do you know the discriminant
#help-49
restive gazelle
restive gazelle
yes
civic slate
Yeah so that still just makes r = (-b plus or minus sqrt(b^2 - 4ac))/2a no?
restive gazelle
yes
civic slate
So was my answer right from the start?
restive gazelle
Yeah so that still just makes r = (-b plus or minus sqrt(b^2 - 4ac))/2a no?
getting this is not sufficient to talk about the nature of roots
your screenshot has a constraint b^2-4ac>0
we need to use it
civic slate
I thought all the discriminate being positive means is that our equation will have two real roots
Am I forgetting something from algebra?
restive gazelle
I thought all the discriminate being positive means is that our equation will ha...
yes this is what we need to state
restive gazelle
you just gave the general formula but okay
do you have any idea of how to do the next part?
civic slate
hm
Well
We would always get a0nr^n + ... + a(n-1)2r^2 + a(n)r = 0
After subbing in e^rt
restive gazelle
yes
civic slate
Okay
restive gazelle
the bounding number is n
but there can be repeated roots
which reduces the number of distinct r
civic slate
How might you go about finding those solutions though
restive gazelle
there isnt a general formula for an nth degree polynomial
you find solutions numerically often
if u r lucky, u can factor it
restive gazelle
okay
civic slate
Okay
So I kind of understand this one
For (i), if we do n = 0, it becomes y prime + p(x)y = q(x) and can be solved using the integrating factor, giving us y = (integral from a sub zero to x of u(s)g(s)ds + c)/u(x). If we do n = 1, we get y prime + p(x)y = q(x)y.
I was thinking that I could maybe divide by y to get (y prime)/y = q(x) - p(x), and do something from there
maybe get like y = y'/(q(x)-p(x))?
lyric charm
btw you can and should write an apostrophe for prime
like y'
and also _ for subscripts like a_0
civic slate
I forgot the apostrphe existed lol
holy
I live in Japan now so I kind of have been forgetting English honestly
civic slate
btw you can and should write an apostrophe for prime
Do you have any ideas
For the problem
lyric charm
unfortunately no, i forgor too much about DEs
lyric charm
been long enough ig
civic slate
<@&286206848099549185>
Oh wait, if n = 1, we can obtain y' + (p(x)-q(x))y = 0
And solve regularly from there
restive gazelle
yes
civic slate
Wait so for the integrating factor what would that simplify to?
Actually I guess you just cant
It doesn't matter I guess
I will just call it u(x)
restive gazelle
you can just do separation of variables
thin sedge
nah like how would you integrate $e^{(p(x) - q(x))}$
grand pondBOT
డ్ర్యాగ్లొక్స్
thin sedge
it's better to just have a u(x) sub as op said
restive gazelle
nah like how would you integrate $e^{(p(x) - q(x))}$
what are u even saying
shadow rose
Again, variable separation would be the best bet
restive gazelle
the goal here is to express in terms of P(x) and q(x)
thin sedge
what are u even saying
integration factor
civic slate
No the integration factor is that, but you dont integrate the whole thing just the exponent
restive gazelle
yes of course the two methods are equivalent
civic slate
Which cant be simplified really so I will just leave it as u(x)
thin sedge
No the integration factor is that, but you dont integrate the whole thing just t...
omg wait that is right
sry it's 3 am my mind is gone
civic slate
After I finish writing down my thoughts I will start ii, which I think I will actually need help with
Okay
By substitute does it mean replace y^n with y^1-n? or does it mean something else
shadow rose
It means something else
civic slate
What does it mean
shadow rose
First of all, you need a y^1-n to appear somewhere
civic slate
I dont get why he assigned all this before even the first lecture bruh
thin sedge
What class is this for?
civic slate
First of all, you need a y^1-n to appear somewhere
Thats what would happen if you replace y^n with y^1-n bruh
civic slate
What class is this for?
DiffQ, I am taking it through Stanford Online High School though so its kind of fast paced I think
shadow rose
As in not replacing y^n, but dividing it
civic slate
Or maybe its slow and I am tweaking
solid iris
substitute means rewrite the ODE for y as an ODE for v
civic slate
And how would I do that...
shadow rose
And how would I do that...
First, divide by y^n so that y^1-n appears
The first step is to always make sure that the substitution can be seen in the ODE
civic slate
Oh, giving us y'*y^-n + p(x)y^1-n = q(x)?
civic slate
Oh, dv/dx = (1-n)y^-ndy/dx
shadow rose
With respect to x
shadow rose
Oh, dv/dx = (1-n)y^-ndy/dx
The y also has to be differentiated with respect to x
thin sedge
Oh, dv/dx = (1-n)y^-ndy/dx
Chain rule
shadow rose
Ok, now substitute everything into the ODE
civic slate
better?
shadow rose
Everything good so far?
civic slate
YEah
shadow rose
Sub both into the ODE
civic slate
I think this would make a lot more sense if a lecture had happened yet lol
I think I understand though
Is it dv/dx * 1/(1-n) + p(x)v = q(x)?
shadow rose
Yeah
shadow rose
Because if you do integration factor, the coefficient of dv/dx should be 1
shadow rose
Could you write that out and send the picture if possible?
civic slate
Wait, I didnt even have to do that though
Doesn't it just say to prove its a linear equation?
shadow rose
Yeah
civic slate
So I could just say v'/(1-n) + p(x)v = q(x), follows the form a_nv^n + ... + a_0v = 0, and is therefore linear
shadow rose
Just leaving it in the form v'/(1-n)+p(x)v=q(x) already shows that it is linear
civic slate
Man
That was a waste of time
Okay now time for iii
I also dont have an eraser so I am hoping he is okay with me just putting it in a box and saying dont mind this
I got a question
For the integrating factor for iii, how do I choose what a_0 is
shadow rose
So you did the substitution right?
civic slate
I got my substituted equation as v' + 2(wierd e)v = -2(omega)
shadow rose
So your integrating factor is e^(integral of coefficient of v)
civic slate
Yes, but what are the bounds
shadow rose
There are no bounds
civic slate
Oh
shadow rose
Its just indefinite integral
civic slate
Whoops
shadow rose
Also you dont need the arbitrary constant
shadow rose
Epsilon
shadow rose
Okay, so just e^(weird e)v^2
Just the coefficient of v
Not the entire term
And you're integrating with respect to x
shadow rose
Just cross it out
civic slate
Yes but I have to send this in
I hope he is fine with it I guess lol
Okay
So I got e^2(epsilon)x * y = integral of -2e^2(epsilon)x * omega
shadow rose
Yup
civic slate
Okay, I got v = - omega over epsilon + c
Is that right?
Which mean y = plus or minus sqrt( 1 / (-omega/epsilon + c))
shadow rose
Should be
shadow rose
I got my substituted equation as v' + 2(wierd e)v = -2(omega)
Check your right hand side here
I dont think theres supposed to be a minus sign
civic slate
Why
shadow rose
You did multiply everything by -2, right
civic slate
I got it from the negative two in the integral
shadow rose
Did you multiply by -2 after you sub everything in?
shadow rose
The original question there was a minus sign on the right hand side
shadow rose
Also your arbitrary constant should have an e^-2(epsilon)x
civic slate
Okay yeah it’s positive
Holy shit bro
I am going to tweak out
Okay I am buying an eraser
Okay but after that it’s right right?
shadow rose
Yeah
civic slate
Isn’t it kind of crazy that this is all before even one lecture?
How long does it take to get to this point usually? Is it really only like one day?
civic slate
But for a problem like this, the target is below 10 minutes
Yeah I think now that I understand the method I could do it in under 10 minutes
I honestly did the first substituted equation part in like 2 min
I just didn’t understand the full process
But after that pain I do now
shadow rose
Just keep practising, you'll get the feeling of it in no time
shadow rose
Oh you're in Japan?
civic slate
Well, mugging do be like that
Nah getting mugged is so rare in Japan
I hope that the food for dinner tonight is good
shadow rose
Nah getting mugged is so rare in Japan
No, I dont mean getting mugged as getting robbed
civic slate
I swear my host family hates good things
shadow rose
Mugging in my culture means studying
So like you go to a cafe or a library and you mug means you study a lot
civic slate
The food is always like really bad, and I am not saying that because it’s Japanese as that’s all I eat, but it’s like only the super traditional parts of Japanese cooking
Which you would think is good but it really is not
shadow rose
Really? Interesting
shadow rose
Nah getting mugged is so rare in Japan
I keep forgetting that mugging is local slang and that internationally it means robbing
civic slate
Really? Interesting
Yes the mountain potato tastes like something I can’t even describe with the worst texture ever, and my host dad legit knows, he said “challenge” and handed me it
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midnight plankBOT
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sand flume
Can someone point me in the right direction. We have 1/2f(c) < f(x) < 3/2f(c) for x_0 - delta < x < x_0 + delta. But I am not sure what to do after
midnight plankBOT
Can someone point me in the right direction. We have 1/2f(c) < f(x) < 3/2f(c) fo...
lyric charm
f(x) > f(c)/2 in an interval of length 2delta
sand flume
isn't the whole thing of length 2 delta but one part of it just delta?
sand flume
f(x) > f(c)/2 in an interval of length 2delta
?
lyric charm
it does not matter what its length is that much
the idea is, your function f is bounded below by the function that takes value f(c)/2 on that interval and 0 elsewhere
and the integral of this bounding function is obv positive
sand flume
so your saying that if the integral of the function is bounded below by a positive value it could never be equal to 0
supple sigil
if it's bounded below by a positive constant on a non null set
in your case, the open interval (c-δ, c+δ)
sand flume
so am I understanding everything correctly. We assume that int from a to b f(x)dx = 0 and say assume that there is a point f(c)>0. Then we say we have 1/2f(c)< f(x) < 3/2f(c) for c- delta < x < c + delta. we have int from c - delta to c > 1\2f(c) *delta
supple sigil
What can you say about the integral of f from c-δ to c+δ?
sand flume
between 1\2f(c) detla and 3/2f(c) delta
lyric charm
between 1\2f(c) detla and 3/2f(c) delta
one of these matters way more than the other
sand flume
the 1/2f(c) but I dont' see how we can compare it to the whole interval since it is only defined on c-delta.
Also for the proof itself is the idea just to show that if f(x) has a point greator than 0 and the int f(x) = 0 then we get a contradiction. But can't you just use the fact that we can represent area with integrals and if we have the function positive somewhere the area must be positive
supple sigil
Maybe a picture will help. The only main idea is that the integral of f over [a,b] is at least the area of the shaded rectangle, which is positive
supple sigil
Also for the proof itself is the idea just to show that if f(x) has a point grea...
This is essentially the argument but made rigourous
midnight plankBOT
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Available help channel!
delicate meadow
Find the sum of the expression (2n-1)!!/(2n + 2)!! from n = 1 to n = infinity.
dreamy lichen
$\frac{1}{2\cdot4}+\frac{1\cdot3}{2\cdot4\cdot6}+\frac{1\cdot3\cdot5}{2\cdot4\cdot6\cdot8}...$
so this sum?
grand pondBOT
MathIsAlwaysRight
delicate meadow
Yes.
dreamy lichen
maybe factoring out the 2s from the denominator would help
hmm, not that much. Maybe converting it all to normal factorials then
delicate meadow
Yeah.
(2n)!! = (2^n * n!)
(2n-1)!! = (2n)!/(2n)!! = (2n)!/(2^n * n!)
So the expression equals:
1/2 * 1/4^n * (2n)!/(n! * (n+1)!)
lusty python
**MathIsAlwaysRight**
idk if this helps
but first fraction can be written into
$\frac{1}{2^2 \cdot 1 \cdot 2}$
grand pondBOT
1 divided by 0 equals Infinity
lusty python
second fraction can be written into
$\frac{1 \cdot 3}{2^3 \cdot 1 \cdot 2 \cdot 3}$
grand pondBOT
1 divided by 0 equals Infinity
lusty python
third fraction can be written into $\frac{1 \cdot 3 \cdot 5}{2^4 \cdot 1 \cdot 2 \cdot 3 \cdot 4}$
grand pondBOT
1 divided by 0 equals Infinity
dreamy lichen
$\frac{\operatorname{nCr}\left(2,1\right)}{2\cdot2^{3}}+\frac{\operatorname{nCr}\left(4,2\right)}{3\cdot2^{5}}+\frac{\operatorname{nCr}\left(6,3\right)}{4\cdot2^{7}}+\frac{\operatorname{nCr}\left(8,4\right)}{5\cdot2^{9}}$
grand pondBOT
MathIsAlwaysRight
dreamy lichen
with a bit more algebra, it can be brought to this
lusty python
**MathIsAlwaysRight**
that emerges a clearer pattern
lyric charm
Find the sum of the expression (2n-1)!!/(2n + 2)!! from n = 1 to n = infinity.
some generatingfunctionology is nearly certain to be lurking nearby
lusty python
$\sum_{n=1}^{\infty} \frac{\binom{2n}{n}}{(n + 1) \cdot 2^{2n + 1}}$
lyric charm
$\sum_{n=1}^{\infty} \frac{\binom{2n}{n}}{(n + 1) \cdot 2^{2n + 1}}$
\binom{n}{k} exists btw
dreamy lichen
2n+1, not n+2
grand pondBOT
1 divided by 0 equals Infinity
lusty python
**1 divided by 0 equals Infinity**
hope this can help some helpers out here
grand pondBOT
lusty python
only missing $x^{2n + 1}$
grand pondBOT
1 divided by 0 equals Infinity
dreamy lichen
we could plug in 1 if that was the issue
lyric charm
value at x=1
dreamy lichen
the issue is 2n+1 instead of n+1, but thats prolly fixable
dreamy lichen
**1 divided by 0 equals Infinity**
oh, technically this is 2n+2
lusty python
how about the $\frac{1}{4^n}$?
grand pondBOT
1 divided by 0 equals Infinity
lusty python
oh, technically this is 2n+2
i don't think so
dreamy lichen
how about the $\frac{1}{4^n}$?
$\sum_{n=1}^{\infty}\frac{\binom{2n}{n}}{\left(2n+2\right)4^{n}}$
grand pondBOT
MathIsAlwaysRight
lusty python
my bad
lusty python
the next one 6 -5
rewrite 1?
lusty python
4-3/2.4+ 3(6-5)/2.4.6
i think that's what i said
silk forge
ye
lusty python
**MathIsAlwaysRight**
this is almost $\arcsin(1)$
grand pondBOT
1 divided by 0 equals Infinity
dreamy lichen
oh thats smart
dreamy lichen
this is almost $\arcsin(1)$
forget it
we could prolly integrate and that'd get the 2n+2, but thats overcomplicated
the telescope is much simpler
lusty python
4-3/2.4+ 3(6-5)/2.4.6
hm
dreamy lichen
4-3/2.4+ 3(6-5)/2.4.6
@delicate meadow this is the simplest solution probably
$\frac{4-3}{2\cdot4}+\frac{\left(6-5\right)\cdot3}{2\cdot4\cdot6}+\frac{\left(8-7\right)\cdot3\cdot5}{2\cdot4\cdot6\cdot8}+\dots$
grand pondBOT
MathIsAlwaysRight
dreamy lichen
now split the terms and it goes 🔭
lusty python
$\sum_{n=1}^{\infty} \frac{\prod_{m=1}^{n} (2m - 1) \cdot ((n + 3) - (n + 2))}{\prod_{m=1}^{n+1} (2m)}$
grand pondBOT
1 divided by 0 equals Infinity
midnight plankBOT
@delicate meadow Has your question been resolved?
delicate meadow
Thanks/
dark cloud
Is there a way to improve my method of solving equations?
aka. if I am to convert 3 minutes 5 seconds to a fraction of an hour,
I would do 1/20 + 5/3600, and then end up with 740/14400 which i'd then try using long division and spend the next 5 minutes making errors
midnight plankBOT
Is there a way to improve my method of solving equations?
aka. if I am to conver...
dark cloud
if the answer is just doing more problems i'll accept it
runic hamlet
I'd do the same probably
a few factors you can quickly see
cancel one of the zeros
divide by 2
and then you are already done in this case
oh I didnt check your fractions
to compute 1/20+5/3600 you really dont need to get a common denominator of 14400
not sure where that is from
its just 180/3600+5/3600
equivalently, convert the 3 minutes into 180 seconds first
paper prism
Actually 720 can serve as a common denominator if you do the 5s on the second frac
runic hamlet
or that
lilac finch
oh
midnight plankBOT
@dark cloud Has your question been resolved?
worthy tundra
if theres only minutes and seconds, i might convert minutes into seconds first
then seconds to hour
3 min = 180s
180 + 5 = 185s
185 / 3600
at least your base is always 3600
dark cloud
equivalently, convert the 3 minutes into 180 seconds first
oh shoot
dark cloud
Actually 720 can serve as a common denominator if you do the 5s on the second fr...
right
sand flume
For number 8 how can I justify the usage of the weighted mean value theorem if it says this is valid if g is strictly positive or negative. Question 8 says that g is just continuous. Also how can I deal with endpoints. We have m≤ f(x) ≤ M and I found m≤0≤M if I can use the weighted mean value theorem. I was thinking of doing mg(x)≤ f(x)g(x). m int g(x) ≤ int f(x)g(x). Letting m = f(x) we have f(x) ≤0. But I feel like the way I set the min is iffy.
midnight plankBOT
For number 8 how can I justify the usage of the weighted mean value theorem if i...
sand flume
midnight plankBOT
@sand flume Has your question been resolved?
sand flume
<@&286206848099549185>
sand flume
Wdym?
Isn't the only way to proceed for this proof with that theorem
But even with it I can show that the endpoints are 0 just everything in between
supple sigil
if f is nonzero at some point c, say f(c) > 0, then you can do a similar trick as what we did earlier with a suitably constructed bump function g
to get int fg > 0
sand flume
But g can be anything
solid iris
if youve already done problem 7 then you can use it to very quickly do problem 8
runic hamlet
there is an obvious choice of g here which is helpful
sand flume
So we can use a positive g
sand flume
if f is nonzero at some point c, say f(c) > 0, then you can do a similar trick a...
Are you saying that it is like this but we are scaling the vertical axis by g https://cdn.discordapp.com/attachments/1018703621841494076/1459162613282897943/Screenshot_20260109_203041_CamScanner.jpg?ex=69624694&is=6960f514&hm=2b8ceddb016cd8748645dbf8406b0bd203437c52e016cd61a22f87cc0399db35&
runic hamlet
something like that in some sense, sure
but think easier
how can you reduce it to problem 7
sand flume
If g(x) was 1
But aren't we then just proving it for a specific case of g not g in general
solid iris
no, P7 assumes nonnegative integrand
runic hamlet
we are allowed to choose any g
whenever we are allowed to do that, then we probably only need one specific choice
sand flume
What if we said g(x) = 0. f wouldn't play a role anymore.
solid iris
setting g=0 gives 0=0… no new info
sand flume
g should probably depend on f
Aren't we alerting the problem statement then?
runic hamlet
what
sand flume
I'm confused on why g needs to depend on f. All the problem says is that g is continuous on a,b
runic hamlet
we are supposed to show something about f
so probably our proof should depend on f
our proof probably proceeds by choosing a smart g
so our g should probably depend on f
those are general proof strategies
at this point the specific problem doesnt even matter yet
sand flume
So by smart g we are choosing either a strictly positive g or strictly negative g right? Since then it is similar to problem 7
runic hamlet
how can we make f*g nonnegative
supple sigil
Are you saying that it is like this but we are scaling the vertical axis by g ht...
Well I can again draw what I had in mind, I'll spoiler it
supple sigil
This is the general idea anyway, it's quite similar to the last question as you said
sand flume
how can we make f*g nonnegative
If we had an f(c)>0 and g negative
sand flume
which sign(s) should g have
G would be positive
runic hamlet
positive*negative=negative
so that would be bad
when is g positive and when is it negative
your answer should depend on f
cause f*g depends on f
sand flume
I'm not following why g being negative is bad? Wouldn't our inqualtiy just be int a to b fg < int from c- delta to c + delta fg < 2 delta 1/2 f(c) < 0 and then we get a contradiction since it is negative
runic hamlet
I want to reduce it to problem 7
I do not want to do any epsilonics
problem 7 already did epsilonics
sand flume
Ok so your saying if we want it to be strictly like 7 g is positive
runic hamlet
no, g is sometimes positive and sometimes negative
I want f*g>=0
f can sometimes be positive or negative
I need to cancel that out
how can I choose g
sand flume
G should be negative everywhere f is negative and positive everywhere f is positive
solid iris
now think of a VERY easy choice of g to ensure that
runic hamlet
and can you find a continuous function g which does that
sand flume
Just f
runic hamlet
yes!
g=f
apply problem 7
done
one line
the choice g=f appears over and over again
in all kinds of problems
sand flume
Wait can you explain again why this still keeps the result general
runic hamlet
by choosing g=f we get that int f^2 =0
but f^2>=0
so f^2=0 by problem 7
so f=0
we were allowed to choose any g we wanted
and we chose a g we liked
sand flume
I don't see how that aligns with every function g
solid iris
P8 gives us a property that is true for all g. logic allows us to pick a specific g and conclude the property holds for it too
sand flume
Oh Im misreading. I thought it was saying show that for every function g f(x) must be 0 if the integral is 0
Ok thx guys for the help
.solved
midnight plankBOT
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Available help channel!
small warren
is my reasoning correct here?
midnight plankBOT
is my reasoning correct here?
small warren
I found that angle ACB is congruent to angle DCE because of vertical angles
I also know that angle B is congruent to angle D because of alternate interior angles.
So I concluded that both triangles are similar, and I found the measure of CE by dividing the side that has a green and red angle with the smaller triangle and making it equal to the side with the green and unknown.
I made CE = x by the way.
Or x = CE
Line BD is a transversal that passes through line AB and DE, and AB and DE are parallel as given from the original diagram.
wanton spade
is 5/3=4/CE or 5/3= CE/4 🤔
sudden cove
ratio you are looking for is ||5 : 3 :: 4 : x||
small warren
ratio you are looking for is ||5 : 3 :: 4 : x||
Yea!
small warren
ratio you are looking for is ||5 : 3 :: 4 : x||
What does :: mean?
wanton spade
your reasoning apart from that is right
sudden cove
What does :: mean?
"5 is to 3 as 4 is to x"
atomic hinge
Basically =
small warren
Here's my new result.
small warren
your reasoning apart from that is right
Thanks! 
wanton spade
👍
small warren
Basically =
Oh cool
Tysm for helping me everyone! 🩷
This was on Khan Academy he didn't say to try and solve it yourself but I wanted to see what the answer could be before he just gave me all the steps!
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midnight plankBOT
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Available help channel!
river robin
You are interviewing 10 job candidates. They appear in a random order. After each interview, you must immediately decide: “hire” or “reject” (you cannot go back to someone you have already rejected). Your goal is to hire only the best candidate from the entire group. What is the best strategy? Support your answer with calculations.
midnight plankBOT
You are interviewing 10 job candidates. They appear in a random order. After eac...
nova yoke
any thoughts?
nova yoke
suppose you interview someone and they're worse than some previous candidate, should you hire or reject them?
(it's not stated, but i assume you're able to determine if someone is better or worse than ones you've already seen)
river robin
suppose you interview someone and they're worse than some previous candidate, sh...
But you don’t know if the next person will be better
nova yoke
that's true, but you know for sure this one isn't the best, if you already saw a better one
so the probability that this one is best = 0
seems to me you should definitely reject in that case
visual tiger
But you don’t know if the next person will be better
The point is to hire the best person. So if they're worse than someone you already saw, they can't be the best
nova yoke
now as for what you should do if it's the final person and they're not the best, i don't know, it doesn't say whether or not you're forced to hire someone
visual tiger
In any case you lose if you didn't hire the best person imo
nova yoke
yea i agree, so probably doesn't matter
visual tiger
I’d reject the first one
There is some idea to this strategy
To never hire the first person, to serve as a base case for the others
But imagine if the 2nd person is better than the first, do you hire them?
Or do you wait again to see the 3rd person?
And if you do wait, do you hire the 3rd person if they're better?
Etc
The question to establish the strategy is: how many people do you automatically reject at the beginning?
nova yoke
intuitively it seems likely that it's not optimal to hire the first guy, nor to wait until the last, but you need to back this up with a solid conditional probability calculation
river robin
This is so difficult
visual tiger
It seems difficult, but the calculations aren't the worst
Suppose your strategy is to reject the first k persons
And the best is in i-th position
river robin
If you’re on the 3rd person you still have higher probability of getting the best than the first two that were
visual tiger
If you’re on the 3rd person you still have higher probability of getting the bes...
You never know until you compute it
visual tiger
And the best is in i-th position
So here are the variables, k and i
Depending on those, what is the probability you hire the best?
river robin
1/10
lyric charm
how is the goodness of a strategy evaluated btw
nova yoke
If you’re on the 3rd person you still have higher probability of getting the bes...
yea i would be a bit careful with this reasoning, you might think that a strategy of skipping the first five should give you 50% success, but it will be less than that because even if the best guy is in the second half, you're not guaranteed to choose him
lyric charm
how is the goodness of a strategy evaluated btw
is it simply the chance that it picks the best candidate
visual tiger
is it simply the chance that it picks the best candidate
It should be, right @river robin
lyric charm
also once we hire someone we stop, or do we?
visual tiger
also once we hire someone we stop, or do we?
It would be too easy if we didn't stop wouldn't it
nova yoke
It would be too easy if we didn't stop wouldn't it
hire everyone!
visual tiger
Hire the first person, and hire someone else if the newest person is better than the one currently hired
lyric charm
so first guy gets rejected nmw as hiring him only gives us 10%
ooga booga strategy is to hire as soon as you see someone better than the first guy
visual tiger
But imagine if the 2nd person is better than the first, do you hire them?
Which raises this series of questions
visual tiger
Suppose your strategy is to reject the first k persons
So let's try one of those strategies : reject the first k persons, and the first one you see that is better than the first k, you hire
visual tiger
ooga booga strategy is to hire as soon as you see someone better than the first ...
So it includes the ooga-booga strategy (k=1)
But also includes others
So fix some value of k, and suppose the best is in position i (of course the one applying the strategy doesn't know that)
What's the probability of hiring them if, for example, k >= i?
river robin
So let's try one of those strategies : reject the first k persons, and the first...
Why exactly if the person is better than the first ones
visual tiger
Why exactly if the person is better than the first ones
If they're worse than the first ones, that means they're not the best
So since you only want the best, the only way you'll be willing to hire them is if they're better than the previous ones
If you still choose to not hire them, then that's a different strategy with a different value of k
visual tiger
So fix some value of k, and suppose the best is in position i (of course the one...
That's what we're doing
We're applying the principle of total events
the best must be at some position i between 1 and 10
lyric charm
i think mamusia might have me on ignore btw
river robin
i think mamusia might have me on ignore btw
no
lyric charm
ok
river robin
Idk where to start can’t you just do the thing @visual tiger
lyric charm
"do the thing"
visual tiger
I won't give you a full solution just like that, but I can guide you to it
visual tiger
What's the probability of hiring them if, for example, k >= i?
Try answering this first warmup question
Your strategy is "skip the first k people, and then hire the first one better than them"
And the best is in position "i"
What happens if k >= i?
river robin
What happens if k >= i?
You fail?
visual tiger
You fail?
Yeah, you automatically fail
Because you skipped the best
Now, let's suppose k < i
So, let's imagine what must happen for you to hire the person in position i
If you didn't hire anyone up to the person in position i-1 included
Then once you reach position i, you're gonna hire them
Because they are better than the first k persons
(Since they're the best)
visual tiger
If you didn't hire anyone up to the person in position i-1 included
So you need to not hire anyone between positions k+1 and i-1
visual tiger
The 10 people you're interviewing come in a certain order
So 'position k+1' means the k+1-th person you interview today
If k = 2 for example
And i = 5 let's say
That means you skip the first 2 people you interview today
And the best person happens to be the 5th person today (even though you don't know that)
So, after interviewing and passing the first two people
You still have to interview the 3rd and 4th person before you interview the best
And you mustn't hire them
river robin
Ok
visual tiger
so k+1 is the first person you interview after automatically skipping
And i-1 is the last person you interview before seeing the best
river robin
Ok
visual tiger
None of those people between k+1 and i-1 should be hired if you wanna hire the best
visual tiger
Your strategy is "skip the first k people, and then hire the first one better th...
So according to your strategy, they can't be better than the first k
Otherwise you would hire them, and so failing because you didn't hire the best
river robin
Mhm
visual tiger
Are there some questions about what I said up until now?
Or do we continue
If everything's fine, then I think I should show you the hint that will help you find the probability to hire the best in that scenario
I'll spoiler it if you want to think about it on your own
Here's the hint: ||Think about the best person out of the bunch you interview before the best, including the people you skipped at the beginning. So that's the best out of the first i-1 people||
||In which position(s) can he be if you want to hire the best of the best in position i?||
river robin
k - k+8?
river robin
Yes
visual tiger
That doesn't make a lot of sense
First of all, the best out of the first i-1 people can only be one of those i-1 people
So he can only be between 1 and i-1
atomic hinge
-# I know the answer, but I don't understand why the best strategy is in the "Reject until we get a better person" strategies
visual tiger
Let's imagine the specific case where k = 2 and i= 5 again
visual tiger
-# I know the answer, but I don't understand why the best strategy is in the "Re...
We can talk about it more in another channel maybe
visual tiger
Let's imagine the specific case where k = 2 and i= 5 again
So, the best of the best is in position 5
And we want to know when do we interview the best out of the first 4 people
visual tiger
So
Can we give them a name so I don't have to repeat "the best out of the first 4 people" all the time?
Let's call them Charlie
Now, what happens if Charlie is the very first person interviewed?
Does person n°3 get hired? Does person n°4 get hired?
river robin
N3
river robin
Does person n°3 get hired? Does person n°4 get hired?
The first one
visual tiger
The first one as in my first question "n°3 gets hired" or the very first person gets hired?
river robin
The first one as in my first question "n°3 gets hired" or the very first person ...
The first one
Ok aren’t you tired of this? @visual tiger
visual tiger
I'm not
If you're not interested in trying even a little bit you can just look up the Secretary Problem
But I doubt you'll learn as much as trying while being guided like we're doing
visual tiger
The first one
So you think n°3 gets hired?
river robin
I genuinely don’t understand you
visual tiger
I genuinely don’t understand you
Alright, tell me what part you didn't understand?
Is it just "n°X" that confuses you? (It means "person number X")
visual tiger
Now, what happens if Charlie is the very first person interviewed?
So I was asking, in the case "Charlie" (the best person out of the first 4 interviewed) is interviewed first, does the third person get hired? Does the fourth person get hired?
river robin
And Charlie was first?
visual tiger
Yeah right now we're supposing Charlie was the first interviewed
river robin
So I was asking, in the case "Charlie" (the best person out of the first 4 inter...
Well neither…
visual tiger
Well neither…
yeah, neither get hired
because both of them are worse than Charlie, who is in the sample of people rejected
so the 5th person, being the best, is the next being interviewed
and he's gonna get hired
meaning we win!
Same question when Charlie now is the 2nd person interviewed. Do we manage to hire the best?
river robin
Yes
visual tiger
Yes
Exactly, it's the same thing, Charlie is still in the sample of people being rejected
and so 3rd and 4th people, being worse than charlie, don't get hired
and best coming 5th is hired
Same question when Charlie now is the 3rd person being interviewed
river robin
Well then we hire 4th person?
visual tiger
Well then we hire 4th person?
let's look at it in order
First we skip the first two
and then 3rd person is Charlie
Remember we hire the first person that is better than the rejected sample
Does Charlie get hired or not?
river robin
Yes
visual tiger
Yes
Right, so Charlie being better than the rejected sample (being the best out of the first 4), is hired
so 4th person doesn't get hired
and more importantly the best in 5th, doesn't get hired
so we fail
Now finally, last possibility
Charlie now is the 4th person being interviewed
Do we manage to hire the best?
visual tiger
No.
This answer is correct, we don't manage to hire the best unfortunately
We skip the first 2 people
and now incomes 3rd person
Do they get hired or not? It doesn't matter
because even if they don't get hired, Charlie comes next
and charlie will get hired
meaning the best doesn't get hired and we fail
So in conclusion
The best only gets hired when charlie is in the rejection sample
So the probability the best gets hired
is exactly the probability Charlie is in the rejection sample
= k/(i-1)
(there are k rejection spots out of the total i-1 people before the best)
So, with our strategy "reject the first k people, then hire the first one you see that is better"
$P(\text{hiring the best one, given best = i}) = \begin{cases}0&\text{if }k \geq i\\frac{k}{i-1}& \text{if }k < i\end{cases}$
grand pondBOT
Rafilouyear2026
visual tiger
So law of total probabilities should help you finish this
\begin{align*} P(\text{hiring the best one}&) =\ &\sum_{i=1}^{10}P(\text{best} = i)P(\text{hiring the best one, given best}=i)\end{align*}
atomic hinge
-# Double backslash for new line
visual tiger
I know that, it's not gonna help, unless
atomic hinge
Oh I misread it
grand pondBOT
Rafilouyear2026
visual tiger
well it works
midnight plankBOT
@river robin Has your question been resolved?
river robin
Well uh
gleaming egret
alr anyone need help
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midnight plankBOT
i dont understand this stuff at all
violet tide
can anyone explain this
nova yoke
consider what happens to the x^4 term of (2+x)^7 when you multiply it by (3 + 1/x^3)
violet tide
280x^4
nova yoke
ok, and when you multiply that by (3 + 1/x^3)?
nova yoke
yea so you have a 280x contribution from the x^4 term of (2+x)^7
violet tide
oh so the answer is 280
nova yoke
and as you showed earlier, you also have a 1344x contribution from the x^1 term
violet tide
ye 1344
nova yoke
so both of those added together will give you the full x term
violet tide
there will be two contributions
what does that mean?
nova yoke
one from the x^0 term of (2+x)^7, and one from the x^3 term
(because x^3 multiplied by 1/x^3 gives you a constant term)
violet tide
did i do this correctly
there might be some calculation issues idk tho
cuz i didnt use calc for the choose bit
nova yoke
yep, add that to the 560 and you're good
oh wait
sorry that already includes the 560
violet tide
i added560+384
nova yoke
i lost track haha
violet tide
yea lol
nova yoke
yep you're good, that's correct
violet tide
no problem thanks so much for helping me
nova yoke
yw
violet tide
i was getting help before and i swr they were yapping for minutes
appreciate u so much
.close
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small jasper
<@&268886789983436800> spammed across channels
midnight plankBOT
lyric charm
a better question is: what was the original question you were doing?
small jasper
lyric charm
what makes it specifically malaysian
small jasper
idk from the few times I've interacted with them that's my hunch
small jasper
op seems to be gone but I'm assuming it's this
so setup was fine
anyway I'm out
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robust field
Open
lyric charm
,rccw
grand pondBOT
lyric charm
I forgot binomial
well then go reread your notes on the binomial thm
the highest degree term in this is going to be 3^n x^(2n)
that lets you find n almost immediately
robust field
Ok
wanton juniper
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civic slate
Can someone help me solve just the first one, and then after I will do the second one myself? I am just kind of confused as we still havent had a lecture for the class yet so I am still confused.
midnight plankBOT
Can someone help me solve just the first one, and then after I will do the secon...
civic slate
Like am I supposed to be taking the partial derivative first and then solving by seperation of variables or something?
<@&286206848099549185>
Bruh
No one likes me
golden crescent
No one likes me
js wait bro
civic slate
js wait bro
; (
Its been twenty minutes man
I have been waiting
Okay nvm I guess
.close
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scenic wharf
For (e) part , I have an alternate method :-
if n ∈N , then 0<(1/n)≤1
And 0<(1/n)^i ≤(1/n) ≤1 ------(1)
where i is a natural number .
If ε> 0 , there exist a natural number k, such that (1/ ε)< k
Or (1/k )< ε
If n≥k , then (1/n)≤ (1/k) -------(2)
Combining (1) and (2) , we get
(1/n)^i < ε or
|(1/n)^i - 0| < ε ,
So 0 is the limit of this sequence
midnight plankBOT
For (e) part , I have an alternate method :-
if n ∈N , then 0<(1/n)≤1
And 0<(1/...
scenic wharf
Kindle check my method
dreamy lichen
For (e) part , I have an alternate method :-
if n ∈N , then 0<(1/n)≤1
And 0<(1/...
Why does your method not even mention b?
seems like you assume b = 1/n
but not every number between 0 and 1 can be expressed as 1/n
for example 0.75 cant be expressed like that
and interestingly, your method proves (1/1)^n -> 0 as n -> inf
which is false
dreamy lichen
well, any method you use has to be able to find N such that for n > N, b^n < epsilon
when we have n in the exponent, using log is the most obvious way to extract it
but you could probably cook up some inequality which would avoid logs
but the books method is certainly the most straightforward one
scenic wharf
Can I make changes in my method , to make it correct?
dreamy lichen
i dont think so
scenic wharf
but you could probably cook up some inequality which would avoid logs
Can u give some idea
dreamy lichen
bernoullis inequality maybe
okay nvm that wont work
we need an upper bound on it, not lower bound
midnight plankBOT
dreamy lichen
maybe if we use b = 1/(1+x) for x > 0
then apply bernoullis on that
yeah, that should work
dreamy lichen
maybe if we use b = 1/(1+x) for x > 0
this is valid, because for any 0 < b < 1, you can explicitly solve for x
meanwhile when you took b = 1/n, it wasnt valid because you cant always find such n
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vapid mulch
how tonsolve this
midnight plankBOT
how tonsolve this
olive matrix
one sentence at a time
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dark cloud
midnight plankBOT
dark cloud
I don't get how this is solved
i've only ever used Event/Set and it's worked out fine, but i have no clue how this works
honest willow
Which grade r u in @dark cloud
dark cloud
finished 12th last year
lilac finch
i've only ever used Event/Set and it's worked out fine, but i have no clue how t...
they multiplied the probabilities
dark cloud
why'd they do so?
lilac finch
when you want two probability two events occur together you mltiply their individual probabilities
unless stated the events are non independent
dark cloud
okay I don't really get it
lilac finch
hmm..
dark cloud
these formulas were there prior asw, but I don't get it
dark cloud
But there's also the 0.01
huh? what's the answer then
atomic hinge
0.81+0.01=0.82
dark cloud
why'd we add 0.01
atomic hinge
0.81 is the probability the bike will find a parking space but the car doesn't
0.01 is the probability the car will find a parking space but the bike doesn't
lilac finch
these formulas were there prior asw, but I don't get it
we can also find using this
$P(A \cup B) - P(A \cap B)$
grand pondBOT
oppenheimer
dark cloud
is there a video I can watch for this
lilac finch
**oppenheimer**
that is subtracting both find parking from either (one of them + both) finds parking
lilac finch
when you want two probability two events occur together you mltiply their indivi...
i'm not sure how i can rephrase this
atomic hinge
why'd we add 0.01
Because the question asks for the probability that one of them find a space, but there's two different ways this can happen:
- The bike finds a space but the car doesn't (0.81)
- The car finds a space but the bike doesn't (0.01)
dark cloud
I GET IT
dark cloud
that is subtracting both find parking from either (one of them + both) finds par...
right
dark cloud
Because the question asks for the probability that one of them find a space, but...
nice
I take it this is like really basic probability right?
could I understand how it works from like the first youtube video I get
atomic hinge
The cases:
- The bike finds a space but the car doesn't (0.9*0.9=0.81)
- The car finds a space but the bike doesn't (0.1*0.1=0.01)
- They both find a space (0.1*0.9=0.09)
- Neither finds a space (0.9*0.1=0.09)
So you could also do 1-0.09-0.09=0.82
atomic hinge
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buoyant bolt
can anyone help with the steps? garde 10
midnight plankBOT
can anyone help with the steps? garde 10
restive gazelle
recall the total angle of a circle and what percent of it the tiger's occupy
slate ferry
so the angle that the arc AB makes
wanton spade
major because you generally assume minor arc without anything present
wanton spade
so the angle that the arc AB makes
might be it's length where you take some radius r
i could be very off
slate ferry
I dont know what major means...
wanton spade
the bigger arc AB
slate ferry
oh i see
slate ferry
might be it's length where you take some radius r
we cant find r, can we?
slate ferry
recall the total angle of a circle and what percent of it the tiger's occupy
or perhaps its just the angle the bigger arc makes
wanton spade
we cant find r, can we?
doesn't seem like it, maybe there's more to the q
we'll have to wait for op
slate ferry
@buoyant bolt
lilac finch
I'm guessing major arc
m AB means measure of arc AB
slate ferry
(لو بتعرف تتكلم عربي, اعرف اساعدك(
wanton spade
m AB means measure of arc AB
oh 😭
slate ferry
m AB means measure of arc AB
and does that mean---> the angle 😭
buoyant bolt
اعرف عربي
lilac finch
They did not specify major/minor
lilac finch
and does that mean---> the angle 😭
Length
slate ferry
انت حاولت ايه؟
buoyant bolt
مدري
lilac finch
Length of arc i assume
slate ferry
iam.
lilac finch
-# i didn't know you speak arabic
slate ferry
-# neither did i
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mint ravine
The area of the region closed by |x+2| + |y-a| = b is 20 units^2. The product ab is 30. What is the value of a? Express your answer in simplest radical form.
midnight plankBOT
The area of the region closed by |x+2| + |y-a| = b is 20 units^2. The product ab...
mint ravine
I think set x=-2 and maybe do something idk
small jasper
That could help with finding "endpoints" for the graph
this is a standard graph to know the shape of
You may also find it helpful to ||rewrite the absolute values as piecewise functions||
mint ravine
oh k
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near geyser
z^2z' is not differentiable anywhere
midnight plankBOT
z^2z' is not differentiable anywhere
near geyser
Can i apply d/dz here?
sharp coral
how can you apply the derivative to a non-differentiable function?
sharp coral
you're asking how to prove that this function is not differentiable?
sharp coral
do you know the cauchy-riemann equations?
near geyser
Yeah
(x+iy)^2(x-iy)
(x^2-y^2+2ixy)(x-iy)
if I apply derivative with respect to z'
i got z^2=0
So z^2z' is differentiable at z=0 only am I wrong?@sharp coral
sharp coral
cauchy riemann equations aren't sufficient for differentiability at an isolated point
sharp coral
you use the limit definition directly
midnight plankBOT
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near geyser
you use the limit definition directly
How do I it?
sharp coral
have you done limits in R^2 before?
midnight plankBOT
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robust field
Markscheme says use 200k x 120 x 10
midnight plankBOT
Markscheme says use 200k x 120 x 10
robust field
But why the question ask energy lost in 1 second
What
1 second moved 14m
Shouldn’t it be 200k x 14 x 10
???
small crypt
ryt
small crypt
dude
cinder flax
?
cinder flax
but it leaves with diff speed
small crypt
yeah
small crypt
what paper was this
cinder flax
?
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small crypt
hey
small crypt
youre giving boards?
cinder flax
yea
small crypt
jee
cinder flax
u hv a ques?
small crypt
yup
small crypt
dont google it