whats the <= in the POSET of real numbers?
#help-36
shrewd mist
scarlet sequoia
whats the <= in the POSET of real numbers?
take a guess
shrewd mist
take a guess
less than or equal to?
shrewd mist
ohhhh, and its totally ordered as for each pair x, y E S, either x <= y or y <= x
scarlet sequoia
like the existence of sup and inf for any non-empty bounded subset
scarlet sequoia
ohhhh, and its totally ordered as for each pair x, y E S, either x <= y or y <= ...
yes
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stiff mulch
Hello
stiff mulch
Anyone can do limit?
spare hazel
share the question
stiff mulch
9/4
terse crypt
remember the sqaure roots
stiff mulch
Uh huh
spare hazel
just 3/2
terse crypt
,w lim x to infty (\sqrt{9x^3 - 5} - \sqrt{x^2 + 3})/(\sqrt{4x^3})
soft zealotBOT
spare hazel
square root 4
tranquil pine
bro a great way to solve these kinds of questions is to ask yourself......how can i make something $\frac[C}{x}$ where C is some number
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Edmund Cloudsley
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(You may edit your message to recompile.)
spare hazel
then divide both numerator and denominator by sqrt(x^3)
terse crypt
U can do that if doing lim x to infty
soft zealotBOT
Edmund Cloudsley
tranquil pine
just try to get the x separated
tranquil pine
then divide both numerator and denominator by sqrt(x^3)
yup basically
spare hazel
bc the logic is x tends to infinity and 1/x^n tends to zero
tranquil pine
precisely
that's like the core idea behind questions of this type
just try to make it some number divided by x^{whatever}
and ez its done
spare hazel
yep
stiff mulch
I see
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@thorny folio Has your question been resolved?
tulip thicket
bro idk what the matrix is stfu
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@thorny folio Has your question been resolved?
runic tulip
i dont know matrices either
how about closing this and starting a new channel to push it back to the top
hallow jungle
so i have been made aware of a proof that for any integer b, b - 1 divides b^n - 1, concluded from the fact that the polynomial x^n -1 has 1 as a root. now my question is, how is this a valid proof, considering that, as far as i am aware, K[X] is euclidiean if K is a field and K[X] being euclidiean, proves that if p(a) = 0 then x - a divides p for p in K[X].
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so i have been made aware of a proof that for any integer b, b - 1 divides b^n -...
hallow jungle
so we know for any fixed b we can write b^n - 1 = (b - 1) * q(b) for some polynomial q but how do we know that q(b) is integral
or does the above implication about roots and polynomial rings over fields also hold for polynomial rings over rings?
desert mantle
you can still do polynomial division as long as the leading coefficient of x-1 is invertible in your ring
hallow jungle
and both resulting factors will still be elements of Z[X] in that case?
hmm
oh
yeah that figures from the algorithm i suppose
thanks!!
.close
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fierce sorrel
On a rectangular gameboard that is divided into n rows of m squares each, k of these squares do not lie along the boundary of the gameboard. Which of the following is a possible value for k?
final saddleBOT
On a rectangular gameboard that is divided into n rows of m squares each, k of t...
fierce sorrel
A. 15
B. 25
C. 35
D. 49
E. 52
I was thinking of something that is divisible by four but
I have no clue
pliant shore
On a rectangular gameboard that is divided into n rows of m squares each, k of t...
can you come up with an expression for k in terms of m and n?
trail mango
the multiple choice makes this problem way too easy
fierce sorrel
would it be 52
fierce sorrel
can you come up with an expression for k in terms of m and n?
i dont know tho
nm =x-k
pliant shore
i dont know tho
well, what are the dimensions of the rectangle which has all the squares that aren't on the boundary?
trail mango
the multiple choice makes this problem way too easy
hm ig it's not as easy as i thought at first
pliant shore
yeah there seems to be more than one correct answer
trail mango
but still helps a lot
fierce sorrel
n*m -4?
pliant shore
also like do you count rectangles as squares, who knows
fierce sorrel
huhh
pliant shore
you're close though, it's (n - 2)(m - 2)
fierce sorrel
then i have no clue
pliant shore
try drawing a board of whatever dimensions out and you'll see
fierce sorrel
try drawing a board of whatever dimensions out and you'll see
oh
so its only the
inner squares
pliant shore
inner squares
yeah
fierce sorrel
that aren't in the boundary
pliant shore
exactly
fierce sorrel
then it has to be divisible by 4?
pliant shore
oh yeah then all of A, B, C, D, E are possible
just let n = 3 so that n - 2 = 1
pliant shore
then it has to be divisible by 4?
no
trail mango
hm i just decided A was possible and then declared that was the answer
pliant shore
oh wait no 52 is possible according to this logic
onyx peak
pliant shore
yeah I just realised everything is possible
pliant shore
oh yeah then all of A, B, C, D, E are possible
just let n = 3 so that n - 2 = 1
this is why
yeah
onyx peak
you can simply just take whatever rectangle has that many tiles and make it a boundary
fierce sorrel
oh wait no 52 is possible according to this logic
the formula u gave?
onyx peak
you don't need formulas for this
pliant shore
oh yeah then all of A, B, C, D, E are possible
just let n = 3 so that n - 2 = 1
yes and then look at this
onyx peak
you can simply just take whatever rectangle has that many tiles and make it a bo...
Take any rectangle with 52 tiles (1 x 52 one is sufficient), and make a boundary for it
pliant shore
oh yeah ofc
trail mango
Take any rectangle with 52 tiles (1 x 52 one is sufficient), and make a boundary...
yea lol
onyx peak
imagine this question (maybe phrased with more jargon) appearing on an interview for some top uni
onyx peak
those are more interesting problems
stone wagon
test
stone wagon
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daring tusk
💀
vital crag
🐧
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leaden breach
i’ve got pretty much more than half of it solved but i’m pretty sure i need to make a formula to get to the answer since i have to subtract the area of the triangle that is inside the semicircle
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i’ve got pretty much more than half of it solved but i’m pretty sure i need to m...
frail raft
hi
leaden breach
hello ive got this done
i’m stuck on the last part where i have to make the formula that can get the shaded area by subtracting the lower part of the triangle
tranquil pine
I think the key here is to recognize the area of the red triangle is 1/4 of the area of the big triangle
Then you can use the red triangle to find the area of Sector FA, and double that to get the shaded area
slender rampart
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balmy raptor
I’m trying to understand this, I don’t understand why the substitution is valid since there is no square root in the integrand.
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I’m trying to understand this, I don’t understand why the substitution is valid ...
formal trail
the square root was never a requirement for trig substitution
slender rampart
do u know what the derivative of arctan is
balmy raptor
One sec
balmy raptor
do u know what the derivative of arctan is
Yes I used to just accept it as a fact since I wasn’t taught the method of trigonometric substitution at the time
Now I’m just wondering how it is derived or rather why it is derived like this
I see that √(a^2+x^2
Can be made into the triangle
slender rampart
which bit do you not understand
balmy raptor
And I understand basic trig so that x=a*tan(theta)
But this all equals to the √ of a^2+x^2
Yet in the example the integrand is just a^2+x^2
Intuitively I would do some algebraic manipulation so that I can substitute these into some form of √(a^2+x^2)
But in the example and other examples online including some of my math homework that that doesnt have to be the case
You can just substitute x=atan(theta) into just a^2+x^2
slender rampart
ultimately what you're trying to do with trig substitutions is manipulating the integrand into some form of derivative of arcsin or arctan
balmy raptor
I understand that just to finish the problem I will just undo the derivative, but our prof have taught us to always think of it like triangles throughout his lecture, so I’m just trying to conceptualize these derivatives and ultimately the integrals
Which is my question above about the triangle
slender rampart
tangent is opp/adj
so arctan x/a is theta
and you derive arctan x/a
it will be 1/x^2+a^2
balmy raptor
Yes but my question is since this triangle is set on the hypotenuse of √a^2+x^2 why is it applicable for a^2+x^2, which I imagine is also the same reason the derivative of arctanx/a is 1/x^2+a^2
Which refers back to this
slender rampart
tan doesn't need hypotenuse
balmy raptor
Yes but the triangle does
slender rampart
sorry i do not understand ur question
balmy raptor
Because the hypotenuse is the sqrt of √a^2+x^2 right
That is what the triangle is based off of
The triangle is not based off of just a^2+x^2
So then any trig sub derivation would be applicable to the √of a^2+x^2
Right?
slender rampart
if its a^2+x^2 then its tan
balmy raptor
Why
slender rampart
because 1/a^2+x^2 is the derivative of arctan(x/a), which is opp/adj
balmy raptor
I’m getting help from a tutor from my school, thank you for your time
muted pelican
Find derivative for 𝑓(𝑡) = (1 − √(𝑡 + 4) )^-1
final saddleBOT
Find derivative for 𝑓(𝑡) = (1 − √(𝑡 + 4) )^-1
fickle bronze
does this look similar to anything you know?
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muted pelican
does this look similar to anything you know?
i know i have to apply the chain rule and other derivative rules but im not sure why
im not sure what to begin with here
i am aware how to use those aforementioned rules but im stuck here
fickle bronze
you can rewrite the whole thing inside the bracets as u if you want and then take the derivative of that => (u)^-1
and then you have to multiply that by the derivative of u
brazen stump
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celest crane
What have you tried?
brazen stump
240/360 x pi x 9
celest crane
What is the formula for the surface area of a cone?
brazen stump
pi x r x l?
celest crane
And what value did you find for the radius?
brazen stump
9 is the radisu?
celest crane
No. That will be the slant length.
You have to imagine the two sides od the cutout sector being closed and the two edges being joined together.
Like so.
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brazen stump
.close
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keen anchor
for this question how do you take a factor of k-1 out for 4
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for this question how do you take a factor of k-1 out for 4
keen anchor
to get 12
I thought if you have a 4^-1 then that would be 1/4 so we would get 12 instead
vital crag
for this question how do you take a factor of k-1 out for 4
the factor is 4^(k-1),
,tex .exp rules
soft zealotBOT
riemann
keen anchor
so they factor out a 4^(k-1) so wouldn't that give 4^k and 4^-1 so wouldn't we get 3/4 still???
vital crag
no
vital crag
**riemann**
Use the first row here with a=4 and x=k-1 and y=1
to factor out 4^(k-1) from 4^k
keen anchor
I'm sorry my brain is fried rn
I still do not see it
like the way you show it
makes sense
but how can we just assume that x = k-1
or is it because hte value of 4 has 1 already?
vital crag
**riemann**
this is a general rule for all a>0, all x and y (except for 0 in some cases)
keen anchor
there are no help chan nels available rn I think
keen anchor
this is a general rule for all a>0, all x and y (except for 0 in some cases)
yeah but the thing is I do not see the x+y straight up
like how can I rmemeber this
wet spire
k
keen anchor
from my understanding when we have 3(4^(k+1)) that's the only way we can do this
vital crag
yeah but the thing is I do not see the x+y straight up
they telll you in the beginning that x+y=k
you work backwards to figure out y
x=k-1 from here
keen anchor
yeah but the bckwards from what I understand is 1/4 which is why I am still confused
like the way I see it we're diividing by 4
vital crag
yeah but the bckwards from what I understand is 1/4 which is why I am still conf...
it's not
it's not
do you understand that $k = (k-1) + 1$?
soft zealotBOT
riemann
keen anchor
yes but I don't understand where teh +1 comes form
vital crag
k - (k-1) = ?
keen anchor
yeah
vital crag
work backwards
keen anchor
how did you get the +1 in k=(k-1)+1
vital crag
k - (k-1) = ?
.
keen anchor
so we get a -1
keen anchor
k-k+1
oh
1
ok I understand that part now but where do we get the 1 like I understand the equation but how do we get the k = (k-1)+1 <- the one right there where the thing is pointed arrow
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drifting burrow
I'm a little confused about these, only one I get is (c) is inconsistent because 0 ≠ 1. (a) is supposed to be one unique solution, which I can kinda see from 1 = *, but how can you know what the other two variables are? since it's in 3d, can't you have the possibility of one variable being equal to a constant? for the others, idk
final saddleBOT
I'm a little confused about these, only one I get is (c) is inconsistent because...
@drifting burrow Has your question been resolved?
drifting burrow
<@&286206848099549185>
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@drifting burrow Has your question been resolved?
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@drifting burrow Has your question been resolved?
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@drifting burrow Has your question been resolved?
warm python
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warm python
Have to prove/ give a counter example
I was thinking ${(1,0,0),(0,1,0)}; {(0,1,0),(0,0,1)}$ as a counter example
soft zealotBOT
Veni, vidi, perii is not f(wai)
vivid parcel
It’s not clear that that would be a counter example. Since then {x,y,z,w} = {x,y,w} which is an orthogonal set
But you’re on the right track
warm python
Oh true
onyx peak
**Veni, vidi, perii is not f(wai)**
Or to fix this, you can either just claim those are multisets, or change one of (0, 1, 0) to (0, 2, 0)
warm python
think R^2
Yeah, just realised
Tq
Maybe (1,0),(1,1),(0,1),(-1,1)
tan(q)!
.close
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tranquil pine
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tranquil pine
With my working out, I only thought that there would only be 1st and 2nd quadrant answers since 2sin2x=+1
Even though the domain extends to 2pi radians
Why would there be 3rd and 4th quadrant answers?
The answer gives 3rd and 4th quad answers, but wouldn’t 2sin2x be negative in these quadrants?
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normal bone
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near folio
Suppose that I have a polytope
$\mathcal{S} = \left{ \text{diag}(k) \cdot x \mid k \in [\eta_1, 1] \times \cdots \times [\eta_n, 1] \right}$
where all $\eta_i \in(0,1]$ (known)
I need to define sufficient conditions so that
$ \mathcal{S} \subset C$
Where C is a convex set. Is the only method to do it verifying that all the vertices of S are in C? Is there any more efficient method?
soft zealotBOT
Drun
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@near folio Has your question been resolved?
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@near folio Has your question been resolved?
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@near folio Has your question been resolved?
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@near folio Has your question been resolved?
unreal sequoia
hey can I have help?
final saddleBOT
hey can I have help?
soft zealotBOT
unreal sequoia
barren hound
,rccw
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unreal sequoia
When there are 80 red cards, there are 4 blue cards and 2 green cards. Find the number of red cards when there are 8 blue cards and 10 green cards
that's another problem
so yeah I can tell if the functions are even, odd, etc
is that what they meant by symmetry?
tiny ridge
OMG I LOVE PRECALC
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@unreal sequoia Has your question been resolved?
unreal sequoia
OMG I LOVE PRECALC
can you help me with my homework?
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@unreal sequoia Has your question been resolved?
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@unreal sequoia Has your question been resolved?
hallow shell
For x<0. g is imaginary
So would think the domain is 0 to inf. Also the range
Nvm the range
I think that one is incorrect
Nvm i think its correct
Since x goes towards infinity, it will reach x values that are indeed 11-th square roots of another number
And the ^9 would make this explode to infinity
I think thats correct but also not entirely sure, so take it with a grain of salt
The third one, g(x) can indeed be even for x values that are perfect 11-th roots
But I havent looked into the odd part yet
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@unreal sequoia Has your question been resolved?
faint smelt
For x<0. g is imaginary
Wrong
faint smelt
Nvm i think its correct
(It's not)
hallow shell
I dont see it, can you elaborate
hallow shell
1. Stop talking about things you don't know
.
faint smelt
Hint for
a) Is g(x) defined for all values of x? If not, what are the exceptions?
b) Can we find an x that makes any value of g(x) valid? If not, what are the exceptions?
c) revise the definition of an odd and even function ( an odd function satisfies g(-x) = -g(x) and an even functiob satisfies g(-x) = g(x) )
d) What happens when x is a large positive or negative number? You may utilize the fact that g(x) is (odd/even/etc.)
faint smelt
All negatives?
Yes
hallow shell
No, just prove that its not imaginary for -1
faint smelt
No, just prove that its not imaginary for -1
It's clearly not
hallow shell
(-1) ^ 9 is -1, correct?
faint smelt
,calc 2(-1)^(9/11)
soft zealotBOT
Result:
-1.6825070656624 + 1.0812816349112i
hmm
hallow shell
Breh
faint smelt
-1^9 is -1
hallow shell
"Stop talking about stuff you dont know"
hallow shell
I literally said its imaginary
faint smelt
I literally said its imaginary
it's not
hallow shell
You literally see i in there
faint smelt
Stop spreading misinformation
hallow shell
i is imaginary and represents √-1
faint smelt
Hint for
a) Is g(x) defined for all values of x? If not, what are the exceptions...
@unreal sequoia
hallow shell
Irrelevant
How?
faint smelt
Even the basic calculator knows it
If you're replying on a calculator for stuff like this then you're pretty incompetent
hallow shell
Bro...
worldly vale
Complex roots are multivalued in general, quit bickering
faint smelt
The principal root in this context would be the real root
hallow shell
Whats the principal root of -1
faint smelt
Whats the principal root of -1
Asking that shows your lack of knowledge
hallow shell
Am trying to understand your thinking
faint smelt
The principal 11th root of -1 in this case of g(x) is -1
hallow shell
Thats incorrect tho
faint smelt
Thats incorrect tho
It is correct, given the context
hallow shell
Nvm
faint smelt
So yeah if you aren't going to help then leave this channel
I believe I provided sufficient hint for the poster
hallow shell
I told you root, not 11th root. And you changed the context to 11th root...
hallow shell
So yeah if you aren't going to help then leave this channel
Ye, cya
faint smelt
I told you root, not 11th root. And you changed the context to 11th root...
There's no other roots involved here
hallow shell
There's no other roots involved here
Then wouldn't it be a good answer to generalize it when asked a general question
faint smelt
Then wouldn't it be a good answer to generalize it when asked a general question
No because in this case we only care about the 11th roots, other questions aren't related to the person who asked for help.
hallow shell
Oh damn, look at that.
@faint smelt
It does sometimes make people feel crazy after jumping to wrong conclusions
But thats ok tho, is natural
I do feel like that atm ig
faint smelt
<@1145519864879009842>
Irrelevant to the question
I could pull this trick too
It's all a matter of choosing the prinpical root, and you clearly aren't helping the original poster.
hallow shell
Ok cool
faint smelt
So if you aren't going to say anything helpful I'll just block you for now
or maybe not
hallow shell
I could pull this trick too
But how are you plotting the imaginary part
faint smelt
But how are you plotting the imaginary part
There's no imaginary part
Desmos defaults to the real root
Which is what the question intended
hallow shell
Result:
```
-1.6825070656624 + 1.0812816349112i
```
Like here there was an imaginary part, if am not wrong
faint smelt
That's because TeXit's computation system priortizes a different 11th root
I do not need to explain further as this is irrelevant to the question.
hallow shell
Ok, i get that cya
faint smelt
Hint for
a) Is g(x) defined for all values of x? If not, what are the exceptions...
(Forwarded for the OP to see)
fallen vigil
i did not understand what their proof was doing
could someone tell me in layman terms what the idea behind the proof is in chunks
rose hamlet
the number of ways a positive integer 𝑁 can be expressed as a product of relatively prime factors can be represented using the formula involving the highest common factor (HCF) or greatest common divisor (GCD).
terse crypt
Is it that HCF x GCD = that integer
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@rose hamlet Has your question been resolved?
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dusk topaz
I badly need help, because its due tommorow huhu, our teacher tasked us to make a problem about optimization that is related to our course which is computer engineering, can you guys help me? I need 3 questions huhu thank you so much
final saddleBOT
I badly need help, because its due tommorow huhu, our teacher tasked us to make ...
dusk topaz
The task is to make a question huhu
slender rampart
what does huhu mean
dusk topaz
Oh it's like an expression sorry for that😭, okay so as I was saying
Our calculus teacher like wanted us to make 3 questions about optimization that is related to computer engineering
final saddleBOT
@dusk topaz Has your question been resolved?
slender rampart
what kind of questions
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@dusk topaz Has your question been resolved?
cosmic zinc
Hi my teacher sent this question and I don’t really know how to finish it if anyone could help
final saddleBOT
Hi my teacher sent this question and I don’t really know how to finish it if any...
@cosmic zinc Has your question been resolved?
edgy burrow
Hi my teacher sent this question and I don’t really know how to finish it if any...
this can be done by using both cosine rule and area of a traingle using sine
pliant shore
Hi my teacher sent this question and I don’t really know how to finish it if any...
what have you tried so far?
cosmic zinc
this can be done by using both cosine rule and area of a traingle using sine
I put it into area of triangle and didn’t get the answer so idk
edgy burrow
cosmic zinc
what have you tried so far?
Area of triangle, I put in the numbers alr given and tried to make it to what I said it was
pliant shore
Area of triangle, I put in the numbers alr given and tried to make it to what I ...
which part are you doing?
cosmic zinc
B
pliant shore
Then I didn’t know what to do
can you solve for $x$ in the equation for the area?
soft zealotBOT
south's secret twin brother
pliant shore
and then sub into the cosine rule
edgy burrow
i would solve for just xy but yeah
cosmic zinc
Wdym solve for xy
edgy burrow
rearrange the area formula to get xy on its own
soft zealotBOT
south's secret twin brother
cosmic zinc
then $y = \frac{120}{(\sin 28) x}$ for example
Alright
cosmic zinc
Okok
cosmic zinc
or actually just sub into part b
Ah ok I solved it now tysm
pliant shore
no worries!
meager sky
I need to find all prime ideals of $\mathbb{Z}{12}$. Do all ideals identify with one element? Like I'm thinking that the set $${2a | a \in \mathbb{Z}{12}}$$ is a prime ideal as, if ab in P (ab is even) then a or b is in P (even). If I wanted to, would it be possible to enumerate each element in Z12 to see if it makes a prime ideal? And whats what would be the standard notation for example the ideal that I mentioned?
final saddleBOT
I need to find all prime ideals of $\mathbb{Z}_{12}$. Do all ideals identify wit...
soft zealotBOT
Bean Man
meager sky
I know that this is different because Z is commutative but yea
Forget my last message, prime ideals are defined with commutative rings
latent fractal
I need to find all prime ideals of $\mathbb{Z}_{12}$. Do all ideals identify wit...
That ideal is called ⟨2⟩, which is defined exactly the way you put it as "all the multiples of 2". I'm not 100% sure what they mean by the word "identify", but it sounds like they're asking if every ideal can be written as ⟨a⟩ for some a in Z_12
that notation extends so that for a ring R, we have e.g. ⟨x, y, z⟩ = { a x + b y + c z | a, b, c in R }
(these are "right ideals", but the different types of ideals coincide in commutative rings)
meager sky
That ideal is called ⟨2⟩, which is defined exactly the way you put it as "all th...
Okay and I can check that <4> is not a prime ideal as 4 \in <4> = 2 * 2 but 2 \notin <4>?
latent fractal
Okay and I can check that <4> is not a prime ideal as 4 \in <4> = 2 * 2 but 2 \n...
yep, that's right
meager sky
yep, that's right
and <5> isn't because its the whole ring?
latent fractal
R itself may or may not be a prime ideal in R, depending on your definition
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meager sky
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meager sky
<10> = <2>, If it asked to say all prime ideals, is the ideal defined by its elements or its generator (I know that if it has the same elements then it has the same generators but I'm not sure how to ask this)
Since Z12 is cyclic the ideals are symmetric around 6 right?
The set generated by 1 = 11, 2 = 10, 3 = 9, etc?
.close
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hot reef
I got this question as hw
final saddleBOT
I got this question as hw
acoustic finch
Specifically 26ii
Do you know what it means to rationalise the denomenator?
acoustic finch
Well if that's the case, you can make it -3/4 -√(3)/4
hot reef
Ye but still
It's not the format the question is looking for
Which just stumps my little brain
Idk how to get it to that, it just seems impossible
acoustic finch
Oh my bad
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hot reef
<@&286206848099549185>
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steep sigil
I am supposed to replace terms in "2002 × 3001 − 2001 × 3002" with n (if that makes sense) in order to solve it faster. For example, in an easier problem (which I understand) "2014^2 − 2012^2", if we find a middle ground between 2014 and 2012 (2013) we can replace that middle ground with n. It will then look like "(n+1)^2-(n-1)^2", after we simplify we replace n with the value we chose, 2013. (I would make this shorter but idk how to call this so I just gave an example)
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I am supposed to replace terms in "2002 × 3001 − 2001 × 3002" with n (if that ma...
vital reef
yeah i get what you mean
i mean it doesn't really matter what the value of n is here, you can simplify no matter what
so why don't you just let n = 2001
then you can simplify the resulting quadratic expression into a linear one
@steep sigil
steep sigil
ok let me try
ty btw :D
@vital reef ok this is easier than I thought, I have a few more subquestions for this exercise so if you could wait for me to finish them too?
vital reef
sure
steep sigil
@vital reef yep I got stuck at 1000 × 1003^2 − 3000 (1001 + 1002)
<@&286206848099549185>
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vital reef
<@251927217800478720> yep I got stuck at 1000 × 1003^2 − 3000 (1001 + 1002)
um
i said let n = 2001
(n+1)(n+1000) - n(n+1001)
expand then simplify
the n^2 term will cancel
steep sigil
no I mean 1000 × 1003^2 − 3000 (1001 + 1002) is a different subquestion on the same exercise
sorry for the confusion
vital reef
try letting n = 1000
see what happens
your value of n doesn't really matter in terms of canceling things about algebraically, it's just that certain choices of n will make it easier
like for instance if you have 2014^2 - 2012^2, you don't HAVE to choose n = 2013. you could just as easily choose n = 2012, 2014, 1, 3000, or 9000000
steep sigil
yeah I am tested in a few hours about "making stuff easier" so this helps
but yeah, ty for your help!
.close
strong oak
Does anyone know what this function is called?
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Does anyone know what this function is called?
vital crag
gaussian
strong oak
Hi, so I did search up for the general form of that function and it seems different?
tiny gorge
only by a scale factor
vital crag
You have 3 undefined free variables
strong oak
only by a scale factor
Can you explain that further?
vital crag
you have to define what A, k, h are
if not, then it's still a gaussian
scale of gaussian = constant multiple of gaussian
tiny gorge
gaussian is typically normalized so it integrates to 1
if you multiply that by the appropriate scale factor then it will be gaussian
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rugged escarp
$x = \sqrt(a^2 - y^2), 0 <= y <= a/2$
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$x = \sqrt(a^2 - y^2), 0 <= y <= a/2$
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wakamole
rugged escarp
The given curve is rotated about the y-axis. Find the area of the resulting surface.
$\int_0^\frac{a}{2}2\pi y \sqrt(f'(y)^2)$
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wakamole
rugged escarp
$y = \sqrt(a^2-y^2) = (a^2-y^2)^\frac{1}{2}, dy = \frac{2a-2y}{2a^2-2y^2}$
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wakamole
forest ember
Wrap the \sqrt with {}
rugged escarp
$\int_0^\frac{a}{2}2\pi y \sqrt(1 + \frac{2a-2y}{2a^2-2y^2}^2)dy$
$(\frac{2a-2y}{2a^2-2y^2})^2 = \frac{4a^2-4ay+4y^2}{4a^4-4a^2y^2+4y^4}$
rugged escarp
$\int_0^\frac{a}{2}2\pi y \sqrt(\frac{4a^4-4a^2y^2+4y^4}{4a^4-4a^2y^2+4y^4}+\frac{4a^2-4ay+4y^2}{4a^4-4a^2y^2+4y^4}dy$
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wakamole
rugged escarp
<@&286206848099549185> did i do something wrong here?
whole halo
@rugged escarp you here?
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rugged escarp
<@578410730659905536> you here?
hey
i found out what i did wrog but used chatgpt
so i didnt know you can actually put the equation as x or y
i have 2 formulas for x axis and 2 for y axis
$SA = \int_a^b 2\pi f(x) \sqrt(1 + f'(x)^2)dx$
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wakamole
rugged escarp
$SA = \int_a^b 2\pi y\sqrt(1 + f'(y)^2)dx$
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wakamole
rugged escarp
$SA = \int_a^b 2\pi x \sqrt(1 + f'(x)^2)dx$
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wakamole
rugged escarp
$SA = \int_a^b 2\pi f(y)\sqrt(1 + f'(y)^2)dx$
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wakamole
rugged escarp
the first two for xaxis and 2last 2 for y axis
whole halo
@rugged escarp first, do you want advice on writing the latex properly
rugged escarp
ok
rugged escarp
<@578410730659905536> first, do you want advice on writing the latex properly
sure
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mtt
whole halo
first, you can right-click and copy discord text
this will show symbols that get hidden due to discord formatting
for example if you copy this message, youll see the _s I used to italicize the this
Ctrl + C will not do this, it will not copy the _s
it only copies what you directly see
you can do that in case you see something new done with text that you need to look into
soft zealotBOT
wakamole
whole halo
as you can see you did not copy the text correctly
whole halo
you need to choose this option
rugged escarp
for example if you copy this message, youll see the _s I used to italicize the this
oo
_i didi it
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wakamole
whole halo
next is that theres a few fixes to do
first, you dont always have to show latex math with
$ ... $
$\text{if your whole message is going to be like that, theres a faster way to trigger the command}$
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mtt
whole halo
,,\text{that's with two commas at the beginning, then you begin in latex's `math mode'}
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mtt
rugged escarp
,,2+2
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wakamole
whole halo
next is how to properly use commands like \sqrt
whole halo
yes instead of ()
latex usually reads \s and {}s
\sqrt{ ... }
\frac{ ... }{ ... }
\int_{ ... }^{ ... }
when you place {}s around something, you are "wrapping" it in braces
if you place ()s around something, you are "wrapping" it in parentheses
whole halo
**wakamole**
now if you want parentheses around something, like here, you use ()
,,\int_0^\frac{a}{2}2\pi y\sqrt{1+\left(\frac{2a-2y}{2a^2-2y^2}\right)^2}dy
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mtt
whole halo
here, Ive inserted \left( and \right) to add in parentheses
the \left and \right's purpose is to make the two parentheses grow bigger if theres something big inside
otherwise they remain the same size
,,\int_0^\frac{a}{2}2\pi y\sqrt{1+(\frac{2a-2y}{2a^2-2y^2})^2}dy
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mtt
rugged escarp
i actually derived wrong
whole halo
i have 2 formulas for x axis and 2 for y axis
now back to the formulas
the formula you need here is $\int_a^b2\pi x\sqrt{1+f'(y)^2}\dd{y}$
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mtt
whole halo
if you slice up a surface into rings,
and in this case, the rings are horizontal,
the radius is x
on the surface, the rings arent all like cylinders
theyre slanted
how much of a slant they have is with that sqrt(1 + f'(y)^2)
rugged escarp
**wakamole**
only these 4
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reef night
problem: Let $\mu$ be a regular Borel measure on a compact Hausdorff space $X$; assume $\mu(X)=1$. Prove that there is a compact set $K\subset X$ (the carrier or support of $\mu$) such that $\mu(K)=1$ but $\mu(H)<1$ for every proper compact subset $H$ of $K$.
my solution so far: Let $K$ be the intersection of all compact $K_{\alpha}$ with $\mu(K_{\alpha})=1$. Then $K=\bigcap_{\alpha\in A}K_{\alpha}$ is an intersection of closed sets and thus closed. It is also a closed subset of a compact set (pick any $K_{\alpha}$), and thus compact. Let $H\subsetneq K$ be compact. If $\mu(H)=1$ then $H=K_{\alpha}$ for some $\alpha$, which implies $K\subseteq H$. So we must have $\mu(H)<1$.
now, what i need to do is prove $\mu(K)=1$. to do that, my textbook suggests showing that if $K\subset V$ then there is some $K_{\alpha}\subset V$, then using regularity of the borel measure. from there, we use regularity of measure to say that $\mu(K)=\inf{\mu(V):K\subset V}$ and $\mu(V)=\sup{\mu(K'):K'\subset V}$, $K'$ arbitrary open, but some $K_{\alpha}\subset V$ so $\mu(V)=1$ for all $V\supset K$ so $\mu(K)=1$. my problem is, i cant find out why we always have $K_{\alpha}\subset V$.
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problem: Let $\mu$ be a regular Borel measure on a compact Hausdorff space $X$; ...
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goat
reef night
any help would be very appreciated, this is my last question and homework is due in 4 hours
<@&286206848099549185>
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reef night
no, not yet
reef night
oh i see you look at complements of everything
reef night
<@1293262180195504191> Has your question been resolved?
yes i have resolved my question
reef night
how do i close this
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whole halo
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left fern
hello
left fern
if we got x * square root (y - square root (z)) can we distrubite using x directly
smth like this
idk how to rotate
tranquil pine
,rccw
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tranquil pine
when the x goes inside the square root it becomes x^2
and then you can just do normal distribution
left fern
ohhh thx
left fern
your x is weirder than mine
cursive goes hard
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simple wave
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tranquil pine
yo can anyone help me with 2.1 tangent lines and velocity
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yo can anyone help me with 2.1 tangent lines and velocity
vital crag
What's the question
vital crag
Khan academy
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odd parrot
yo can anyone help me with 2.1 tangent lines and velocity
averagevelocity = secant line (slope)
instantaneous velocity = tangent line (derivative)
odd parrot
instantaneous velocity = tangent line (derivative)
your teacher would probably want you to use this formula
odd parrot
your teacher would probably want you to use this formula
sometimes you'll be asked to approximate a tangent line by drawing it and finidng points on that tangent line
but otherwise
2.1 is pretty much just the exact same thing in different words
find the average rate of change of ...
find the instantaneous rate of change of ...
remote mauve
I’m a little confused. For y = tanx, b = 1 right, so if you use pi/b to find the asymptotes, how does it end up as x = +- pi/2 instead of well x = +- pi/1? do you take the result and split it in half for the full period?
final saddleBOT
I’m a little confused. For y = tanx, b = 1 right, so if you use pi/b to find the...
remote mauve
And if that’s the case, how come in y=2tan(1/4x), b = 1/4, so 2pi/1/1/4 = 2pi/1 * 4/1 = 8pi. And if you take half of 8pi for each asymptote it would be x= +-4pi, but the asymptote are actually at x= +-2pi
pliant shore
I’m a little confused. For y = tanx, b = 1 right, so if you use pi/b to find the...
pi/b is not the asymptote of tan x
it is true that tan x has a period of pi, so the asymptotes are pi units apart
and by function transformations, tan(bx) has a period of pi/b, so the asymptotes will also be pi/b units apart
the period info doesn't tell you where the asymptotes are
cunning stirrup
in fact, ur own notes say that the asymptotes are at x=+- pi/2b
pliant shore
yeah cause we start from the fact that tan(pi/2) is undefined
and work our way from there to figure out the other asymptotes
pliant shore
So I take the period and divide it in two?
no
remote mauve
Oh
pliant shore
the asymptote is the value of x such that f(x) goes to infinity
or -infinity
if it's a vertical asymptote ofc
cunning stirrup
if we have tan(something), the asymptote is whenever that something approaches pi/2 or -pi/2
or of course any of the multiples
remote mauve
Hmm
cunning stirrup
so to find the something, you just do something=pi/2 or something=-pi/2
that something could just be x, it could be bx, it could be bx+a
remote mauve
So if say I have something as 1/4 I take pi/2/1/4 to get pi/2 * 4/1 = 4pi/2 = 2pi as the asymptotes?
cunning stirrup
sure, as your own notes say, if we are looking at tan(1/4 * x), then we for the asymptotes we got 1/4 * x=pi/2 => x=2pi
and the same for 1/4 * x = -pi/2
for -2pi
pliant shore
yeah, and then function transformations
remote mauve
Ok
pliant shore
if you compress (pi/2, infinity) by a factor of b horizontally
you get (pi/(2b), infinity)
and now you can add arbitrary multiples of pi/b to this
to get the other asymptotes
remote mauve
How would I then find the point above the x-intercept
Like say from (0,0) how do I go the next point in the positive direction
If the graph is y=2tan(1/4x)
pliant shore
If the graph is y=2tan(1/4x)
okay so that means b = 1/4, and it's a horizontal stretch by a factor of 4
so the period of this is 4pi
hence from (0, 0) you can go to (4pi, 0)
remote mauve
I get how periods work, but how do I find the next points to graph the squiggly line
Like in between the asymptotes
Do I go half the distance from the x-int to the asymptote and the up 1(a) for the y coordinate?
And since one of the asymptotes is at x = 2pi, and from (0,0) half the distance from the x-int to the asymptote is pi, and then you take 1(a) which in this case a=2 so you go up 2, so the next point from (0,0) is at (pi,2)?
<@&286206848099549185>
umbral narwhal
How would I then find the point above the x-intercept
wdym?
point above the x intercept?
remote mauve
Like how do I go from (0,0) to (pi/2) for the graph y=2tan(1/4x)
umbral narwhal
thought I could help man
umbral narwhal
Like how do I go from (0,0) to (pi/2) for the graph y=2tan(1/4x)
nice question th
strange gyro
I’m a little confused. For y = tanx, b = 1 right, so if you use pi/b to find the...
Is this the question?
remote mauve
Not anymore
How do I go from (0,0) to (pi/2) for the graph y=2tan(1/4x)
The graph is y=2tan(1/4x)
@strange gyro
strange gyro
<@916943277897449472>
Idk, I didn’t use any of these
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woeful sable
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woeful sable
How
tranquil pine
cant read
woeful sable
$lim_{x \to \infty} f(x) = \infty
lim_{x \to -\infty} f(x) = -\infty
lim_{x \to \infty} \frac{f(x)}{x} = \frac{1}{2}
lim_{x \to -\infty} \frac{f(x)}{x} = -1
Find [fof(x)/x] when x go to +infinity and -infinity
I find 1/4 when x go to +infinity and the-1/2 when x go to -infinity is that correct?
tranquil pine
$lim_{x \to \infty} f(x) = \infty
lim_{x \to -\infty} f(x) = -\infty
lim_{x \to ...
hm i still cant read
i think u need to put $ somewhere
around
$lim{x \to \infty} f(x) = \infty
lim{x \to -\infty} f(x) = -\infty
lim{x \to \infty} \frac{f(x)}{x} = \frac{1}{2}
lim{x \to -\infty} \frac{f(x)}{x} = -1$
soft zealotBOT
skiou_hhd
tranquil pine
hm can u change it again ? idk how to write latex
soft zealotBOT
skiou_hhd
woeful sable
$lim_{x \to \infty} f(x) = \infty
lim_{x \to -\infty} f(x) = -\infty
lim_{x \to \infty} \frac{f(x)}{x} = \frac{1}{2}
lim_{x \to -\infty} \frac{f(x)}{x} = -1
Wait
soft zealotBOT
skiou_hhd
woeful sable
@tranquil pine
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unkempt seal
what did i do wrong?
final saddleBOT
what did i do wrong?
unkempt seal
checked the answer, was close but they didn’t have the 1/2 y^2 on LHS
candid hull
yeah cause e^t y is already the integral of e^t dy/dt + e^t y
you don't have to integrate twice
candid hull
yeah, and integrating that LHS (e^t dy/dt + e^t y) gives you e^t y already
unkempt seal
oh
thanks!
i thought i contracted that part to the product rule and had to integrate again
candid hull
the whole point is that e^t dy/dt + e^t y is d/dt (e^t y) already
as you said, product rule
unkempt seal
yeah i see now, thx again
clever solstice
greetings mathemagicians
final saddleBOT
greetings mathemagicians
clever solstice
can you help me to understand this
its basically that thing in statistics where you pick objects from a pool N
and don't put them back and the variation matters
how is that and factorial connected?
factorial is basically permutation?
this is what factorial basically means?
3 * 2 * 1
basically all possibilities
SO if we pull from a pool of N = 3
and we pick 3 times without putting anything back
we just get all the possible permuations which are possible so 6 in this case
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glad otter
Could someone confirm my proof for sqr(4) and if it's correct.
final saddleBOT
Could someone confirm my proof for sqr(4) and if it's correct.
glad otter
The arrows are just supposed to show the steps.
A few more questions which I would like to get confirmation on:
1). Suppose n is a natural number such that n > 2. T/F: n is prime if n is odd.
A) False. 9 is odd, but is not a prime number.
2). Suppose n is a natural number that n > 2. T/F: n is prime only if n is odd.
A) True. A can be true if B is true, if 9 is odd, then it is possible it is prime.
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tulip coyote
A few more questions which I would like to get confirmation on:
1). Suppose n i...
For the second one, I would more say that even numbers are divisible by 2, so e.g. any integer n > 2 that is even is automatically not prime, and has 2 as a factor
Do you have the proof of sqrt{2} being irrational?
glad otter
For the second one, I would more say that even numbers are divisible by 2, so e....
Lemme think about this one again.
tulip coyote
Could you also explain what you did here?
glad otter
Do you have the proof of sqrt{2} being irrational?
Sadly no. It's been a while since the last class we went over this in -- so a lot of my math was done by the class trying to understand the TAs.
tulip coyote
Damn 
let me try and fetch it from somewhere then-
glad otter
Could you also explain what you did here?
I think what I did here was if we know a = 2 * (literally anything), we can call that a different value such as b. If b is even, then it would be written as b = 2 * (another thing), which i denoted as c
tulip coyote
Yoinked (from https://www.homeschoolmath.net/teaching/proof_square_root_2_irrational.php for this one)
glad otter
Yoinked (from https://www.homeschoolmath.net/teaching/proof_square_root_2_irrati...
I mean I did something similar to that before the TA got me scared.
4b^2 = a^2
a = 4k
b^2 = (4k)^2
4b^2 = 16k^2
but I'm unsure whether where to go from here
tulip coyote
I think what I did here was if we know a = 2 * (literally anything), we can call...
Hmmm, I mean, at the point you have, you would already know that a = 4b = 2(2b) though? Are you trying to imply that b itself is even at that point?
(And
4b^2 = a^2
a = 4k
a^2 being a multiple of 4 here doesn't quite mean that a is a multiple of 4 as well)
glad otter
I think I see what you mean. a = 4b = 2(2b) just tells us that a is a multiple of 4, it doesn't imply that b is even.
glad otter
(And
> 4b^2 = a^2
> a = 4k
a^2 being a multiple of 4 here doesn't quite mean tha...
Yeah. I was pretty unsure what I was doing at this point so I and the majority of kids just did what the TA's where telling us what to do. Which was my previous equation.
tulip coyote
Yep, you know that a is a multiple of 4, and subsequently is even, but you don't yet know that b is even or a multiple of 4 yet
glad otter
Yeah, gimme a minute. this kind of math hurts my head -- can you explain what you mean on my second question though
A can possibly be true only when B is true. If B is an odd number, which can possibly be true. That if n is odd
A can possibly be true?? That's what gets me with all these types of questions. The meaning of them is like, there is a chance it could be true, so how do we even know if its true to begin with.
tulip coyote
If you've made it clear somewhere that A is e.g. the statement "n is prime" and that B is e.g. the statement "n is odd" somewhere, you are right, but at least imo it doesn't read too well to phrase it that way
I would rather e.g. explicitly state it out, that "any odd integer greater than 2 could possibly be prime", but more than that-
It's worth explicitly reasoning why any prime that's strictly greater than 2 must be prime
One way to do that is to e.g. take the equivalent contrapositive statement, of the original
"any prime number [strictly greater than 2] is odd"
which is
"any even number [strictly greater than 2] is not prime"
with the latter quite easy to show
You could also phrase it similarly by contradiction: assume you have an n > 2 that is both prime, and also even, so then n is divisible by 2, yet n is not equal to 2 - would you look at that, a factor that isn't 1 or n itself, so n really isn't prime...
glad otter
So that the contrapositive of the statement is "If n is even, then n is not prime" which if we assume n > 2, n is prime and also even. If it is even, it must be divisible by 2 which means that it has a divisor other than just itself?
tulip coyote
If you do the contrapositive, you just need to show that if you have some even n and it's strictly greater than 2, then in particular, it's divisible by 2, which as you've noted, is neither n itself (nor is 2 equal to 1) and so it's not prime
tl;dr "not odd (so even) implies not prime"
That's different to doing it by contradiction: "assume prime, but also not odd, but [...] contradiction"
twilit copper
hi
glad otter
I'll likely have to come back to this chat. However, the statement is False because not all odd numbers are prime even though its true that the prime numbers are greater than 2 would be odd. Hopefulyl I'm understanding this a bit
glad otter
If you do the contrapositive, you just need to show that if you have some even n...
What book/course did you learn this from that you would suggest.
tulip coyote
Not all odd numbers are prime, but all primes that are at least 3 must be odd as per above
tulip coyote
What book/course did you learn this from that you would suggest.
And I don't think I have any suggestions off the top of my head atm 
glad otter
And I don't think I have any suggestions off the top of my head atm <:sadcat:547...
I heard of one -- hopefully I'm allowed to send links. https://discrete.openmathbooks.org/dmoi3/ch_intro.html
I've been just trying to follow this. I guess can you check two more things before I go to bed
I said this is True.
This would be False. If you gimme a minute I'll explain why for each
glad otter
This would be False. If you gimme a minute I'll explain why for each
This is false because since its "sufficient" it is telling us that n is not a prime, hence it should make n even. if non-prime integers are odd, then it being not prime would not be enough for n to be even
X: n is not prime.
Y: n is even.
glad otter
I said this is True.
Actually don't remember anything I did for this.
final saddleBOT
@glad otter Has your question been resolved?
glad otter
<@1155032488880640031> Has your question been resolved?
no
@tulip coyote hey, would you be fine looking over them. if you cant/asleep. thank you so much for the helps.
glad otter
Choose the ❌ react
omg. the new discord update doesn't show them unless I click on the three dots
my bad
tulip coyote
This is false because since its "sufficient" it is telling us that n is not a pr...
And well, that is "alright", but you could've used your previous example: 9 is not prime but isn't even, so being not prime certainly is not sufficient to be even
tulip coyote
Actually don't remember anything I did for this.
And as for this one...
final saddleBOT
glad otter
.reopen
final saddleBOT
✅
tulip coyote
A few more questions which I would like to get confirmation on:
1). Suppose n i...
...it's effectively the previous question, part 2), we had up here, and its contrapositive statement: "n is not prime implies n is even" is the same as "n is odd implies n is prime"
glad otter
And well, that is "alright", but you could've used your previous example: 9 is n...
Ahh. I appreciate it a lot man. I'll do better at explaining my reasonings such as this. Still -- thanks a millions for the help. Goodnight <3
.close
glacial rock
can anyone help me solve this since im not familiar with x^2 + x - 2 to divide with the 2x^4
final saddleBOT
can anyone help me solve this since im not familiar with x^2 + x - 2 to divide ...
glacial rock
is it just gonna be longer?
grave pumice
is it just gonna be longer?
Yeah it will be
pliant shore
can anyone help me solve this since im not familiar with x^2 + x - 2 to divide ...
in your polynomial long division, write 2x^4 + 0x^3 + 0x^2 + 0x - 6
that will help a lot
have you done polynomial long division before, by the way?
final saddleBOT
@glacial rock Has your question been resolved?
tiny ridge
I think i see what the question is asking its just the question is too wordy
Do u know how to use long division?
@glacial rock
glacial rock
have you done polynomial long division before, by the way?
yes i do know how to use long division but i just got abit nervous when i saw trinomial
tiny ridge
its ok i can walk u through it
glacial rock
wait im doing finding inverse of f(x) = 3x -2
tiny ridge
i dont think it says u should
glacial rock
https://www.youtube.com/watch?app=desktop&v=8lT00iLntFc&ab_channel=TheOrganicChemistryTutor
w video though
glacial rock
i dont think it says u should
its another question :)
tiny ridge
oh
glacial rock
is there any online graph sketching i can use?
glacial rock
oh great