SWR
#help-49
grand pondBOT
junior flower
dreamy lichen
Lol
dreamy lichen
this makes more sense prolly
twilit field
I mean It's just a family of points
grand pondBOT
MæthIsAlwaysRight
dreamy lichen
finally
grand pondBOT
(why am i here )= idk
dreamy lichen
yep
dreamy lichen
Tbh I can't describe it any better than through the set
twilit field
I mean 1 is co prime with all x, $x \in \R$
grand pondBOT
(why am i here )= idk
twilit field
so x=1 is an element of this set
After that it becomes more interesting
wait
oops
1 is co prime with all $\N$
grand pondBOT
(why am i here )= idk
past kestrel
👀
summer terrace
is this the set which generates the spirals!? i think ive seen this somewhere
dreamy lichen
is this the set which generates the spirals!? i think ive seen this somewhere
Not really
it's this
past kestrel
lol qr code
dreamy lichen
shouldve spoilered it
dreamy lichen
although it probably doesnt reveal anything
junior flower
i also thought qr code
twilit field
Then 2 is co prime with all odd naturals
past kestrel
yes
junior flower
hm what is the intended sketch for this?
dreamy lichen
I also wonder
dreamy lichen
although it probably doesnt reveal anything
imagine sketching this
twilit field
All even numbers are co prime with all odd numbers
dreamy lichen
All even numbers are co prime with all odd numbers
hmm
dreamy lichen
so 3 is coprime to 6?
twilit field
no
just realised
I feel like this requires some number theory
or Am I mistaken >
dreamy lichen
it doesnt really generate any nice set
dreamy lichen
although it probably doesnt reveal anything
if you were to plot it, it would look somewhat like this
twilit field
it doesnt really generate any nice set
I mean that does suggest a neat distribution for co-primes
junior flower
although it probably doesnt reveal anything
i have a fun exercise. how can you easily identify which numbers are prime from this picture?
dreamy lichen
interesting exercise
junior flower
(assuming things are labelled so you do not need to count indices to know where you are in the grid...)
twilit field
If It isn't too much of a hassel, could i have more problems like that too
junior flower
like what?
twilit field
**MæthIsAlwaysRight**
Like this
dreamy lichen
im too bad at labeling
junior flower
(assuming things are labelled so you do not need to count indices to know where ...
the (or an?) answer is ||to check if n is prime, look at cell (n,n) on the diagonal. if all the cells above and to the left are black, n is prime because that means it's relatively prime to all numbers below it||
hollow oyster
almost all green channel
dreamy lichen
If It isn't too much of a hassel, could i have more problems like that too
idk, it all looks pretty much the same
(x, sin(nx))
(x, x^z)
(x, n^x)
(x^2, x^4) might be more interesting
or let's say
(1/x, x^2)
twilit field
I mean $(x^2, x^4) will be.very similar in shape to y=x^2
except that it;s only defiend on $\W$
grand pondBOT
(why am i here )= idk
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twilit field
or let's say
(1/x, x^2)
Here, which set does $x$ belong to
grand pondBOT
(why am i here )= idk
twilit field
or let's say
(1/x, x^2)
Let $x=u$, so we have (1/u,u^2)$. Now let the abcissa $(1/u)$ be x, so $y=1/x^2$?
grand pondBOT
(why am i here )= idk
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twilit field
I'm trying to express this in rooster notation
midnight plankBOT
I'm trying to express this in rooster notation
grand pondBOT
(why am i here )= idk
hollow oyster
ig it works wai
strong lava
Or have I given too few sample elements
the rules aren't written in stone
this is ok
as an exercise, try to write it with only one ... rather than 2
hence enumerating that set
twilit field
the rules aren't written in stone
this is ok
as an exercise, try to write it wit...
What do you mean , try writing it with one?
strong lava
you're doing a doubly infinite list, it has no start and no end
Find an enumeration that has a start, and no end
twilit field
Hmm
dreamy lichen
you're doing a doubly infinite list, it has no start and no end
Find an enumerat...
hmm why is that necessary?
twilit field
${0 , \pm 2\pi, \pm 4 \pi \dots}$
grand pondBOT
(why am i here )= idk
strong lava
hmm why is that necessary?
it shows this set is countable
also it might be that roster notation requires it, idk the definition
hard umbra
dreamy lichen
it shows this set is countable
if that shows it's countable, then the other way of writing it also shows it's countable
dreamy lichen
it requires the same amount as writing it in the other way
strong lava
as you soon as you talk about writing something, it's that you have found an enumeration of it, so that's kinda a bad argument
dreamy lichen
also, why care about countability here?
hard umbra
strong lava
I said, as an extra exercise, write it in a different roster notation which is more usual
twilit field
**(why am i here )= idk**
Is this fine?
strong lava
ofc this is basic set theory at the end of the day you care about nothing here
dreamy lichen
both are fine imo
strong lava
also, why care about countability here?
and why not secretly prepare someone to that kind of exercise?
hard umbra
twilit field
**(why am i here )= idk**
So is this the way you were expecting?
strong lava
pretty much, just my expectations didn't have +-
strong lava
{0, 2pi, -2pi, ...}
grand pondBOT
(why am i here )= idk
midnight plankBOT
midnight plankBOT
Available help channel!
twilit field
If a function is odd/even about a certain point other than 0, would it be wrong to say that the function is odd/even about that point
midnight plankBOT
If a function is odd/even about a certain point other than 0, would it be wrong ...
last slate
i personally would avoid doing so
hard umbra
not that ive seen anybody actually say that but its not an incorrect concept
maybe i have seen people say it actually 
i wouldnt say its common usage though
timid tangle
Why not for odd
Because we can't prove it by contradiction
hard umbra
hollow oyster
Because we can't prove it by contradiction
huh?
last slate
Because we can't prove it by contradiction
what
last slate
i wouldnt say its common usage though
i think it would be morally acceptable if specified clearly
hard umbra
like say if a function satisfies
[ f(-x + a) = -f(x + a) ]
or something, i.e., ``odd around $a$'', then you could get away with saying that the function $f(x + a)$ is odd without introducing new terminology
grand pondBOT
twilit field
I mean take $y=(x-1)^3$ . That is odd about x=1, right?
grand pondBOT
(why am i here )= idk
timid tangle
last slate
i think it would be morally acceptable if specified clearly
it would be best to avoid ambiguity
hard umbra
I mean take $y=(x-1)^3$ . That is odd about x=1, right?
yeah thats reasonable to say
last slate
**(why am i here )= idk**
odd around the point (1, 0) perhaps
hollow oyster
no like how that comes into play in this case
last slate
how does this correlate to odd/even functions
timid tangle
There are properties for even and odd functions
hard umbra
**(why am i here )= idk**
but say we let $f(x) = (x - 1)^3$, you could get away with saying ``$f(x + 1)$ is odd'' to communicate the same idea without introducing new terminology
grand pondBOT
hard umbra
so 
twilit field
Nah, I was just wondering if this was standard in maths as it can be useful in integral calculus
hard umbra
i dont think its commonly used, but like
im sure people would understand what you mean
if you do say something like that
hard umbra
or you could just be more vague and say like "due to the symmetry of f around a"
last slate
or you could just be more vague and say like "due to the symmetry of f around a"
that sounds like a terrible idea
midnight plankBOT
midnight plankBOT
Available help channel!
twilit field
Deriving the equation $y=mx+c$ from $ax+by+c=0$
midnight plankBOT
Deriving the equation $y=mx+c$ from $ax+by+c=0$
grand pondBOT
(why am i here )= idk
modern sapphire
c/b but ok...
twilit field
my bad
grand pondBOT
(why am i here )= idk
dreamy lichen
this works, as long as b != 0
grand pondBOT
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twilit field
so let that be $\zeta$
grand pondBOT
(why am i here )= idk
twilit field
so we have $y= \frac{-a}{b}x+ \zeta$
dreamy lichen
x is missing there
grand pondBOT
(why am i here )= idk
midnight plankBOT
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inner lance
i have problem. I have 4 numbers 0,2,10,42. thats a series. what is a pattern here? can anyone help me?
midnight plankBOT
i have problem. I have 4 numbers 0,2,10,42. thats a series. what is a pattern ...
merry pewter
can try oeis
visual tiger
i have problem. I have 4 numbers 0,2,10,42. thats a series. what is a pattern ...
have you tried looking at differences?
inner lance
i think that arythmetical-geomtric series
sudden yacht
its diffrent
Nope, it's 4
sudden yacht
but a 2 to x power works in my question
Indeed
inner lance
what?
merry pewter
but a 2 to x power works in my question
what's your question
inner lance
Nope, it's 4
i dont understand it
does it mean does dividing other numbers in this series?
merry pewter
2,8,32 = 2, 4*2, 4^2 2
sudden yacht
i dont understand it
2, 8, 32 is a geometric sequence
Therefore the next number on that sequence would be ...
inner lance
2, 8, 32 is a geometric sequence
Therefore the next number on that sequence woul...
i didnt know how to solve it
thx @sudden yacht
Guys
I have second question
or nvm
midnight plankBOT
@inner lance Has your question been resolved?
wide marsh
can someone check my work? the question is to find the area of the inner loop of this limacon
midnight plankBOT
can someone check my work? the question is to find the area of the inner loop of...
No need to ask “Can I ask…?” or “Does anyone know about…?”—it’s faster for everyone if you just ask your question! See https://dontasktoask.com/
fathom knoll
ok
wide marsh
i did ask my question lmao
fathom knoll
you said to check your work? where is it?
feral sedge
to whoever sent that factoid just let the guy post the image
wide marsh
feral sedge
the answer is right
wide marsh
how do i find the interval which the curve is traced once
midnight plankBOT
how do i find the interval which the curve is traced once
wide marsh
wouuld it just be the interval of cos(theta) aka 2pi multiplied by 4 since theta is divided by 4
so 0 to 8 pi
wait nvm the interval for cos theta is pi lol
so is the answer just 4 pi
eager summit
Well not so fast. 4pi is the period of the curve, not the length of the curve over 1 period.
wide marsh
yeah i figured that was completely wrong lol
eager summit
So you correctly identified the interval
How do you compute the length of a curve over some interval?
wide marsh
just the arc length formula right
eager summit
Bingo
wide marsh
wait are you sure the curve is traced once from 0 to 4 pi though
i would use desmos to check but they only support up to cos^2
cause i think my logic i used is wrong
eg the period of cos^2 x is different from cosx
eager summit
If you’re checking on Desmos, try writing (cos(theta/4))^4. Desmos is picky* about exponential notation with trig functions
eager summit
Now that you’ve confirmed the interval you found is correct, can you set up the integral for the arc length?
wide marsh
do you just integrate from 4pi to 0 and then inside the integrand is sqrt (r^2 + (dr/dtheta)^2)
eager summit
Well I would integrate from 0 to 4pi, but yeah exactly
I think if you integrate in the other direction you’ll get the negative of the length?
wide marsh
Well I would integrate from 0 to 4pi, but yeah exactly
wdym
like i meant 4pi=b and 0=a
or is that wrong
eager summit
Okay what you’ve just now said is right.
Colloquially when someone says “integrate from p to q” they use p as the lower bound (bottom of the integral) and q the upper.
eager summit
No biggy, you had the right idea all along 🙂
wide marsh
do i just argue what i said earlier where the period of r=costheta is pi, so the period of theta/4 would be 4 times that
eager summit
Hmm let me think of a more concrete argument…
eager summit
So from the graph it’s clear that it starts and ends at the polar coordinate (1,0) (same as the Cartesian coordinate) right?
wide marsh
yeah
eager summit
So then you simply need to ask when does your function once again output the radius 1. Then, the argument that you’ve made suffices
wide marsh
right
ok that makes sense
tysm
now i just have to deal with this annoying integral yay
thanks for the help!
.close
midnight plankBOT
eager summit
thanks for the help!
And that will differ depending on what polar function you’re trying to integrate over, but the type of argument will be similar. But you’ll notice with flower petal shaped functions such as r=cos(n*theta), where n>1 and n is an integer, that we go through the starting point multiple times.
But nice work! Do a reopen if you struggle with that integral
wide marsh
ok! hope you dont mind if i end up pinging you haha
eager summit
Please do 🙂
eager summit
ah the integral is not bad at all
Usually once you’ve gotten to this point in calculus, the difficulty is in setting up the integral. Practice problems are usually rigged to result in an integral with a lot of simplification 🙂
So much so that if you see a really difficult to compute integral it may be a red flag that you set it up wrong.
wide marsh
Usually once you’ve gotten to this point in calculus, the difficulty is in setti...
omg wait i have a rpoblem
i thought i was integrating it fine but now im subbing in the bounds into the antiderivative and im getting 0
midnight plankBOT
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wide marsh
is it just 0 cause half the curve is above the x axis and half is below lol
midnight plankBOT
is it just 0 cause half the curve is above the x axis and half is below lol
wide marsh
maybe i should integrate from 2pi to 0 instead and multiply the antiderivative by 2?
eager summit
No the arclength integral is not a signed length like the standard integral is a signed area.
So far your work looks great, still looking for a small mistake of some kind
wide marsh
No the arclength integral is not a signed length like the standard integral is a...
oh i see
altohugh technically if i integrated from 0 to 2 pi and multiplied by 2 the answer would still be correct due to symmrtry right
so far i have 4/3
welp wolfram says its 16/3 lol
wide marsh
oh
ugh youre right lol
we covered an example in lecture where it was sqrt of sin^2 so it became abs of cosx
and then we had to adjust the bounds of integration to make cosx positive
wow truly an annoying question lol
eager summit
Im wrong about the signedness of the arclength integral. Exactly because of that square root, it is in essence a signed length. So your analysis of why you got 0 is absolutely correct, it is the symmetry.
But yeah doing the doubling over half the bounds trick is absolutely the way to get around this.
Sorry for saying the wrong thing there. Had to fact check myself 😅
wide marsh
no worries lol
im not too good at these questions myself
but i dont think you can just double the answer from 0 to 2 pi because part of that curve is negative within that integral
so i guess i just have to find where cosx is positive again aka -pi/3 to pi/3
ugh no it would be where cos^3(theta/4) is positive
so -2pi to 2pi
i really dont understand how this q is only worth 5 marks
eager summit
So from 0 to pi it’s above the x axis. From pi to 2pi it’s below, from 2pi to 3pi it’s above (looping around) and from 3pi to 4pi it’s below.
wide marsh
So from 0 to pi it’s above the x axis. From pi to 2pi it’s below, from 2pi to 3p...
oh youre looking at the polar grpah right
i was looking at the cartesian graph cause my prof did that in that one example
ill see if doing it this way can get the right answer cause id rather not have two separate integrals lol
hmm i got 8/3 lol
i am kinda losing my sanity
midnight plankBOT
@wide marsh Has your question been resolved?
wide marsh
<@&286206848099549185>
midnight plankBOT
@wide marsh Has your question been resolved?
shadow schooner
i think your pain and suffering come from this
ah sorry someone already pointed it out
wide marsh
yes i know it’s abs value
but i have to change the boundaries to only represent where it’s positive right
i tried doing 2pi and -2pi which is what desmos shows and multiplying the anti derivative by 2 but i’m still off by a factor of 2
the answer by wolfram is 16/3
shadow schooner
yeah i would try doing something like that
shadow schooner
the answer by wolfram is 16/3
my guess is that there must be an error of some sort. i’ll try the integral and see
wide marsh
my guess is that there must be an error of some sort. i’ll try the integral and ...
yeah, i’m assuming i made some mistake because wolfram and now desmos are saying 16/3
midnight plankBOT
wide marsh
.reopen
midnight plankBOT
✅
eager summit
hmm i got 8/3 lol
Sorry; I’m in and out tonight. Let me see your work that led to this. You’re off by a factor of 2 right?
wide marsh
Sorry; I’m in and out tonight. Let me see your work that led to this. You’re off...
no worries and yeah, i got 8/3
shadow schooner
no worries and yeah, i got 8/3
wide marsh
omg wowwww
i put the question into wolfram and did show steps and the first step was to sub u for theta/4 which is what you did, i just didn’t know why
is there a reason why you did that other than to make the integral easier to work with?
shadow schooner
is there a reason why you did that other than to make the integral easier to wor...
yeah i didn’t want to think about “complicated” angles
when substituting the values after using the ftc
wide marsh
yeah i didn’t want to think about “complicated” angles
so basically if you were to not do that step and instead integrate from 2pi to 0 with the original theta/4 would you get the same answer
so you multiply the integral by 2 instead of 4
instead of 8 i mean
shadow schooner
so basically if you were to not do that step and instead integrate from 2pi to 0...
you would need to multiply the integral by 2
to get the 16/3 i think
wide marsh
oh i see
hm i have another question
why did it not work when integrating from 2pi to -2pi? and instead you have to integrate only half of that and multiply the answer by 2
cause the interval from 2 pi to -2pi is positive so why didn’t it work?
shadow schooner
cause the interval from 2 pi to -2pi is positive so why didn’t it work?
nah without noticing the function was even f(x)=f(-x) you would get the same result but yet again you would have to deal with more terms when evaluating the integral with the ftc.
you forgor a 2 imo, a classic blunder
happens to everyone
wide marsh
you forgor a 2 imo, a classic blunder
i did? where? i checked my work so many times and thought it was fine
shadow schooner
i did? where? i checked my work so many times and thought it was fine
idk, you didn’t send when you did it from -2pi to 2pi
shadow schooner
here i multiplied by 2
you don’t need the extra 2
you forgot a times 4 in the integral of cos(theta/4). i did the same error when retracing your steps
wide marsh
you don’t need the extra 2
oh wait you’re right 2pi -(-2pi) is 4 pi so the period is the same
i just freaking integrated it wrong
jesus christ i despise math
ok thanks lol second set of eyes is very helpful
shadow schooner
oh wait you’re right 2pi -(-2pi) is 4 pi so the period is the same
that’s the reason my u sub is good in that case
me no think when integrating the cos
midnight plankBOT
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sudden harbor
midnight plankBOT
sudden harbor
Do my answers look correct
Are you applying the exponents after solving the function
slender walrus
function inverse, as opposed to multiplicative inverse
slender walrus
apply definition of the function inverse
g^-1(2) is asking for what value when plugged into g gives 2
slender walrus
yes
slender walrus
and then solve for y
sudden harbor
slender walrus
yes
sudden harbor
so is that my answer
slender walrus
h^-1(x) = that,
but yes
midnight plankBOT
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viral dagger
ABCD are points on a circle, AC and BD are perpendicular and E is the intersection point, AE=8 BE=4 CE=6, find the area of the circle
midnight plankBOT
ABCD are points on a circle, AC and BD are perpendicular and E is the intersecti...
viral dagger
using power point, DE=12 but how am i supposed to find the area of the circle after it
i cant tell if im stupid or im missing something
neat canyon
using power point, DE=12 but how am i supposed to find the area of the circle af...
is DE + BE diameter?
viral dagger
er this is an olympiad problem so no calculators
summer terrace
idk about that, but check it out, interesting
viral dagger
it is
summer terrace
theres also a theorem called The intersecting chords theorem, i just learned about it
you could calculate DE using it
its not the main thing here, but anyway just saying
viral dagger
using power point, DE=12 but how am i supposed to find the area of the circle af...
yes thats what i used
merry pewter
yes thats what i used
it is a fact that the R = abc/4A, where R is the circumradius, A is the area, and a,b,c are the lengths of the sides of a triangle
so construct a triangle that has the circle as its circumcircle
viral dagger
oh
viral dagger
.close thanks
fallen crescent
Hello
midnight plankBOT
@fallen crescent Has your question been resolved?
last slate
How to take intersection of set of two combinations?
wdym?
midnight plankBOT
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drifting root
can i get a fact check on this is the right way to set this out
midnight plankBOT
can i get a fact check on this is the right way to set this out
@drifting root Has your question been resolved?
drifting root
.close
midnight plankBOT
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midnight plankBOT
red helm
Ohh
midnight plankBOT
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last slate
last slate
<@&286206848099549185>
pulsar tree
This is your question?
last slate
This is your question?
yes
midnight plankBOT
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rugged yew
Idk
midnight plankBOT
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placid dragon
how to approach this?
midnight plankBOT
how to approach this?
summer ravine
hi giga chad
placid dragon
hi giga chad
sup
worldly coral
guys can you please help me simplfiy this Factorization of algebraic expression using algebraic identities,i have been trying but could not solve questions, any simply tricks and tips to help me understand better
placid dragon
guys can you please help me simplfiy this Factorization of algebraic expression ...
huh?
summer ravine
,w ((10^9/(10^9+1))+(8/9))/2
grand pondBOT
worldly coral
huh?
i meant i unable to factorize the algebraic expression using algebraic identities
summer ravine
,w ((10^9/(10^9+1))+(8/9))/2 - 0.1865028428383742
placid dragon
i meant i unable to factorize the algebraic expression using algebraic identitie...
show the expressions
placid dragon
na just tell me how to solve them
bro what? Id even know the expressions
how can i solve them
placid dragon
333667/
500500
coorrect
midnight plankBOT
@placid dragon Has your question been resolved?
dreamy lichen
Is there any way to quickly find out whether this is convex / concave for x > 0, without having to compute the derivatives
midnight plankBOT
Is there any way to quickly find out whether this is convex / concave for x > 0,...
dreamy lichen
,w d/dx x(sqrt(1+1/x)-1)
grand pondBOT
shell wigeon
Maybe not "quick" enough, but you can rationalize that
dreamy lichen
$\sqrt{x^{2}+x}-x$
grand pondBOT
MæthIsAlwaysRight
shell wigeon
Yeah
dreamy lichen
**MæthIsAlwaysRight**
Okay, how would I know whether this is convex or concave?
grand pondBOT
Nel
dreamy lichen
okay
shell wigeon
The left factor is decreasing, similar to 1/sqrt(x)
dreamy lichen
right
shell wigeon
The right factor, x, is a straight line
I'm not sure how to explain but if you multiply a straight line with a decreasing function, that makes it concave
dreamy lichen
1/x^2 * x doesnt seem to work
shell wigeon
Yes it's not just a decreasing function, that's what I'm having trouble articulating
Its reciprocal needs to grow slower than x
I believe that's the condition
dreamy lichen
reciprocal of that actually grows faster
i might be looking for something what doesnt even exist
i should probably just stop being lazy and actually do the 2nd derivative thing
sounds like a plan ig
ty
.close
midnight plankBOT
shell wigeon
reciprocal of that actually grows faster
(up to a constant factor) it grows as fast as x
But yeah I'm just wrong
It just needs to be concave
concave * straight = concave
Or not
@dreamy lichen I found a way; reopen and ping if interested
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dreamy lichen
<@951190447185874994> I found a way; reopen and ping if interested
@shell wigeon
midnight plankBOT
<@525582731569266698>
shell wigeon
I may be wrong, I've been trying to look for a proof, but it has to do with function composition
1/x is convex (on x>0)
and non-increasing
(and so is 1+1/x)
sqrt(x) is concave
I think the composition sqrt(1+1/x) is therefore convex
dreamy lichen
right
shell wigeon
And then if that's true, sqrt(1+1/x)+1 is also convex, and the reciprocal is concave
dreamy lichen
I think the composition sqrt(1+1/x) is therefore convex
Does this surely work btw?
is there some theorem that says that
shell wigeon
That's what I've been trying to find
zealous schooner
$(f\circ g)''=(f'\circ g\cdot g')'=f''\circ g\cdot (g')^2+f'\circ g\cdot g''$
grand pondBOT
kheerii
shell wigeon
I found something similar: if f is concave and g is convex and non-increasing, gºf is convex
dreamy lichen
just found this
f(x) = h(g(x))
so our situation is
g is convex and non-increasing
h is concave (and increasing)
oh
shell wigeon
No that doesn't work
dreamy lichen
No that doesn't work <:pandahmm:858834883760750602>
have you found a counterexample
it would be nice if it worked
shell wigeon
No but our situation isn't in the list
grand pondBOT
MæthIsAlwaysRight
zealous schooner
and sqrt(1+1/x) + 1 also does, I presume
we can consider sqrt x + 1 which has f'' < 0 and 1+1/x which has f'' > 0
dreamy lichen
oh
zealous schooner
something like this probably works
dreamy lichen
yeah
dreamy lichen
**MæthIsAlwaysRight**
this might also work
by this rule
sqrt(x / (x+1)) should be concave
So $\sqrt{1+\frac{1}{x}}$ must be concave too
shell wigeon
Where does that x/(x+1) come from
grand pondBOT
MæthIsAlwaysRight
dreamy lichen
Where does that x/(x+1) come from
I made it up
dreamy lichen
**MæthIsAlwaysRight**
wait this is bullshit actually
shell wigeon
Lmao
dreamy lichen
it's 1/(1+1/x)
but probably not so helpful
I would only need rule that says 1/concave is concave
1/x is convex and non-increasing
uh
shell wigeon
I would only need rule that says 1/concave is concave
Not true unfortunately
1/sqrt(x) is convex
dreamy lichen
yeah
But why
oh
f is convex
then why the fuck is sqrt(1+1/x) concave
oh it's not
okay my approach works then
it's stupid but works
x/(x+1) is concave
thererefore sqrt(x/(x+1)) is concave too
thererfore 1/sqrt(x/(x+1)) is convex
shell wigeon
x/(x+1) is concave
It's not
dreamy lichen
Oh I mean x/(x+1)
dreamy lichen
thererfore 1/sqrt(x/(x+1)) is convex
and therefore sqrt(1 + 1/x) is convex too
Those rules are actually hella helpful
I need to remember them
shell wigeon
and therefore sqrt(1 + 1/x) is convex too
Oh yeah that works
dreamy lichen
oh
shell wigeon
None of the rules show that's concave
dreamy lichen
well
sqrt(1+1/x) + 1 must be convex
so it's convex-convex situation
oh, you're right
shell wigeon
1/x convex non-increasing
I also tried showing that sqrt(x^2+x) is concave, but it's pretty much the same as computing the second derivative
dreamy lichen
$x\left(\sqrt{1+\frac{1}{x}}-1\right)$
grand pondBOT
MæthIsAlwaysRight
dreamy lichen
it suffices to show that this is concave
shell wigeon
Yes but that's a product, not a composition
dreamy lichen
unfortunately
shell wigeon
I think the "fastest" way is just to show sqrt(x^2+x) is concave
The function is sqrt(x^2+x) - x, and -x doesn't change the convexity
dreamy lichen
The function is sqrt(x^2+x) - x, and -x doesn't change the convexity
right
dreamy lichen
I think the "fastest" way is just to show sqrt(x^2+x) is concave
that sounds doable
it's composition of concave-non-decreasing and convex
rules dont say anything about that
shell wigeon
Probably because it's the same problem
dreamy lichen
yeah
dreamy lichen
I think the "fastest" way is just to show sqrt(x^2+x) is concave
how would that be shown?
through 2nd derivative?
shell wigeon
I wonder if there's some argument like the sqrt of any quadratic is straight or concave
grand pondBOT
dreamy lichen
this has to be le 0
it seems like it depends on number of roots / determinant
Which makes a lot of sense when you think about it
shell wigeon
Yeah, single root => straight, two roots => concave
dreamy lichen
Yeah, single root => straight, two roots => concave
no root => convex
dreamy lichen
the term down there is either undefined, or positive
so it depends on sign of 4c - b^2
determinant
shell wigeon
discriminant
dreamy lichen
yeah, that
shell wigeon
But yeah, c=0 in our case
shell wigeon
Concave
dreamy lichen
oh, yeah
shell wigeon
dreamy lichen
enough math for today, lol
shell wigeon
Well, that's not exactly quick
dreamy lichen
Well, that's not exactly quick
it's not, for the first time
but now im able to determine convexity / concavity of quite a lot simpler functions
shell wigeon
I suppose if you have more sqrts of quadratics, sure
dreamy lichen
it could probably be generalized
shell wigeon
Maybe
dreamy lichen
thanks for your help btw
shell wigeon
dreamy lichen
.close
midnight plankBOT
midnight plankBOT
Available help channel!
rancid whale
How do I solve this system of equations for I1, I2 and I3?
midnight plankBOT
How do I solve this system of equations for I1, I2 and I3?
@rancid whale Has your question been resolved?
rancid whale
<@&286206848099549185>
cloud hull
is this for solving circuit
rancid whale
yes
cloud hull
what is that symbol inside parenthesis alongside R
cloud hull
I assume currents will be solved in terms of E, R, r right
rancid whale
yes
rancid whale
r u familier with linear algebra btw
i dont know what to say. if you re going to solve using matrices i didnt learn them yet in my country.
like the latest thing i learnt are logarithms
cloud hull
alr what about solving system of 3 equation in general with substitution and elimination
rancid whale
i can do that
cloud hull
start by substituting ("packaging") variables to a1, a2, a3 to make math cleaner (otherwise it'll be seriously ugly)
Deal with equation 2 & 3 to remove I2 and get equation with only I1 & I3
Then deal with equation 1 and 2 and remove I2 (by multiplying equation 1 with R2, then adding the 2 equation)
Then solve for I1, I3. Use that info to solve I2
I'm also working throguh the problem in meantime, but that is the approach (pretty much Elimination method)
midnight plankBOT
@rancid whale Has your question been resolved?
exotic bay
i opted for option b looking at the intersections at x axis but apparently the answer is a and it is because of exponents which i dont understand
midnight plankBOT
i opted for option b looking at the intersections at x axis but apparently the a...
celest oriole
its because no of roots
grim vector
Are u sure its a) ?
exotic bay
Are u sure its a) ?
yeah
exotic bay
its because no of roots
uh how would i figure out the number of roots
celest oriole
its because no of roots
they started from -y axis and ended in - y axis meaning the exponent is odd
celest oriole
uh how would i figure out the number of roots
power of x
exotic bay
i could see 3 so b should also work right?
celest oriole
b would have worked if the graph continued in + y direction
exotic bay
oh wait nevermind i mean i put a and its wrong and the answer is b
celest oriole
so basically the starting and ending points will end on the opp side if the no of roots is even
and same side if its odd
lmao sorry then its the opp
celest oriole
yeah i think its the opposite?
yea its the opp mb
celest oriole
but i dont get the logic behind it?
idk that sorry. guess thats how graphs work
grim vector
So its b) ? Right ?
exotic bay
So its b) ? Right ?
yes
grim vector
but i dont get the logic behind it?
Look at limits
celest oriole
by exponents they mean power of x ig
grim vector
Ok, so since it goes down both sides, the sum must be even
celest oriole
Ok, so since it goes down both sides, the sum must be even
yea exactly
exotic bay
Ok, so since it goes down both sides, the sum must be even
yeah i got that
and if one goes up and the other goes down then the sum must be odd?
grim vector
Take x^3 as example
exotic bay
alright i think i got it
grim vector
And x^2 too
boreal wadi
idk if im missing a step or sum and i tried other weird ways to solve and still nowhere close
midnight plankBOT
idk if im missing a step or sum and i tried other weird ways to solve and still ...
grim vector
Add the 4 sides since its asked for perimeters and put x=2 at the end to calculate it.
I dont see where that log come from
Its not require here
boreal wadi
Add the 4 sides since its asked for perimeters and put x=2 at the end to calcula...
ik im slow but uhh wym?
i did add the four sides
the 30,31,20 right
grim vector
No the four sides, with the x terms too
Like 4x^2 + 8x
Added to the 3x^2 -5x + 20
Added to 7x+30
Added to 31
Which gives 7x^2 +10x + 81
Let X = 2
And calculate it
boreal wadi
oh yea i got confused cuz the other one had exponent
but wait lemme retry rq with 10x as 20
ohh.
so i got steps right i js messed up with the 10
ok thanks
.close
midnight plankBOT
grim vector
yw
midnight plankBOT
Available help channel!
twilit field
This matrix would be in REF, right?
midnight plankBOT
This matrix would be in REF, right?
twilit field
Using the defn, I feel it is
prime hornet
can you show the definition?
twilit field
Can be fixed by multiplying across by 2/7
twilit field
Nope
twilit field
A lot more by the looks of it
prime hornet
it shouldn't be much more
twilit field
This is my current matrix
twilit field
I feel like I first divide $R_3$ by $-10$, and then $C_6 \leftrightarrow C_7$
grand pondBOT
(why am i here )= idk
twilit field
Interchange columns 6 and 7
twilit field
Okay. Noted
prime hornet
Gaussian/Gauss-Jordan Elimination and row reduction are the same thing btw
I always call it row reduction though haha
prime hornet
I like the division of R_3 by -10
right, what should you do after this?
twilit field
I have to make the 6 in the last row 0, and the 2 \to 1
prime hornet
yes
twilit field
I'd probably start by dividing by 3
sharp coral
Gaussian/Gauss-Jordan Elimination and row reduction are the same thing btw
technically the difference is that gaussian elimination gets to ref, whereas gauss-jordan row reduces upwards and therefore gets to rref
but in practice they're interchangeable
prime hornet
technically the difference is that gaussian elimination gets to ref, whereas gau...
I know, but I have literally never seen anybody distinguish the two in practice
and I prefer to call it row reduction in any case
twilit field
Here I feel like $R_1+R_4$ would be a very bad idea
grand pondBOT
(why am i here )= idk
prime hornet
yes, it probably would be one
use R3 to clear R4 instead
that is the point of this algorithm
twilit field
Oh right
subtle blaze
I hate having to deal with fractions
prime hornet
mhm
subtle blaze
I’ll just leave everything in integers
And use LCM to match up the first non zeroes
And do the division at the end
subtle blaze
This is my current matrix
So here is multiply R3 by 3 and R4 by 5
Then I can easily add them together with the 30 at the front
twilit field
I think I'm pretty close now
twilit field
So here is multiply R3 by 3 and R4 by 5
I'll keep that in mind in the future
Reducing this to RREF is going to be fun
I'd probably start by clearing out the -1
any suggestions on how to further reduce this?
sharp coral
the gauss-jordan algorithm would suggest you start with the rightmost pivot, eliminate upwards, and repeat going left
midnight plankBOT
@twilit field Has your question been resolved?
twilit field
get to RREF?
Yup!
Sorry for replying late, was getting breakfast
I think coming back to such a huge matrix after becoming more comfortable with 3 by 3 matrices may be a good idea
sorry
can I close this for now
.close
midnight plankBOT
midnight plankBOT
Available help channel!
uncut stratus
this is confusing
midnight plankBOT
this is confusing
lavish venture
key word
"must"
does it have to be true that h is increasing for 5<x<10
can you think of an instance where that might not be true
uncut stratus
the function f is always decrasing from 0 < x <10
is that why
i mean negative ROC
lavish venture
mhm yea same thing
since f is always decreasing
h doesnt necessarily have to increase from 5 to 10 just because g does
if f is decreasing more than g is increasing
then h will be decreasing
uncut stratus
so since f is decreasing more
h would be equal to A
h is decreasing on 0 < x <10
lavish venture
but you don’t know that
they never told you if it was decreasing more
it could be
might not be though
uncut stratus
oh
uncut stratus
i think its C
lavish venture
no
neither increasing or decreasing means its AROC is zero
which means f is decreasing just as much as g is increasing
uncut stratus
oh wait i read D wrong
lavish venture
which again assumes you know that
uncut stratus
from 0 -5 it is fs decreaing right
lavish venture
yes
uncut stratus
and then the second segment cant be determinded
lavish venture
because both f and g are decreasing
uncut stratus
so it would be D
lavish venture
mhm
lavish venture
you’re welcome
lavish venture
so for november there is a point of inflection where the function changes from concave down to concave up meaning the rate at which the slopes changed went from decreasing to increasing thus november would be a relative minimum for the slope
does this make sense
if you’re having trouble visualizing concave up vs down
think of drawing a tangent line to the curve
if the line is above the curve then the function is concave down
if the line is below the curve the function is concave up
uncut stratus
it makes sense to me
i understadn the concave up vs down thing
but in this case
would novvember be correct
im not sure
lavish venture
well read what i wrote
if it’s a relative minimum for the rate of change
does that mean it’s the highest or potentially the lowest
uncut stratus
august?
lavish venture
no
that’s the highest temperature
not the greatest rate of change of temperature
think of the slopes
not the y value
uncut stratus
feb?
lavish venture
nope
uncut stratus
im dumb wtf
uncut stratus
at august?
lavish venture
what does the slope change from
uncut stratus
alright
lavish venture
the function changes from increasing to decreasing yes?
increases to a maximum
then decreases
uncut stratus
yes
lavish venture
so we can apply this same line of reasoning for f’
where f’ is the slope function
or "derivative"
think of like rate of change of f
so
f’ or the rate of change
will be a maximum when it increases to a maximum then decreases
yes?
what do you know about concavity
what does concave up/down mean about a function
in terms of the functions slope
lavish venture
right
so with that being said
that means the slope or rate of change reaches a maximum when it changes from concave up to concave down yes?
because it increased to a maximum
then decreased
uncut stratus
yes
lavish venture
so which month is that
lavish venture
mhm
uncut stratus
to the whole thing
lavish venture
yes
uncut stratus
thank you
uncut stratus
pre calc
ap
i dont get how this is wrong
10 is the greatest value
so how is it wrong
lavish venture
look at the interval length
lavish venture
right so the amount it increases is just the interval length times the rate of change
over that interval
midnight plankBOT
@uncut stratus Has your question been resolved?
twilit field
why would one want to use guassian elimination here?
midnight plankBOT
why would one want to use guassian elimination here?
twilit field
I mean I can instead write $x_1-2x_2+2x_3-2 = a(x_1+x_2+x_3-1)+b(2x_1-x_2+3x_3-3)$
grand pondBOT
(why am i here )= idk
twilit field
and then compare both sides, right?
lavish venture
there are multiple ways of solving systems of equations
🤷🏼♂️
you could do whatever you’d like but they’re just telling you to solve it with gaussian elimination
twilit field
No, I mean how would the guassian algorithm help us express one equation, in terms of the other two
Like I can solve for x_i using the guassoan algorithm
Just a hint, please
lavish venture
so you can say the first+third equation is the second equation
twilit field
I mean that's true just by observation
Okay, so I keep track of the operations I perform
lavish venture
with gaussian elimination all you’re doing is creating an upper triangular matrix
twilit field
and from that I arrive at a conclusion ?