just a mon
#help-28
thick hedge
*min
Let $A$ be open , let $x \in int(\partial A)$. Then $V_{\varepsilon}(x) \subseteq \partial A$. But $x \in \partial A \implies \forall \varepsilon, V_{\varepsilon} \cap A^c ≠ \varnothing$. This would mean $V_{\varepsilon}(x) \nsubseteq \partial A$ a contradiction
glossy valveBOT
wai
split swift
how are you getting the last non-containment?
thick hedge
V_{\varepsilon} intersects A^c and is non-empty, and thus is not entirely contained in A, even for that epsilon for which we have proper containnment by the fact that it's open
hallow walrus
Bro isn’t it 7 30 AM in India
thick hedge
Bro isn’t it 7 30 AM in India
around , yes
hallow walrus
Insane grind
thick hedge
Insane grind
I need that B i grade n RA 😭
split swift
take A = Q as a subset of R; the boundary of A is R, so int(boundary(A)) = R; for any x in int(boundary(A)) = R, every neighbourhood of x is a subset of the boundary of A
I agree that every neighbourhood intersects the complement, but I don't think this gives that it's not contained in the boundary
thick hedge
take A = Q as a subset of R; the boundary of A is R, so int(boundary(A)) = R; fo...
hmm?
split swift
I'm giving a counterexample(?) to your claim
the boundary of A isn't (necessarily) a subset of A, so intersecting the complement shouldn't be enough
thick hedge
take A = Q as a subset of R; the boundary of A is R, so int(boundary(A)) = R; fo...
yes, but no neigbourhood is contained in A
split swift
that's because the original statement in the exercise is indeed true
but I'm saying that your final implication isn't sound
it's a counterexample to $U\ni x$; $U\cap A^c\ne0;\rightarrow; U\not\subseteq\partial A$, not to the theorem
glossy valveBOT
Desync
thick hedge
but I'm saying that your final implication isn't sound
okay, so what should my argument have been
thick hedge
you haven't used that A is open
as A is open, a neigbourhood must be contained in A
split swift
but our neighbourhood is in the interior of the boundary
a priori, this could be entirely disjoint from A
actually, for open A, boundary(A) = clos(A) \ A, so A is indeed disjoint from its boundary
if we pick some open U in boundary(A) = clos(A) \ A, then U is disjoint from A, so U is in the interior of A^c; but int(A^c) is disjoint from clos(A), so this contradicts that U is a subset of clos(A)
I think the proof for A closed should be very similar
thick hedge
if we pick some open U in boundary(A) = clos(A) \ A, then U is disjoint from A, ...
I have to use the metric space/ defn in R
of boundary
split swift
for A closed, boundary(A) = A \ int(A) subseteq A; again, let U in boundary(A), so U is a subset of A, and since U is open, U is a subset of int(A), but boundary(A) is disjoint from int(A), and this is a similar contradiction to before
the thing to remember with boundaries is that it doesn't have to be contained in either A nor A^c
if you have a half open interval in R, for example, one part of the boundary is going to be in the set (the closed end) and the other is in the complement
split swift
nw
thick hedge
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manic bramble
2 different books on quaternions have a formula to create the individual quaternion components, based on 3 euler angles, shown as: alpha, beta, gamma.
manic bramble
one starts with separating the terms with +, and the other with -. Is one of them wrong, or are they both valid?
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@manic bramble Has your question been resolved?
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ruby ermine
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unique cloud
who comes up with something so poorly worded? so ab/ba should be equal to one and bc/cb should be different from one?
unique cloud
A problem is hard if its hard, doing this sneaky stuff is like a video game adding aritificial difficulty by making your controls reverse for a second
nevermind its still guuci, a,b,c can each be 10 values, so all you gotta do is try 1000 values until you get what you want
full whale
my interpretation is that $\frac{ab}{ba} = \frac{bc}{cb} \neq 1$
glossy valveBOT
Tairi
full whale
because otherwise the problem can be trivialized by letting a, b, and c all be the same digit
I had hoped to probe OP to see if he knows to turn the problem into this
unique cloud
@full whale omg true I hope OP didn't see it sorry
full whale
nah it's fine
so OP, if ab is a 2-digit number with tens digit a and units digit b, how would you represent the value of the number ab in terms of the digits a and b?
terse adder
Let $$ be an operation on a set $S$. An element $\lambda \in S$ is a \textbf{left identity} for $$ if $\lambda * a = a$ for all $a \in S$; and $\rho \in S$ is a \textbf{right identity} for $*$ if $a * \rho = a$ for all $a \in S$.
(b) Show that if the operation $$ has both a left identity $\lambda$ and a right identity $\rho$, then $$ has an identity element.
glossy valveBOT
edg
terse adder
here's my current attempt in writing the proof
\begin{proof}
Recall the definition of an identity element $e$ in a set $S$ under the operation $$ has the property $a * e = e * a = a$. Suppose that the operation $$ has the following properties $a * \lambda = a$ and $\rho * a = a$ for all $\lambda, \rho, a \in S$. Since we can show that $\lambda * a = a = a * \rho$, then $*$ has an identity element if it has a left and right identity in $S$.
\end{proof}
oh wait a minute, let me adjust something
\begin{proof}
Recall the definition of an identity element $e$ in a set $S$ under the operation $$ has the property $a * e = e * a = a$. Suppose that the operation $$ has the following properties $\lambda * a = a$ and $a * \rho = a$ for all $\lambda, \rho, a \in S$. Since we can show that $\lambda * a = a = a * \rho$, then $*$ has an identity element if it has a left and right identity in $S$.
\end{proof}
glossy valveBOT
edg
terse adder
am i right in stating that $\lambda * a = a = a * \rho$ to identify that $*$ has an identity element if it satisfies this
glossy valveBOT
edg
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terse adder
essentially you need to show that lambda = rho
oh okay thanks for the clarification, ill work on it
last bolt
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last bolt
I just know C is right So it doesn't qualify to be the answer
copper tangle
f(x) mod 2 = 1, right?
last bolt
Yes
copper tangle
so hey, at least A is wrong
last bolt
How?
last bolt
Oh! you are absolutely right!
Yaa A is incorrect
What about g(x)
g(x) can't have an integer root similarly
What about rational root
copper tangle
g(x) can't have an integer root similarly
why? also mod 2?
last bolt
Yes
any odd number u plug in u get odd number
Any even number u plug in u again get an odd number so never zero
copper tangle
Well, g(x) has at least one real root, I'm trying to remember if there is anything that would tell us anything about rational roots (obviously you could just use RRT and plug all possible values, but that seems... hard)
you could once again use modular arithmetic to try and remove some of the candidates tho
last bolt
Yaa,it's hard to plug all values we need to check 1/3,5/3,7/3 all with +- sign so 6 values we need to check
last bolt
Ok
delicate torrent
to basically find the roots (rational/integral), basically find all the factors of 105. if there is an integral root then it must be a factor of 105.
if there is a rational root then it must be a fraction made up of factors of 105
this should help the trick
last bolt
to basically find the roots (rational/integral), basically find all the factors ...
That's basically rrt
delicate torrent
That's basically rrt
try to use that
copper tangle
Oh good, cuz I didn't yet
Yeah I was really excited I could use all the stuff I recently learned for this problem when I saw it, but I really couldn't
delicate torrent
ooh
last bolt
Thanks there's one more problem shall I send it?
delicate torrent
Thanks there's one more problem shall I send it?
alr
copper tangle
I mean sure
delicate torrent
do it
last bolt
delicate torrent
hmm
copper tangle
Because sum doesn't change degree, unless with exactly equal (but opposite in sign) leading coefficient
delicate torrent
sum of 3 polynomials is just sum of 4 of them subtract 1 one them
im literally thinking rn
delicate torrent
wait
delicate torrent
Because sum doesn't change degree, unless with exactly equal (but opposite in si...
I see something
Notice if there exists one set of polynomials satisfying the following condition
delicate torrent
Then you can multiply those 4 polynomials by a real number $r \neq 0$ and it would still satisfy the conditions
glossy valveBOT
1 divided by 0 equals Infinity
delicate torrent
So the answer is either $\infty$ or $0$
glossy valveBOT
1 divided by 0 equals Infinity
last bolt
So the answer is either $\infty$ or $0$
Yes
copper tangle
What?
delicate torrent
How do we prove there exists no such set?
This is the main question now
copper tangle
Oh yeah I see what you mean
Well if a polynomial of even degree has no real solutions, then it's either everywhere negative or everywhere positive
delicate torrent
Aka in a form of $f(x)^{2k} + c$ where $c > 0$ and $k \in \mathbb N$
copper tangle
If you add two polynomials with the same sign, then you get another polynomial of the same sign
glossy valveBOT
1 divided by 0 equals Infinity
copper tangle
So the root has to come from adding polynomials with opposite signs
This seems actually possible for a pair of polynomials
delicate torrent
No way there's infinitely many polynomials like that
copper tangle
So it seems that the impossibility must stem from the fact that all their pairwise sums mustn't have real solutions
copper tangle
No way there's infinitely many polynomials like that
That's not what I meant
Suppose there are three such polynomials such that their sum has a root... ?
can we show that at least one of the sub-sums must also have a root?
last bolt
P1(x)+P2(x) must be even degree
Now P3(x) may be even or odd degree
If it's odd degree
P3(x)+P4(x) needs to be even degree
Implies P4(x) is of odd degree with opposite leading coefficient
If it's even degree then all of them are even
copper tangle
no?
last bolt
So,we have 2 cases even even odd odd
Or all even
But in even even odd odd
1+3 will have real root
So only case 2
Where all are even degree
So,it's clear that all of them must be even degree
last bolt
Oh I messed it up
last bolt
e.g. P3 = 7x^3 + 5x^2 + 3x + 1, P4 = x^4
I always assumed in my mind by mistake that odd one is higher
So,we have 2 cases
Even even odd odd
All even
copper tangle
interestingly, we could still consider the polynomials as a product of monomials, but with all roots complex? I'm not sure how much that helps
Probably not a lot, because they could have a lot of roots
and we have to add them
last bolt
Yaa that doesn't help according to me
copper tangle
oh we do have a coefficient that corresponds to the sum of roots. If real coefficients, they all conjugate each other
last bolt
Yaa
copper tangle
One thing I'm guessing is that we could use lagrange interpolation to get pretty much any shape we want
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@last bolt Has your question been resolved?
copper tangle
I reckon I could construct a set of three polynomials with this property
Now we need one more
last bolt
One thing I'm guessing is that we could use lagrange interpolation to get pretty...
Actually,I am a high school student.I don't understand lagrange interpolation
copper tangle
Doesn't matter. It just says we can draw a polynomial through a bunch of points on a plane.
last bolt
That's obvious
mossy vortex
Hi
last bolt
If we have n points,the maximum degree of a polynomial required is n-1
mossy vortex
I am doing limit questions.anyone interested?
copper tangle
I think I figured it out @last bolt tell me If I'm wrong
So suppose we have polynomials X and Y
if neither has roots, and X+Y does, then they must be on opposite sides
Now suppose that we have polynomials A, B, C, D
The question asks that A+B, A+C, A+D, B+C, B+D, C+D don't have roots, but A+B+C, A+B+D, A+C+D, B+C+D do
spare hedge
I am doing limit questions.anyone interested?
Send
copper tangle
Let's say A+B has sign + because it doesn't matter (by symmetry)
spare hedge
this is a help channel tho
Can you tell me one qn
copper tangle
or whatever
spare hedge
copper tangle
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copper tangle
wow
... then C must have sign -
=> A+D and B+D must have sign +
And working down the table we get:
A -
B -
C -
D -
A+B +
A+C +
A+D +
B+C +
B+D +
C+D +
A+B+C both?
A+B+D both?
A+C+D both?
B+C+D both?
This is just from the assumption that e.g. if (A+B)+C has a root, then e.g. A+B and C must have opposite signs
you can fill it out
But notice how now we are saying that all A is fully negative, B is fully negative, but A+B is fully positive
This, I claim is a contradiction
last bolt
Hi
Lemme see
Wlog let's say A+B>0
implies C<0
D<0 for A+B+C,A+B+D to root. Now,B<0,A<0 for B+D and A+D to not have root.But,A+B>0 so it's not possible.
Thank you so much you figured it out
Thanks
copper tangle
This is also true for all continuous functions (defined on the same domain)
last bolt
This is also true for all continuous functions (defined on the same domain)
Why they need to be defined on same domain?
last bolt
idk how to add them otherwise
I mean their domain should have intersection atleast
copper tangle
idk it could get complicated
last bolt
I mean their domain should have intersection atleast
Because otherwise it won't make sense to add them
But we can add,if domain has intersection
And define a new domain as intersection of domain of the added functions
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proven tide
hi this is a request for help with creating a table to describe dice roll probabilities in a game (the mid-2010s space combat tabletop wargame Firestorm Armada if anyone's interested)
the game uses 6-sided dice. they miss on a 1-3, score a success on a 4-5, and on a 6, they score 2 successes and another die is rolled for a chance at more successes under the same rules. some of the maths behind this system is discussed here: https://www3.risc.jku.at/education/courses/ws2016/cas/exploding.html
the game requires you to roll successes greater than or equal to a target number to hit. I'd like to make a table (excel spreadsheet?) showing the probability of meeting or exceeding target numbers between 1 and 30 for dice pools of up to 30 dice. I'd like to shade the cells by colour based on how high the probability is, e.g. red if probability of equalling or exceeding the target number < 0.5, yellow if 0.5<=p<90, green if p=>90 or something.
is this a python angle and then copy the results into a spreadsheet? is there a clever way to do this entirely in excel?? can this be run up in R or something? xD
really grateful for any help!
proven tide
also, the website quotes the expected value of a single die as 0.8 taking into account the fact the probability of any successes at all on an individual die is 0.5 but 6s "explode" into 2 successes and an extra die added to the pool. but just using 0.8 in the maths breaks down in edge case situations, like the probability of meeting or exceeding target number 1 with 1 die isn't 0.8 it's 0.5
can just manually handle/ignore those edge cases obviously but
trail barn
also, the website quotes the expected value of a single die as 0.8 taking into a...
That's not what expected value means. For a different example, if we were summing the values on the dice then the expected value of a dice would be 3.5, a number which has nothing to do with any probabilities
Here an expected value if 0.8 means that if you roll e.g. 20 dice, then on average you would have 0.8 * 20 = 16 successes, but it doesn't say anything more than that regarding the spread
proven tide
That's not what expected value means. For a different example, if we were summin...
oh oops jashdjksa
trail barn
The section above that on generating functions does actually tell you how to find the probabilities
Uh although I'm not sure which python library you would need to handle tbe symbolic algebra (or whether this can be done in R)
proven tide
I acc can't even tell what this graph means
like what do the colours correspond to?
trail barn
The colours represent different numbers of dice
It's labelled wrong
The x axis is the number of successes
And the probability is the probability getting that exact number of successes
proven tide
ohhh yeah i thought x axis was k
ok no that makes more sense
how does the x in g(x) like. have any relevance to the rest of the variables. like what does g(x)^k look like in practice
not sure im understanding how generating functions work
ok I clearly need to do more reading on this ashdjash gonna close the channel. thanks for pointing me towards that paragraph though!
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hoary bone
gritty rose
!msgdel
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hoary bone
Oh alr
hoary bone
Can yall help me in maths
find an aviailable channel and post ur question there.
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tardy pike
If you're finding a root, you need some type of f(x) to work with. So if you're given something like sin(ln(x))-x^2= e^x, the first thing would subtract e^x on both sides to get
sin(ln(x))-x^2 -e^x= 0.
Finding solutions to that is the same as showing solutions to sin(ln(x))-x^2= e^x exist. Then define
f(x) = sin(ln(x))-x^2 -e^x
Then you appliy IVT to that function f.
If they say something like "show a solution exists in the interval [a, b]". Then you know you're working with x=a and x=b. Otherwise you have to pick values you know.
pseudo trout
Wym by pick values i know? Make my own interval you mean
tardy pike
yeah, if you aren't given one
pseudo trout
Do i just trial and error for that
tardy pike
yeah, but with values you know
like, for above, we know ln(1)=0, and sin(0)=0. so x =1 could be one endpoint of your interval
pseudo trout
How would i find the other
pseudo trout
Cant see how that can be tricky id just plug in values say its continuous and say theres a sign change no?
tardy pike
Write, but you have to show the sign changes. that could be tricky
pseudo trout
How so
tardy pike
Well, for example, for the one above I'm not sure off the top of my head what I would pick
pseudo trout
Yeah but that means hes gonna ask me to find the interval and not just give me it? Thats where i assumed the hard part woukd be
tardy pike
I don't know whether he will give the interval or not. Usually it is given. He could be saying it will be hard just because it will be a complicated looking expression. but the idea is the same
pseudo trout
I don't know whether he will give the interval or not. Usually it is given. He c...
Hes not usually that nice but hopefully
tardy pike
For this one, since you have a sin(ln(x)), you would want to think about things like e^pi, e^-pi, e^-pi/2, values you can say something about
pseudo trout
For this one, since you have a sin(ln(x)), you would want to think about things ...
Ohh that makes sense
Prolly the negative powers rho no?
tardy pike
But I don't think this one has any real roots. Is this the example he gave?
tardy pike
hmmm. maybe he just made it up on the spot.
tardy pike
No, you wouldn't find the exact value. You would typically only estimate the value if you have a calculator.
Otherwise you are just showing that a solution exists in the interval.
Another thing, make sure you know the conditions for IVT and apply it correctly
Like, if you have something like f(x) =sin(e^x)-x, you should say f is continuous on the interval.
In that case you could say sin(e^x) is continuous since it is the composition of continuous functions, and then f is continuous since it is the difference of continuous functions.
pseudo trout
Do i need to justify why its continous?
tardy pike
yes, to apply the theorem
pseudo trout
Oh
tardy pike
if you do think you will have to estimate solutions, then the method you're using is called bisection method
pseudo trout
Ill say that sinlnx and x^2 and e^x are continous and the difference between continuous functions are continuous?
tardy pike
Yeah, and sin(lnx) is continuous because it is the composition of continuous functions
pseudo trout
What other things dont affect the continuity between two funcs
Do multiplying and dividing affect it?
Prolly dividing since it makes an asymptote
tardy pike
if you divide by zero you have a discontinuity, yes an asymptote
and ln(x) is only contininuous for x>0
pseudo trout
So continuous except dor where the function i divided by is equal to zero
pseudo trout
and ln(x) is only contininuous for x>0
Do i need to mention that since if im given an interval it'll definitely be x>0 because ln wouldn't exist orherwise
tardy pike
for whatever interval you're given you can probably just say "continuous in the interval"
because that is what you need for the theorem
pseudo trout
But I'd have to justify if it contains various funcs?
tardy pike
right, but you know sum/difference and product/difference of cts functions are cts
and compostion of cts functions are cts
so you don't have to do much to justify it, just say it
tardy pike
gl, look for practice problems and try a few
tight cloak
How can I approximate 10^(2/5) without calculator
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How can I approximate 10^(2/5) without calculator
gritty rose
10^2 = 100 and 100 = 5 * 20
tight cloak
An engine takes in 5 moles of air at 20°C and 1 atm, and compresses it adiabatically to 1/10th of the original volume. Assuming air to be a diatomic ideal gas made up of rigid molecules, the change in its internal energy during this process comes out to be X kJ. The value of X to the nearest integer is ......
The original question was a physics problem
tight cloak
10^2 = 100 and 100 = 5 * 20
I missed typing the parenthesis
civic bay
well 10^(2/5) is the fifth root of 100, so to narrow down some integers it helps to look at perfect 5th powers. clearly 2^5 = 32 and 3^5 = 243 so the fifth root is somewhere between 2 and 3
you’d wanna know whether 2.5^5 is smaller or larger than 100 so you know which of those two integers to round to
2.5^5 is the same as (5/2)^5 = 5^5 / 32 = 3125/32 and that’s clearly less than 100
so this tells you that 10^(2/5) is somewhere between 2 and 3, and greater than 2.5, so it rounds to 3
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tight cloak
so this tells you that 10^(2/5) is somewhere between 2 and 3, and greater than 2...
I actually thought of something else
10^0.4=10^1-0.6 = 10/10^0.4 = 10/4 =2.5
log 2=0.3
last bolt
I strongly feel that there must be an elegant solution that doesn't involve shifting terms to LHS and applying D>=0
last bolt
For set B
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last bolt
<@&286206848099549185> pls help
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winter rune
does anyone have a nicer soln?
My soln was to consider an aribtrary subset S of even cardinality of [n]
than i consider the following bijective function
consider the first natural i < n, such that the |2*|[i]-S|-i| = 2, and consider the subset of [n] which for j <= i, contains i iff S contains i, and for n>=j>=i contains i iff S doesnt contain i
now the cardinality of this is 2 mod 4 iff cardinality of S is 0 mod 4
so now consider the subsets which this function isnt defined for
those are exactly the ones that contain exactly one of every (2i,2i+1) pair
so, they are 2^(n/2) = 2^2k such sets, and their cardinality is 2k, so they are given a - sign in the first summation iff k is odd
so the value is
2^2k(-1)^k
i dont like this proof much because it feels very arbitrary, i came up with this by converting it into a paths problem, and so was wondering for a nicer soln
winter rune
(in the form of paths problem i am basically considering 2 diagonal lines 2 units away from the origin, and reflecting paths around the first point of intersection)
wise tartan
maybe roots of unity filter on (1+ix)^n ?
oh does the proof have to use combinatorical reasoning
winter rune
maybe roots of unity filter on (1+ix)^n ?
that is nice
winter rune
oh does the proof have to use combinatorical reasoning
that was the only thing i was trying tbh.
winter rune
maybe roots of unity filter on (1+ix)^n ?
just ((1+i)^n+(1-i)^n)/2 would do the trick here
i dont think you need the x
wise tartan
yeah thats just a placeholder
wise tartan
no sorry
winter rune
oh, thanks for the binomial one
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slate flower
can someone give me a hint on this question please
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can someone give me a hint on this question please
onyx glen
can i suggest redrawing this diagram so that ABC actually looks equilateral?
and maybe marking angles B and C with their known values based on that
slate flower
i mean, its a problem in a pdf file and i drew it in my excercise book with the angles
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nova cloak
Let X represent the difference between the num-
ber of heads and the number of tails obtained
when a coin is tossed n times. What are the pos-
sible values of X?
nova cloak
i said X = H - T
n = H + T
X = 2H - n
and we can solve for X if n is given right?
sour shore
eh r u supposed to give a range or what
full whale
I think they want you to show the list of possible values of X instead
also, you would still need H to solve for X, no?
nova cloak
also, you would still need H to solve for X, no?
we can put different Hs
i think the range is
from -n if we put H = 0
and we can toss the coin n times right?
full whale
mhm
sour shore
YYep
sour shore
yh
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sour shore
i kind feel like theres smth missing tho
full whale
so what is your final solution set?
nova cloak
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✅ Original question: #help-28 message
nova cloak
what happened
full whale
i mean just to make sure you fully got it, unless you wanna finish the question off yourself
if so, then I apologize and you can reclose
nova cloak
so what is your final solution set?
i think it should be X in {-n, 2 - n, 4 - n, ...., n}
full whale
mm, well that works. I was thinking of concisely representing this information
nova cloak
can i also ask one more question?
full whale
but yes, absolutely correct
sour shore
oh yeah ntg missing thts ryt
nova cloak
nova cloak
we just should compute the complement of P{meu(N) = 0} right?
full whale
the hint implies so, yes
nova cloak
the hint implies so, yes
so its just 1 - 6/pi^2 or we should take lim
full whale
I have a feeling that's it but something also tells me that it's not
I'm gonna say that's it based on the hint
nova cloak
I have a feeling that's it but something also tells me that it's not
i asked someone before and they said we cant until the
events are indep
but here events are dependent
nova cloak
look
if we
make an event
A_p = {N is not divisible by p^2} for primes p in {1, 2, ..., 10^k}
then A_p is dependent to A_q
full whale
I suppose A_q here is the complement (N is divisible by p^2)?
nova cloak
I suppose A_q here is the complement (N is divisible by p^2)?
yes
full whale
then I don't think "dependent" is the right word to use here? like, obviously if an N fits A_p it won't fit A_q. I believe you meant mutually exclusive?
nova cloak
then I don't think "dependent" is the right word to use here? like, obviously if...
sorry i was eating, i meant to say its another prime not complement
A_q is an event that N is not divisible by q
they dependent
full whale
so you are saying that N not being divisible by p^2 is dependent on N not being divisible by q^2, where q is another prime. am I understanding you correctly?
nova cloak
so you are saying that N not being divisible by p^2 is dependent on N not being ...
yes
nova cloak
how so?
say that they are dependent under P_k, where P_k is the uniform distribution on {1, 2, ..., 10^k}
full whale
I'm still not following the logic of why they are dependent
but what does your answer key say is the answer, if you have it?
nova cloak
but what does your answer key say is the answer, if you have it?
ye the thing is it doesnt have answer key
full whale
fair enough
I stand by my initial answer, in any case
I don't see why this should be dependent on anything as k goes to inf
nova cloak
I don't see why this should be dependent on anything as k goes to inf
well as k goes to inf lim P_k(A_p) = prod(1 - 1/p^2)
$$\lim_{k \to \infty} P_k(A_p) = \prod_{p \text{ prime}} \left(1 - \frac{1}{p^2}\right)$$
glossy valveBOT
Viͥђaͣnͫ
full whale
I agree
nova cloak
$$\lim_{k \to \infty} P_k(A_p \cap A_q) = \left(1 - \frac{1}{p^2}\right)\left(1 - \frac{1}{q^2}\right)$$
glossy valveBOT
Viͥђaͣnͫ
nova cloak
but you may get second opinions if you wish
idk if u have any other perspective on this question?
full whale
I currently do not have any
nova cloak
I currently do not have any
shall i keep this open?
full whale
up to you
onyx glen
yesterday you told me you couldnt upload images through discord 
nova cloak
yesterday you told me you couldnt upload images through discord <:thonk:40657573...
yes i cant
nova cloak
yesterday you told me you couldnt upload images through discord <:thonk:40657573...
am using my mac rn cant on phone
onyx glen
im not saying you are a liar, i just thought it was strange to see this inconsistency is all
nova cloak
im not saying you are a liar, i just thought it was strange to see this inconsis...
my country is just messed up i cant upload photos through even my laptop if it doesnt have vpn
anyway if u have any sol for this question i would appreciate that
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@nova cloak Has your question been resolved?
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@nova cloak Has your question been resolved?
keen canopy
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keen canopy
why can't we just do this:
1=(pi)(X)^2
X=0.56419
4=3*(pi)(0.56419M)^2
M=1.154
I only get 2 if I do
4=(pi)(0.56419M)^2
M=1.9999
M=2
2 is the answer
but I don't get it why I get it if I exclude the 3
I want to avoid all those long workings shown by them in the explanation
hot herald
Why would you include 3?
hot herald
that factor increase is what gave you the new area of 4
pseudo cape
The wording is really weird, for them, an increase from 1 to 4 is 'an increase by a factor of 3'. But an increase from 1 to 2 (of the radius) is an 'increase by a factor of 2'
keen canopy
The wording is really weird, for them, an increase from 1 to 4 is 'an increase b...
yeah
pseudo cape
I'd guess the 3 was a typo and it meant to say 'factor of 4'
hot herald
Wording is fine
full whale
I don't see a 3 in the included solution
hot herald
increase by a factor of 3:
Initial + 3 * initial = 4 initial
pseudo cape
increase by a factor of 3:
Initial + **3 * initial** = 4 initial
Sure but then, if r_2 = 2r_1, why does it not say that the factor of increase for the radius is 1?
hot herald
Wording is potentially ambiguous,
but here the final area is provided
pseudo cape
Yep, really, it doesn't matter. What matters is we start with 1m^2 and end up with 4m^2
hot herald
so it doesn't really matter and you can completely ignore that part
pseudo cape
.That being said, if you want a quicker solution, you just notice that first we have [1 = \pi r_1^2] and afterwards [4 = \pi r_2^2,] that means $r_2$ needs to be double the value of $r_1$ so that we can get that factor of $4$
glossy valveBOT
Kepe
keen canopy
I see
pseudo cape
You can also say that you divide the last equation by the first to obtain [4 = \l(\frac{r_2}{r_1}\r)^2 \leadsto 2 = \frac{r_2}{r_1}]
glossy valveBOT
Kepe
pseudo cape
(you divide the left side of the bottom by the left side of the top and similarly for the RHS)
np
keen canopy
I see
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verbal sky
how do they do F = ma without any knowledge of the F3 force?
they dont even do F = ma, they just set F1 = F2, idk why
makes no sense to me
somber pike
What is the F3 force supposed to represent here?
verbal sky
What is the F3 force supposed to represent here?
the particle is being moved up the plane
so a force needs to pull it up
somber pike
But it's decelerating
verbal sky
But it's decelerating
yeah
so it's being pulled up, slower and slower
F3 still exists or else it would not be getting pulled up at all
somber pike
The only forces acting on the object are gravity, normal force and friction force
There's no indication that there should be an F3
verbal sky
There's no indication that there should be an F3
so how does the particle move up the plane
somber pike
The object started out with a given speed, and is now decelerating by that fraction of g
verbal sky
The object started out with a given speed, and is now decelerating by that fract...
okay this makes sense
how did u infer this from the quesiton?
somber pike
Well if there was another external force acting on the object, it would say that in the question
Like for example if it were a car, there could be a force F3 that represents the engine power or something
verbal sky
Like for example if it were a car, there could be a force F3 that represents the...
okay thanks
.close
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neon basin
small seal
yo quick question, can i add sins?
for example
sin(2) + sin(3) = sin(5)
idk it might be a dumb question because sin is a function wtv
small seal
aight
i'm trying to figure out how to write the trigonometric functions in terms of irrational numbers, could you help me?
like sin(20) for example
sonic stratus
Tbh figuring out the value of sin and cosine on its own its a bit of a hustle
Like, you can, but it requires some good geometry knowledge and being able to precisely draw angles.
small seal
hm well yeah i kinda gotta learn it so
sonic stratus
Supposing we are talking about 20 radians and not degrees
small seal
how do you know
sonic stratus
Mostly cause we always assume so unless told otherwise.
small seal
okay
sonic stratus
now, 20 / pi is about 6.3662
If you know how radians work, >we are basically finding how many revolutions we have
Since we have an even number from here, we can conclude that this angle forms the same as 0.3662 * pi
small seal
the unit is even
sonic stratus
yeah
small seal
so if it was 5.something it'd be another story?
sonic stratus
when you have an odd you keep it with a 1
ex: 1.3662 * pi
Thats because for rotations, 2pi = 0pi
small seal
yuh
sonic stratus
Since we have an even number from here, we can conclude that this angle forms th...
This is about 65.91 degrees
you use the
radian * 180 / pi formula for that
small seal
wait so for sin(20) we are assuming it is in radians and then converting it into degrees so it's sin(65.91)?
sonic stratus
wait so for sin(20) we are assuming it is in radians and then converting it into...
We are deriving the value of sin 20 through geometry
Like, unless you have access to some other value of some trig function, you cant really calculate it
small seal
wait so where did you get the 65.91 from i don't really get it
aight
i see
0.3662pi(180/pi)
small seal
so is there a reason why you don't just write 0.3662(180)
eh honestly it's not really relevant
so evaluating sin(20),
small seal
ok
sonic stratus
The length of the red line / the length of the radius is an approximation to the value of sin(20)
small seal
aight
sonic stratus
The other option is them giving you information on some other angle / trig function
or using a calculator.
small seal
so step one was converting radians to degrees
then graphing those degrees and taking the opp/hypotenuse
to be the evaluation
sonic stratus
yep.
small seal
much easier if it's already in degrees basically
sonic stratus
tbh, the values of sin(x) are defined mathematically like this.
Since this is where it comes from
sonic stratus
much easier if it's already in degrees basically
Tbh there are other mathematical ways to find the values
sonic stratus
yep, taylor series can allow you to find numerical approximations
gimme a sec
$\sin x = \sum_{n=0}^\infty (-1)^n\frac{x^{2n+1}}{(2n+1)!}$
glossy valveBOT
sonic stratus
but you can see how calculating the later terms of this can be quite tedious.
small seal
tbh i don't really know how the infinity sigma works so
sonic stratus
when you want an aproximation you chose some number
small seal
like 3? 😊
sonic stratus
for a good approximation you probably want to choose something bigger
Well, looking at it, 3 is close enough
sonic stratus
the thing about this approximation is that its "centered" near 0.
you can probably see that its technically a polynomial.
small seal
for a good approximation you probably want to choose something bigger
i mean
i just wanna convert it into some irrational number
so i can work with it
preferably not some weird ass decimal sequence
sonic stratus
oh, that
some have closed off values using pi
some dont.
youre generally better off using (sin20) as a coeficcient on its own
and then approximating at the end.
small seal
okay,
so what i'm working with here
is some sin(xpi/a multiple of 6) and some cos(xpi/a multiple of 6) and i am trying to evaluate them
sonic stratus
so, this:
$\left( cos\left( x\frac\pi{6k} \right), sin\left( x\frac\pi{6k} \right) \right)$
glossy valveBOT
sonic stratus
where k is some natural number
small seal
yep
sonic stratus
yeah, to get a closed off value youll have some problems as far as i know
May i ask why?
small seal
just an assignment
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@small seal Has your question been resolved?
mighty rose
Why is this not correct?
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Why is this not correct?
ancient zinc
What is the problem
mighty rose
find the area of the surface between $f(x)$, $g(x)$, and $y=0$
glossy valveBOT
Bakoles
ancient zinc
So I don’t think your integration bounds match the area you’ve drawn
mighty rose
i'm pretty sure it matches the area i drew with red
ancient zinc
Yeees mb actually
I think it gets confusing with the drawing. What you’ve subtracted is not the precise area under the graph f
mighty rose
why not?
ancient zinc
brisk obsidian
The setup looks correct. What makes you believe that it is wrong?
mighty rose
the thing with this graph is that the tangent "cuts" f(x), so both (-2, 2/3) and (2/3, 1) are kinda valid but idk which one to choose
mighty rose
The setup looks correct. What makes you believe that it is wrong?
asked chatgpt and gemini, both agree that i'm wrong
ancient zinc
Actually no I don’t see anything wrong
mighty rose
this yellow region is also bounded by the three curves
ancient zinc
Yeah it could be that region as well
mighty rose
but there can't be 2 solutions right?
brisk obsidian
Can you post a copy of the math problem you were asked to solve?
Did it say bounded above by f(x) and bounded below by g(x) and y=0?
brisk obsidian
That's leaving a lot to interpretation because Bakoles' bounded area also works as well.
ancient zinc
Yeah I think both answers are valid
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pine dove
basically on the answer key
of a similar problem
it looks like they just approxmiated the y values
like how do i approxmiate the y values without getting the estimate wrong
like that is bs
im so mad rn
bro i actually don't know how to do this
help
i know its just width x (area of all the rectangles )
but idk how they got all those y values so they could calculate the area
tender nacelle
They just looked at the graph
tender nacelle
Show your work, ideally a drawing of your rectangles too
The low estimate will take y-values on the left side of the rectanges (since f(x) is increasing, taking left side values will yield lower y values, hence low estimate). You split your domain (x = [0, 10]) up into 5 equal-width rectangles, and read the y values at their starts to get their heights
pine dove
oh i think i did it wrong ngl
is this how im supposed to do it?
for lower value estimate
its like
20 width
unreal wren
So your intervals should be?
unreal wren
Well, your leftmost interval should be [0, 20)
How about the other ones?
They must have a width of 20
pine dove
(20,40)?
pine dove
So this is wrong
if its just x values then it should be that
but intervals im thinking are x and y
idk any coordinates that aren't x and y
i haven't learned intervals so im guessing
unreal wren
if its just x values then it should be that
What do you mean by this
"Just x values"
pine dove
What do you mean by this
like the x values that contain any rectangle
unreal wren
These intervals I'm mentioning are also only x values
pine dove
yes
im not sure what intervals are
im guessing its the domain of rectangles
basically
unreal wren
im guessing its the domain of rectangles
Yes something like that
Well, then what is the interval for the next rectangle, just to make sure you get the concept
pine dove
(20,40)?
unreal wren
You can observe that as X gets larger f(x) also increases
pine dove
yea
unreal wren
That means, for the height of each of these rectangles we will take f(lower bound of the interval)
pine dove
oh
unreal wren
oh
Get it? 
Because the lower bound of the interval is the smallest value of X which is still in the interval
And it produces the smallest f(x) in the interval
pine dove
yea
unreal wren
yea
Now you can try to give the question a second try
And you can continue asking questions here if you get stuck
thick hedge
<@&268886789983436800>
spring vigil
let
f(x) = 1 / 2 ( e^x + e ^ -x)
find maclaurin series
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let
f(x) = 1 / 2 ( e^x + e ^ -x)
find maclaurin series
spring vigil
can someone help me go over this
slate glacier
maybe cosh(x)?
spring vigil
maybe cosh(x)?
i wanna do it the regular maclaurin way where u get derivatives
i dont know that hyperbolic sin stuff
slate glacier
That's the simplest way if you wanna do it through that, otherwise sorry but I'm not familiar with the traditional way, you're gonna have to wait
spring vigil
do u know the mclaurin for e^x
so i did the first 5 derivatives of e^x and then did the f(0) = 1 for all
so its all e^x and 1s
spring vigil
how about e^(-x)
wait i wanna understand if i did it right for the pattern
woeful pasture
yes
sonic stratus
yes, but theres a pretty basic intuition about e^-x using the taylor series of e^x
woeful pasture
this is for e^x?
spring vigil
yea
woeful pasture
and get this eventually
spring vigil
ok so i understand the e^x
sonic stratus
what do you think the series for e^-x will look like?
spring vigil
what do you think the series for e^-x will look like?
i would do the same
sonic stratus
do we agree that: $e^x = \sum_0^\infty \frac{x^n}{n!}$?
glossy valveBOT
sonic stratus
now, we can do this also for composite functions
glossy valveBOT
hearty plinth
is this magic languange
spring vigil
is this magic languange
looks like it
sonic stratus
now you should be able to find the taylor series of e^-x really quick without having to do a single calculation
sonic stratus
Look at this image and the one below it and tell me if you agree with the idea
spring vigil
i dont rly understand what ur trying to say tho
spring vigil
but from what u say
e^x (whatever x is)
we can plug it in
whatever x
to
x^n / n!
i see that
sonic stratus
The fact that the function e^x and that sum are equal means that if you plug x with a number, the results will be equal
spring vigil
The fact that the function e^x and that sum are equal means that if you plug x w...
wdym
sonic stratus
for example:
e^1 is obviously = e
mb
that means that $\sum_0^\infty \frac{1^n}{n!}$ is also $= e$
glossy valveBOT
sonic stratus
we plug the same number into the x of the e^x and the x in the infinite sum, you get the same result.
do you understand this idea?
sonic stratus
you can plug anything in, not only numbers
sonic stratus
this should make sense now
rough tundra
just be careful, that is (-x)^n/n! not just -x/n!
sonic stratus
we just plugged (sin x) as a composite function
this is why taylor and maclaurin series are so useful
spring vigil
just be careful, that is (-x)^n/n! not just -x/n!
thats what i have on my soln
typo
mb
so now
the q is still
1/2 (e^x + e^-x)
we got e^x and e^-x
sonic stratus
now we have:
$$e^x = \sum_0^\infty \frac{x^n}{n!}$$
$$e^{-x} = \sum_0^\infty \frac{(-x)^n}{n!}$$
glossy valveBOT
spring vigil
yes
sonic stratus
you probably remember the property of sums that if you have two sums adding together, and they have the same bounds, then you can add the general term.
spring vigil
spring vigil
you probably remember the property of sums that if you have two sums adding toge...
i do not
need an example im sure ive done it
sonic stratus
$\sum_0^3 n^2 + \sum_0^3 (n+3) = \sum_0^3 (n^2 + n + 3)$
glossy valveBOT
spring vigil
i just put a + sign
sonic stratus
sort of
spring vigil
would it be -2x^n?
x^n + x^n = 2x^n?
but the other one hsa a - so just put the - on
sonic stratus
$$e^x +e^{-x} = \sum_0^\infty \frac{x^n}{n!}+\frac{(-x)^n}{n!}$$
glossy valveBOT
sonic stratus
theres a thing about it, this sum can be simplified
spring vigil
sonic stratus
.
sonic stratus
there are two main cases, and thats why im trying to show you
again, try to calculate for n=0
spring vigil
2
sonic stratus
what about n=1?
spring vigil
1 + -x
spring vigil
x + -x
sonic stratus
and what is that equal to?
spring vigil
0
sonic stratus
try doing N=2 and N=3 next
spring vigil
for n = 2
would that be
x^2 + x^2 / 2!
sonic stratus
yep, which is?
Also, dont simplify the 2 with the /2!, just for the sake of visualizing the sequence
sonic stratus
2!
sonic stratus
you mean 3?
spring vigil
n = 3
spring vigil
so how is it relevant? how can we use it to make the summation
sonic stratus
so we have
N = 0: 2x^0 / 0!
N = 1: 0
N = 2: 2x^2 / 2!
N = 3: 0
N = 4: 2x^4 / 4!
and if you allow me, ill write something to N = 0
spring vigil
for n =4 btw
its
4x^4/ 4! right
but isnt it
4 x 3 x 2 x 1 then cancel 4 on 4x^4
the first 4
so we got 6 on denom
sonic stratus
so we have
`N = 0: 2x^0 / 0!`
`N = 1: 0`
`N = 2: 2x^2 / 2!`
`N = 3: 0`
`N = 4: 2...
now, lets erase all the 0s from the sum since they dont apport anything
sonic stratus
ill add a few more N's too
N = 0: 2x^0 / 0!
N = 2: 2x^2 / 2!
N = 4: 2x^4 / 4!
N = 6: 2x^6 / 6!
N = 8: 2x^8 / 8!
spring vigil
cant i juts do + ... after the n = 4
ok
so 2 stays the same
x^n changes
n! changes
2x^n / n!
sonic stratus
do you agree that we can take the factor 2 out?
since all of the terms of the sum have it or are either 0.
spring vigil
yes? but how again just so we r on the same page
u put the 2 on the left of the summation
then keep the x^n / n!
right
sonic stratus
remember that the problem is asking for that sum divided by two
sonic stratus
let
f(x) = 1 / 2 ( e^x + e ^ -x)
find maclaurin series
.
so the 2 we pulled out and the 1/2 cancel out
we have this
$$\frac{x^0}{0!}+\frac{x^2}{2!}+\frac{x^4}{4!}+\frac{x^6}{6!}+\frac{x^8}{8!}...$$
glossy valveBOT
sonic stratus
and this can be written as a new sum of its own from 0 to infinity
sonic stratus
yep, we are just a step away from that same solution
spring vigil
the e^x + e^-x is not the same
didnt we do
2x^n / n!
damn i missed the extra 2 on n
sonic stratus
he wrote on a few lines the same process we followed
sonic stratus
`N = 0: 2x^0 / 0!`
`N = 2: 2x^2 / 2!`
`N = 4: 2x^4 / 4!`
`N = 6: 2x^6 / 6!`
`N =...
.
for the first value, remember that 2x^0 / 0! is equal to 2
spring vigil
yea i missed addding the
2 on the exponent n and the n!
sonic stratus
so what will the final answer look like in sum notation?
sonic stratus
not quite
sonic stratus
.
spring vigil
im not seeing it
i gotta clock out from my job soon and go home ☠️
is there any way i can reserve this help-28 slot
ill come back to it
sonic stratus
you can just open it later
spring vigil
ok
spring vigil
but how is it not this/
?
welp gotta pause this
thank you for your help it was actually rly helpful, with what we discussed so far @sonic stratus
spring vigil
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scarlet dome
Can anyone solve that
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pine dove
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slate glacier
I mean if you're talking about calculus then yeah, the smaller the breadth of your rectangles, the more precise the answer
slate glacier
Okay, but the more the number of rectangles, the more precise would be the area under the curve
pine dove
pine dove
Okay, but the more the number of rectangles, the more precise would be the area ...
yea
idk how to calculate the precise y value
civic bay
if they've given you the function f(x) then you can, otherwise you eyeball
slate glacier
Wait you're doing this with calculus yeah?
civic bay
how is this optimization it looks like you're just computing a right riemann sum?
what was the original problem
pine dove
idk what a rie rieman sum is
(a) By reading values from the given graph of f, use five rectangles to find a lower estimate and an upper estimate for the area under the given graph of f from x = 0 to x = 100.
civic bay
yes so summing up the rectangle areas to get the area under a curve is called a riemann sum, in this case you'll notice that using a left sum would underestimate the area and using a right sum would overestimate
civic bay
you're correct in using the five rectangles like you drew, i guess you're forced to eyeball values
pine dove
ok yeah
i hate this
its so easy to get it wrong when i eyeball values
solved
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fierce roost
can someone check this answer?
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can someone check this answer?
fierce roost
i think its wrong but i have no idea why its wrong
shy stump
can you show your work?
slate violet
i think its wrong but i have no idea why its wrong
check your derivative of (cos x)^3