ok, good luck
#help-33
primal drift
are you not in the course?
runic temple
no
primal drift
runic temple
whoa
primal drift
whoa is right im gonna be spending hours on the next couple
but hey last day of this course is always exciting
also for the answer to question 2 was it like 7.2 ish
@runic temple
primal drift
with so root 26 times root 8 times 1/2 but that was wrong
since root 8 is the base of a to B
runic temple
you can find AC using pythagorean
primal drift
I thought BC was the height though
runic temple
cool
primal drift
and I guess you fit the bill by just simplifying the problems
so you would be perfect for this
runic temple
i did participate in AMC in high school
marsh citrusBOT
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Available help channel!
vocal dew
can someone help me with this? i dont understand the 1 equation 3 unknown
tired ore
can someone help me with this? i dont understand the 1 equation 3 unknown
try to get a common denominator on the right side
marsh citrusBOT
@vocal dew Has your question been resolved?
stark trail
marsh citrusBOT
stark trail
Is this a valid application of the central limit theorem
I'm just a bit worried because of the sqrt(n) in the denominator in the probability of the last line
stark trail
How are you sure, sir?
broken zealot
clt just says that the distribution looks like a normal after sending n to infinity, so that means the cdf will look like a normal's cdf
just ignore the inequality with the infinity as it is true with probability 1 anyway
stark trail
No I meant sure about the other thing
I understand ignoring the infinity side
like
CLT would say this is integral from 0.01/(sigma sqrt(n)) to infinity of the standard normal pdf
but I just replaced the lower bound with 0 because that felt right
although I'm not sure if it is
As n to infinity it approaches 0
broken zealot
why not
stark trail
you have to get
(X_nbar - mu)/(sigma/sqrt(n)) in the middle term
if you do this then you get the left bound as 0.01/(sigma sqrt(n)) not 0.01
broken zealot
wait i think i've done this problem, is this 395
stark trail
yes
broken zealot
can you send the original
broken zealot
oh ok then yeah
stark trail
becomes integral infinity to infinity which is0
broken zealot
should be i -> 0, ii-> 1/2, iii-> 1
stark trail
So is this valid fix
the lower bound is then sqrt(n) * 0.01/sigma which tends to infinity
stark trail
yes, since the left side goes to infinity
Do you see my doubt about why this is valid or not?
0.01/sigma sqrt(n) is not a fixed value a
so I'm just wondering about if pulling the limit inside is fine to make it inf
the statement is like
broken zealot
the distribution tends to a normal as n grows large, the actual interval doesn't matter it can still depend on n
stark trail
for all a,b with -inf <= a <= b <= inf
lim n->inf P( a <= ... <= b) = integrate standard normal pdf from a to b
but a depends on n
in the above
broken zealot
yeah, a and b can depend on n
broken zealot
the distribution tends to a normal as n grows large, the actual interval doesn't...
_ _
it's better to think of the clt as saying that the distribution converges to something, not a specific probability statement
stark trail
yea
broken zealot
unless a,b oscilate or something
stark trail
Okay sir
broken zealot
does that seem reasonable
stark trail
Well it always seemed reasonable to me I just don't want to lose points for applying the theorem in a way that wasn't explicitly written
broken zealot
yeah i think it is ok
clt is usually phrased in a stronger way
or maybe not usually, but often idk
stark trail
.close
marsh citrusBOT
stark trail
also gl on your finals
ty sir
broken zealot
oh i see
marsh citrusBOT
Available help channel!
native gull
marsh citrusBOT
naive stump
no calculator
are u familiar with compound/double angle results?
if yes, consider rewriting cos10 = cos(2x5)
wary kite
marsh citrusBOT
@native gull Has your question been resolved?
small stream
Let the function ( f(x) ) have the second derivative on ( (0, +\infty) ) and ( f''(x) < 0 ). Let ( u_n = f(n) ), then the sequence ( {u_n} ) diverges is the ( ) of ( u_1 > u_2 ) \
(A) Sufficient and necessary condition\
(B) Sufficient but not necessary condition\
(C) Necessary but not sufficient condition\
(D) Neither sufficient nor necessary condition
marsh citrusBOT
Let the function \( f(x) \) have the second derivative on \( (0, +\infty) \) and...
small stream
I'm cooked. No where to start
brittle cliff
What exactly is the question
brittle cliff
You sure it’s ui>uj and not the other way around?
fair kindle
Ooh lemme modify
some modifications still need to be made, "diverges is the () of" check this part
elfin berryBOT
riyobi
crystal lintel
it makes sense. the correct option is supposed to go in the ( )
fair kindle
it makes sense. the correct option is supposed to go in the ( )
okay that makes sense
brittle cliff
**riyobi**
This is entirely different now though
marsh citrusBOT
Please show the original problem, exactly as it was stated to you, with the entire original context. A picture or screenshot is best. If the original problem is not in English, then post it anyway! The additional context might still be helpful. Do your best to provide a translation.
small stream
This is entirely different now though
I used gpt to translate, it changed some meaning now I've modified
crystal lintel
you should still post the original too
brittle cliff
I used gpt to translate, it changed some meaning now I've modified
So this is correct now?
small stream
Yes
crystal lintel
weird question then
fair kindle
Yes
I recommend you graph the question and try to look for clues
it doesn't look too hard, so I think you will be able to answer it
always do that by the way, it's a global hint, always try to visualize questions if possible
small stream
But f(x) is not a f with explicit expression, how to graph it?
fair kindle
make something up which satisfies given conditions
brittle cliff
But f(x) is not a f with explicit expression, how to graph it?
Well what does the condition f“<0 mean geometrically
small stream
Let's say f=-x^4
fair kindle
Let's say f=-x^4
second derivative is positive on that one
don't even need to calculate you can imagine it xD
also the function you wanna come up with doesn't have to be elementary or something you know
you can just put axes and draw anything random with your pen but make sure it satisfies the conditions
try drawing multiple ones and ask yourself questions
can I draw one which diverges satisfying the condition ? can I draw one which diverges without satisfying the condition ? and so on
small stream
,w plot -e^x
elfin berryBOT
small stream
Is this one's un diverging?
fair kindle
clearly diverges
as you go to infinity
it blows up to -infinity
is u1 bigger than u2 here ?
small stream
Yes
fair kindle
from this graph. generally if u1 > u2, does your function have to diverge ?
small stream
Yes
fair kindle
that's great
but does it have to be true ?
is it necessary
aka can you come up with another example in which it diverges but u1 < u2
past atlas
small stream
Can't think of other examples
fair kindle
Can't think of other examples
unleash your imagination xD
doesn't have to be a sine or a cosine or an e
just draw something with a pen
start with a function that has any value
make it rise
then keep it's second derivative negative
anything
again it doesn't have to have a symbol. or be something you have seen before
as long as you know what it means for the second derivative to be negative geometrically
you can just graph something without knowing what it is
okay how about this, what does it mean for the second derivative to be negative geometrically ?
like how would you describe it on a graph
small stream
,w plot -sinx-2cosx
elfin berryBOT
small stream
,w plot sinx + 2cosx
elfin berryBOT
fair kindle
okay so for this function for example you just drew
when is the second derivative positive
and when is it negative
and don't use any calculations
just look at it and tell me
if you don't know what it means geometrically you can ask by the way this is not an exam
small stream
For this, let u1=0.1 and u2=0.2, we get u2>u1
fair kindle
when is the second derivative positive
can you answer this question
it's important
you won't answer properly if you can't answer it
small stream
Yes ofc i know
fair kindle
then tell me
sinx + 2cosx
when does it have a positive second derivative and when doesn't it, you can name points, you don't have to be too precise
and make zero calculations
just tell me from looks
if f(x) = sinx + 2cosx
what is the sign of f''(0) and f''(pi)
small stream
fair kindle
okay
small stream
Anyway, i knew, if idk I'll just ask
fair kindle
it's true then
okay so you know that it's related to concavity
all i'm asking is that you draw a concave down function that has u1 < u2
and see if you can make it divergent
fair kindle
non of these work by the way and i'm sure you know why
small stream
It's not divergent
fair kindle
no they don't have negative second derivative over (0,inf)
that's the condition in the original question
they go concave up in some regions
you wanna draw something that is always concave down
but starts rising
you can draw it using paint or anything , again it doesn't have to be something you would graph on wolframalpha
small stream
,w plot -(x-1)^2
elfin berryBOT
elfin berryBOT
small stream
Yes
fair kindle
does it diverge ?
small stream
Yes
small stream
D
fair kindle
are you sure
is it sufficient ?
like if that condition is true does it make the function diverge ?
small stream
Yes?
fair kindle
you tell me xD
we have established it's not necessary correct ?
now tell me whether or not it's sufficient and the question is solved
small stream
Necessary is : u1>u2 implies divergent?
fair kindle
yes
wait
no that's sufficient
sufficient means the condition is sufficient to make the function diverge
small stream
😭 it's so hard to remember it correctIy
fair kindle
necessary means that without it, it's impossible for the function to diverge
fair kindle
😭 it's so hard to remember it correctIy
it can be annoying at first, the wording of logic
but just translate it literally
something that is sufficient means its mere existence is enough to make a statement true but it doesn't have to be there for the statement to be true, if it has to be there then it's necessary
the existence of rain is sufficient to deduce the sky has clouds, but it's not necessary to have rain in order to have clouds
small stream
But then we'll need to come up with another example to see if u1>u2 implies non divergent or not. If we can come up with one example, then it's not sufficient
fair kindle
that's a great note !
my question is can you come up with an example ?
if you can then you are right it's not sufficient
small stream
I can't, but that doesn't mean it's not true because maybe others can🤣
fair kindle
I can't, but that doesn't mean it's not true because maybe others can🤣
well in that scenario we prove that we can't, so no one else can 😂
small stream
Wait how did we prove
fair kindle
we didn't ... yet xD
I asked you if it was true or not and you said yes, you didn't provide evidence
can you prove that it's sufficient
fair kindle
lemme give you an example with a problem associated with it
you have a function f(x), f(0) = 0 and f'(0) = -1 , its second derivative is always negative, would that function be less or more or intersecting the line y = -x
small stream
Isn't the f(x) itself y=-x
fair kindle
Isn't the f(x) itself y=-x
this has zero second derivative
the other one has negative
so it should be falling faster
small stream
Yes
fair kindle
if you take the sequence y(n) = -n, that will diverge
well then what of the sequence you get from a function lower than that ?
small stream
Due to squeeze thm or something, it'll divergent too
fair kindle
not really squeeze theorem but it's a similar idea
proving the actual thing in your question however is slightly harder
because we don't have f'(x) but rather we have two points far apart
small stream
fair kindle
now check this out xD
f(1) > f(2)
a line passing through the points (1,f(1)),(2,f(2)) would have negative slope
which is f(2)-f(1) / (2-1)
the mean value theorem tells us the function f(x) has a point c in (1,2) which has exactly that slope
and since the function has negative second derivative
then f'(c) > f'(2) since c < 2
from here, for x > 2 the line passing through the two points must be larger than the function value.
and since that line is decending at a constant rate, again the sequence taken from it diverges to minus infinity
hence the sequence taken from the function below it also diverges to - inf
small stream
not really squeeze theorem but it's a similar idea
Does this idea have a name
I want to look it up as a reference
fair kindle
i'm not really sure if it does, it goes straight back to the definition of limits
the limit of a sequence an = -inf if for every B you can always find an N such that if n >= N then an < B. if bn < an after some threshould. then if you can find an N to make an < B then you can use the same N to make bn < B since it's lower.
this would prove by the definition of limits that if bn < an and an goes to -inf, bn must go to -inf as well
note than N is an integer and B is any real number
marsh citrusBOT
@small stream Has your question been resolved?
marsh citrusBOT
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Available help channel!
frosty bronze
can someone help me check if this is the correct answer for this question?
lunar ember
30%
naive fox
!noans
marsh citrusBOT
The purpose of this server is to help you learn, not to hand out answers. Do not ask someone to give you the answer directly.
marsh citrusBOT
30%
As a helper, please do not give out answers that could be copied as a homework solution. Have the student work through the problem themselves and guide them along the way.
lunar ember
Total density= 0.50
naive fox
Welp, wrong one used.
frosty bronze
0.45 is correct
naive fox
Total density= 0.50
That is not true. Total density must be 1.
lunar ember
What is this density ?
naive fox
It is probability density
From axiom of probability, the integral for the entire domain must be 1.
marsh citrusBOT
@frosty bronze Has your question been resolved?
idle vessel
Why do palindromic polynomials with odd degrees necessarily have -1 as a zero?
hollow glen
plug in 0 and see what happens, look at the first and last term, second and second last, etc
idle vessel
0?
hollow glen
oops -1 my bad
marsh citrusBOT
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Available help channel!
valid kiln
Hello everyone, I need help with this one. Is the correct answer for s=1 and t=0 or s=5 and t=-1? And could u explain me how did u get the answer
marsh citrusBOT
Hello everyone, I need help with this one. Is the correct answer for s=1 and t=0...
pallid tapir
Hello everyone, I need help with this one. Is the correct answer for s=1 and t=0...
you can always check yourself
does (s,t) = (1,0) or (s,t) = (5,-1) yield 1?
valid kiln
does (s,t) = (1,0) or (s,t) = (5,-1) yield 1?
what does yield 1 mean?
valid kiln
does it yield the gcd of the two numbers
could u explain me what does yield mean in this case?
pallid tapir
yield as in
give you
return
make the function/expression in question evaluate to
valid kiln
make the function/expression in question evaluate to
i do not understand, could u explain it
i tried checking with the calculator, and both gives 793
pallid tapir
great, so they yielded 793
pallid tapir
Hello everyone, I need help with this one. Is the correct answer for s=1 and t=0...
it also means that your choice of integers are not unique
so if 793 is their gcd, then both pairs are fine
valid kiln
so if 793 is their gcd, then both pairs are fine
also i wanted to know if the question is + t3172 and for example i found t=-1 so is it wrong or correct since t is negative, which is different from the question
or can i write + (-1)3172 ? but t is still -1 right?
pallid tapir
your task was to find a pair of integers (s,t) satisfying the above. you've found two pairs. -1 is an integer -- there is no problem
valid kiln
alright thank you
valid kiln
your task was to find a pair of integers (s,t) satisfying the above. you've foun...
can i write my answer like this
marsh citrusBOT
@valid kiln Has your question been resolved?
marsh citrusBOT
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Available help channel!
minor nebula
help
late geode
!status
marsh citrusBOT
What step are you on?
1. I don't know where to begin.
2. I have begun but got stuck midway.
3. I got an answer but I was told that it's wrong.
4. I got an answer and would like my work checked.
5. I have a question about someone else's work/solution.
6. I have completed the problem and don't need help anymore. Thank you.
7. None of the above
sorry wrong qs but yes idk where to begin
i tried 360-(21x-1 ) to find RSTP
wait
i think
ok nvm
.close
marsh citrusBOT
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Available help channel!
flat meteor
can someone help? need to prove
stoic slate
flat meteor
can someone help? need to prove
rustic field
just asumme a<b<c WLOG
marsh citrusBOT
@flat meteor Has your question been resolved?
marsh citrusBOT
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Available help channel!
empty compass
how do I solve this
marsh citrusBOT
how do I solve this
whole thorn
any progress or ideas?
empty compass
any progress or ideas?
I think I need to find the slope but I don't know where to start
whole thorn
maybe first find the point of intersection???
empty compass
ok I'll give it a try
marsh citrusBOT
@empty compass Has your question been resolved?
wheat moth
somone help
marsh citrusBOT
somone help
normal moss
Don't ping random people/mods.
normal moss
Last warning to stop pinging random users
normal moss
We have #❓how-to-get-help for info on this.
normal moss
I don't care. Find time.
wheat moth
I don't care. Find time.
typical mod
eager tinsel
just say it
you just have to wait for helpers to come to your channel
wheat moth
alr
wheat moth
you just have to wait for helpers to come to your channel
ok fine
heres the sum again
someone welp
eager tinsel
ok fine
if they don't come in 15 mins
you can ping the Helpers
but only after 15 mins
wheat moth
if they don't come in 15 mins
you can ping the Helpers
ok fine
normal moss
It says all this in the first paragraph of the channel I linked.
thorny bison
Hint 1 - || open the square what do you get ||
marsh citrusBOT
@wheat moth Has your question been resolved?
thorny bison
Hint 2 - || in the numerator u get the expression x⁴ +x³ +2x² +1 which can be written as x⁴ + x² +1 + x(x² + x)
Now divide it by the denominator to get two telescopic series ||
wheat moth
Hint 2 - || in the numerator u get the expression x⁴ +x³ +2x² +1 which can be wr...
ok alr
wheat moth
Hint 2 - || in the numerator u get the expression x⁴ +x³ +2x² +1 which can be wr...
im gettin 3/2
thorny bison
im gettin 3/2
What the answer?
thorny bison
.reopen
sharp peak
What level of math do I need to be able to do computer science?
marsh citrusBOT
What level of math do I need to be able to do computer science?
sharp peak
Like pre calc? Or just algebra 2
sharp peak
What specifically do you wanna do
Either game engineering or software engineering
timid merlin
Well, u can enter the university with only the high-school math u know
It would be nice if u know pre calc
hollow glen
a bit of linear algebra too
wheat moth
What the answer?
ig thats the answer
sharp peak
Oh so just regular linear and quadratic formula algebra works for computer science also?
wheat moth
what did u gry
dapper ravine
Either game engineering or software engineering
For games you're gonna need calculus and trig
thorny bison
ig thats the answer
Show ur work 
thorny bison
Show ur work <:bnuuy:1270384601654886580>
In dm
hollow glen
oh no linear algebra is much more than just linear polynomials
dapper ravine
For general software eng you just need good mathematical reasoning skills, but being good at precalc should give you a leg up
thorny bison
what did u gry
I tried what I told you in the hint 
dapper ravine
Oh so just regular linear and quadratic formula algebra works for computer scien...
Linear algebra is vectors, matrices, linear transformations
hollow glen
you should know about matrices and how to represent transformations with them and stuff
sharp peak
Linear algebra is vectors, matrices, linear transformations
Oh so I guess I haven't reached it yet since im still in 9th
My school learning system is a bit slow
marsh citrusBOT
@sharp peak Has your question been resolved?
sharp peak
Yes
alpine wharf
I need to solve this using trig identities to prove the equation is true but I'm rlly stuck on the step at the bottom. What do I do next?
marsh citrusBOT
I need to solve this using trig identities to prove the equation is true but I'm...
brave marsh
Just split the fraction here
alpine wharf
1/cos - cos^2/cos..?
uncut oasis
There you go
alpine wharf
.close
marsh citrusBOT
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Available help channel!
dusty folio
The 3rd question
marsh citrusBOT
The 3rd question
dusty folio
<@&286206848099549185>
marsh citrusBOT
Please only use the <@&286206848099549185> ping once if your question has not been answered for 15 minutes. Please do not ping or DM individual users about your question.
uncut oasis
The 3rd question
Alright
First thing, 1.024 is 1000+24 or 1 +24x10^-3?
(just to make sure I get the notation)
dusty folio
1.024 × 10^-7
uncut oasis
Lemme rephrase my question
uncut oasis
1.024 × 10^-7
Does the dot signal thousands or decimals?
dusty folio
Decimals
dusty folio
May I ask why writing it in terms of power of 4
timid crater
because if you write it in terms of the power of 4, its easier to solve it than multiplying all the decimal points of 0.04
uncut oasis
May I ask why writing it in terms of power of 4
The expression with x you've got is 4 times a multiple of 10
timid crater
or smth like that ig
timid crater
The expression with x you've got is 4 times a multiple of 10
thats also a reason too
uncut oasis
4 is a natural number, which is usually the most convenient number set to work with in these kinds of questions
uncut oasis
So first I should write 4 instead of x .. am I right?
No, rewrite 0.04 as (10^-k times 4)^x
uncut oasis
Then what should I do?
Once you've got everything written in terms of 4 and 10, try to figure out the value of x
Should be pretty straightforward from there
Especially if they asked you to deduce it
uncut oasis
How to close the channel??
.close
dusty folio
.close
marsh citrusBOT
uncut oasis
No problem
marsh citrusBOT
Available help channel!
eternal sequoia
How in the world do I solve this lol
marsh citrusBOT
How in the world do I solve this lol
night mica
$\left( \frac{D^6k^6-2k^{12}}{D^{13}}\right)$
elfin berryBOT
everg
eternal sequoia
**everg**
Right, though I have no clue how to integrate
the t is still stuck to this function of D
@night mica oops
night mica
maybe Separation of variables could work
eternal sequoia
I can't
the v is preventing it
im wondering if I could try to substitute D for a trig function of some sort
marsh citrusBOT
@eternal sequoia Has your question been resolved?
eternal sequoia
is it even possible?
spiral plank
can someone check my work, i have no idea how to do differential equations
flat meteor
just asumme a<b<c WLOG
Could u explain more deeply? I didn't get what to do after assumption
formal spruce
can someone check my work, i have no idea how to do differential equations
In the 1st step, you should be dividing by x on the right hand side
spiral plank
omg bruh
wow the whole thing is wrong classic
ill be back
des are literally giving me brain worms
spiral plank
In the 1st step, you should be dividing by x on the right hand side
is this ok
indigo karma
seems ok to me idk
formal spruce
seems ok to me idk
Same here
formal spruce
this is the next q, is this a non separable one
Non separable?
Is that something specifically intructed?
spiral plank
Non separable?
like is it ia linear first order de
oh wait am i supposed to use substitution for this one where u=x+y
blazing ferry
i think is ok
formal spruce
can someone check this one too please
It should be fine
spiral plank
im back with more questions
marsh citrusBOT
im back with more questions
spiral plank
this would be a first order de right? does that mean i need to find the integrating constant
pallid tapir
isn't the integrating constant like, your initial value thing? that y(0) = 9?
spiral plank
right i did
spiral plank
so is the integrating factor just e^(-cost)
pallid tapir
sounds real
pallid tapir
isn't g(t) piecewise? that'd only be one piece
spiral plank
yeah, this is the first one
anywayas i figured out the error i didnt divide c by the if
pallid tapir
ah, I see it
spiral plank
wouls the answer be y=1+8/e
spiral plank
lol i typed in my answers and they were both wrong
what am i doing wrong i dont get it
pallid tapir
wouls the answer be y=1+8/e
did you substitute 8/e back into the original expression for y in the first case?
your y in this message is just a constant
spiral plank
oh wait
no i did not
so should the answer for the first one be y=1+ (8/e)/(e^-cost)
pallid tapir
I believe it
spiral plank
let’s try
pallid tapir
which you can simplify to 1+8e^{cos(t) - 1}
spiral plank
oh the first one was right the second one wasnt lol
pallid tapir
there's another point for the second piece
the problem is your initial value thing
mainly, 0 is not in the interval [pi, 2pi]
spiral plank
ohhh
2pi is equal to 0 though right
so should i use that instead
wait thats literally the same answer
pallid tapir
2pi isn't equal to 0 but I know what you mean
I still don't think so
so you can find two equations y1 and y2 corresponding to the two pieces of g
defined on their own respective domains
y is just the piecewise definition of y1 and y2, you know
spiral plank
right
pallid tapir
since y is asked to be continuous, you want these two pieces y1 and y2 to be attached
for all points in the interior of the domain of y1 and y2, the corresponding piece, hence the function, is continuous at that point
it gets dodgy at the endpoints, in particular the place where y1 ends and y2 begins
that's what gives you your initial value condition for the second piece
spiral plank
so like, the intial value information given can only be used to solve the first de for the interval from 0 to pi
and i shuold make my answer for y2 equal my ansewr for y1 when t=pi cause thats when the piecewise connects them?
pallid tapir
exactly
pallid tapir
technically it's more like, the limit as x approaches pi from the right (coming from the domain of y2) should equal y1(pi)
but it's all the same really
spiral plank
technically it's more like, the limit as x approaches pi from the right (coming ...
so for my answer for y2 i should keep the part with (10/e)/(e^-cost) right
but instead of -1 at the front i should change it to reflect continuity?
pallid tapir
I think the constant is the thing that ought to change right?
spiral plank
wdym
pallid tapir
all of the DE shenanigans should still work up to this point, with arbitrary constant C
using your new initial value premise that y2(pi) = y1(pi) should give you a value for c that's probably different from 10/e
pallid tapir
nice
pallid tapir
I literally did the same thing btw
just now
and yeah, I agree with your choice of c
pallid tapir
very cool
pallid tapir
ive reverse my math brainrot by like 20% just now
this is good news lol
pallid tapir
you as well
spiral plank
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past atlas
Hi, I was wondering how you would turn this problem into a function:
past atlas
I am struggling with how to translate what is being explained into math
I imagine 27 is the y-value.
pallid tapir
you mean how do you generalize its solution?
past atlas
yeah if that is the correct terminology
I guess that sounds right because really I am wondering about any problem like this
particurarly if you had n amount of workers
but honestly just a function for this specific problem would probably help me understand how to think about this better
pallid tapir
right, it's like the difficulty of modeling and generalizing something, sure
past atlas
i have trouble deciding what the speed of the workers should be and what difference it makes that the workers are not working on the same dress
yeah i suppose that is what you'd call it
pallid tapir
it's probably a lot uglier to assume that there isn't any kind of linear collaboration going on, if you will
like, all but one person could be done and have nothing to do
devout pebble
i have trouble deciding what the speed of the workers should be and what differe...
Ig the speed of the workers can be in terms of fractions of a dress per day
pallid tapir
we can always describe the i'th worker's productivity as being a rate of r_i products per time unit
which turns out to be the actually useful thing
past atlas
$D(t) = \frac{r_i}{t}$ ?? idk how to turn it into math at all lol
elfin berryBOT
Neo
past atlas
I genuinely can't remember how to model even basic things
pallid tapir
so assume there is $s$ total product to be made and there are $n$ workers making the product, for which the $i$'th worker makes product at a rate of $r_i$ product per second, say. assuming productivity scales linearly, then the amount of time it takes to make $s$ product, $t$, satisfies $$r_1t+\ldots+r_nt=(r_1+\ldots+r_n)t=s,$$ yielding a time of $t=\frac s{r_1+\ldots+r_n}$
elfin berryBOT
Flip
past atlas
sorry I'm having trouble understand the i'th worker, is the i'th worked the sum of all the n workers?
the rate of all the n workers?
pallid tapir
I'm not immediately sure how you'd do this if s were a positive integer and each worker can only work on at most one item at a time
past atlas
ah that's okay to be honest just understanding this in of itself would be very helpful
pallid tapir
nah, if I name my employees w_1, w_2, ..., w_n, then I'm saying that w_i works at a rate of r_i
past atlas
but i am interesting in learning how that works too, that only one worker can work on an item at a time
oh okay so r is the rate
and r_n is the rate for that n worker?
pallid tapir
I think it'd have to do with determining the first person to have nothing to do on their own, then hijacking the slowest worker's project if they're faster
then keep going like that
pallid tapir
oh okay so r is the rate
idk if this helps, but: I didn't define r. I defined n productivity rates, denoted r_1, r_2, ..., r_n
n is a reserved character denoting the number of workers, so we can't use that as an index
past atlas
man im lost, but have i at least understood things right that the r in r_n is the rate for the n'th worker?
pallid tapir
instead of exhaustively saying the first worker works at a rate of r_1, the second works at a rate of r_2, and so on, I just say that the i'th worker has a rate of r_i
past atlas
since there are many different workers?
and you chose i arbitrarily or is that convention if n is already being used for something else?
pallid tapir
ah, sorry, that's conventional. if I declare n items and say that the i'th item has attribute r_i, it's implied that i is like, a dummy index, that I use to quickly declare the n attributes. by context, the dummy variable i will take on integer values between 1 and n as I define each of the n attributes
if you like programming, it's like saying R = [r_i for i in range(1, n)]
past atlas
i think maybe im being unclear
basically think of me as a high school student who is bad at math
i barely understand any of the terminology used on here but i am slowly learning
i somehow got into uni despite this
I am also Swedish so I don't know what range means
I think it has something to do with the number line or sets
pallid tapir
ah word, my bad lol
pallid tapir
if you like programming, it's like saying `R = [r_i for i in range(1, n)]`
this "range" is different from the "range" of a function
past atlas
R are the real numbers?
pallid tapir
I also wrote that to just be a different R, I'm sorry lol
past atlas
no that's alright, i'm probably just out of my depth even with this question
pallid tapir
I think we should ignore that example lol
past atlas
maybe this would be better
actually no it wouldnt be
damn I suck at math
i will get back to you when i can at least formulate a question in some manner
thanks for trying to help
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marsh citrusBOT
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slim summit
hey, I'm trying to understand the relationship of this trig identity:
C_1cos(theta*t) + C_2sin(theta*t) = C_3cos(theta*t+ϕ)
but I don't really see how Acos(theta*t+ϕ) is a sum of these two vectors? I know that A = sqrt(A^2 + B^2) and that phi = arctan(B/A)
but when I try doing measurements it doesn't add up am I doing something wrong?
marsh citrusBOT
hey, I'm trying to understand the relationship of this trig identity:
C_1cos(th...
slim summit
for context, it's for differential equations for Dampened harmonic motion
marsh citrusBOT
@slim summit Has your question been resolved?
quaint kettle
can anyone tell me a very logic reason why am i bad at math and why the others is really simple to solve math equations?. Give me tips too to improve more
marsh citrusBOT
can anyone tell me a very logic reason why am i bad at math and why the others i...
pallid tapir
can anyone tell me a very logic reason why am i bad at math and why the others i...
I think it's all about practice, mentality (attitude), and intuition personally
next raft
go study
late geode
what are examples of things you're struggling with?
it may be an issue of how you're being taught things
quaint kettle
like the mindset tho, like math is just stressful subject and i dont wanna learn about it
and u get humiliated sometimes
late geode
that's gonna be an issue
next raft
sounds like a bad learning environment u are in
quaint kettle
yups
leaden monolith
math is a skill and as with all skills it comes with practice and experience
you're only bad at math because you haven't been exposed to it enough
next raft
study with your friends and classmates
quaint kettle
can u like give me a good mindset how should i picture math
leaden monolith
can you ride a unicycle
next raft
so abstract lol
next raft
i think u just learn it slowly using the method of your choice
dont think too much about this
quaint kettle
i think u just learn it slowly using the method of your choice
thankss
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simple kayak
Hello i'm having trouble in trying to visualize this box fractal and i can't determine what the number of self-similar pieces are
marsh citrusBOT
Hello i'm having trouble in trying to visualize this box fractal and i can't det...
simple kayak
I assume the scaling factor is 1/3 so all i got for now is
D= log(n)/log(s)
D= log(???)/log 3
When i try drawing the box, it ends up looking like a cross and i'm not sure if that's correct
wheat moth
I tried what I told you in the hint <:wut:1269167408896016394>
wut did u get
i tried this
marsh citrusBOT
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marsh citrusBOT
@simple kayak Has your question been resolved?
small stream
Let the function ( f(x) ) be once differentiable on the interval [0,1], and ( f(0) = 0 ), ( f(1) = 3 ), and ( \int_{0}^{1} f(x) e^x , dx = e ). Prove that: (1) There exists a ( c \in (0,1) ) such that ( f(c) = c^{2} + 2c ). (2) There exist distinct ( c_1, c_2 \in (0,1) ) such that ( f'(c_1) - f'(c_2) = 1 ).\
Any hints
marsh citrusBOT
Let the function \( f(x) \) be once differentiable on the interval [0,1], and \(...
elfin berryBOT
riyobi
marsh citrusBOT
@small stream Has your question been resolved?
marsh citrusBOT
@small stream Has your question been resolved?
fair kindle
@small stream Insane question by the way xD
I would give you a hint but, this style of hints is a bit too advanced compared to questions we have done before.
consider the function $g(x)=f(x) - x^2 - 2x$
elfin berryBOT
Mohamed Mohsen
fair kindle
this would be my hint
explore this function a bit and see if you can relate it to your question
small stream
I'm really fed up with this kind of mean value theorem problem😭
g(0) = f(0) - 0^2 - 2(0) = f(0) = 0
g(1) = f(1) - 1^2 - 2(1) = 3 - 1 - 2 = 0
fair kindle
this problem is very neat though
fair kindle
g(0) = f(0) - 0^2 - 2(0) = f(0) = 0
g(1) = f(1) - 1^2 - 2(1) = 3 - 1 - 2 = 0
beautiful observation
keep going
what more can you deduce ?
what does this function look like ?
small stream
hmnn
The exact shape of g(x) depends on the properties of f(x) so we dont know
[\int_0^1 f(x) e^x , dx = \int_0^1 \left( g(x) + x^2 + 2x \right) e^x , dx ] \Expanding this: [ \int_0^1 f(x) e^x , dx = \int_0^1 g(x) e^x , dx + \int_0^1 x^2 e^x , dx + \int_0^1 2x e^x , dx ]
[\int_0^1 x^2 e^x , dx = \left( 2 + e \right)]
[\int_0^1 2x e^x , dx = \left( e + 1 \right)] [e = \int_0^1 g(x) e^x , dx + (2 + e) + (e + 1)]
[\int_0^1 g(x) e^x , dx = -2e - 3]
elfin berryBOT
riyobi
fair kindle
how did you calculate the x^2e^x integral
you are on the right track, I just think the calculation is not correct
you forgot the slash
at the end
[\int x^2 e^x , dx = x^2 e^x - \left( 2x e^x - 2 e^x \right) ]
small stream
[\int x^2 e^x , dx = x^2 e^x - \left( 2x e^x - 2 e^x \right) ]
fair kindle
oh yea it doesnt xD
elfin berryBOT
riyobi
fair kindle
**riyobi**
substitute the bounds
when you put 1
you have e
when you put 0, you get 2
but then you need to subtract not add
so it's e - 2
wait how
small stream
Nvm, mb🥲
elfin berryBOT
Mohamed Mohsen
small stream
0(x=1)
small stream
-2
fair kindle
yea it's -2 at zero
don't forget that
so when you use the fundamental theorem of calc
it's 2
because you subtract the second bound
adding the two integrals is adding e - 2 + 2
what is that ?
small stream
[e = \int_0^1 g(x) e^x , dx + e ]
elfin berryBOT
riyobi
small stream
Uhh wait what's the antiderivative of g(x)ex
fair kindle
bro xD
fair kindle
**riyobi**
use this picture
small stream
Or we can get g(x)ex is odd function
fair kindle
what is the integral of g * e^x
i'm confused, you already solved it
why did we go back
small stream
what is the integral of g * e^x
0
small stream
But what can we derive things from it
fair kindle
oh boy you can derive alot from that
we have a function that has g(0) = g(1) = 0
and its integration times a positive function ( e^x in that case) is also zero
this must have one of two meanings
either the function g(x) = 0 or it has a zero in the interval (0,1)
tell me why
small stream
Symmetry
fair kindle
the function inside can't be fully positive or fully negative
yes
there must have been some ... not really symmetry but you need to have the function be positive in places and negative in places
or you need the entire function to be zero itself
but you know our function is differentiable and that means it's continuous
ofc so is g because you just added a polynomial to f
this means by the intermediate value theorem, there must be c inside (0,1) such that g(c) = 0
small stream
Maybe some thm
As I said, it's some thm, it's IVT 😭
fair kindle
forget about the theorem xD you can't get there without your understanding
you understand what you are doing then you write it using proper theorems
i wasn't thinking ivt when i got this idea
i was thinking the integral can't be zero if g had a singular polarity
small stream
I understand now
fair kindle
great and what does it mean for g(c) to be equal zero ?
small stream
(1) is proven
fair kindle
exactly
now think about part 2 and tell me what you think
I think it's easy so give it a shot now after you got part 1
part 1 was the hard work
small stream
Since g(0)=g(1)=0, we know there's a c s.t. g'(c1)=0
fair kindle
okay keep going
small stream
And since (1) we know that there's g'(c2)=0 ?
small stream
g"(m)=0 , umm wrong direction ig
fair kindle
i think it's a good practice to avoid naming the second derivative at all since the question mentioned a differentiable function only so it doesn't have to have a second derivative
small stream
[ f'(c) = \frac{f(b) - f(a)}{b - a} ] [ f'(c) = \frac{f(1) - f(0)}{1 - 0} = \frac{3 - 0}{1} = 3 ]
elfin berryBOT
riyobi
small stream
Seems like I need a little hint
fair kindle
**riyobi**
we are getting there xD
you are on the write track just using wrong points
you have 0,c,1 and we got c from part one, you can use those to make two points
small stream
[ f'(c_1) = \frac{f(c) - f(0)}{c - 0} = \frac{c^2 + 2c - 0}{c} = c + 2, ] [ f'(c_2) = \frac{f(1) - f(c)}{1 - c} = \frac{3 - (c^2 + 2c)}{1 - c}. ]
[ f'(c_1) - f'(c_2) = \left( c + 2 \right) - \frac{3 - (c^2 + 2c)}{1 - c}. ]
elfin berryBOT
riyobi
small stream
I guess this will result in 1
Yes it is
Wow, this question is not that difficult as I imagine
Thank you so much🙏💕
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marsh citrusBOT
fair kindle
Thank you so much🙏💕
you are most welcome ^ ^
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misty plume
Can anyone explain me what is mod 6
marsh citrusBOT
Can anyone explain me what is mod 6
cloud field
remainder after division by 6
marsh citrusBOT
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dull mesa
Hello all 😊
This is more of a theoretical question, but - We can obtain the amount of possible combinations of k elements from a set of n total elements with repetition allowed using the binomial coefficient (n + k - 1, k).
However, do we have a method / algorithm for obtaining the actual combinations?
For permutations, one could "itemwise" iterate over all possible options until all are obtained, but how could this be done for combinations?
Is there a more efficient / straightforward approach than generating all permutations and sorting + filtering out already existing combinations?
hollow glen
i think because order does not matter, one could just make up some order for these n elements and list up only the sorted combinations, which would then look like regular counting, just that whenever a digit wraps around, it has to start from the value of its left neightbor (after it has been incremented), for example
1 ... 1 1 1
1 ... 1 1 2
...
1 ... 1 1 n
1 ... 1 2 2
1 ... 1 2 3
...
1 ... 1 2 n
1 ... 1 3 3
...
1 ... 1 n n
1 ... 2 2 2
sleek lake
yeah you "iterate" the binomial coefficient
marsh citrusBOT
@dull mesa Has your question been resolved?
sleek lake
i don't know exactly how, it's not that easy, here's python's "combinations" for example
blissful terrace
I throw defective die 10 times, probabilty of even face occuring 5 times is twice of even face occuring four times , probabilty of even face occuring?
marsh citrusBOT
I throw defective die 10 times, probabilty of even face occuring 5 times is twic...
blissful terrace
so what i did is
let p of even throw is x
So
(x)⁵(1-x)⁵ = 2(x)⁴(1-x)⁶
But its wrong
Idgt
tired knoll
Hello, can you please provide a picture of the question you're working on
marsh citrusBOT
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blissful terrace
marsh citrusBOT
blissful terrace
how do i start
marsh citrusBOT
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dawn pendant
marsh citrusBOT
dawn pendant
can you go another channel im already here
viral wyvern
marsh citrusBOT
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marsh citrusBOT
@dawn pendant Has your question been resolved?
marsh citrusBOT
@dawn pendant Has your question been resolved?
wheat moth
bruh
dawn pendant
?
marsh citrusBOT
@dawn pendant Has your question been resolved?
round furnace
Sketchy yes but mathematically it seems to check out, your conclusion also seems valid
But sketchy can be said about the entire exercise because never have I ever seen a simple mass spring damper system ma+cv+kx=0 being written in such a cursed way with momentum p🤮🤮
marsh citrusBOT
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versed niche
Money money money
dawn pendant
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timid prairie
Hello does anyone know how this equation is formed?
marsh citrusBOT
Hello does anyone know how this equation is formed?
timid prairie
shadow ibex
2a + 2pi*(b/2) = 400
...
🤔
We need some excuse to express b in terms of a
Oh, maximized area
No 💩
A = ab + pi*(b/2)^2
shadow ibex
A = ab + pi\*(b/2)^2
A(a) = a[2(200 - a)/pi] + pi(200-a)^2
wheat moth
hm
timid prairie
why can the 2/pi be seperated from the rest of the equation?
<@&286206848099549185>
still temple
I am not a calculus expert, but try to do 2 divided by 3.14 which is pi in number
timid prairie
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wary bluff
How is this hint useful?
marsh citrusBOT
How is this hint useful?
wary bluff
|p(k+1)| = |p(k) U p(k+1)| ?
acc nah idk
ngl I was thinking of something like this
Inductive step: WTS p(k+1) = T(k+1)
T(k+1) = T(k-2) + T(k-4) - T(k-7)
and by inductive hypothesis we know:
p(k-2) = T(k-2), p(k-4) = T(k-4), p(k-7) = T(k-7)
or maybe for p(k+1)
let A = p(k-2) and B = p(k-4) and then A n B = p(k-7) somehow?
ye shit idk
<@&286206848099549185>
marsh citrusBOT
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@wary bluff Has your question been resolved?
marsh citrusBOT
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vestal forge
I'm not quite sure how to formally do this but if you look at the two sets:
\begin{align*}
A={(a,b)\in\mathbb{N}^2|3a+5b=k-2} = {(a,b)\in\mathbb{N}^2|3(a+1)+5b=k+1}\
B={(a,b)\in\mathbb{N}^2|3a+5b=k-4} = {(a,b)\in\mathbb{N}^2|3a+5(b+1)=k+1}
\end{align*}
elfin berryBOT
waler
vestal forge
Notice that if you take any solution (a,b) from set A, and 'transform' it to (a+1,b), you gain a solution to the (k+1)-cent problem where you must use a 3-cent stamp. That is a solution that has at least one 3-cent stamp.
So essentially, for every element in A, you would be able to build a solution for the problem "create a postage of (k+1) cents using at least one 3-cent stamp and some arbitrary amount of 5-cent stamp".
Similarly for the set B, taking any elements from set B, you would be able to build a solution for the problem "create a postage of (k+1) cents using at least one 5-cent stamp and some arbitrary amount of 3-cent stamp".
and if you now consider the union of A and B. Using any element from this set you can build a solution to the problem "create a postage of (k+1) cents using at least one 3-cent stamp or at least one 5-cent stamp". Which is equivalent to using either 3-cent stamp or 5-cent stamp to make a postage of (k+1) cents.
Which means that card(AUB) = p(k+1).
elfin berryBOT
waler
vestal forge
So you can now prove that p(k+1) = p(k-2)+p(k-4)-p(k-7) using this and the hint given.
Though a problem is this is still very crude and I'm not sure what would be a more fomal way to represent it
quasi gust
yea wait mb
still temple
yea whatever
marsh citrusBOT
still temple
now open a new channel
😭
quasi gust
done