yeah sure
#help-28
sour shore
Yes
foggy vapor
since a and b are integers |a-b| must be an integer
what is the constraint on |a-b| to have it be an integer?
given this
sour shore
C would be 3?
foggy vapor
yeah
sour shore
Bcz we'd need the denominator to be 1
sour shore
Yes
sour shore
But we need the smallest possible
foggy vapor
what is |a-b| given c = 3?
glossy valveBOT
foggy vapor
what equations
spice raven
this
foggy vapor
we did that
foggy vapor
.
spice raven
sour shore
Oh ok i got it
sour shore
Everything was about the smallest possible thingy
foggy vapor
yeah
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foggy vapor
no worries, good luck!
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hollow oracle
I will go to the beach, unless it rains.
Formalize this using logical symbols.
I have:
B: I go to the beach.
R: It rains.
R → ~B
Now, it seems that in classical logic unless corresponds to a conditional as I wrote it.
However, I also feel that
~B → R,
given that it was said (with certainty) I will go, meaning that, in principle, nothing else besides raining would keep me from going to the beach.
hollow oracle
Hence, in this case, I see it as an equivalence.
The AI agrees with me: in classical logic it means a single conditional, but generally speaking one could interpret it as a biconditional.
I'd like to know what you guys think, though.
foggy vapor
"B unless R" is "If not R, then B" in my mind
fast peak
and there are reasons why we dont use it in math
zenith kernel
B unless R is R->(not B) to me though
hollow oracle
"B unless R" is "If not R, then B" in my mind
Well, I am arguing that both are correct. It just so happens that I thought the first assessment would be the clear one.
zenith kernel
Your hypothesis cannot be true anyway, there are other events stop you go to beach. Like the beach vanishes .
foggy vapor
Your hypothesis cannot be true anyway, there are other events stop you go to bea...
Naturally
hollow oracle
see also this
I think it depends a lot on the language (as it is supposed to).
In my sentence, I think raining feels necessary and sufficient for me not to go to the beach (even if in principle many other reasons could keep me from going to the beach).
fast peak
feels necessary
reason enough to realize that it is a somewhat pointless discussion
hollow oracle
In any case, it seems debatable, which means I may not be completely wrong. lol
fast peak
there cant be a nice result to this
hollow oracle
Ok. I guess I am happy with the lack of consensus, or the consensus that it depends.
.close
glad marten
Can somebody help me figure out how I got 2 and 4 a wrong the last pic is the answer key
foggy vapor
Can somebody help me figure out how I got 2 and 4 a wrong the last pic is the an...
,, x^2 -4x \0r\ne(x-2)(x+2)
glossy valveBOT
foggy vapor
What about 2 ?
You know how like
Uh
,, \lim_{x\to\pm\infty}\4{a_nx^n + \dots + a_1x^1 + a_0}{b_nx^m + \dots + b_1x^1 + b_0} = \4{a_n}{b_n} \tss{assuming} a_n, b_n \ne 0, , n=m
glad marten
I know if n=m then it’s a/b and if it’s n<m y=0
But I still don’t know how we get 3 since 7/1=7
glossy valveBOT
foggy vapor
But I still don’t know how we get 3 since 7/1=7
Yeah well thats why your solution is incorrect isnt it
Replace 7 with -3
Also your fraction doesnt have x = 7 as a vertical asymptote
glad marten
Ohh I mixed up the vertical and horizontal
So when x is going tj infinity it is end value aka horizontal ashmphentote
I cant spell
trim silo
here anyone persuing or completed btech
I want study partner if anyone interested then dm me
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desert monolith
can anyone explain to me how to do this problem like I have 0 clue
undone arch
i have a sol that uses 9 16's
wise tartan
yeah i got it with 5
desert monolith
is there any method or just trial and error ?
torn sand
you can write some code if you want
grave elm
yeah, code or trial and error
undone arch
you can write some code if you want
that's basically glorified trial and error
desert monolith
thanks
grave elm
by square rooting (which doesnt use any extra 16), you can make 16, 4, 2
so you can actually use these 3 numbers
oh
im stupid
that brings me to 5
yep, i have solution with 5 16's
desert monolith
this is a math olympiad question so, i need show work how do i do that ?
undone arch
i have 5 too
grave elm
this is a math olympiad question so, i need show work how do i do that ?
damn
well
you'd need to show that it's impossible to do it with 4
wise tartan
this is a math olympiad question so, i need show work how do i do that ?
is it like proof writing or multiple choices
undone arch
With 3 or fewer 16's, the largest value you can reach is 64, but you cannot subtract exactly 3 without introducing extra 16’s.
With 4 16's, you still cannot form both 64 and 3 simultaneously using allowed operations without running out of numbers.
Hence, 5 is the minimum.
grave elm
With 3 or fewer 16's, the largest value you can reach is 64, but you cannot subt...
how do u know itll be done using 64 and 3?
what if there was something else?
writing a rigorous proof for this will be pain
undone arch
16*sqrt(16) - (sqrt(16) - 16/16) = 64 - (4 - 1) = 64 - 3 = 61
grave elm
yeah, thats my sol too
undone arch
can you give us the source material? @desert monolith
grave elm
I'll assume that non-whole numbers aren't allowed.
We could define cost(n) to be the cost of some number to make
Then
cost(16) = cost(4) = cost(2) = 1 and no other number has cost 1
Now the following numbers, the results of +, -, / and sqrt of the above have cost 2:
32, 256, 1, 8, 64, 12, 14, 6,
and idk how to continue...
to make any odd number other than 1, we need at least 3 16s
nah, i dont believe there is a sufficiently nice proof of optimality
wise tartan
to make any odd number we need to use 1
grave elm
indeed
wise tartan
so we can look at the cost of 62 and 60
grave elm
(256-12)/4 is another way to make it btw
grave elm
With 3 or fewer 16's, the largest value you can reach is 64, but you cannot subt...
which contradicts this line of thought
though its just a multiple of 4
if there was a solution of cost 4, it must consist of cost 3 and cost 1, or of cost 2 and cost 2
by bruteforce, it cant be cost2 and cost2, because the only way to make odd is by including 1 or division. And there is nothing 1 less / 1 more than 61, nor is there any multiple of 61
so the only viable option would be cost 3 and cost 1
desert monolith
one thing what is "cost" ?
grave elm
one thing what is "cost" ?
number of 16's to make a given number
so e.g. cost(4) is 1
because you only need one 16 to make 4 (as sqrt(16))
or cost(12) is 2, because 16 - sqrt(16) = 12
grave elm
so the only viable option would be cost 3 and cost 1
this means that 61 would have to be written:
cost(3) + cost(1)
cost(3) - cost(1)
cost(3) * cost(1) (nope, cost1 is only 16, 4, 2; 61 isnt multiple of either of those)
cost(3) / cost(1)
sqrt(cost(3) * cost(1)) (but this is impossible i believe)
that means that we'd need to make one of:
61, 62, 60, 65, 57, 45, 77, 61, 122, 244, 976
with a cost of 3
which is still terrible to check
crimson igloo
crimson igloo
is this answer B
robust slate
,w x'=2x+y+2t-1, y'=x+2y+4e^t
glossy valveBOT
foggy vapor
the answer is indeed B
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grim monolith
I know that the answer is graph A but I don't know why. I know it should have 4 pedals because its over 0 and pi instead of 0 and 2pi
grim monolith
I also recognize that it's a rose
queen crater
Well, between 0 and pi/4 it's positive, that draws the first petal (right above the x axis)
Then you can just follow the curve
It goes through the origin, corresponding to going negative from pi/4 to 2pi/4
Then positive again, and then negative again
grim monolith
Well, between 0 and pi/4 it's positive, that draws the first petal (right above ...
how do you know that? did you plug something in for theta?
queen crater
Just imagine the graph of sin(4x)
grim monolith
Just imagine the graph of sin(4x)
k, I know it oscilates between y=-1 and y=1
queen crater
When is sin(x) positive? When does it cross the x-axis?
grim monolith
When is sin(x) positive? When does it cross the x-axis?
doesn't it cross the x axis for every npi/4?
queen crater
sin(4x), yes
grim monolith
so basically If I know where sin4x is positive and negative i'll be able to identify the correct graph right away right?
and that goes for any rose?
(depending on its equation)
sullen pumice
hi
austere cove
Hi @sullen pumice please ask your question
sullen pumice
So I have $W(2)-W(1)\sim N(0,1)$ and $W(1)\sim N(0,1)$. Kinda missed something on probability theory I'm sure but why is $W(2)|W(1)=1\sim N(1,1)$ again? I can't figure it out.
glossy valveBOT
bluepianist
sullen pumice
sorry if this sounds really basic im just bad at statistics
austere cove
W here is a distribution? Which one? Weibull?
sullen pumice
oh sorry W here is the standard Brownian motion one
austere cove
Sorry, I've never encountered this exact distribution, can you define the notation for me?
austere cove
oic, yes
hmm... I'm also not sure, my assumption of your notation is W(2) | W(1) is that we have W(2) is conditional on some specific realization of W(1) which means that we have a specific "bridge" we need to build to a specific value of W(1), but I don't see how that would force this to be similar to N(1, 1) specifically.
@sullen pumice if this is not the proper definition of W(2) | W(1) though please let me know
sullen pumice
yeah it is, conditional as usual
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austere cove
I'm thinking that perhaps there is a typo here, wouldn't W(2)|W(1) be sim W(1) = N(W(1), 1)?
@sullen pumice
where is this originally from?
austere cove
So I have $W(2)-W(1)\sim N(0,1)$ and $W(1)\sim N(0,1)$. Kinda missed something o...
I meant this
We have pretty easily that $W(2) = N(0, 2)$ from the definition of $W$, so it wouldn't make any sense at all for $W(2)|W(1) = N(1, 1)$, but it would make sense for $W(2) | W(1) = N(W(1), 1)$, for whatever realization of W(1).
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OmnipotentEntity
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coral canyon
apparently the second answer is -10 what am i overlooking or doing wrong
coral canyon
i thought we change it the second time to get another possible answer
main grail
okay, the thing is
you have to change the sign of the expression inside the absolute values when you want to get the 2 possible values of x
the reason why your "trick" worked on problem 36 is because
if you changed |x+1| = 42 into
-(x+1) = 42, this is equivalent to (x+1) = -42
but in general, your 2 cases are:
|x+1| = x+1, and
|x+1| = -(x+1)
coral canyon
i see what you mean, when i did -(x +3) instead i got the right answer, i guess im confused cuz i was taught to make what its equaled to negative for the second solution and doing it that way, all of my answers have been right so far
main grail
I believe that your teacher taught you folks this trick because it seems relatively easy to remember
but it breaks down for equations that have terms not under absolute values combined with terms w/ absolute values
such as number 39 and number 40
coral canyon
oh wait i actually see what i did wrong
i made 5 negative instead of adding two then making 7 negative
i missed a step
its supposed to be x + 3 = 7 and x + 3 = -7
grim skiff
its supposed to be x + 3 = 7 and x + 3 = -7
I was going to say that you needed the isolate the absolute value part first
Meaning that you needed to add 2 first
coral canyon
thank you guys i see what i did wrong now
main grail
no problem
also, have you ever heard of professor leonard on youtube?
coral canyon
i have not
main grail
He's a great maths lecturer that covers topics ranging from Cal 1 to Cal 3, and even has content on algebra
so if you're planning to take Calc, I recommend you check him out
crisp ermine
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crisp ermine
would 7a be 45 or 225 degrees
sour shore
how did u get tht.
sacred yarrow
would 7a be 45 or 225 degrees
There’s only one answer
sacred yarrow
But I’m still curious about how you got 225
sour shore
But I’m still curious about how you got 225
yh lol
vale remnant
i don't think my teacher ever covered this in class, she gave us a review packet for the final, I don't understand what any of this means lol. Could someone explain it to me? (AP Precalc)
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i don't think my teacher ever covered this in class, she gave us a review packet...
@vale remnant Has your question been resolved?
onyx glen
p composed with one of these guys gives the same function as p itself. which one?
dull stratus
a composition function has the output of a certain function, say g(x), become the input for an outside function like f(x)
it can be expressed as f o g (x) or f(g(x))
so the problem is telling you that p(x) = p(_(x)) and its asking you which function out of f, g, h, and k would fill in the blank to make the statement true
zealous gyro
^ one of the 4 options takes the place of the underscore in jeremiah's message if that's not clear
@vale remnant
vale remnant
<@967594027543326800>
i'll be honest, im so lost, my teacher barely covered compositions of functions using graphs, she mainly did problems like like find f(g(x)), f(x)=x^2+1, g(x)=4x-1 and transformations of graphs. (That's what im seeing on the worksheets she gave). Could you recommend me any videos for these type of problems?
dull stratus
if i were you id just like talk to claude or chatgpt or you can watch organic chemistry tutor and prof leonard are great as well
dull stratus
i'll be honest, im so lost, my teacher barely covered compositions of functions ...
essentially, in this problem you just have to be able to tell that the function f(x) = x from the linear line plotted on the graph. after you know that, its clear that having that as your inside composition function will not change the output of p(x) since your new input is literally the exact same as your "old" input
p(f(x)) = p(x) since f(x) = x
zealous gyro
i'll be honest, im so lost, my teacher barely covered compositions of functions ...
start by writing the functions of the options
and then subbing them as inputs to p(_(x)) replacing the underscore for the functions of the options
one of them will return you p(x)
that's your answer
as jeremiah stated above, that happens to be f(x) since the graph of f(x) = x and so p( f(x) ) = p( x ) so LHS = RHS, bam boom bop that's what you're after
@vale remnant
vale remnant
and then subbing them as inputs to p(_(x)) replacing the underscore for the func...
so like for example, if I plugged in 3, p(3)=3^3, p(_(x)) needs to equal 3^3 correct? f(3)=3, p(f(3)) = 3^3? is this why f is the correct graph?
dull stratus
right, however i would advise you to figure out the expression for each function and plug that into the composite function p instead since there can be problems where the functions turn into long polynomials and you don't want to guess and check values in those scenarios
so for example
the graph of h is h(x) = 1
therefore, p(h(x)) = p(1) which does NOT equal p(x)
vale remnant
therefore, p(h(x)) = p(1) which does NOT equal p(x)
ohh i see now
dull stratus
i gotu goat
faint moon
The data in dark blue is my recorded data. The light blue connected dots is me trying to model that data using the function on the left. How can I make it more accurate?
unreal wren
The data in dark blue is my recorded data. The light blue connected dots is me t...
Your data seems pretty linear, so you might want to just use mx + b
dusk tusk
Was gonna say this ^
faint moon
Your data seems pretty linear, so you might want to just use mx + b
Because I got my model from an exponential decay equation since thats what I'm trying to represent
dusk tusk
Thank you
Currently your equation is exponential so it is simply a matter of adjusting the base equation to fit a linear shape. I believe you can also use the natural log
faint moon
But my issue is that I needed to use the model I'm using because I'm representing exponential decay
So it feels kind of like I'd be disproving my original claims
dusk tusk
So it feels kind of like I'd be disproving my original claims
tis how experiments go
bright bronze
So it feels kind of like I'd be disproving my original claims
maybe your data is in logarithmic form, might be worth a look to see whether it is
faint moon
maybe your data is in logarithmic form, might be worth a look to see whether it ...
how could i check that?
bright bronze
from its source
faint moon
its primary data
bright bronze
where did you get the data? is it from your own experiments or did you use some online dataset
faint moon
my own
faint moon
but ik that its exponential decay
cuz each point is the mass of solid chocolate after a certain time of melting
dusk tusk
perhaps adjust your scale?
faint moon
how would that help
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long wind
I forgot how to do this. Does anyone know how to best set this up
vital shadow
I forgot how to do this. Does anyone know how to best set this up
are you asking how to solve the question or how to set it up in excel?
if you're asking how to solve, think of it as 8% = 25 hours, what would 1% be? if you have 1%, you can find 100%
long wind
are you asking how to solve the question or how to set it up in excel?
I think both, Cause I can't figure out on paper either
And my teacher heavy on excel and efficiency but this was from a unit so long ago I have forgotten everythinhg 😭
vital shadow
if you're asking how to solve, think of it as 8% = 25 hours, what would 1% be? i...
does this answer your question?
vital shadow
ye
long wind
Thank you@! Have a lovely day please
vital shadow
np
long wind
.solved
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supple lodge
hi guys i'm just asking can we write like the second picture even though both function are not defined at x=0?
supple lodge
if we cannot write it like this , how should we do it then
sonic stratus
yeah, technically it should be $\left(0, \frac\pi2\right]$
glossy valveBOT
sonic stratus
.close
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supple lodge
.close
sonic stratus
i already did myself dont worry, lol
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junior ruin
can you help me turn dB SPL to μPA? i've found this formula using mathgpt but i don't really know how to use it
onyx glen
!noai
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Please do not trust ChatGPT or similar AI tools for mathematical tasks, as they often generate output which "sounds correct" but has numerous factual or logical errors. Use of these AI tools to answer other people's help questions is strictly against server rules (see #rules).
onyx glen
just sayin'
junior ruin
alright, i didn't really trust it. do you have any idea how to solve it?
btw i think outrageous_armadillo_79763 is trying to scam me or something like that. can i report him somewhere?
onyx glen
btw i think outrageous_armadillo_79763 is trying to scam me or something like th...
modmail
junior ruin
modmail?
onyx glen
DM @midnight mauve
junior ruin
is that some kind of scam or virus? he wants to pay me to do data managment math
junior ruin
DM <@833145701701124116>
alright. thanks
i sended screenshots and the user ID to it.
can you help me with the math?
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random lichen
Hi so like
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Hi so like
random lichen
I was checking the model answers for this
I got the right calculation at the end but i got 77 rather than 103?
polar bison
Hi so like
1/2 ab sin c
random lichen
random lichen
1/2 ab sin c
Yes ik that part
I got to the end
But idk why when.i put it in the calc i got 77.3 rather than 103
is it like a calculator issue?
random lichen
is this gcse maths?
tes
polar bison
oh
well, i will suggest you look at the marking scheme points and compare your workings with it, since its highly likely you messed one of the calculator steps
random lichen
augh
Oh could it be because i rounded too early?
nvm same outcome
i basically did this
it says mark 1
hot herald
to avoid that, use cosine rule again
random lichen
But since its for angle i have to rearrange it right
hot herald
OR apply sine rule to get the smaller angle (where there won't be ambiguity)
and angle sum of a triangle to get the remaining angle
random lichen
OR apply sine rule to get the smaller angle (where there won't be ambiguity)
and...
so for safe measures i shld do that instead right
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void nova
Hi
void nova
.close
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stiff pier
leaden ermine
Hi
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wicked galleon
a²-b² = (a-b)(a+b)
Does the same formula apply for a²+b²?
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a²-b² = (a-b)(a+b)
Does the same formula apply for a²+b²?
twilit leaf
rapid laurel
honoured one asking the real questions
foggy vapor
I mean its a curious question so I respect it
leaden ermine
I mean its a curious question so I respect it
Is it or are u curious
foggy vapor
a²-b² = (a-b)(a+b)
Does the same formula apply for a²+b²?
Short answer: it doesnt exist as far as you're concerned at your stage
Long answer: there exists an analogous version where a^2 + b^2 = (a+ib)(a-ib) and i is the imaginary unit i = sqrt(-1)
wicked galleon
I see
Thank you guys
And hey, Lex
Thank you for helping me every time
You're a life savior
Huge respect
You are the best
foggy vapor
No worries 
wicked galleon
.close
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neon basin
Lex is great yeah
leaden ermine
Lex the best
foggy vapor
Compliment overload 
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verbal sky
for c
you have
2e^0.03t - 40e^-0.03t = 99
is there another way to solve this rather than multiplying everything by e^0.03t
cerulean cargo
aren’t u going to use a calculator eventually anyways
verbal sky
aren’t u going to use a calculator eventually anyways
yeah
but i remember a question like this and they factorised it somehow
i think
cerulean cargo
technically u can factor out e^-0.03t of the LHS
and then divide
since in general when factoring u factor out the thing with the smallest degree, it should make sense that u would factor out the terms with the negative exponent
knotty cloud
this is a true or false question, and I solved it to get false (actual answer is also false) but my method was incorrect
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knotty cloud
I did y = sqrt(r^2 - x^2) + 3
then y^2 = r^2 - x^2 + 9
y^2 + x^2 = r^2 + 9
so radius of the circle is r + 3
and so the area of the quarter circle in the question is (pi•(r+3)^2)/4 = (pi•(r^2 + 6r + 9)/4 but thats wrong
knotty cloud
so (y - 3)^2 + x^2 = r^2?
leaden ermine
i meant, to only consider y=sqrt(r^2-x^2) which is just the top circle part
the integral of 3 from 0 to r should be easy
integrating sqrt(r^2-x^2) from 0 to r, you get the quarter area of a circle with radius r
leaden ermine
so (y - 3)^2 + x^2 = r^2?
that would be correct btw
the problem though is, you are moving a circle with radius r by 3 units along the y-axis up
and when you integrate that from 0 to r, then thats not exactly a quarter
considering only sqrt(r^2-x^2) has the advantage that the circle is nicely centered at the origin
brisk minnow
Hi for this, The idea is to show, given I know the least upper bound property, can I prove the greatest lower bound property.
lime ether
you're trying to show that given that R has the LUB property then it has the GLB property?
brisk minnow
brisk minnow
you're trying to show that given that R has the LUB property then it has the GLB...
I guess yea
Lemme send the exercise
rare dock
lime ether
lmao
lime ether
Yes I agree
do you have to use the hint btw?
there are two approaches im familiar with
this one and another one i have in mind
which i think is a little more intuitive
brisk minnow
It’s not a compulsory exercise, I’ll be happy to understand either
lime ether
ok how about we try the one that i find a bit more intuitive
so our goal is to show that if E is any nonempty subset of R that is bounded below then there exists say l in R such that l is the greatest lower bound for E (im assuming you know how GLBs are defined)
brisk minnow
Yes
lime ether
consider the set B of lower bounds for E, to show that B has a supremum would be to show that E has a greatest lower bound no?
do you agree
brisk minnow
Hold on I’m trying to digest sory
lime ether
B is the set of lower bounds for E so the supremum, if it exists, would be the largest such lower bound for E and hence the infimum of E
i find this pretty intuitive
brisk minnow
Set of lower bounds
What do we mean by this
Ohh I see
Given a set E we can have it’s lower bounds, then the biggest of these lower bounds will be the infimum of E
Yes
That makes sense
lime ether
yea like for example if E = (0, 1) then B would be all the numbers less than or equal to 0
brisk minnow
Yes I agree
lime ether
Given a set E we can have it’s lower bounds, then the biggest of these lower bou...
yes exactly
so we want to show that if B is the set of lower bounds then sup B exists and sup B = inf E
brisk minnow
And and from the idea that we assume the least upper bound property exists then the sup of B exists
lime ether
so to show sup B exists what are the two things we must show?
lime ether
And and from the idea that we assume the least upper bound property exists then ...
yes but we have to show two things about B to argue sup B exists first
what are those two things
brisk minnow
Well that it’s an upper bound firstly and if there are any other upper bounds , then they are all greater than the supremum
lime ether
nono like
if E is any subset of R, what conditions must be satisfied for us to argue that sup E exists
brisk minnow
Oh it’s bounded
brisk minnow
Ah and non empty?
lime ether
well yea we assume that E is bounded below so B is nonempty
brisk minnow
Oh just from assumption
lime ether
E being bounded below translates to "there exists b in R such that b <= x for all x in E"
brisk minnow
ahh i see
lime ether
Oh just from assumption
yes
brisk minnow
Alright
lime ether
now how do we show B is bounded above
lime ether
E being bounded below translates to "there exists b in R such that b <= x for al...
hint hint
brisk minnow
Hm since it’s non empty, even if it’s only one set containing b itself, then b must be the upper bound?
lime ether
yea like for example if E = (0, 1) then B would be all the numbers less than or ...
for example -1 is in B here but its not an upper bound for B?
lime ether
you mean x in B
brisk minnow
Yes sorry, got myself confused
brisk minnow
But how does link to the idea of B being non empty
lime ether
and we know that B is the set of lower bounds for E so for all b in B and for all x in E, b <= x
lime ether
But how does link to the idea of B being non empty
well it linked to how we defined B
brisk minnow
and *we know* that B is the set of lower bounds for E so for all b in B and for ...
Ohh I see
lime ether
do you see how to argue B is bounded above
its bounded above by elements of E
E is bounded below by elements of B so B is bounded above by elements of E
brisk minnow
Yes because we know that b is set of all LOWER bounds so for all x in E, we must have that b<=x for all b in B
lime ether
yes
so any x in E is an upper bound for B
and E is nonempty
so such an upper bound exists
now we just showed that B is nonempty and bounded above
so by the LUB property we can say what about B
brisk minnow
Wait wdym
lime ether
🤔
whats the conclusion we can draw from B being a nonempty bounded above subset of R?
brisk minnow
It’s the GLB of E?
brisk minnow
Well it’s least upper bound is the glb of E?
lime ether
but what is it
lime ether
what does the least upper bound property say for any nonempty bounded above subset of R?
brisk minnow
what does the least upper bound property say for any nonempty bounded above subs...
Well all its elements are less than the least upper bound
lime ether
well it says the least upper bound exists
brisk minnow
😭 isn’t that a given?
brisk minnow
Well if we know it contains a lub then surely it exists no?
lime ether
our goal was to show that
- the set B of lower bounds for E has a supremum
- sup B = inf E
before we get to 2 we have to show that sup B even exists
to show it exists we simply had to show that B is a nonempty, bounded above subset of R
brisk minnow
Yea I understand ur intuition behind it, but to show it exists, did we not do so already,we see it non empyty and bounded above
lime ether
Well if we know it contains a lub then surely it exists no?
this is circular?
how do you know it contains a LUB?
lime ether
Yea I understand ur intuition behind it, but to show it exists, did we not do so...
yes but this is what i was trying to get you to say
brisk minnow
Oh
lime ether
sup B exists since B is nonempty and bounded above
brisk minnow
Ah alright fairs
lime ether
this is what the LUB property says
brisk minnow
Oh it’s the “there exists” part I was missing
lime ether
yea
brisk minnow
Oh I see
lime ether
now we need to show that sup B = inf E
i.e. sup B is a lower bound for E and for any lower bound l we have that l <= sup B
this should be very simple
give it a shot
brisk minnow
Well since E is bounded below by set of lower bounds B and we know B is is bounded above by supB and it exists, hence supB is the greatest such lower bound of E?
Unless I’m missing smth obvious
lime ether
yea i mean thats basically it but i think you should elaborate
like
why is sup B is a lower bound
brisk minnow
Well the set B is set of all lower bounds and that set is bounded above by supB
lime ether
these are things that you should be able to write down easily in pure mathematical terms
lime ether
Well the set B is set of all lower bounds and that set is bounded above by supB
yea sure
continue
brisk minnow
Hence E is bounded below by SupB
brisk minnow
Ahhh 😂
lime ether
"clearly..."
brisk minnow
Clearly E is bounded below?
lime ether
even though it might seem obvious or simple you should be able to formulate it yourself
lime ether
Clearly E is bounded below?
well this is given but why is it bounded below by sup B
write out what sup B being a lower bound means
then show its true
brisk minnow
Alright
Well being a lower bound of E means for all x in E, there exists a b in B such that x>=b.
Now
For supB
lime ether
nono that means E is bounded below
what does it mean for a particular number to be a lower bound
brisk minnow
x>=supBfor all x in E?
brisk minnow
Well we know supB is lub of B so any x in E will be >=supB
lime ether
yes nice
since every x in E is an upper bound for B you know sup B <= x for all x in E
so sup B is a lower bound for E
then showing that its the greatest such lower bound is immediate from the definition
brisk minnow
since every x in E is an upper bound for B you know sup B <= x for all x in E
Oh this is what I was missing, the idea of showing it’s bounded by supB
brisk minnow
Hidden in plain sight fr
lime ether
to show sup B = inf E you must show that sup B is a lower bound for E and that it is the greatest such lower bound
brisk minnow
to show sup B = inf E you must show that sup B is a lower bound for E and that i...
Ah yes
Fairs
lime ether
the second part is definitionally true essentially
brisk minnow
Yea that works
This definitely seems more intuitive than what’s to come from negating the sup to get inf
lime ether
yea i think intuition is completely lost when you negate the set
brisk minnow
Alr tysm, I’ll probably call it a day, I’ll write this proof down and then see if I can do the negation one tmrw
lime ether
like ok i remember when i first was learning this and was asked to conjecture what inf E would be in terms of sup E i actually did come up with inf E = -sup -E when visualizing it but i think this approach we did now is much better
lime ether
Alr tysm, I’ll probably call it a day, I’ll write this proof down and then see i...
good luck sir
brisk minnow
like ok i remember when i first was learning this and was asked to conjecture wh...
Yea I drew a set to visualise this
But I agree ur approach is much nicer
.close
full forumBOT
lime ether
yea i think i was visualizing (1, 2) goes to (-2, -1) then negating the sup
full forumBOT
Available help channel!
shy pilot
How do this, I forgor
full forumBOT
How do this, I forgor
spice knot
you can try sketching all of the conditions then combining them
for example you know that f(2) = f(4) = 0
how does that look on the graph
shy pilot
Just two dots/points of the function f(x), one on (2,0) and the other at (4,0)
But that’s all I know
shy pilot
The derivative of f(x)
spice knot
so f'(x) being positive would mean?
spice knot
yes its really common to write positive as > 0
shy pilot
so f'(x) being positive would mean?
f(x) is also positive?
spice knot
no
shy pilot
Oh 😞
spice knot
what does f'(x) actually mean for your function
shy pilot
That the derivative of f(x) is positive when x < 3?
spice knot
surely theres something to the derivative other than being called the derivative
shy pilot
Tangent to the function?
spice knot
f(3) is a number, right?
shy pilot
Right
shy pilot
Well f’(3) doesn’t exist for all I know
spice knot
thats a good catch
so f' may not exist
now for other values of x, we usually expect this to exist
lets take f'(0) for instance
we know f(0) is a number
is f(0) always 0?
shy pilot
by what im given, f'(x) > 0 for x < 3 and f'(x) < 0 for x > 3, i think no?
spice knot
I said f(0), not f'(0), but sure lets go with that
shy pilot
oh
spice knot
so you still gotta be careful with what youre reading
shy pilot
mb
spice knot
theres a few things that can be easily mixed up here
shy pilot
is f(0) always 0?
i dont know, its not given nor is f(x)
spice knot
yea
shy pilot
wait
spice knot
alr lets take this as an example of an f(x)
shy pilot
oh nvm
spice knot
can you see this
shy pilot
yes
spice knot
now think about if you zoom in on any portion of this
the more you zoom in, the more it looks like a line, right?
shy pilot
right
spice knot
now on this line,
the slope of that line is called the derivative
the derivative is also a number, its only tangentially related to the original f(x)
now with this,
lets say we call this point (a, b)
b = f(a), right?
shy pilot
yes
spice knot
nice
now the straight line is called the tangent line
its a line that is tangent
we know that if we zoom in, the function should look more and more like the tangent line, right?
shy pilot
yes yes
spice knot
so that menas the derivative of the function should match the tangent line's slope too, right
shy pilot
at that point tho?
spice knot
yep
everywhere else, the derivative of the function wouldnt match the tangent line
you can sort of tell because the function is only parallel with the line at (a,b )
shy pilot
i see that
spice knot
alr, now consider this
shy pilot
ew
spice knot
as you can see this line is not at all tangent to the function at all
we'll just use it as a reference
whats the slope of that horizontal line?
shy pilot
is that a sin(x) or cos(x) 🤔
spice knot
the function does not at all need to be named
this is a general example that should work for any differentiable function
shy pilot
whats the slope of that horizontal line?
0, or wherever on the y-axis it is
spice knot
youll need to be more certain than that
shy pilot
wait, is it 1?
shy pilot
then its 0
spice knot
do you remember the slope formula?
"because its not 1" is the shakiest reason Ive heard
do you remember the slope formula?
spice knot
you said the slope was 1
shy pilot
or the change in y over the change in x
spice knot
where was that from
shy pilot
you said the slope was 1
cause you said no when i answered 0 so i got confused 😭
shy pilot
isnt that just "no"?
spice knot
thats not a no
shy pilot
or at least saying that the answer is incorrect?
shy pilot
"are you sure?"
spice knot
0, or wherever on the y-axis it is
alr Im gonna point out what you said
is that 1 answer or 2
spice knot
it does not at all make sense why the slope would depend on the y-axis
it should just be the direction of the line
spice knot
so we got a horizontal line of slope 0 down there
now think about what we said before
can you see the two times the function is parallel to the line?
burnt karma
can you see the two times the function is parallel to the line?
yeah
shy pilot
can you see the two times the function is parallel to the line?
the crest and the trough?
spice knot
yep
burnt karma
spice knot
lol
now at these two points, what would the derivative of the function be?
@burnt karma no spoilers please
burnt karma
now at these two points, what would the derivative of the function be?
<@9811140...
10
spice knot
too engaged?
spice knot
https://tenor.com/view/black-gif-24454390
"wdym you still remember what happened? I changed the history books"
burnt karma
why are you answering questions directed at the OP anyway?
i just opened chat didn't see
shy pilot
i just opened chat didn't see
burnt karma
spice knot
can you see the two times the function is parallel to the line?
ah yes, with a question worded like this, its not at all one of those leading questions
@shy pilot now look at the areas in between
now we previously knew that the derivative isnt 0 in the other areas, because they wouldnt be parallel to a line of slope 0
consider what a positive/negative derivative would mean for zooming in on the function
would the red region have a positive or a negative derivative?
shy pilot
im assuming negative
spice knot
yep
burnt karma
im assuming negative
yeah 🙂
spice knot
you can imagine it slopes down, or it slopes negative
thats not exactly the right way to say it of course, but to be honest it works really well
delicate iron
does someone want to catch me up im curious about this one
i might be able to help
spice knot
you can imagine we're using a version of the slope that only uses one point
derivative you can imagine is slope at a point
thats what we're going to stick to for now
delicate iron
okay okay
spice knot
@shy pilot so going back to this, what do you think this would say about f(x) for any value of x less than 3
shy pilot
f(x) or f'(x)?
shy pilot
hmm
spice knot
now f(x) on the other hand is what we need to graph
a positive f'(x), see any way you can visually describe f(x) knowing that?
shy pilot
a positive f'(x), see any way you can visually describe f(x) knowing that?
i dont, know ☹️
shy pilot
right
spice knot
what does positive slope look like then
shy pilot
going up
spice knot
yea thats all
the function is increasing
f'(x) > 0 means the function is increasing
shy pilot
yes but thats for f'(x)
delicate iron
derivative means slope at a singular points so if you were to graph a functino f(x) then f'(x) would just look like the corresponding slope for every imput
spice knot
yes but thats for f'(x)
what are you talking about
f'(x) > 0 means f(x) is increasing
f(x), otherwise known as "the function"
spice knot
you dont need to use a special test to know this information
f'(x) > 0 on its own means that f(x) is increasing
because if you zoom in, the function slopes upwards
upwards things are increasing
shy pilot
so then f(x) is increasing when x < 3 and decreasing when x > 3?
spice knot
yea
shy pilot
but i dont know at what rate
spice knot
lets go through all the requirements first
we still have the two dots from earlier
now Im going to have / represent that the function is increasing
because itd be sloping upwards
and \ for decreasing
shy pilot
right
spice knot
shy pilot
wait
since f'(3) does not exist, does that mean there is a vertical asymptote at x = 3?
spice knot
doesnt have to
didnt you have a
numbered list
that shows all the ways it could not have a derivative?
shy pilot
yes
spice knot
couldve been any of the others
shy pilot
cusp
hole
vertical asymptote
corner
shy pilot
couldve been any of the others
oh, right
spice knot
oh thats what this is supposed to be
shy pilot
yes 😭
spice knot
a hole is the easiest way to get the derivative not to exist
since f(3) doesnt exist, neither would f'(3) or f''(3)
civic oracle
spice knot
we're gonna have a again be the (a, b) from before
shy pilot
ah
shy pilot
ohhhh
spice knot
but slope is always connected between two points
when those two points are picked,
the slope is $\frac{f(a+h)-f(a)}{(a+h)-a}=\frac{f(a+h)-f(a)}h$
glossy valveBOT
mtt
spice knot
the derivative then considers what slope we seem to get as we pick h that is closer to 0
from either side, Im gonna add
make sense?
shy pilot
yur
spice knot
but notice here it always pick (a, f(a)) as one of the points
now the derivative could easily have picked anything else other than (a, f(a))
it couldve done x = a - h or x = a + 1/2 h
but by choosing (a, f(a)), we need f(a) to be defined
so if we had a hole like this,
is it possible that the slope can be intuited or filled in? yes
but is f(a) always required to be defined right for the derivative? also yes
that includes examples like this too
the derivative isnt defined at a, even though we can see the nearby slopes are all 0
shy pilot
right 🤔
spice knot
you can see as a result that if you pick any other point other than a,
the slope isnt gonna be 0
shy pilot
mhm
spice knot
so all you gotta do to get f'(3) to not exist is to put a hole there
spice knot
some y-coordinates wont work
we know f(2) is at 0
so from there, consider where the hole might be
can it be above or below the y-axis?
shy pilot
can't it be anywhere on the y-axis so long as it's where x=3?
shy pilot
/ = increasing, \ = decreasing
spice knot
see any problems then?
shy pilot
oh mb
i was looking at it inccorectly
like completly
it has to be above otherwise if its below and it crosses (2,0) it would be a negative slope
spice knot
yep
so we expect the hole to be above the y-axis
other than that, the hole can be anywhere
shy pilot
and nothing else helps figure out where it would be?
spice knot
nope
theres one last condition here
it wants you to interpret f''(x) similar to how we interpreted f'(x) as slope (increasing/decreasing)
shy pilot
the second derivative is positive, so its increasing when x =/= 3?
spice knot
now youre mistaking f''(x) with f'(x)
f''(x) being positive would mean f'(x) is increasing instead
shy pilot
would it not have any?
shy pilot
what do you look for for concavity
this right?
spice knot
alr
now do you know why thats called concave?
or why that has to do with increasing/decreasing f'(x)?
shy pilot
to say when in changes direction?
spice knot
thats not english, try that again
spice knot
it doesnt need to have an inflection point
shy pilot
idk then
all i know about the second derivative is that its the derivative of the first derivative and that its used to determine concavity when curve sketching
spice knot
you seem to know an intuitive lot about this concavity thing
lets say for example we have this concave up function
shy pilot
right
spice knot
now pay attention to the derivative at each point
shy pilot
where its increasing the derivative is also increasing, where its decreasing the derivative is also decreasing?
spice knot
are you sure the derivative is decreasing here?
the function is decreasing, but what about the derivative?
theyre only tangentially related, that usually means "not that related, only barely"
spice knot
are you sure the derivative is decreasing here?
what is this a graph of?
f(x)
or f'(x)?
shy pilot
im going to assume f(x)?
stone matrix
isnt its slope of the derivative also decreasing?
Is -2 greater than -1? Same thing with the slope
spice knot
assume?
spice knot
lets say for example we have this concave up function
look at what I asked here
spice knot
yea there you go
the function is decreasing, but what about the derivative?
theyre only tangentially related, that usually means "not that related, only barely"
spice knot
does that mean f(x) is decreasing or f'(x) is decreasing?
shy pilot
in the case of f(x), which is a positive parabola, they are the same
spice knot
f(x) is not the same thing as f'(x)
spice knot
f'(x) is not the same thing as f''(x)
well it looks like you dont really know what the shape of f'(x) means
this is the graph of f(x)
read that again ok?
f(x)
now heres the graph of f'(x):
spice knot
did you forget what we were talking about?
what was the purpose of me showing you that?
shy pilot
you tell me 😭
spice knot
?????
shy pilot
to show concavity?
spice knot
ok lets try something funny
Im going to repeat myself
theres one last condition here
it has something to do with f''(x) being positive
already we know that f' > 0 means f is increasing
so we expect that f'' > 0 means f' is increasing too
shy pilot
which i said....
spice knot
:) you didnt
you said f' was decreasing
then when I zoomed in on it, you insisted
as i was saying
we expect that f'' > 0 means f' is increasing too
shy pilot
i did?
spice knot
please stop talking, Im giving background information
you can afford to hold off the questions until after I repeat myself
spice knot
a negative slope is a negative slope, which is decreasing no?
and also:
shy pilot
which is in the same sentacne at "the second derivative"
spice knot
isnt its slope of the derivative also decreasing?
additionally:
shy pilot
additionally:
those are in reference to the example you gave of f(x), not the f(x) of my problem
spice knot
which is in the same sentacne at "the second derivative"
well usually if you say "A is B, and it is C", then "it" commonly refers to A bro