the key thing to note here is that a judicious choice of p will tell you exactly what q is
keen pawn
p=ax
robust isle
just x is enough but yes that'll do
robust isle
keen pawn
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rancid minnow
\zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s}
amber waspBOT
\zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s}
grizzled pagodaBOT
fake Toby fox
$\zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s}$
rancid minnow
Ok thanms
Thanks
How do I change background
I want this to be black and background white
grizzled pagodaBOT
rancid minnow
$\zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s}, tc white$
grizzled pagodaBOT
fake Toby fox
rancid minnow
, Tc color white
grizzled pagodaBOT
You have switched to the white colourscheme.
rancid minnow
$\zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s}$
grizzled pagodaBOT
fake Toby fox
rancid minnow
No
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rancid minnow
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latent schooner
I don't get what it means here, The limit exists iff all sequences {p_n} converge to p, what is {p_n}, is {p_n} some collection of x?
shrewd elm
it's a sequence of points in E\{p} like it says
meaning, points in E other than p itself
mint nacelle
it's a sequence of points in `E\{p}` like it says
hi Japanese classical grammar
latent schooner
so it is in the set of function input domain except p?
latent schooner
alr, so can any sequence doesn't have a limit of q, causing the function dont have a limit q at point p
is there any counter example for this proposition to visualise it
shrewd elm
if there is some sequence (p_n) in E\{p} that converges to p, such that f(p_n) doesn't converge to q, then you can conclude that f(x) does not converge to q as x->p
say for example you define f(x) = 1 if x is rational, and f(x) = 0 if x is irrational
then f(x) has no limit as x->0 (or as x-> any other point for that matter)
because you can choose a sequence of rationals (p_n) and find that f(p_n) -> 1
and a sequence of irrationals (p_n) such that f(p_n) -> 0
it's a toy example but it illustrates why it works
latent schooner
oh, so does this definition implies continuity?
shrewd elm
it's not the definition, it's an equivalent condition
that's why the title says proposition
it doesn't necessarily imply continuity btw
no one says that f(p) = q
latent schooner
oh right
shrewd elm
it's just a statement about the limit as x approaches p
but if f(p) = q then yea that implies continuity
latent schooner
so it is stating the uniqueness of limit
shrewd elm
well you don't really need this proposition to prove that limits are unique but yea if the limit exists it's unique
this just gives you a different way to test if the limit exists
using sequences instead of epsilon/delta arguments
latent schooner
alr, is there any benefit of using this way instead of epsilon delta definition?
anyways, i think those question proving limit would be strictly require "by definition"
thx for help
.close
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shrewd elm
alr, is there any benefit of using this way instead of epsilon delta definition?
sometimes it's easier to argue using sequences
and it's definitely beneficial later when you get to things like measure theory where you want to make arguments involving countable operations
but don't worry about that now haha
latent schooner
alright
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copper bramble
I was trying to use proof by contradiction to proove that \lim_{n \to \infty } \sin(n) does not exist. I saw some proofs online that started out by stating: \lim_{n \to \infty } \sin(n) = g => \lim_{n \to \infty } \sin(n+1) = g. What is the reasoning behind that implication? Is it that if n approaches infinity then if we increment the argument of sin we still get a sin of an infinitely large number so the limit stays the same? Is there a way to prove it using trig identities?
swift pebble
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shy willow
shadow stump
in general [ \lim_{n \to \infty} f(n) = \lim_{n \to \infty} f(n+1) ] if such a limit exists
grizzled pagodaBOT
cloud
copper bramble
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hollow quartz
If I need to translate please let me know
amber waspBOT
If I need to translate please let me know
fierce edge
If I need to translate please let me know
yes translate
amber waspBOT
@hollow quartz Has your question been resolved?
hollow quartz
ABC is a triangle. M is a point of the segment BC
N is the projection of B on (AC) parallel to (AM)
P is the projection of B on (AB) parallel to (AM)
1-a) Show that MA/BN=CM/CB and that MA/CP=BM/BC
b) Conclude that 1/AM=1/BN+1/CP
second question is easy
hollow quartz
yes translate
here you go tnx
amber waspBOT
@hollow quartz Has your question been resolved?
hollow quartz
<@&286206848099549185>
amber waspBOT
@hollow quartz Has your question been resolved?
robust bane
I remember something similar to this in Euclidean Geometry, right?
amber waspBOT
@hollow quartz Has your question been resolved?
rancid parrot
Could someone please help me with the problem below?
amber waspBOT
Could someone please help me with the problem below?
rancid parrot
I want to know
If how I solved for y is ok
Or should have I plugged into my x into another equation then solved for y?
undone merlin
You can rewrite the system as a single vector equation
you should have something like v' = Av where v is a vector, and A is a matrix.
rancid parrot
Idk if I am allowed to do that
Because we weren't taught to
I want to know that if the only correct way to do this is to plug in my x into the other equation then why is that the only correct way?
undone merlin
If you can't use linear algebra, you can differentiate one equation, and substitute in the values to get something in the form x'' = f(y).
rancid parrot
Im still worried about why the way that I did it might not be correct
rancid parrot
I basically solved for x and y
I solved for x first
Plugged it into
y'=4x-3y
Then solved for y
I want to know
Is it ok that I did it this way?
Does it matter?
undone merlin
How did you solve for x?
rancid parrot
By eliminating y
undone merlin
Ok. Differentiate eqn.1, substitute in eqn.1 and eqn.2 for x' and y' respectively, rearange for y = ... and substitute this back into eqn.1. Now you have a second order ODE in x only.
Is that what you did?
rancid parrot
I substituted the x I got into eqn.1.
amber waspBOT
@rancid parrot Has your question been resolved?
old goblet
I am having trouble with this math problem on my study guide. I am not even sure if what I am doing is correct and if it is, then im not sure what to do next.
split sail
try the divergence test
old goblet
the nth term divergence test?
split sail
yes
old goblet
since n grows faster than ln(n), then it equals infinity meaning that since it doesn't equal 0 then it diverges?
wow i just did all that work for nothing lol okay
split sail
since n grows faster than ln(n), then it equals infinity meaning that since it d...
why write allat just do lhopital
but youre right
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slender blaze
amber waspBOT
slender blaze
I need help with this identity
slender blaze
what if i want to work only with one side
thorny terrace
ah forgot
hmm
ah ok i think i got it
use ur pythagorean trig identieis
look at 1/sin^2 (a)
can u rewrite that as another trig function
thorny terrace
what is that equal to, in terms of cot
slender blaze
cosec^2 - 1
thorny terrace
no, like define csc^2 in terms of cot
slender blaze
1 + ctg^2?
slender blaze
perfect square
thorny terrace
yes
thorny terrace
np
slender blaze
.solved
thin falcon
Hello can anyone check if my work is correct and if I need to simply anymore? It’s precalc btw ty
pallid canopy
What's the question
thin falcon
The top is the main function and im basically just plugging it in.
Into that function
thorny terrace
yea its all right
theres not much to check lol
maybe u should simplify more but thats on ur teacher
thin falcon
How much more simplified should it be ?
thorny terrace
like 1/2 + 2
thin falcon
I’m preparing myself for precalc so I’m on my own kinda lol
thorny terrace
u can simplify that
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Available help channel!
compact wagon
amber waspBOT
river whale
is the answer the first one
compact wagon
yea
but i just started learning this so im kinda confused how were supposed to find i
the degree of the numerator is less than the denominator therfore ha: y=0..
river whale
but i just started learning this so im kinda confused how were supposed to find ...
what i did was see that as x gets bigger and bigger as it approaches infinity, y would become 0 too
the only option that shows that is the first one
river whale
what i did was see that as x gets bigger and bigger as it approaches infinity, y...
the reason for this is, as you already said, the degree of the numerator is less than the denominator
compact wagon
.close
amber waspBOT
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low wave
i dont understand what the last column is asking for
amber waspBOT
i dont understand what the last column is asking for
pallid canopy
Does the third column imply the fourth column
low wave
Does the third column imply the fourth column
idk
low wave
That's what it's asking
How do I do this could you dumb it down for me
I’m so confused
What do I write down
brisk leaf
Morning,
I think what the column is asking is, if the 3rd column true/false will that imply that the fourth column is true/false aswell.
sinful sapphire
it's like "less than or equal"
brisk leaf
I think you can analyze with an example.
For the first column
If p and q are speaking the truth
That means the statement p or q will be true aswell
sinful sapphire
no come on that's not what it's for
low wave
am I supposed to look at the 3rd and 4th column, and if so what am i looking for?
sinful sapphire
you're looking for TF combination, that's the only one that gives F
low wave
So it would be F T T F?
sinful sapphire
no why F TT F
low wave
okay wait the thing is asking
sinful sapphire
it's TFFT
sinful sapphire
yes
low wave
anything else like F T or TT or FF is T then right
sinful sapphire
yes
sinful sapphire
no
low wave
what does it mena
low wave
okay i just want to know how did you understand what it 's asking for
sinful sapphire
there's nothing to understand, it's not supposed to mean something, it's just →
the definition is that it equals F if you have TF, and otherwise it's T
i think of it as "less than or equal sign" like i said
if P or Q is true then P and Q is true, (otherwise it's true or false)
azure goblet
Area in terms of x and y (x-6)(y-4), as a function of x alone is y = 509-4x-2910/x, critical number is 26.972, x = 26.972 and y = 17.982, how do I find the maximum printed area?
azure goblet
I tried (26.972-6)(17.982-4) so (20.972)(13.982) is roughly like 293 but for some reason that's wrong
split sail
Does the total area given here take into account the margins?
azure goblet
I don't think so, I believe that's before the margins
azure goblet
Sounds good
split sail
Yup done
Is this how it looks like
The red line corresponds to the side margin of 3 inches
And the blue line corresponds to the top and bottom margin of 2 inches
@azure goblet you there mate?
azure goblet
Yup I'm here
split sail
Ok great
Now we know that the total area is equal to 485 square inches
This means that
$$x \cdot y = 485$$
grizzled pagodaBOT
Edmund Cloudsley
split sail
Correct?
azure goblet
Right
split sail
Now what the question is asking us to do is to maximise the printable area
For that we would first have to create a function, let’s call it A(x) that gives us the area of the printable function
azure goblet
So is that when we add the margins to the x and y and set it equal to 485?
split sail
If we look at the diagram
$$A(x) = \text{length of printable area} \times \text{width of printable area}$$
$$A(x) = (y-2(2)) \times (x -2(3)) $$
$$A(x) = (y-4)(x-6)$$
grizzled pagodaBOT
Edmund Cloudsley
split sail
Did you understand how we got the area function?
azure goblet
You just multiplied the side margins for y, and then the top and side margins right?