#help-39
versed mica
🔥
versed mica
he gave us the wrong fucking exam.
how does this happen
pearl pondBOT
@turbid lark Has your question been resolved?
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void estuary
pearl pondBOT
void estuary
how do I solve this?
hidden hawk
how do I solve this?
You dont need to do all the algebra
mental moon
I don't get it
me neither
hidden hawk
I don't get it
So you know FTC?
void estuary
no
compact ridge
,w (d/dx (1+cos^2 x)/(cos^7 x)) - (sin2x + 2tan x)/(cos^6 x + 6 cos^2 x + 4)
hidden hawk
Fundemental theorem of calc
void estuary
Fundemental theorem of calc
yea
jolly parrotBOT
hidden hawk
So you just need to differentiate the answers to see if you get a desired form
But not in totality
void estuary
💀
hidden hawk
Ykwim
void estuary
cannot I solve the integration somehow?
hidden hawk
You could but it looks like a pain
Easier to eliminate possible answers
You know its not (4) because derivative of ln(u) gives du/u will give you wrong coefficients
pearl pondBOT
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void estuary
$1^2 + 5^2 + 9^2 +..... + (77)^2 = ?$
pearl pondBOT
$1^2 + 5^2 + 9^2 +..... + (77)^2 = ?$
jolly parrotBOT
Prathmesh
void estuary
how to calculate this?
toxic lichen
rhe numbers being squared, are they in AP with step size 4?
$\sum_{k=1}^{19} (4k-3)^2$
jolly parrotBOT
Ann
void estuary
rhe numbers being squared, are they in AP with step size 4?
yes
jolly parrotBOT
Ann
void estuary
yea
toxic lichen
should be doable then
unless you're about to tell me this method is explicitly forbidden
void estuary
nahh nahh it's completely fine 😭
toxic lichen
do you see how to continue
void estuary
do you see how to continue
yea I can solve ahead
pearl pondBOT
If you are done with this channel, please mark your problem as solved by typing .close
vernal bear
would you tell me how did you figure this out?
Common difference
pearl pondBOT
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shadow reef
pearl pondBOT
vernal bear
Which one?
shadow reef
The bottom two
vernal bear
What did you try?
shadow reef
(0,14) and 0
vernal bear
I mean how did you get that?
shadow reef
I found the derivative, got critical points, and plugged in test points
vernal bear
Idk if it's -13 or -4/3
Substitute
shadow reef
(0,14) and (0,14] are both incorrect
shadow reef
No im on my last attempt
toxic lichen
ah crap
pearl pondBOT
@shadow reef Has your question been resolved?
stoic imp
why in a partial order relation instead of asking for symmetric property we ask for the antisymmetry property (is almost the same compared to an equivalence relation with that distinction)
tropic saddle
we ask for transitivity
and antisymmetry
in an order you dont want to have two elements a,b such that a is less than b and b is less than a
that makes no sense
if you can compare a and b, you only want one of a<b, a=b or a>b
antisymmetry excludes that we can have a<b and a>b
stoic imp
what about total order how does it differ from partial order
cursive wraith
A total order is a partial order where you must have EXACTLY one of a < b, a = b or a > b
Vs partial order when you need to have at most one of these
Meaning in a partial order, it's possible some elements don't interact at all
Imagine the order defined by "a is a direct ancestor of b"
Well, two siblings are not put in relation by this order
So it's not total
cursive wraith
Yeah
Btw, depending on people, partial order can either just mean "order" or "order that isn't total"
So depending on who you ask, total orders are in particular partial orders
stoic imp
well in my intro to math class they only mention order, so i am assuming is partial and not total, would need to check the definition from the reader pdf
i was just trying to find out what totality was because it was mentioned by a helper to me the other day
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fathom scaffold
$f(x) = x^3 e^x$, find the 2025th derivative of f(x) evaluated at 0
jolly parrotBOT
Bakoles
fathom scaffold
Any ideas? Supposedly an easy problem for my math teacher
runic shoal
have you learnt leibniz rule?
fathom scaffold
Is it in calc 1?
runic shoal
yes
fathom scaffold
Nah i didn't
runic shoal
(at least in my calc 1)
fathom scaffold
What's the rule like
inland ivy
I would personally use Maclaurin series, if you've learned those yet
fathom scaffold
I would personally use Maclaurin series, if you've learned those yet
Will study them sometime next week
But this problem was given to us at the start of the semester
rotund ferry
But this problem was given to us at the start of the semester
What were you supposed to use to solve it 
fathom scaffold
According to her, it's a highschool math olympiad problem
vernal bear
According to her, it's a highschool math olympiad problem
eh tell her to teach olympiad technique
rotund ferry
Have you tried calculating the 1st, 2nd and 3rd derivative?
fathom scaffold
Yes but it got messy
rotund ferry
Yes but it got messy
Can you show a picture
dusky widget
Simply derivate 2025 times and you should get an answer easily after plugging x = 0.
||joke||
nocturne grail
**Bakoles**
maybe it repeats somewhere
derive it 20 times and see if it has symmetry 🤩
(semi-joke)
fathom scaffold
Mmm lemme try that
toxic lichen
consider this + the fact that e^x is its own derivative + the fact that x^3 has easy to calculate derivatives
warm dagger
According to her, it's a highschool math olympiad problem
Nah she tripping
Olympiad problems don’t have calculus
rotund ferry
Nah she tripping
Olympiad problems don’t have calculus
They do actually
runic shoal
consider this + the fact that e^x is its own derivative + the fact that x^3 has ...
isnt this leibniz rule?
toxic lichen
isnt this leibniz rule?
sure
runic shoal
isnt this leibniz rule?
op said they didnt learn leibniz rule yet
rotund ferry
At what level?
Generally any high school Olympiad tbh
toxic lichen
does op know product rule
warm dagger
I mean like IMO
iron basin
op said they didnt learn leibniz rule yet
all roads lead to rome
rotund ferry
I mean like IMO
Not just IMO
fathom scaffold
does op know product rule
Ye
rotund ferry
Ye
I've tried calculating the derivatives and there's a pattern
Are you sure it ended up messy?
Can you send a photo of your work @fathom scaffold ?
I gtg tho sorry
warm dagger
Not just IMO
IMO allows the use of calculus and complex analysis but none of the problems require it
fathom scaffold
Are you sure it ended up messy?
I mean it's just a third degree polynomial no?
But it's a lot of addition
rotund ferry
IMO allows the use of calculus and complex analysis but none of the problems req...
This isn't true..
warm dagger
I googled it :/
rotund ferry
I mean it's just a third degree polynomial no?
Yeah
rotund ferry
I googled it :/
Oh, so google is never wrong now huh?
warm dagger
No
It’s just that I was pretty sure this was true beforehand
I think this is just in the us then
rotund ferry
I mean it's just a third degree polynomial no?
It's a third degree polynomial, but we want to calculate it at x=0 so only care about the terms without x
Everything should simplify
iron basin
Everything should simplify
honestly its kind of gruelling without leibnitz directly or indirectly
you get a combinatorial answer
rotund ferry
What you should do now, is calculate a couple of derivatives, then figure out the term with no factor of x
Like, what is the coefficient of that term
vernal bear
What you should do now, is calculate a couple of derivatives, then figure out th...
crazy
vernal bear
imean the options we left with crazy
fathom scaffold
Are you calling me crazy or my solution crazy..?
Wait
I see something
The last term is of the form n!/(n-3)! i think
Where n is the degree of the derivative
Is that true?
fathom scaffold
The last term is of the form n!/(n-3)! i think
I mean first i found that it looks like n(n-1)(n-2)
rotund ferry
Lemme check
fathom scaffold
Since for the 5th derivative, the term is 60, and 60 is just 5!/2!
fathom scaffold
Nice
fathom scaffold
So the answer should be 2025x2024x2023?
calm wing
So the answer should be 2025x2024x2023?
yes
fathom scaffold
Okay thank you so much
calm wing
good job 
pearl pondBOT
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unique viper
Let M be a square matrix of sidelength nm.
If M, when realised as an n by n block matrix of m by m matrices, is diagonal with entries A1, A2 ...
is it true that det(M) = det(A1)det(A2) ... det(An)?
unique viper
Not quite finished typing out this question
Now I have
I would assume the proof would be inductive, with M = A1 the base case
Any ideas as to how to formulate this?
unique viper
Fair enough. The case for n=2 is stated as fact on Wikipedia
unique viper
Oh interesting. This same proof would be generalisable for the inductive step (I mentioned above) I believe
fading nexus
Why is (X+2)(x-3) not next to the two?
pearl pondBOT
Why is (X+2)(x-3) not next to the two?
glass meadow
Why would it be?
fading nexus
Because we are making the denominator that and we must
Multiply the top by the same thing
glass meadow
The denominator is already that
burnt copper
💀
burnt copper
Understandable
glass meadow
That's not the same question though
fading nexus
I mean it’s the answer key for lesson 3.5 hw
for question 5
It must of been a thpo
Typo
glass meadow
Why is (X+2)(x-3) not next to the two?
But this is question 3
glass meadow
Go sleep then
vernal bear
I mean I’m in school
Sleep
pearl pondBOT
@fading nexus Has your question been resolved?
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analog badger
i need help working through this
pearl pondBOT
i need help working through this
analog badger
so i know that ill have some sort of generating functon, ill raise it to the power of 5 (for 5 children), and then find the coeffecient of the term raised to the 12th (for 12 toys). im not sure how to get the function in the first place tho given the limit of 3 toys
but before that, just to confirm, if there wasnt such constraint, the normal function would just be 1+x+x^2+... = 1/(1-x) (getmetric series) right? and then id do (1/(1-x))^5
then after expanding, find coeff at x^12
but since that isnt the case, how do i approach this
vernal bear
but before that, just to confirm, if there wasnt such constraint, the normal fun...
cap it at x^3
analog badger
so change it to 3 children?
or wait
3 toys
but that answers giving 3 toys to 5 children, here theres 12 toys meaning more than 1 kid can get 3 toys
so like 4 kids can get 3 toys
but the way u said it, at most 1 kid can get 3 toys
right?
vernal bear
so like 4 kids can get 3 toys
hmm
feral sedge
i dont see how
vernal bear
but the way u said it, at most 1 kid can get 3 toys
1 kid can get atmost 3
feral sedge
each one will have an x^3 term right
vernal bear
each one will have an x^3 term right
5 *3 = 15
analog badger
1 kid can get atmost 3
yes, thats not what the problem is asking. as per the rules in the problem, 4 kids can get at most 3, ur solution states that only 1 kid can get at most 3
pls explain if im mistaken in my understanding
4 kids get at most 3 since theres 12 toys, so we if we max out 4 kids, that 4 kids with 3 toys and last kid gets no toys
vernal bear
so till, 2 toys each we can use function and thenneed to make cases
vernal bear
4 kids get at most 3 since theres 12 toys, so we if we max out 4 kids, that 4 ki...
question says atmost 3 so a kid can get no toys
analog badger
well yeah, but capping at x^3 excludes too many cases
such as the case that more than 1 kid gets 3
im sure there is a solution where it can be solved with a single function
feral sedge
each 1/(1-x) represents one child, right
it starts with this
analog badger
it starts with this
yes, if i am not mistaken, then that is 1 child without respect to the constriant
analog badger
coeffecint of that term yes
feral sedge
the coefficients represent the number of "ways" to give one child n toys, which is ofc 1
then when i multiply
analog badger
for one child yeah
feral sedge
(1+x+x^2+x^3+x^4+...)(1+x+x^2+x^3+x^4+...)(1+x+x^2+x^3+x^4+...)(1+x+x^2+x^3+x^4+...)(1+x+x^2+x^3+x^4+...)
i look at the x^12 term
analog badger
but when you expand, its different
feral sedge
it gets +1 in it's coefficient for each way to choose 5 powers here adding up to 12
that is, each way to choose how many ways to assign the toys
ok, so if we don't want to consider giving a child more than 3 toys, what can we do
analog badger
if we dont consider the constrinat, then the solution is simply just finding the coeff of x^12 in the expanded form of (1/(1-x))^5
feral sedge
yes
feral sedge
(1+x+x^2+x^3+x^4+...)(1+x+x^2+x^3+x^4+...)(1+x+x^2+x^3+x^4+...)(1+x+x^2+x^3+x^4+...
but look at this form
analog badger
ok?
feral sedge
if we dont want to have the option to give children more than 3 toys, how can we change the form
analog badger
just stop at x^3 instead of x^12
feral sedge
well it wasn't really stopped at x^12 before
analog badger
oh yea nvm, find coeff of x^3
feral sedge
we could if we wanted to but then it would be different to 1/(1-x)
feral sedge
oh yea nvm, find coeff of x^3
the coefficient we're finding should be the total number of toys
analog badger
so still coeff at x^12
analog badger
yeah
vernal bear
such as the case that more than 1 kid gets 3
how does it miss the individual sums are for respective children, we then just raise it to number of children
feral sedge
so we have to remove the higher powers from the children
analog badger
so we have to remove the higher powers from the children
wdym
analog badger
so we dont even end up using 1/(1-x) then
right?
bc instead of using the series, we stop at x^3?
feral sedge
yeah
vernal bear
so we dont even end up using 1/(1-x) then
it is when the terms tend to infinity
analog badger
ohhh so thats what u were saying heisenberg
wait so what exactly is the difference between stopping the series somewhere, vs tending towards infinity
like what does that represent
vernal bear
it tends to an infinite gp
analog badger
but like in terms of toys and children, what does that look like?
vernal bear
you can say infinite toys
pearl pondBOT
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orchid hawk
this is from a math textbook. it’s in ukrainian, so i’ll translate:
“The teacher has asked to calculate the limit lim_(n → ∞) 1.
The student Basil the Confused has solved the task in this way:
…
Do you agree with Basil’s solution?”
obviously this solution is false, lim_(n → ∞) 1 = 1 ≠ 0. however i am doubtful where exactly did he go wrong. is it because lim_(n → ∞) n ⋅ 1/n = ∞ ⋅ 0 and ∞ ⋅ 0 is undefined?
plush bramble
lim (a + b) = lim(a) + lim(b) is true when both sequences converge to a finite limit and it's a sum of just 2 terms
narrow forge
I think it is more properly called indeterminate.
orchid hawk
I think it is more properly called **indeterminate**.
yeah, i forgot the right word
wolfram alpha however shows “(undefined)”,
,w infinity times zero
jolly parrotBOT
orchid hawk
lim (a + b) = lim(a) + lim(b) is true when both sequences converge to a finite l...
makes sense
versed mica
you can't just substitute in every scenario
orchid hawk
this is not how limits work
yeah that’s the point of the exercise, to try to catch basil’s error
versed mica
no im saying your idea of the limit being infinity times 0
pearl pondBOT
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faint fossil
Hello
warped violet
Hello
faint fossil
I need help with this pls
Oh eait
Oh wait.
I think
We met before
ur the goated helper 🐐
warped violet
We have
faint fossil
Teach gave us this
To prepare for second test
And my brain nearly exploded cause it's totally different from what we did in class
The only one that's ez is the first question it says
We need to prove that a
Is 4√2 -4
We simplify a
warped violet
I see
warped violet
👍
faint fossil
One question tho
Do we break the √72 into √12 x √6
Or
Do I break it into smth else
warped violet
It depends
Usually we want to break it into the largest number of which we can take the square root
And then whatever is left
faint fossil
Yes
warped violet
Exactly
warped violet
Do we break the √72 into √12 x √6
Of course you can continue with this as well
12 = 4 * 3, 6 = 3 * 2
faint fossil
I did that
faint fossil
I got it
warped violet
Okay, now we have $6 \cdot \sqrt{2} - \sqrt{2} \cdot \left(\sqrt{8} + 2 \right)$
jolly parrotBOT
USS-Enterprise
warped violet
Now what
warped violet
Sure
faint fossil
And
warped violet
Another way is factor out sqrt(2)
Whatever you see / like
$6 \cdot \sqrt{2} - 4 - 2 \cdot \sqrt{2}$
jolly parrotBOT
USS-Enterprise
warped violet
Right
faint fossil
Yes
warped violet
Yes
warped violet
Correct
faint fossil
So I want another one and for u
Second question says
Compare between the numbers 4√2 and 4
Wait let me translate smth
warped violet
faint fossil
<:what:561568026210730004>
I think
warped violet
$4 \cdot \sqrt{2} \neq 4$
faint fossil
We compare who is bigger and smaller?
jolly parrotBOT
USS-Enterprise
faint fossil
Since we studied that
warped violet
How would we approach that?
faint fossil
First thing we see
If one is minus
And there is no number of those 2 minus they are both postive
We just calculate √2 and do 4-√2?
And it's obviously
Smaller than 4
So a is a negative number
Right?
I think that is what they want from us
warped violet
Uh give me a second
faint fossil
Since there is a question
warped violet
You said we are comparing 4*sqrt(2) to 4?
warped violet
Why would we do 4 - sqrt(2)?
faint fossil
Wdym
warped violet
We just calculate √2 and do 4-√2?
.
warped violet
If 4 - 4sqrt(2) is positive it must mean that 4 is larger than 4sqrt(2)
Because both are positive numbers
That's your reasoning right
warped violet
Well yes, same thing but other way around. If that is positive then 4*sqrt(2) is larger
faint fossil
Yes
warped violet
$a - b > 0 \implies a > b; a, b > 0$
jolly parrotBOT
USS-Enterprise
warped violet
This by the way works for any number, not just positive ones. -2 - (-6) = -2 + 6 = 4, which is >0, means -2 is larger than -6
faint fossil
Yes
faint fossil
Ok
warped violet
Another way is also just look at 4sqrt(2) and 4
one is 4 multiplied by something, the other is just 4
The only way the 4 multiplied by something is larger than 4 is if that something is larger than 1 right
And sqrt(2) > 1
warped violet
Exactly
warped violet
😄
faint fossil
Now
Question 3
We got x and y are number:
x=1/√2 and y=1/2-√2
Prove that
x-y=a/4
warped violet
$x = \frac{1}{\sqrt{2}}$
jolly parrotBOT
USS-Enterprise
faint fossil
Yes
warped violet
$y = \frac{1}{2 - \sqrt{2}}$
jolly parrotBOT
USS-Enterprise
faint fossil
Yes
We prove that x-y=a/4
So we should first get the underline of both x and y equal
Or
We could flip them and multiply them by (-1)
Right?
warped violet
Underline?
faint fossil
Yes
warped violet
We could flip them and multiply them by (-1)
How would we flip them
faint fossil
I think we scrap the flip thingy
warped violet
faint fossil
Doesn't matter 😅
faint fossil
We common denominator
warped violet
$x = \frac{1}{\sqrt{2}}$
jolly parrotBOT
USS-Enterprise
warped violet
$y = \frac{1}{2 - \sqrt{2}}$
jolly parrotBOT
USS-Enterprise
faint fossil
Yes
warped violet
$a = 4 \cdot \sqrt{2} - 4$
jolly parrotBOT
USS-Enterprise
faint fossil
Yes a/4
faint fossil
Yes.. after we simplified
jolly parrotBOT
USS-Enterprise
faint fossil
Yup
warped violet
So
We can calculate each side separately
Start with a/4
$\frac{4\cdot \sqrt{2} - 4}{4}$
jolly parrotBOT
USS-Enterprise
warped violet
Is what
jolly parrotBOT
USS-Enterprise
warped violet
How much is this
faint fossil
4
faint fossil
3
warped violet
Exactly
warped violet
We can only cancel (delete)
When multiplication is the main operation
$\frac{4\cdot \sqrt{2} - 4}{4}$
jolly parrotBOT
USS-Enterprise
faint fossil
Yes
warped violet
Can we change something so that multiplication becomes our main operation
On the top (numerator)
faint fossil
Uhm
jolly parrotBOT
USS-Enterprise
jolly parrotBOT
USS-Enterprise
warped violet
Now we can't subtract
faint fossil
Yes
jolly parrotBOT
USS-Enterprise
warped violet
Notice if you multiply it out you get the same thing as above
faint fossil
Yes
warped violet
We can take like terms (in this case a), and "group" them
warped violet
3a -a
We did the same thing here
3a - a = a * (3-1) = a * 2 = 2a
But with sqrt(3) we get stuck on the 2nd step
Because we don't know what sqrt(3) - 1 is
faint fossil
Yes
jolly parrotBOT
USS-Enterprise
warped violet
What can do on the top
faint fossil
4*(√2-1)?
jolly parrotBOT
USS-Enterprise
warped violet
Notice how the main operation is now multiplication
faint fossil
So x-y=√2-1
jolly parrotBOT
USS-Enterprise
warped violet
Factor out 2, we get (2 * (5-1))/2
warped violet
Which is (2 * 4)/2 = 4
faint fossil
4
warped violet
Yes
warped violet
So x-y=√2-1
Correct
warped violet
$\frac{1}{\sqrt{2}} - \frac{1}{2 - \sqrt{2}}$
jolly parrotBOT
USS-Enterprise
warped violet
Oh
faint fossil
So it becomes
warped violet
Rationalize?
faint fossil
I think
faint fossil
Idk how to say it in the mother america language
Ye that's an option that I thought of but
Like flip them so it becomes
-√2 - -(2-√2)
-√2 - (-2) +√2
Oh
But that would make √2
Go
warped violet
No, we can't do that
faint fossil
Ok
warped violet
That is like saying 1/2 = 2/1
😅
We can only multiply the top and bottom by some number with fractions
faint fossil
Oh wait I'm so srry
warped violet
That's all we are allowed to do to keep it unchanged
faint fossil
I messed that up with the powers
Since when u flip numbers with power
The power that is -becomes +
And the opposite too
I'm so srry
faint fossil
Like 2^-2 =1/2^2
warped violet
Oh!
Yes
But here we would have
(sqrt(2))^-1 = 1/sqrt(2)
But that doesn't really help us here
faint fossil
Let's rationalise then
faint fossil
Ok so
faint fossil
Multi by √2
warped violet
$y = \frac{1}{2 - \sqrt{2}}$
jolly parrotBOT
USS-Enterprise
faint fossil
Ok wait
warped violet
Exactly
faint fossil
And
warped violet
*difference of squares*
This
faint fossil
Power 2 cancels square root
warped violet
$a^2 - b^2 = (a - b) \cdot (a + b)$
jolly parrotBOT
USS-Enterprise
warped violet
We have the (a - b)
So we just need to multiply it by (a + b)
To get a^2 - b^2
And a^2 - b^2 assures us we never end up with a root right
faint fossil
Yes
warped violet
As you said the power 2 cancels (reverses) the square root
faint fossil
So we get
warped violet
$\frac{\sqrt{2}}{2} - \frac{2 + \sqrt{2}}{2}$
jolly parrotBOT
USS-Enterprise
warped violet
√2/2 - 2+√2/2
We have this
but the second part is in brackets
Very important
The minus is distributed to both terms
faint fossil
Yes
warped violet
And then -2/2
faint fossil
But now we can't get x-y =√2-1
jolly parrotBOT
USS-Enterprise
jolly parrotBOT
USS-Enterprise
warped violet
Which is not true
warped violet
Give me a second to go over everything again, otherwise it means that x - y DOES NOT equal a/4
faint fossil
Has a mistake
faint fossil
Give me a second to go over everything again, otherwise it means that x - y DOES...
Ok
Wait
Do we have to put the brackets
With no brackets we get the correct answer
√2-1=√2-1
jolly parrotBOT
USS-Enterprise
faint fossil
Hmm
warped violet
The brackets must be there
warped violet
No I looked over again
faint fossil
Changes
warped violet
Did it with photomath
Everything we did is correct
Also
What does the question say
Exactly
Maybe we just needed to see if x - y = a/4
And we have showed it does not
faint fossil
Nope it says prove that
Teacher said
If we find prove that
It means
It must =
It says x-y and not y-x
It says a/4
Hmm
warped violet
Yeah, we did it right
faint fossil
Maybe
warped violet
It is a mistake
faint fossil
The teacher forgot to add brackets herself when she was making this question?
warped violet
$x - y \neq \frac{a}{4}$
jolly parrotBOT
USS-Enterprise
warped violet
I threw the entire problem into photomath
warped violet
And it returns the same result we got
faint fossil
We will do the correction
warped violet
Let me know
warped violet
I am positive we are correct
faint fossil
Let's move on
faint fossil
I am positive we are correct
Me too I redid it tn
faint fossil
Unfortunately I can't 🙁
It's all right
faint fossil
U helped alot
warped violet
Ask the teacher about it tomorrow
faint fossil
Tysm
faint fossil
Cya
warped violet
👋
faint fossil
.close
pearl pondBOT
pearl pondBOT
Available help channel!
runic forum
hey there, is the hyperbola y=-2/(x+1) a continuous or non-continuous function
runic forum
dont understand why some sources say its either
glass meadow
Probably because it's continuous on (-inf, -1) U (-1, +inf)
(which is where it's defined)
runic forum
So is it technically continous or discontinuous, since theres two asymptotes
disclaimer i havent really studied calculus
glass meadow
There is only one vertical asymptote at x = -1
thorn musk
There is only one vertical asymptote at x = -1
What question?
glass meadow
Let the function g(x) = -2/(x+1) for all x except -1, and g(-1) = 0
g is defined on R
g is not continuous
Let f(x) = -2/(x+1)
f is defined on R \ {-1}
f is continuous
runic forum
doesnt y not = 0
glass meadow
Why would that matter?
runic forum
because that is an asymptote
glass meadow
e^{-x^2} (a Gaussian function) also has a horizontal asymptote y=0
It's still continuous
(and defined on all of R)
runic forum
so what does continuous exactly mean
glass meadow
A function is continuous if it's continuous at every point of its domain
A function f is continuous at a point x if f(x) equals the limit of f at x
runic forum
So is this function technically not continuous
glass meadow
Which one
runic forum
y=-2/(x+1)
glass meadow
First of all, that's an equation, not a function
If you want a function, you can write y(x) = -2/(x+1), so you can refer to it as the function y
Now, y is defined on (-inf, -1) U (-1, +inf)
The limit of y at any point of its domain does equal its value at that point
So y is continuous
You could say that y has a discontinuity at -1, but -1 is not part of the domain of y, so it doesn't make y discontinuous
Similarly, I could define the function h with domain (-inf, -1) U (-1, +inf), where h(x) = 1
h is obviously continuous, even if it's not defined at -1
runic forum
so what would it be in simple terms
runic forum
i dont understand limits tho, thats the problem
runic forum
The original question was asking: Is the hyperbola y=-2/(x+1) continuous?
glass meadow
And the answer is yes
runic forum
.close
pearl pondBOT
pearl pondBOT
Available help channel!
outer yarrow
Until now, experts believed that a computer program with a runtime of t \in \mathbb{N} steps requires at least \frac{t}{\lg(t)} bits of storage space. This corresponds to a nearly linear relationship between space and time. At the Association for Computing Machinery (ACM) symposium in Prague in June 2025, Williams presented his new result: Every countable problem solvable in t steps requires at most \sqrt{t} \cdot \lg(t) bits of storage space.
"The result shows that we were completely wrong with our previous intuition," Williams told Scientific American. "I thought something must be wrong, because the result is extremely unexpected."
With his proof, Williams invalidates the previous assumption about practically solvable problems. From a mathematical perspective, Williams' result differs vastly from the previously suspected relationship. Instead of the nearly linear relationship between space and time, it is now defined by the square root; as the number of steps t increases, the difference becomes increasingly apparent.
- At which t do both storage space values match?
- From which t onwards has the storage space value decreased by a factor of at least 10 due to the result?
pearl pondBOT
Until now, experts believed that a computer program with a runtime of t \in \mat...
@outer yarrow Has your question been resolved?
outer yarrow
For Base-2 there's a solution, but it's not possible for Base-10 (as far as i know)
But every help I've gotten tells me it has to be Base-10
So I'm definitely missing something.
<@&286206848099549185>
I've been sitting at this problem for 4 hours so far
No move forward yet.
We might suspect one of the t values for 1. is 0
based on this
pastel umbra
Jokes aside, the red graph there is not the same as the stated minimum by Williams:
at most \sqrt(t) \cdot \lg(t)
[Also, yk we have a LaTeX bot here]
outer yarrow
[Also, yk we have a LaTeX bot here]
(yeah sorry i'm new i don't know how to fully use and activate it yet)
but this completely throws me off
all my tries are to no avail
pastel umbra
Jokes aside, the red graph there is not the same as the stated minimum by Willia...
I'd still look at this
outer yarrow
Jokes aside, the red graph there is not the same as the stated minimum by Willia...
does the square only include t or * lg(t) as well?
pastel umbra
The brackets are around the t, not the log
Heck, if you format it with LaTeX, and maybe you're ripping this off of somewhere to ironically not have the context, it's $\sqrt{t} \cdot \lg{t}$
jolly parrotBOT
Waes (Wires)
outer yarrow
Heck, if you format it with LaTeX, and maybe you're ripping this off of somewher...
huh, the exercise i got wrote it exactly like that
pastel umbra
Dann falsch umschrieben hast du es also
Base 2 seems the only reasonable base, and even then this doesn't seem right to me
In any case, I'd bring this up with a supervisor/lecturer/teacher who's given you that passage (assuming you're not self-studying this)
outer yarrow
Base 2 seems the only reasonable base, and even then this doesn't seem right to ...
Habe ich auch probiert, ist aber am Ende nicht die richtige Lösung, es wird immer gesagt lg ist der Logarithmus Basis 10. Andere Schulen haben es ja durch diese Runde geschafft, also muss es irgendwie möglich sein. Nur von dem was ich weiß, sieht es nicht möglich aus. Hier ist die Aufgabe in Originalsprache.
Das sollte "12. Klasse Level" sein aber es sieht nix wie 12. Klasse aus.
outer yarrow
Das hier...
Haben die ein Fehler in der Aufgabenstellung gemacht?
outer yarrow
hiermit vergleichen?
Um ernst zu sein, den Satz hab ich gar nicht verstanden.
Ich hab nicht die Korrelation zwischen der Aufgabe und diesem Satz gefunden.
pastel umbra
Dass es mit dem Quadratwurzel vergleichbar ist, so lautet er vielleicht
outer yarrow
Dass es mit dem Quadratwurzel _vergleichbar_ ist, so lautet er vielleicht
Könntest du das vielleicht bisschen anders formulieren bitte?
pastel umbra
sqrt(t lg (t)) ist also die "echte" Formel
Die ist der Quadratwurzel-Formel (dh. "sqrt(t)") ähnlich
Und zwar ganz gleich für irgendeinen Wert t
dh. Frage 1
outer yarrow
Aber das ändert ja glaub ich nichts daran, das wenn ich die alte und neue Gleichung grafisch darstelle, ich keinen einzigen Schnittpunkt finde, oder?
Im Unendlichen driften die 2 Gleichungen weiter und weiter voneinander entfernt.
pastel umbra
Aber das ändert ja glaub ich nichts daran, das wenn ich die alte und neue Gleich...
Tja doch
pastel umbra
Bei Desmos hier ist das doch log10
Mit dem Basis B ist der x-Wert des Schnittpunkts auch gleich B
Jetzt aber stimmt was nicht für die 2e Frage 😭
outer yarrow
Jetzt aber stimmt was nicht für die 2e Frage 😭
Es ist alles durcheinander immer mit dieser Aufgabe 😭
Und Nachfrage für 1
Kann es sein, das mehrere Schnittpunkte da sind?
Oder ist die Range einfach nur da um zu verwirren
pastel umbra
Bei Differentialrechnung kann mans probieren
Bin aber der Meinung, nicht so, denn x log x als x schon schneller ist
Egal was für'n Basis
pastel umbra
Jetzt aber stimmt was nicht für die 2e Frage 😭
oh nein DOCH DAS PASST perhaps idk
Nur ist es enorm
outer yarrow
Die zweite Frage hat mich verwirrt. So wie ich es sehe ist dann jetzt t für die alte Gleichung geteilt durch t für die neue Gleichung ≥ 10
oder?
pastel umbra
so sehe ichs auch
outer yarrow
Nur ist es enorm
ist es zufällig 48036?
outer yarrow
aber wir rechnen mit B = 10 oder...
pastel umbra
Dann noch grösser
outer yarrow
hilfe hilfe hilfe das ist
pastel umbra
(hab kein esszett lol)
pastel umbra
hilfe hilfe hilfe das ist
Hast du schon aus irgendwo eine Antworte, oder...
eh
was denn
Oh, du vergleichsts mitm originellen
Dann noch Achtung, weil dieser Log auch im Q.W. liegt
(gibts eine Abkurzung wie sqrt auf Deutsch 
)
dh lg t . sqrt(lg(t)) darunter
ugh
outer yarrow
boah mein gehirn ist nicht mehr am funktionieren
pastel umbra
Mein auch nicht
outer yarrow
Oh, du vergleichsts mitm originellen
würde man es anders machen?
pastel umbra
Ich dachte mitm normalen QW
outer yarrow
also alte gleichung ist ja
pastel umbra
aber wie du es machst passt auch glaube ich
pastel umbra
dh lg t . sqrt(lg(t)) darunter
hiervon abgesehen
pastel umbra
Mein auch nicht
Eins weil ich noch keine Ahnung hab worum das hier geht lol
Zwei weil ich's auf Deustch versuche 
outer yarrow
Darf man's nutzen, hier würde ich so denken
alles außer KI (KI = Plagiat)
pastel umbra
Danach ">= 10" mal schreiben
outer yarrow
wie kann ich es von diesem graph dann bekommen 😭
wolfram alpha noch nie wirklich viel benutzt
so ein krampf
pastel umbra
,w solve ( t / log(t) ) / sqrt( t log(t) ) = 10
outer yarrow
das is keine natürliche zahl
jolly parrotBOT
pastel umbra
Hasts du gesehen oder...
outer yarrow
bin ich blind?
pastel umbra
Ich meine vor dieser Conversation
Da ist doch ein Zahl, deine Reaktion hat mir nur verwirrt
outer yarrow
ach, das ist gerunded???
pastel umbra
...yeah