#help-36
final saddleBOT
soft zealotBOT
BigBen
blissful meadow
I don't think that if $P$ is a primitive, then $P(x) = \int_c^x f(t) \dd{t}$.
soft zealotBOT
Azyrashacorki
south dirge
no not necessary, the differnece between the two primitives can be different
what i mean to say is that the constant term from the P(c) and A(c) would be different for both
as the two functions are different
left trail
I don't think that if $P$ is a primitive, then $P(x) = \int_c^x f(t) \dd{t}$.
why not? Isn't that what the first fundemental theorem tell us
left trail
no not necessary, the differnece between the two primitives can be different
wdym by different?
south dirge
consider the function, f(x) = 5x +2
it could have infinte primitive functions
left trail
yes
blissful meadow
$\int_c^x f(t) = F(x)$ is \emph{a} primitive
soft zealotBOT
Azyrashacorki
left trail
$\int_c^x f(t) = F(x)$ is \emph{a} primitive
isn't $P(x) = \int_c^x f(t)dt$ and $P'(x) = f(x)$ just the first fundemental theorem by using the function P instead of A. And either P or A would be a primitive
soft zealotBOT
BigBen
left trail
**Azyrashacorki**
I agree with this but I don't see how this affects P
south dirge
Nvm you are having a doubt in a different spot i think @blissful meadow is on the right track to explain it so ill leave it to him
blissful meadow
The first theorem just tells you more or less that $\int_c^x f(t) \dd{t}$ is a primitive.
soft zealotBOT
Azyrashacorki
left trail
but he says P represents any primitive of f
blissful meadow
From $P(x) = P(c) + \int_c^x f(t) \dd{t}$ you can't conclude that $P(c) = 0$.
soft zealotBOT
Azyrashacorki
blissful meadow
But you \emph{can} conclude that $P(x) - \int_c^x f(t) \dd{t} = P(c)$, a constant
soft zealotBOT
Azyrashacorki
blissful meadow
And thus if you take two primitives $P(x)$, $Q(x)$, you get $$P(x) - Q(x) = P(c) + \int_c^x f(t) \dd{t} - \left(Q(c) + \int_c^x f(t) \dd{t}\right) = P(c) - Q(c)$$, a constant
soft zealotBOT
Azyrashacorki
left trail
**Azyrashacorki**
But this all depends on my confusion though. I am still struggling to see why we are not allowed to say $P(x) = \int_c^x f(t)dt$
soft zealotBOT
BigBen
blissful meadow
A primitive of f(x) is just a function whose derivative is f(x).
$\int_c^x f(t) \dd{t}$ is a primitive indeed. That's what the first theorem tells you
soft zealotBOT
Azyrashacorki
left trail
so then P(x) should be good if P'(x) = f(x)
blissful meadow
But if P(x) is a primitive it needs not be $\int_c^x f(t) \dd{t}$
soft zealotBOT
Azyrashacorki
left trail
why not? If we just apply the first fundamental theorem but with P thats what we'll get
blissful meadow
Because $\int_c^x f(t) \dd{t} + K$ is also a primitive
soft zealotBOT
Azyrashacorki
blissful meadow
And clearly $\int_c^x f(t) \dd{t} + K \ne \int_c^x f(t)\dd{t}$ if $K \ne 0$.
soft zealotBOT
Azyrashacorki
blissful meadow
The first theorem tells you that a primitive exists
And then from that any function with an added constant is a primitive also
But they're not the same
left trail
but how do we know that $P(x) = \int_c^x f(t)dt +k$ and not just $\int_c^x f(t)dt$. Also how can he say that $A(x) = \int_c^x f(t)dt $ if A(x) is a version of P
soft zealotBOT
BigBen
vital crag
but he says P represents any primitive of f
You said it here
vital crag
Maybe your confusion is just the definition of primitive
left trail
Maybe your confusion is just the definition of primitive
the way I understand it is a primitve is a function P where P'(x) = f(x)
vital crag
Right. And what's the derivative of a number K
left trail
A(x) is a specific primitive... the one the first FTC gives
so I could say that $A(x) = P(z)$
soft zealotBOT
BigBen
left trail
Right. And what's the derivative of a number K
0
vital crag
Right and 0 + f(x) = f(x)
vital crag
**Azyrashacorki**
That should explain this K
blissful meadow
**BigBen**
A(x) is a primitive (chosen to be the one given by FTCpart1) and P(x) is any primitive. There's nothing more to it than A'(x) and P'(x) being both f(x).
If you explicitly say that P(x) = A(x), then P(x) is not any primitive anymore ... you've specifically picked one out, which isn't what they're trying to show.
left trail
ok let me know if I am understanding this correctly. We can not just replace A from the first fundemental theorem. And if we say that A(x) = P(x) then P(x) is locked in but x and c are supposed to be an elements in the interval I
The only thing that feels weird is that my understanding seems to place to much an emphasis on the choice of A
since the book could have chosen any letter to represent that function
blissful meadow
You could just go through the whole thing by writing the integral every time
If it is the case that you picked $P(x) = A(x)$, then the theorem would just say that $$A(x) - P(x) = 0 = -P(c) = -A(c) = \int_c^c f(t) \dd{t},$$ which is a constant.
soft zealotBOT
Azyrashacorki
vital crag
A stands for area since definite integrals represent signed area. P stands for primitive.
left trail
You could just go through the whole thing by writing the integral every time
by this do you mean that we would write $\int_c^x f(t)dt$ but plus a different constant for P(x), P(c), and A(x)?
soft zealotBOT
BigBen
blissful meadow
The point of their proof is essentially showing that $\int_c^x f(t)\dd{t} - P(x) = -P(c)$ for any primitive $P$ of $f$.
soft zealotBOT
Azyrashacorki
blissful meadow
They just decided to write A(x) for the first term to avoid writing it multiple times
left trail
so we are saying that some exact primitive is equal to the differences of two primitives who can change based on their inputs
blissful meadow
They're arguing that the exact constant by which they differ is determined by the value of the primitive at the lower bound, yes.
left trail
but is it not just the lower bound but difference between upper and lower bound
blissful meadow
I'm saying the primitives differ by a constant and that constant is determined by P(c)
In particular, the main use of this is that from $\int_c^x f(t) \dd{t} = P(x) - P(c)$, you get $$\int_c^a f(t) \dd{t} = P(a) - P(c).$$
left trail
I'm saying the primitives differ by a constant and that constant is determined b...
oh since c is always going to be the same and x is changing
soft zealotBOT
Azyrashacorki
blissful meadow
oh since c is always going to be the same and x is changing
For a fixed primitive, yes
Well rather c is fixed and P(c) changes depending on the primitive
soft zealotBOT
BigBen
blissful meadow
Yes
final saddleBOT
final saddleBOT
Available help channel!
inland flame
For this i got √2 for foci
If they aak for PF1+PF2 it's like
-√2+√2
=0 uh
timber leaf
,rccw
soft zealotBOT
timber jungle
you dont need to find c here
timber jungle
theres a property of elipse which states that the sum of the distances from the 2 foci to a point on the elipse is equal to the length of major axis
timber leaf
Why'd it become 2a?
Thats the definition as yash just stated
inland flame
theres a property of elipse which states that the sum of the distances from the ...
So f1= -√2 & f2=√2
Both are equal to major axis which is 4 and it becomes 2?
timber jungle
So f1= -√2 & f2=√2
Both are equal to major axis which is 4 and it becomes 2?
f1 and f2 are the position of foci from the center of elipse, and one more thing distance is always positive, the positions can be -√2 and √2 but the distance would be √2
inland flame
H= $x^2/a^2 - y^2/b^2
V= x^2/b^2 - y^2/a^2$
soft zealotBOT
Mugnes
inland flame
Huh?
inland flame
f1 and f2 are the position of foci from the center of elipse, and one more thing...
Oh ok
timber jungle
Wait i thought its x²/a² whyys urs y first
both are correct but it depends if the elipse is verticle or horizontal
inland flame
For horizontal the bottom is b² first right?
timber leaf
Yash got it there
timber jungle
if its vertical you get y^2/a^2 if its horizontal you get x^2/a^2
a goes with basically whats your major axis
and b goes with whatever is left
timber leaf
,rccw
soft zealotBOT
inland flame
I still don't get it js it because there are 2 foci that's why jt becomes 2a?
Is it because PFis 2?
😭😭
The formula for pf is 2a?
timber leaf
Hi
inland flame
But its 2a because there are 2 foci?
timber leaf
A is the major foci i suppose
inland flame
A is the major foci i suppose
But where did the 2 from 2a come from qwq
pliant elk
For this i got √2 for foci
If they aak for PF1+PF2 it's like
-√2+√2
=0 uh
thats the length of major axis i think
ellipse has property that the sum of distances of a point on ellipse from the foci is constant
which is equal to length of major axis
which is 2a
pliant elk
ye
final saddleBOT
@inland flame Has your question been resolved?
ivory vessel
can i have a hint for part a?
ivory vessel
<@&286206848099549185>
flat arch
idk how to help but, pjsk mentioned??!!
ivory vessel
yes i play pjsk
i love to discuss pjsk at any moment but i kinda need to work on this problem right now
flat arch
ok. I'm still in elementary school(learnt up to ap calc bc) so I can't really help u with that
@south salmon why that reaction?????
runic needle
ok. I'm still in elementary school(learnt up to ap calc bc) so I can't really he...
how old are you?
timber leaf
ok. I'm still in elementary school(learnt up to ap calc bc) so I can't really he...
bro you're so cooked ☠️
flat arch
how old are you?
13
timber leaf
elementary is not 13
flat arch
yes it is, grade 7
runic needle
that's middle school
runic needle
so you don't have middle?
flat arch
no not here
dense coral
2012 or 2013 born then 
flat arch
2013
runic needle
sus
ivory vessel
dense coral
good grief 
flat arch
??
runic needle
you're good
dense coral
so this is the feeling people get when they realize they’ve grown older 
flat arch
wdym???
lilac bison
go back to the problem in hand 😭
ivory vessel
dense coral
I just realized that this isn’t discussion 
my bad everyone 
ivory vessel
if you wanna talk you can still go to #discussion tho
ivory vessel
idk how to help
if you don't know how to help then don't engage here
ivory vessel
i don't mean anything by that ofc, but you should avoid cluttering the help channel in case someone else wants to help
flat arch
ok
odd seal
can i have a hint for part a?
integrate the loss over the range of t and find the minima for L?
ivory vessel
can i have a hint for part a?
original question
ivory vessel
integrate the loss over the range of t and find the minima for L?
hmm, i assume that t here is nonnegative?
i don't understand what the uncertainty of t means
odd seal
well, its a normal distribution. I'd expect it to be non-negative, but the tails extend all the way to -infty 
ivory vessel
i suppose there's a nonzero chance that the government owes the retailer instead lol
what did they mean by the uncertainty of t though, like does t follow a normal distribution, or its uncertainty does?
odd seal
the uncertainity does. t is a constant (or thats how it reads to me)
ivory vessel
how is the uncertainty calculated? is it just the difference of t (the estimated sales tax owed) and the true t (the actual sales tax owed)?
i honestly am still not sure what data the question is giving me
odd seal
like t is some value the retailer calculates for themselves, and there are n of them samples. The samples are expected to follow a normal distribution. There are many such retailers, and the tax returns of one such retailer is a random variable from that distribution, which is a singular t sample. a I think is the singular global value that irs or whatever expects a retailer to pay, but since the individual retailer have their own independent sets of circumstances, the single value doesnt work for them and as such you need to penalize the IRS's model that came up with this a. Of course, for certain value of a, this penalty would be the leastcoz it works best with all the retailers
ivory vessel
so t itself is a random variable with mean mu and sd sigma, and the "uncertainty" part just means "because there are many retailers with their own t, we want to model the behavior of t by modeling it as a normal distribution"
odd seal
or maybe on the second thought, each retailer has n of their transactions sampled, based off of which you build a guess a from the government? Like that sounds more logical to me
odd seal
so t itself is a random variable with mean mu and sd sigma, and the "uncertainty...
Now that I thought again, there is a single retailer, and they have many many transactions done over the year with their clients. Each transaction has a different value, and so the exact tax owed per single transaction is different. you, as the government have sampled n such transaction, and based off of that, you guess that the tax to be paid is t. Of course, being built off of the samples, it has some uncertainty. This t therefore is the random variable. The government wants to tell you that you need to pay a amount of tax, and wants that guess to be reasonable wrt the t distribution
ivory vessel
yeah this seems the most plausible to me
ivory vessel
integrate the loss over the range of t and find the minima for L?
now how do i do this? we know that t is now a normal rv, so then if i let Y to be the loss, then Y itself is also an rv right?
now i need to find the pdf of Y
odd seal
not really, my idea was to integrate $\int_{-\infty}^{\infty} L(a,t) p(t) dt$ which would be the expected loss and minimise it by leibnitz or whatever
soft zealotBOT
Bacter14Fr0g
odd seal
well, not exactly integrate, but like simplfy to a managable form after subbing in the exact loss function
final saddleBOT
@ivory vessel Has your question been resolved?
severe canyon
I guess so
But for the second integral you need to flip the bounds, by writing a minus sign in front of the integral
ivory vessel
alright
final saddleBOT
final saddleBOT
Available help channel!
idle fossil
hey this is on the topic of curvature, this question is super confusing, could anyone help?
final saddleBOT
hey this is on the topic of curvature, this question is super confusing, could a...
stiff wind
By curvature, they mean the second derivative i'm guessing? We see y'' = AB^2 exp(Bx+C).
Say A=B=1, then exp(2.42+C)=1.42 and C=-2.42+ln(1.42) i guess
Note that A,B are free to be chosen
final saddleBOT
Available help channel!
muted pasture
Alice goes from A to B. She is only allowed to move upward or to the right along the grid lines. How many such paths are there if Alice must pass through point O?
final saddleBOT
Alice goes from A to B. She is only allowed to move upward or to the right along...
pliant elk
maybe think backwards
start from 0
and count the ways , in which you can go to start point
and count the ways in which you can go to B from O
multiply them
final saddleBOT
forest steeple
you must move 2 in y axis in beginning
final saddleBOT
Available help channel!
gritty flume
final saddleBOT
drowsy epoch
What have you tried
gritty flume
The red
drowsy epoch
How did you get 2x
gritty flume
The thing looks squished
drowsy epoch
you should look at the numbers
after all you can squish a graph as you want but that's not reliable
actually wait i am dumb
final tangle
looks fine, what's the issue?
drowsy epoch
it should be correct
final tangle
well the end result looks fine, the form you stated with: a sin(x)(x-p)+qisn't
severe canyon
Actually I forgot how I got 2x
Hint: periodicity
drowsy epoch
the functions oscillates twice as fast
final saddleBOT
@gritty flume Has your question been resolved?
drowsy epoch
I did 200 divided by 10p
what's your question
drowsy epoch
if you were to look at the origin and go along the graph, at some point you will realize that it's looping, any idea where that could be in the picture
gritty flume
At x
gritty flume
200?
gritty flume
Butn8 said 200
drowsy epoch
who is butn8
gritty flume
who is butn8
I meant “but I said 200
drowsy epoch
ok
drowsy epoch
that distance period is not exactly 200° there is a small offset
but damn those teachers using confusing or ambiguous coordinate systems
gritty flume
So the periods 200?
drowsy epoch
do you know what an offset is
gritty flume
No
drowsy epoch
do you notice on the left there is some distance left
gritty flume
Yes
drowsy epoch
so it can't be quite 200
final saddleBOT
@gritty flume Has your question been resolved?
final saddleBOT
final saddleBOT
Available help channel!
rain sentinel
where do we start with this question?
final saddleBOT
where do we start with this question?
vital crag
Taylor series 
rain sentinel
do we not derive the maclaurin series?
modest badge
the mclaurin series is a special case of a taylor series centered at 0
the general case is the taylor series
rain sentinel
ah
and would we first find f'(x)
of tan^-1x
and is it easier to just memorise this or should i work out the arttan
modest badge
and would we first find f'(x)
yes
blissful meadow
It's probably easier to find a MacLaurin series for its derivative and then integrate that if that's an option for you
blissful meadow
and would we first find f'(x)
But otherwise yes
I just feel like higher degree derivatives will get quite annoying to compute
So in particular, as I suggested, you might want to expand $(\arctan(x))' = \frac{1}{1+x^2}$ and then integrate that term by term
soft zealotBOT
Azyrashacorki
severe canyon
Yeah, I guess you should start by observing that the derivative of $\arctan x$ is $\frac{1}{1 + x^2}$, which you can rewrite as $\frac{1}{1 - (-x ^2)}$
rain sentinel
and expand using the germetric series?
soft zealotBOT
Alberto Z.
drowsy epoch
How do I observe the derivative of arctan(x) is that without prior knowledge
-# use trig
rain sentinel
How do I observe the derivative of arctan(x) is that without prior knowledge
you'd have to work it out
drowsy epoch
you'd have to work it out
hi
stray mica
Hi
final saddleBOT
unborn forge
<@&268886789983436800>
left trail
Couple questions about this. I see that z can represent any point in the interval [x,y] which means that it can take on everything on in [a,b] since x,y represent everything in [a,b]. But why does he choose the represent a point in the interval that the function is defined as $z=\alpha y + (1-\alpha)x$ couldn't he just say g(x) <= then the chord for all x in [a,b]. Also why is he using <= instead of < if we want the point on the chord to be above the graph. Also I want to clarify we don't need every point on the chord to be above the graph we can even see that is false from the photo. Is he just saying that for each point in your function my chord will have a corresponding point that is greater than your function. And since it is greater for all the points. My chord is above the function for all the points
final saddleBOT
Couple questions about this. I see that z can represent any point in the interva...
soft zealotBOT
BigBen
final saddleBOT
@left trail Has your question been resolved?
steep marsh
hi quick question
final saddleBOT
hi quick question
loud sundial
is arc ab 30 or 60
60 by the inscribed angle theorem
loud sundial
is arc ab 30 or 60
timber leaf
60 by the [inscribed angle theorem](https://en.wikipedia.org/wiki/Inscribed_angl...
This model looks cool as hell
final saddleBOT
@steep marsh Has your question been resolved?
gritty flume
can I get help with the standard tri graphs
final saddleBOT
can I get help with the standard tri graphs
lilac bison
thats goated
gritty flume
no like the points
My teacher said I should compare the standard key points to the graph given
steady locust
My teacher said I should compare the standard key points to the graph given
do you know the unit circle?
steady locust
I don’t think so
this
gritty flume
Like th turning points
steady locust
yeah you get the sin/cos values from the unit circle
gritty flume
no I don’t k ow tht
steady locust
the cosine value is the x coordinate, and the sine value is the y coordinate
you have to memorize the unit circle though
yeah the values are all plus or minus 0, 1, 1/2, sqrt(2)/2, or sqrt(3)/2
gritty flume
I don’t understand this
steady locust
I don’t understand this
do you remember learning sin/cos with triangles?
this website can also help https://www.mathsisfun.com/algebra/trig-interactive-unit-circle.html
timber leaf
you have to memorize the unit circle though
i never memorised ☠️
unreal robin
You just memorize a quarter and apply that to the rest with respect to the +- signs
final saddleBOT
@gritty flume Has your question been resolved?
lilac bison
you dont need to memorise anything.
lilac bison
and equilateral triangle leads the way 🙂↕️
steady locust
what is this thing call again?
unit circle
supple imp
unit circle
ty
gritty flume
final saddleBOT
@gritty flume Has your question been resolved?
final saddleBOT
final saddleBOT
Available help channel!
light belfry
Two circle , both of radii a , touch each other and each of them touches internally a circle of radius 2a , then the radiusof the circle which touches all the three circles is :
final saddleBOT
Two circle , both of radii a , touch each other and each of them touches interna...
light belfry
in terms of a
night raft
Drawing a diagram would help
light belfry
i did
night raft
Mind posting what you drawn?
light belfry
ehh wait 3 mins
need to grab ma phone
I just need th perpendicular
@night raft
Ok found smtg
2/3a?
can someone verify
nvm
.close
final saddleBOT
timber leaf
Nice job
light belfry
it's alright
final saddleBOT
Available help channel!
autumn girder
Can anyone tell me there any fast and swift method of solving this without trial and error? I am in urgent need of enlightment
final saddleBOT
Can anyone tell me there any fast and swift method of solving this without trial...
worldly spruce
Can anyone tell me there any fast and swift method of solving this without trial...
I would first rewrite everything in cube roots
Then cube both sides of the equation
To get rid of the roots in a way
twin pivot
a+b=2^(1/3)
upper owl
a+b=2^(1/3)
Mr Bumble prime
timber leaf
a+b=2^(1/3)
dont troll pls
twin pivot
the first term is a and the second b
karmic glen
a+b=2^(1/3)
and a^3 + b^3 = 2
autumn girder
wow i love seeing different approaches, please only speak abt the solution
tranquil pine
<@&268886789983436800>
worldly spruce
wow i love seeing different approaches, please only speak abt the solution
We won't give you the solution
timber leaf
Insulting people is no good
worldly spruce
I gave you a clue
twin pivot
and a^3 + b^3 = 2
yes
worldly spruce
Now do it yourself
worldly spruce
Can y'all stop posting massive gifs
prime skiff
ahh edit
tranquil pine
<@&268886789983436800>
timber leaf
<@&268886789983436800> pedo joke
karmic glen
i like the system of equations more than cubing its cleaner
timber leaf
I would just cube both side
autumn girder
cubing makes the equation easier to deal with in my opinion
unreal robin
You can visually roll out two solutions but idk how useful that is
twin pivot
yes
autumn girder
You can visually roll out two solutions but idk how useful that is
i think that method is time consuming
unreal robin
It took like 3 seconds, but idk how you can use that information
Like finding 2 solutions doesn't mean finding all solutions
autumn girder
It took like 3 seconds, but idk how you can use that information
i think that's too hard
unreal robin
You just gotta cube then
autumn girder
ok
autumn girder
Can someone help me with this?
final saddleBOT
Can someone help me with this?
final saddleBOT
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
Is that not Physics?
wary juniper
Not really
inland apex
physics with application of maths
minor sandal
physics with application of maths
Can u help me , in my channel?
inland apex
Can u help me , in my channel?
wdym?
minor sandal
See Help -21
unreal robin
Can someone help me with this?
Remember that the horse and one guy can move simultaneously
final saddleBOT
@autumn girder Has your question been resolved?
sturdy cypress
oh i get it
wary juniper
If 1 of the brothers is at the fair before the other one, that means it will take more time for the 2nd one to arrive, time which the 1st brother isnt moving, which probably means we havent fully used our time wisely
This would imply that both brothers arrive at the same time
sturdy cypress
at first the horse goes 20 miles minus x
then it goes 20 miles minus 2x
then it goes 20 miles minus x
that doesn't seem to help lol
unreal robin
They just have to keep moving at all times
Which means yeah they'll arrive at the same time
I'm pretty sure there's better algebraic formulation
But essentially, you're accounting for 3 things
- The distance crossed by the first walking guy
- The distance the the horse has to cross to go back
- The distance crossed by the second guy
So for example if the horse moves 10 miles, the walking dude would have crossed 4. The horse will have to go back around 4-6 miles to pick-up the guy
sturdy cypress
x/4 = (20-2x)/14 + (20-x)/10
unreal robin
I think that's some integral formula that I'm not exactly good at
unreal robin
at first the horse goes 20 miles minus x
then it goes 20 miles minus 2x
then it ...
20 - x, then (20-x - y)
For the sake of accuracy
sturdy cypress
why
unreal robin
y is the distance crossed by the guy
sturdy cypress
well it's the same
unreal robin
20 - x because the horse stops midway through
But then the horse goes back that remaining (20-x) minus the distance crossed by the guy which I called y
It's not x
sturdy cypress
ok, but it's the same
unreal robin
I'm pretty sure that's some integral problem so I can't really help unfortunately
marble lintel
Wrong channel
sturdy cypress
7.2727272727 miles
keen storm
@sturdy cypress if u could provide the integral i could try to help
keen storm
oh mb
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red nova
Guys please help me solve this :
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Guys please help me solve this :
south salmon
pentagon with 4 angles known
red nova
pentagon with 4 angles known
Bro it is a regular pentagon
south salmon
you have angles of the heptagon
then you can take the Pentagon with 3 red sides and 2 black sides containg angle x
red nova
Heptagon*
wary juniper
Ok then can you see the isoceles triangle where 1 angle is an angle of a regular heptagon and the other 2 are x
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rain sentinel
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karmic glen
is the answer for a incorrect?
it is incorrect
blissful meadow
I feel like the answers are all shifted or something (although it's true a) diversed
final tangle
are you looking at the correct section in the answer key?
south salmon
yea b should be 1/2
fathom jackal
he is right q 1 is undefind, ncert btw or nah?
south salmon
if this is ncert then makes sense why the answers are wrong
rain sentinel
we get -1/x^3
fathom jackal
I feel like the answers are all shifted or something (although it's true a) dive...
am not understanding how e is 2 wth
bro
@rain sentinel check if u checking the correct answer key
there is no shot the book would have this much error
blissful meadow
The answer key just has the answers shifted
(d) evaluates to the answer given for (e)
fathom jackal
but what about e
blissful meadow
It's not given
rain sentinel
this is the answer key im given
south salmon
return it back
final tangle
they've printed the answer to e) in a)
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crude island
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south dirge
show your working please
crude island
brudda I have no working 😭
i've been thinking about it but genuinely don't know how to show this
let's just say dim V = n though
then we need to find n^2
diagonalizable operators
that are linearly independent
but of course I'm going to avoid explicitly constructing them via matrices or something
muted prairie
is V over an arbitrary field, or R or C?
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@crude island Has your question been resolved?
shrewd ether
if V has a basis $v_1,...,v_n$ than just take operators $E_{ij}$, such that $E_{ij}(v_j)=v_i$ and $E_{ij}(v_k)=0,k\neq j$
soft zealotBOT
Larry
crude island
**Larry**
is this not the standard basis?
of L(V)?
like to simplify let's work with L(C^2)
we basicallly just need to find 4 diagonalizable matrices
that span all possible matrices
or that are linearly independent
But $\begin{pmatrix} 0&0\1&0 \end{pmatrix}$ doesn't seem so diagonalizable to me
soft zealotBOT
holo_morph
muted prairie
Yeah its not
crude island
$\begin{pmatrix} 1&0\0&0 \end{pmatrix}$ and $\begin{pmatrix} 0&0\0&1 \end{pmatrix}$ are 2 of the 4 that we need and they are diagonal
muted prairie
If V is a vector space over C then diagonalizable matrices are dense, which is much stronger
soft zealotBOT
holo_morph
crude island
If V is a vector space over C then diagonalizable matrices are dense, which is m...
idk what that means
I have to use stuff in ladr
chapter 5 and before
crude island
arbitrary
shrewd ether
**holo\_morph**
yeah nvm
muted prairie
Over C most matrices are diagonalizable
"almost every" matrix
not so over an arbitrary field
crude island
not so over an arbitrary field
but yeah this is like R or C
sometimes theorems don't work for weird edge cases like GF(2)
so really just assume R or C
tranquil pine
I will had to do il before and i remember it's easily solve.
take the family of endomorphisms fi,j with fi,i=ui,i and fi,j=ui,j.
Having previously define ui,j as the canonical endomorphisms of L(V).
(Sorry for my english, i normally speak french :).
crude island
yeah i'm not exactly understanding what u are getting at
wide swift
Hello im trying to do some calculus homework and im unsure on how to start on these questions, I already have the answers but I'd like to learn the logic behind them thank you
final saddleBOT
Hello im trying to do some calculus homework and im unsure on how to start on th...
wide swift
so what im trying to do is, H * (L-20) * (W-20) ?
abstract bramble
hmm why - 20
wide swift
thats what we took in class but i dont remember why.
abstract bramble
what's the variable in this problem? what's changing
wide swift
Corners are being cut so a square piece of cardboard can be folded so the length and width from the corners are being cut out in order to get the H
from how i understand it
basically removing equal numbers from the W and L
wide swift
thats what im trying to find
the only given is thats its a 20x20 cardboard square
ill show you what i got
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@wide swift Has your question been resolved?
worthy zephyr
hello, i know im terrible, but i need to learn my course about derivative functions and im really lost, can someone help please?
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hello, i know im terrible, but i need to learn my course about derivative functi...
timber leaf
Sure, can you give an example of the problem?
worthy zephyr
i know its freakin easy but i had a hard day and i cannot understand anything
1)We are given the graphical representation of a function 𝑓.
Construct the sign table of its derivative
2)We are given the graphical representation of a function 𝑓.
Construct the sign table of its derivative.
timber leaf
Do you notice the function go up or down whenever it reaches a peak?
For example, x=-6 or x=-4
worthy zephyr
yes when f'(x) > 0 its positive and when f'(x) < 0 its negative
timber leaf
Then what happen to the derivative sign at these points
worthy zephyr
its basicaly change so f'(x) is positive, then negative, positive, negative....
timber leaf
No, the derivative at peak point is 0
When x=-1 for example, f' approach 0 from negativity and move to positivity
worthy zephyr
yes i understand that
@timber leaf sorry for disturbing you i understood by myself, i maybe should quit smoking
have a great evening
timber leaf
have a great evening
!done
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worthy zephyr
.close
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inner granite
How is my proof here?
final saddleBOT
How is my proof here?
inner granite
The books does things a bit different but I don't see why I couldn't factor the k from (ak - bk) to then use lemma 2
deft ravine
The books does things a bit different but I don't see why I couldn't factor the ...
You would need to prove Lemma 2. Also why is there no Lemma 1
inner granite
You would need to prove Lemma 2. Also why is there no Lemma 1
they were proven a few pages back
so I am just using them now
inner granite
You would need to prove Lemma 2. Also why is there no Lemma 1
there is a lemma 1, I don't need it here
<@&268886789983436800>
<@&268886789983436800>
deft ravine
they were proven a few pages back
Then I don't see a mistake
inner granite
okay, thank you for looking
deft ravine
You're welcome
inner granite
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fallen valve
Does the proof of this require using the standard inner product on F^n?
fallen valve
I seem to have a proof using that, however I cannot come up with one for an abritrayr inner product on F^n
cerulean epoch
all inner products on F^n are equivalent to the standard one. the idea is to find a basis where all the vectors are orthogonal to each other and of norm 1, and you can do this using something called the gram-schmidt process
fallen valve
all inner products on F^n are equivalent to the standard one. the idea is to fin...
ah I see okay. I kinda get it but not really, since if I change the inner product I have no way of showing that $A^\dagger = V \Sigma^\dagger U^*$
soft zealotBOT
LXDL
cerulean epoch
i assume you want to show A^\dagger as defined is the pseudoinverse?
,tex i think you can compute $
U\Sigma V^V\Sigma^\dagger U^ U\Sigma V^*$ and $V\Sigma^\dagger U^*U\Sigma V^V\Sigma^\dagger U^$ directly
soft zealotBOT
fallen valve
i assume you want to show A^\dagger as defined is the pseudoinverse?
yes, that is what im trying to show
fallen valve
,tex i think you can compute $
U\Sigma V^*V\Sigma^\dagger U^* U\Sigma V^*$ and $...
where does this come from? i think there was an exercise in the book that has this, but it uses the theorem
cerulean epoch
well how do you define the pseudoinverse
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@fallen valve Has your question been resolved?
fallen valve
well how do you define the pseudoinverse
They define it using a linear transformaiton like this:
which in turn uses this:
so my goal is to show that letting $B = V\Sigma^\dagger U^*$ to show that $L_B = (L_A)^\dagger$ on a basis
soft zealotBOT
LXDL
soft zealotBOT
LXDL
solemn loom
would this not be correct?
final saddleBOT
would this not be correct?
supple jolt
Well
For the sin one
It seems it is not stretched enough in the x direction
Becuase you intersect the x axis at preety big values
solemn loom
It seems it is not stretched enough in the x direction
so what should it be
im confused
final tangle
seems ok, whos saying its wrong
solemn loom
seems ok, whos saying its wrong
well i think theyre equivalent
but
i put them through desmos
and they look different
final tangle
make sure its set to degrees
solemn loom
supple jolt
Yeah you re right
supple jolt
The grapgh is in degrees
solemn loom
.close
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ivory lynx
hallo
ivory lynx
another question for transitivty property in relation
if (1, 2), (2, 1) appears in the relation therefore (1, 1) must be in the relation set?
stone flint
if R is to be transitive, yes.
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autumn girder
what are the easiest approach to this?
final saddleBOT
what are the easiest approach to this?
onyx peak
what are the easiest approach to this?
Whats the goal here
autumn girder
to simplify the expression
fathom jackal
oh i know this model
look, let the entire expression be t
be equal to t
oh wait nvm
onyx peak
The best you can do is prolly make them have common denominator and add
autumn girder
yeah its easy but it takes up too much space
autumn girder
no it takes too long
if u keep doing it that way it becomes easy but it takes too much space
i'd like to way other approach to this if u could kindly help
autumn girder
thanks
onyx peak
Oh partial fractions maybe
autumn girder
but wouldn't it make the numerators more complicated?
fathom jackal
ahhh ok got it
look
once u took t
u bring deno to other side
u would get an equality
now, at x=a, equality is 0
x=b, equality is 0
autumn girder
u bring deno to other side
do we like get all three fractions under a common denominator first?
fathom jackal
and x=c equality is 0
fathom jackal
do we like get all three fractions under a common denominator first?
yeah
autumn girder
oh ok
fathom jackal
u basically get sum of 3 quadratics = a cubic
now
we find that a,b,c is the root of cubic part
but, a quadratic can have a maximum of only 2 roots
this means that the quadratic must be constant
and equal to 0
hence the answer
autumn girder
k
autumn girder
to assume x-a =0 ; x=a etc
fathom jackal
wdym?
autumn girder
if we take the denominator to the other side
autumn girder
a(b-c)(x-b)(x-c) + b(c-a)(x-a)(x-c) + c(a-b)(x-a)(x-b) remains tho
on one side
and
t times denominator remains on the other?
autumn girder
oh ok
fathom jackal
and on assuming x=a
we get a(b-c)(x-b)(x-c)=0, but we know b and c is also the root of this
3 roots for quadratic? that means quadratic itself is 0
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autumn girder
what are the common ways to prove this?
final saddleBOT
what are the common ways to prove this?
autumn girder
K thanks
trim monolith
any other way beside that
Binomial theorem is the easiest ?
trim monolith
oh ok
You can use expotentital limit
Divide both side by 300 ³⁰⁰
And the solve using the identity (1 + 1/n)^n = e
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hybrid torrent
Hi guys, how would I do this?
final saddleBOT
Hi guys, how would I do this?
drowsy epoch
Split the fraction
hybrid torrent
Do I times it by the denominator?
Sorry I’m really behind in maths, dumb question my bad
hybrid torrent
Well 8/2 =4
timber leaf
?
hybrid torrent
(8-3sqrt3)/2sqrt3 right?
Oh right
timber leaf
Then what do you do if there exists a root in denominator
hybrid torrent
Then what do you do if there exists a root in denominator
Rationalise?
timber leaf
Now you will learn
hybrid torrent
Thank you
timber leaf
Then what do we multiply with
hybrid torrent
By sqr3
timber leaf
Yes
proven echo
what law did u defeat
timber leaf
Huh?
hybrid torrent
what law did u defeat
Don’t know sorry
hybrid torrent
Ok so from here I pretty much have the left hand side equation
I guess I’ll go from there
Thank you very very much
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hybrid torrent
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prime skiff
damn
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timber leaf
Bro wth
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gentle zephyr
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Renato
gentle zephyr
anyone has any idea how can the volume of the body can be calculated as that?
this whole section is related to cavalieri's principle
also, it must be noted that the z = a and the z = l planes are parallel
as far as I know cavalieri says that if two solids have same height and every cross section made at the same level has the same area then the solids have the same volume
young rune
I think this is for objects which have varying cross sectional areas, like that potato
young rune
so you find the volume by integrating the function of cross sectional area at height t: A(t)
gentle zephyr
so you find the volume by integrating the function of cross sectional area at he...
what?
ohh, A(t) is the cross section area at height t
young rune
yessss
young rune
A(t) is cross sectional area and t height, and l and a are probably just the limited space which it occupies
idk if i worded that correctly
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left trail
what does he mean by J is the set of values that g takes on I? Does he mean that J is the outputs of g on I or that J represents the elements in I that represent the domain of g?
final saddleBOT
what does he mean by J is the set of values that g takes on I? Does he mean that...
loud sundial
what does he mean by J is the set of values that g takes on I? Does he mean that...
$J=g(I)$, aka the range of $g$ on $I$.
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Civil Service Pigeon
left trail
ok thank you
I have one more question regarding this proof
So he says that both integrands are continuous on these intervals. For P(x) this is true because f is cont on j but if we look at Q(x) we see that he has the condition x in I. g' is cont on I. g is also cont on I because g' exists. but f is said to be cont on I
oh I see g(t) in J for x in I
.solved
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ivory lynx
i need help on 2.4, i genuinely dont know where to start lol
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i need help on 2.4, i genuinely dont know where to start lol
ivory lynx
i dont know what some integer k means for some dumb reason lol
abstract bramble
i dont know what some integer k means for some dumb reason lol
it just means x = 2^k * y
so it could be x = y, x = 2y, x = 4y, x = 8y, etc
sturdy cypress
it's kind of a more specific >
all the same properties
uh, depends on what's antisymmetric
ivory lynx
uh, depends on what's antisymmetric
well antisymmetric means that if ordered pairs (a, b) and (b, a) are elements of R, then a = b
ivory lynx
it just means x = 2^k * y
so it could be x = y, x = 2y, x = 4y, x = 8y, etc
wait how did we arrive here 😭
oh nvm
sturdy cypress
right so that's also true
ivory lynx
nvm nvm
sturdy cypress
a > b guarantess that b is not larger than a
ivory lynx
a > b guarantess that b is not larger than a
yes, also
ivory lynx
it's kind of a more specific >
how did we arrive here?
sturdy cypress
i made it up
sturdy cypress
x/y = 2^k means at least that x > y
abstract bramble
whawt about negative k
sturdy cypress
oh you're right
I misunderstood integer k as integer 2^k
so it's an equivalence relation?
ivory lynx
it just means x = 2^k * y
so it could be x = y, x = 2y, x = 4y, x = 8y, etc
i assume so
for some integer k... hmm
wouldnt it be easier if i just assume k to be 0 here
i assume, because for some integer k the available properties of the relation given the equation will also carry over to any other integer k
pliant elk
wouldnt it be easier if i just assume k to be 0 here
you would negate all the other cases
that basically means you are taking x=y
ivory lynx
you would negate all the other cases
that basically means you are taking x=y
wdym negate all other cases?
ivory lynx
you would negate all the other cases
that basically means you are taking x=y
so ur pretty much just want to remove 2^k there and basically x = y
pliant elk
so ur pretty much just want to remove 2^k there and basically x = y
but thats an error
do you disagree from the fact that you are making an error by doing that
ivory lynx
do you disagree from the fact that you are making an error by doing that
im tryna understand what you mean cause i saw like 2^k != 0 or smth
pliant elk
hmm
ivory lynx
well shit i prolly misunderstood it HAHAH
pliant elk
im just saying
dont take k = 0
sturdy cypress
just start doing it
ivory lynx
k > 0
sturdy cypress
there's no like trick
sturdy cypress
is it reflexive or not
pliant elk
there is no set method for doing these questions
pliant elk
k is variable, you get to decide it on the spot
calling it an arbitary constant would be better
ivory lynx
its not reflexive, symmetrical and transitive
it is antisymmetrical,
sturdy cypress
no that's specifically not a constant
ivory lynx
i assumed k = 1 here for it to be easier
pliant elk
it might be reflexiv tho
ivory lynx
so x/y = 2;
4/2, 8/4, 16/8
sturdy cypress
I guess it's like you could intepret it either way
pliant elk
it is reflexive tho
sturdy cypress
we could be talking about different questions
pliant elk
k is flexible
ofc
ivory lynx
flexible i assume it can be any integer
pliant elk
flexible i assume it can be any integer
yes
ivory lynx
i dont think its reflexive when k != 0, its only true if and only if k = 0 i think
otherwise it isnt
sturdy cypress
basically when you go from 8/2 to 2/8, k changes from 2 to −2
and that's allowed
it changes on the spot, it's symmetrical
ivory lynx
it changes on the spot, it's symmetrical
same idea, the way we did it is different lol
pliant elk
k can be anything
ivory lynx
theres more leeway for it to be symmetrical
ivory lynx
k can be anything
yeah
and since the appearance of symmetrical
and x != y
its not antisymmetrical
theres also leeway for transitivity
pliant elk
and x != y
x=y you mean ….
ivory lynx
so its also antisymmetrical..
so its not reflexive, but it is symmetric, anti, and trans
sturdy cypress
no
pliant elk
why do you say its not reflexive
ivory lynx
its only really reflexive if k = 0; if less than or more than 0 its not fully
sturdy cypress
im just saying
dont take k = 0
because we were saying this
pliant elk
i meant it like
a general case
sturdy cypress
because it's hard to communicate
sturdy cypress
its only really reflexive if k = 0; if less than or more than 0 its not fully
there's no set k at all, you can't think of it like that
ivory lynx
fair fair
sturdy cypress
(a,b) is in R if you can find a k to satisfy the equation
(a,a) would always work, and have k = 0
ivory lynx
yeah now that i think about it it is reflexive
sturdy cypress
if you learned equivalence relations it's one of those
ivory lynx
symmetric as well and antisymmetric cause x =y
pliant elk
thats not how it works
a relation cant be both symmetric and anti symmetric
the definition of symmetric relation is
(a,a) belongs to the set R for every a belonging to A
where R is the set of relations
and the Relation is over set A
sturdy cypress
that's reflexive
ivory lynx
a relation cant be both symmetric and anti symmetric
they can if a relation is purely reflexive (e.g R = {(1, 1), (2, 2), (3, 3)}
sturdy cypress
ivory lynx
(1, 1) is reflexive, symmetric and antisymmetric (x = y) at the same time
sturdy cypress
a R b means numbers have the same color
ivory lynx
(1, 4), (2, 8), (3, 6)...
pliant elk
they can if a relation is *purely reflexive* (e.g R = {(1, 1), (2, 2), (3, 3)}
dont we take that as a symmetric relation , @sturdy cypress
sturdy cypress
it's both like maladroit said
antisymmetric means there's never a flip, except (a,a) flipping into itself is allowed
asymmetric means it's not allowed
ivory lynx
honestly, with a very hard guess i think they're all of the properties due to leeway
k can be any integer
yeah i think im goods
thanks guys
this was a good discussion for a number lolol
sturdy cypress
it's not anti symmetric
sturdy cypress
because it's one relation
like, you're still trying to fix the k to some number
i think
sturdy cypress
it has (2,4) and (2,2) and (4,2)
ivory lynx
the appearance of (2, 4) and (4, 2) is a nono i guess
sturdy cypress
yeah
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shadow smelt
yo
final saddleBOT
yo
Asking the actual question right away is more likely to get responses.
Asking "Can I ask...?" or "Does anyone know about...?" doesn't give people enough information to decide whether they can help, and answering can feel like a promise to help with the actual question, which they might find themselves unable to.
shadow smelt
how do i construct a 60 degree angle and also bisecting the lines
without using a protractor
lucid nymph
what polygon has 60 degree angles in it
shadow smelt
its a triangle
lucid nymph
do you know how to construct a triangle
shadow smelt
dire glacier
a perpendicular line going through the peak of the triangle is going to cut the a and b in half
or if you cant do perpendicular lines
using only compas
shadow smelt
a perpendicular line going through the peak of the triangle is going to cut the ...
so this how u tell without using a protractor
dire glacier
you do the same thing but under your main line