you will use it once you start dipping into algebra formally
#help-26
mild saddle
elder sentinel
you will use it once you start dipping into algebra formally
what kind of algebra?
mild saddle
normal algebra
elder sentinel
i did an algebra test back then
mild saddle
I'm definitely not talking about linear or abstract algebra. those are years away
regular, high school algebra
mild saddle
if you've done stuff like y = mx + c, then you should know what the gradient is already
elder sentinel
if you've done stuff like y = mx + c, then you should know what the gradient is ...
numbers are involved or not?
mild saddle
but either way, if you haven't, this is how you find the gradient of a graph generally speaking, though of course our method relates it to speed
mild saddle
numbers are involved or not?
the general form of the gradient doesn't use any numbers, but when you look at concrete graphs, of course there will be numbers
elder sentinel
the general form of the gradient doesn't use any numbers, but when you look at c...
i mightve not used it then
well thank you
i get alot now
mild saddle
you're welcome, and good luck in your math journey!
elder sentinel
you're welcome, and good luck in your math journey!
thank you !!
mild saddle
if you're done, you may .close this channel.
elder sentinel
.close
topaz sinewBOT
elder sentinel
.
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wooden kayak
wooden kayak
oops sorry
limber elk
what have u tried or where r u stuck
limber elk
well
wooden kayak
gave me a weird decimal
limber elk
did u try the next one
wooden kayak
but I know they are also square numbers
limber elk
do u notice some sort of pattern
wooden kayak
should be
limber elk
what ur trying to do is write a summation of the square numbers from 16 to 144
how can u express square numbers algebreiacly
wooden kayak
4^2,5^2,6^2 ....
limber elk
can u generalize that w a variable
wooden kayak
i dont think so
limber elk
hm
limber elk
4^2,5^2,6^2 ....
i think this is a good start
because u notice that there is one think thats increasing by 1 every time
which is perfect for a summation
now u just gotta put a variable where that thing is
so u can set it up
wooden kayak
oh the rule?
limber elk
4^2,5^2,6^2 ....
going back to this
ur asked to put 4^2 + 5^2 + 6^2. . . 12^2 into to sigma notation
this is mostly abt pattern recognition
ur summation term is whatever u notice is increasing by 1 everytime
so ur summation term is n^2
then u just gotta put start and end values and boom ur done
wooden kayak
ohhh
ok so if its neither, we dont use the
given formulas
(the arithmetic formula and geometric formula since its neither)
limber elk
given formulas
correct
this question is trying to get u to think away from arithmetic/geometric sequences and more abt how sigma notation works
wooden kayak
understandable
is it ok to skip understanding from my teacher
since I wasnt understanding his proof of it, I just watched the application of it on yt
but I dont really understand the proof he was showing us and manipulation of index etc
limber elk
im not sure i understnd what u mean
u dont rlly need to manipulate the index or anything
for this question
but try and understand the lesson that ur teacher tu
taught
plenty videos on khan acadmey
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sturdy fulcrum
Hello
sturdy fulcrum
I need help. How can I use Seifert Van Kampen's Theorem for CW complexes?
For example, for this one
Is there anything like an algorithm or something? I don't know hor to use the generators 🙁
topaz sinewBOT
@sturdy fulcrum Has your question been resolved?
sturdy fulcrum
I think the idea is to follow this theorem
But I don't know how
<@&286206848099549185>
topaz sinewBOT
@sturdy fulcrum Has your question been resolved?
balmy spruce
@sturdy fulcrum i recommend that you try in #alg-top-geo-top because people how to answer such a question rarely come to help channels
i would've liked to help but i dont have sufficient knowledge hahahah
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neon iron
Why do Taylor polynomials look the way they do?
Any theory as to why there are fractions there?
I dont really want to just remember a form blindly, i want to understand it so i can determine it later on.
It looks like an usual Taylor series with coeffficients an n∊N
But the polynomial form has fractions
Why so?
topaz sinewBOT
Why do Taylor polynomials look the way they do?
Any theory as to why there are f...
radiant marlin
the idea is that a taylor polynomial matches all the derivatives of a function at a point
neon iron
And how does one know when to use it?
The only time i've seen it being used was when proving Einstein's Criterion
And the book that i'm reading this from doesnt try that hard to explain it.
(It only has exercises and solutions so i guess that its not that good for learning alone)
radiant marlin
oh they show up everywhere, whenever you want to approximate a complicated function with an easy polynomial
neon iron
I see. In my book it was used to prove the irreductibility of a polynomial. that has only prime numbers.
Are these polynomial forms calculated just like a taylor series?
Or are they not realated because a Taylor series goes on forever.
radiant marlin
I'm not sure how taylor polynomials relate to irreducibility at all, could you link something that explains that connection
neon iron
I have nothing, i just thought that they could be linked.
Because they are both centered around an x for example.
But the Taylor series doesnt stop in n steps.
radiant marlin
it's just for approximating hard things into something simple, like turning sin(x) into x
neon iron
So i cant use abel hadamard cauchy for taylor polynomials?
radiant marlin
wait are you asking for the difference between taylor series and taylor polynomials?
neon iron
Yes
I see that a taylor series has an infinite degree, while a taylor polynomial has a finite degree.
radiant marlin
yeah that's all it is, they're the same idea
usually it's because the later terms are so small that you don't have to worry about them
neon iron
Due to the factorial and because your starting point raises to power?
radiant marlin
I'm sort of confused at what you're getting at, so I'll just explain the formula
so like the 3rd term in a taylor polynimal at 0 is 1/3! * f'''(0) * x^3
neon iron
Are Taylor polynomials always centered at 0?
radiant marlin
no you can do them anywhere, you pick a point and approximate from there
neon iron
Oh, ok.
radiant marlin
I'll just do the general version, 1/3! * f'''(c)*(x-c)^3
the reason for that 1/3! is because when you take the 3rd derivative, you get the right derivative
taking three derivatives of (x-3)^3 gives 3*(x-3)^2 -> 3*2(x-3)^1 -> 3*2*1
or just 3!, so you need 1/3! to cancel it out
that way the 3rd derivative of this taylor polynimal gives f'''(c), which is the goal
neon iron
Oh, i see.
Ive seen these types of polynomials at the irreducible polynomials and now i'm wondering. What is the relation between Taylor polynomials and irreductible polynomials?
Is it a property of the Taylor polynomials or a property of polynomials in general? (Ive seen the form in the exercises that the book provides, and theres no theroy to cover for that so i'm trying to wrap my head around so i can ask about Sylvester inequalities)
radiant marlin
I wouldn't know, maybe someone else can help
neon iron
Its okay.
Don't stress over it that much, now that i know that Taylor series and polynomials are kind of related i can understand better.
Ill look for the form of Taylor polynomials when i look at these exercises from now.
Though i have one question about polynomials and integration.
If thats not too much.
topaz sinewBOT
@neon iron Has your question been resolved?
neon iron
I know that when a polynomial goes through integration a technique is separating the polynomial in easy fractions
For example A/(x-a)+B/(x-B)+...
And a and b are real roots.
And then there are the quadratics of the form Cx+D/(x^2+cx+d)+...
When delta for x^2+cx+d is less than 0 then there are no real roots.
But how is that form solvable?
Or do we just index the roots and leavve them like that.
For example ai, bi, ci
drifting swift
when you say "solvable" do you mean solvable for integration purposes?
@neon iron
neon iron
1 sec lemme read what i typed because i forgot
neon iron
when you say "solvable" do you mean solvable for integration purposes?
Solvable for integration and polynomial purposes
drifting swift
....
"polynomial purposes"?
ok sorry can you just restate the entire question for me
neon iron
Well, i didnt know of another way to say it
How can it be solved for in exercises where one has to prove the irreducibility of a polynomial function.
I think i got how it can be solved for integrals.
drifting swift
oh, so you're trying to prove that a polynomial is irreducible
are you looking for irreducibility over the real numbers?
or the rational numbers?
or maybe something else entirely?
neon iron
are you looking for irreducibility over the real numbers?
Real numbers
And at the same time understand the way to prove that a polynomial is irreducible.
drifting swift
a polynomial is irreducible over R iff it is quadratic with negative discriminant or linear
anything else can be factored
neon iron
(sorry for the late replies im trying to revere engineer exercise solutions to understand irreducible polynomials)
drifting swift
if all you're working with is the real numbers, proving irreducibility is checking against that criterion i just wrote
neon iron
I see that in an exercise einsteins criterion is also mentioned.
neon iron
Sadly this book has no theory or explanation for any of the exercises. Its from 1991 and its in romanian.
drifting swift
also i think you may have meant Eisenstein
you should post the exercises you are looking at even if they are in another language
drifting swift
also eisenstein is for irreducibility over the rational numbers
not the real numbers as you led me to believe
neon iron
Sending the photos rn
Its 1, 2 and 5. No worries they are not homework. I'm trying to learn on my own to prepare for a contest.
Is it readable enough?
I can take more photos if you need
I chose the book because it had contest exercises at the end, but i dont think i can get to those chapters before the contest starts.
Ive noticed that 2 looks like a Taylor series.
And i was told that it's a Taylor polynomial.
drifting swift
this?
drifting swift
the resemblance of that to a taylor series is probably just coincidental
neon iron
Oh. I dont see how they get the forms for thee polynomial functions so i'm trying to link them to stuff that i know or have seen.
The book is supposed to be for highschool students, but i liked it to Taylor series from university.
That was the only thing that came to my mind after 3 days of looking over it.
neon iron
Maybe for even and odd numbers?
drifting swift
no i think you're just overthinking things at this point
neon iron
Well, its f(m) and f(m+1)
So all terms from 2 are 0 and all terms from 3 are 0, etc
Though that means that all values would be 1
Or no
Im wrong
Yeah, i still dont get it
drifting swift
honestly you've gotten me confused now as well
neon iron
Should i move it to an advanced topic? Such as pre-university competition math?
drifting swift
idk
neon iron
honestly you've gotten me confused now as well
Well, im confused too, so its okay.
hushed coral
could anyone help this stupid 8th grader out. I hv absolutely no idea how to answer these questions. Surely the 12th graders out there 🥺
topaz sinewBOT
could anyone help this stupid 8th grader out. I hv absolutely no idea how to ans...
autumn seal
could anyone help this stupid 8th grader out. I hv absolutely no idea how to ans...
what have you attempted ? Have you been able to solve part a,b?
hushed coral
I’ve tried to put them into a table, but it’s not working for me
autumn seal
Have you done arithmetic sequences before?
hushed coral
nop
autumn seal
what do you notice about the increase in black squares in each successive design?
hushed coral
um, it increases by 3, I imagine?
autumn seal
yup so part a what would the answer be?
hushed coral
15
hushed coral
oki
autumn seal
lmk when you reach part c
autumn seal
yes
hushed coral
yay
autumn seal
👍
hushed coral
Am I correct with this one
topaz sinewBOT
Am I correct with this one
hushed coral
I’m not too sure
drifting swift
,rccw
thorny flameBOT
drifting swift
@hushed coral you've gone a bit roundabout, but you did get the right answer at the end
really you could've just done 250 L / (2 L/min) = 125 min
topaz sinewBOT
hushed coral
thanks
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summer prairie
Hello, how can I solve this integral?
topaz sinewBOT
Hello, how can I solve this integral?
summer prairie
$\frac{1}{2\pi}\int_{-\pi}^{\pi}\left | cos(x) \right |e^{-inx}dx$
thorny flameBOT
EyalTeasher
drifting swift
what have you tried thus far?
summer prairie
I splited the integral to 3 parts for take out the absoulte value
than I tried to replace cos with exponent
drifting swift
may i suggest starting with the symmetric-interval simplification instead?
summer prairie
but it is too long, someone told me I can do it in one line, I can't see how
drifting swift
in one line? hm.
drifting swift
may i suggest starting with the symmetric-interval simplification instead?
maybe this is what they meant
$|\cos(x)| e^{-inx} = |\cos(x)| (\cos(nx) - i \sin(nx))\ = |\cos(x)| \cos(nx) - i |\cos(x)| \sin(nx)$
thorny flameBOT
Ann
drifting swift
-i |cos(x)| sin(nx) is odd so its integral over [-pi, pi] vanishes
so you're left with $\frac{1}{2\pi} \int_{-\pi}^{\pi} |\cos(x)| \cos(nx) \dd{x}$
summer prairie
why the second dissapear? and now should I split it to 3 integrals?
thorny flameBOT
Ann
drifting swift
and to answer your question:
why the second dissapear?
it's because the integral of an odd function over a symmetric interval is zero
symmetric interval in this context means an interval symmetric around zero, i.e. [-a, a]
summer prairie
alright sure
drifting swift
no
you know how there's a symmetric interval thing for even functions too?
$\int_{-a}^a f(x) \dd{x} = 2 \int_0^a f(x) \dd{x}$ when $f$ is even
thorny flameBOT
Ann
drifting swift
multiply by two while integrating over only half of the interval
you'll have $\frac{1}{\pi} \int_0^\pi |\cos(x)| \cos(nx) \dd{x}$
thorny flameBOT
Ann
summer prairie
but how can I get rid of the absoulte value?
drifting swift
and at this point you can substitute u := x - pi/2 and do more symmetry stuff
summer prairie
and at this point you can substitute u := x - pi/2 and do more symmetry stuff
u doing that substitute for get rid of abs right? how can I know the function is positive when I moving it in pi/2?
drifting swift
we're not actually doing any integration yet
... i mean ok
at this point maybe it's easier to just split the absolute value
so you'll have $\frac{1}{\pi} \paren{\int_0^{\pi/2} \cos(x) \cos(nx) \dd{x} - \int_{\pi/2}^{\pi} \cos(x) \cos(nx) \dd{x}}$
thorny flameBOT
Ann
pliant arch
how do u do this without expanding the damn thing
i can use this right since they finished?..
drifting swift
ah, i forgot to close the channel
@pliant arch please post in an available channel for the time being
.close
topaz sinewBOT
pliant arch
<@732801184351387648> please post in an available channel for the time being
oh this not availabel i thought aval/
?
ohh i see
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spice skiff
topaz sinewBOT
spice skiff
the bottom one is english
i got xy = p^(2m+n)/2 but i dont get how to answer with base 3
topaz sinewBOT
@spice skiff Has your question been resolved?
topaz sinewBOT
@spice skiff Has your question been resolved?
prisma badger
$a^b=c^{\log_ca^b}=c^{b\log_ca}$
thorny flameBOT
Euclid31415
prisma badger
i got xy = p^(2m+n)/2 but i dont get how to answer with base 3
You could do this but with c=3
prisma badger
Yes
grave patio
Can someone help please!
topaz sinewBOT
Can someone help please!
neon iron
quiz? 🙄
grave patio
no it’s homework
I don’t want the answer
Can you just guid me on how i am supposed to approach this problem?
hollow hedge
what have you tried?
grave patio
Finding the limit
hollow hedge
the limit of?
grave patio
the positive and negative infinity?
hollow hedge
ehh no, you should scope around the options
notice that they mainly focuses on the fact that f(x)=0 or f(x) is positive and negative
so as long as you see f(x)=0, you have to think of stuff like intermediate value theorem or factor theorem
basically anything that has to do with roots of polynomial
and then notice they say something about a number in an interval that makes f(x)=0
so this is implying that we want to use intermediate value theorem
can you do that for me?
topaz sinewBOT
@grave patio Has your question been resolved?
topaz sinewBOT
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pallid steeple
L is a linear transformation. Is it okay to apply it on both sides of an equation ? For example :
x = L(y)
L(x) = L(L(y)), is this always true ?
topaz sinewBOT
L is a linear transformation. Is it okay to apply it on both sides of an equatio...
pallid steeple
L : E → E which means L(L(x)) is well defined, I'm not sure what else I should check ? I've never encountered this property before
last quarry
just know it's true fr yr case
kind cipher
Okay quick question
topaz sinewBOT
Okay quick question
thorny flameBOT
shriller44
kind cipher
when taking the partial derivative of this
when in its parametric form
if y does not actually influence the equation at all would it be a of this form:
$f(x,y) = 4cos(x)$ for example
thorny flameBOT
shriller44
kind cipher
technically the y is part of it but its not directly involved
like the function is of x and y but adding y inputs does not directly make any difference
kinda like $f(x,y) = 4cos(x) + 0y$
thorny flameBOT
shriller44
kind cipher
im basically trying to follow this concept but
what would i put as f_y(P') if my y doesnt affect the output at all
oh acc wait a partial derivative of a parametric equation
is just applying a partial derivative to each bit
so is this essentially what it would be in my case right
any mistake in this logic
i could have simplified to just say f(x,y) rather in the parametric bit but yeah
topaz sinewBOT
@kind cipher Has your question been resolved?
topaz sinewBOT
@kind cipher Has your question been resolved?
topaz sinewBOT
@kind cipher Has your question been resolved?
kind cipher
calm
anyway , you know yesterday i was talking about my plane in f(x,y)
where is y actually in my equation
just when i was doing that derivation of the normal i just implied it was some +0y as it did nothing
i think how i should i think about it is
the plane is just basic rectangle however you normally describe a plane in R^3
but an additional function has been applied to whatever the input x is
so with no vertex shader , the plane is just a basic rectangle in space
yeah basically takes a f(x,y)
and transforms it
by adding depth to it in the ripples
like its a f(g(x), y)
would that change my partial derivative parametric equation logic at all
i guess no as my multivariate function takes in two inputs and outputs y as it is but x as transformed so partial derivative of y is still 0 in that sense as it doesnt affect x
like intuitively it has to as an input change of y has absolutely no affect how x behaves
topaz sinewBOT
@kind cipher Has your question been resolved?
meager notch
Understood
topaz sinewBOT
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strange mica
I got stuck here. How can i prove for the case where that expression is not 0 that the limit doesn't exist or is not finite?
topaz sinewBOT
I got stuck here. How can i prove for the case where that expression is not 0 th...
strange mica
sour bloom
Demonstrate there are two sequences z1 z2 z3 z4... and y1 y2 y3 y4 converging to 0 giving different limits?
that's one way to deal with it
or find a sequence z1 z2 z3 z4... -> 0 without any limit for L
strange mica
Basically i need to find some concrete expression for g and h?
topaz sinewBOT
@strange mica Has your question been resolved?
sullen hazel
strange mica
It's not important how i draw it, but how i solve it
topaz sinewBOT
@strange mica Has your question been resolved?
strange mica
.close
livid siren
Anyone know how to do a? Would be appreciated:)
topaz sinewBOT
Anyone know how to do a? Would be appreciated:)
@livid siren Has your question been resolved?
topaz sinewBOT
@livid siren Has your question been resolved?
tough nest
@livid siren if i'm not mistaken you can calculate the total waiting time of everything combined and then divide that by the total number of waiting times
$f_1 * w_1 + f_2 * w_2 + ... $ all divided by $f_1 + f_2 + ...$
thorny flameBOT
Katharine
tough nest
that would give you the mean
livid siren
ok ty
tough nest
actually holdon
i think i made a mistake
because i tried it using some good old spreadsheet
and i noticed that i got 12
which looks not correct
😄
i got 9 or so now
still looks sortof wrong but better now
😄
look
to get the mean you multiply each height of the bar by the middle value of the bar
which is column C
C = A * B
and then you divide the resulting sums
and get 9.86
if i'm not mistaken that's how to calculate the mean
i was mistaken in the first time because i didn't use the middle of the bars
@livid siren
hopefully this gives some understanding
topaz sinewBOT
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quartz fjord
Hey I have a joint pdf with regions 0<x<inf and -x<y<x im not sure how i would find the support so i could find the marginal pdfs
topaz sinewBOT
Hey I have a joint pdf with regions 0<x<inf and -x<y<x im not sure how i would f...
@quartz fjord Has your question been resolved?
hollow jay
Hey how did they get rid of Xn and the negative xn/2?
Is this just super simple that I just dont understand, or is there a reason for this?
topaz sinewBOT
@quartz fjord Has your question been resolved?
brisk orbit
topaz sinewBOT
@quartz fjord Has your question been resolved?
next bridge
topaz sinewBOT
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next bridge
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hallow rain
I have no clue about how to solve this
He wants Tan = oppo/adj
But I'm only have hypo
I need to get oppo & adj to solve it, but how can I get it?
What's the method to solve this equation?
topaz sinewBOT
I have no clue about how to solve this
He wants Tan = oppo/adj
But I'm only have...
pseudo bear
@hallow rain Do you know more about ABCD other than that it's a rectangle?
Like any other sides or the slope of the diagonal?
hallow rain
<@762680156917334047> Do you know more about ABCD other than that it's a rectang...
No
hallow rain
Idk if he made a mistake and didn't gave me the whole equation
pseudo bear
OK, so here's how I'd answer it.
hallow rain
Do you have a picture of the problem?
That's was the pic
pseudo bear
tan(angle) = slope.
So, that angle is pointing along the diagonal.
So, the slope of the diagonal is DC/BC.
So, tan(m∠DBC) = DC/BC.
Also, that's opposite over adjacent.
So, that's another way to get it.
If they want an actual number, there's not enough information there to get it.
hallow rain
If they want an actual number, there's not enough information there to get it.
I think he made a mistake and didn't gave me the correct equation, however, Thanks for help <3
pseudo bear
You're welcome.
topaz sinewBOT
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earnest moon
Can someone help me?
topaz sinewBOT
Can someone help me?
earnest moon
I have to differentiate the sec( tan(√x))
topaz sinewBOT
@earnest moon Has your question been resolved?
earnest moon
<@&286206848099549185>
tacit skiff
I have to differentiate the sec( tan(√x))
this is a composite function
do you know chain rule?
earnest moon
Ya I know
tacit skiff
ok, first you need to calculate the derivative of the secant
in which every argument will be (\tan(\sqrt{x})), right?
thorny flameBOT
SubGui
earnest moon
Okay
tacit skiff
after this, multiply everything by the derivative of (\tan(\sqrt{x})), in which you'll need to apply the chain rule again
thorny flameBOT
SubGui
tacit skiff
calculate the derivative of tan, with the argument being (\sqrt{x})
finally multiplying everything by the derivative of (\sqrt{x})
thorny flameBOT
SubGui
earnest moon
Actually I was trying to solve in the bookish way
And in my book there is no mention of derivative of sec
So I have to solve in a way where I don't use derivative of sec
So I transformed sec into 1/cos
But now I am stuck
My answer is not matching with the answer given in book
crimson rampart
,rotate
thorny flameBOT
crimson rampart
You still have to differentiate the $\sqrt{x}$ since you used the chain rule again for $tan(\sqrt{x})$
thorny flameBOT
RESIDGE
crimson rampart
@earnest moon
topaz sinewBOT
@earnest moon Has your question been resolved?
earnest moon
Sorry guys , idk why I was unable to solve such a silly problem
earnest moon
Thank you @crimson rampart
@tacit skiff
topaz sinewBOT
Thank you <@398242893535510530>
<@520057286220316683>
earnest moon
.close
topaz sinewBOT
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lapis oak
topaz sinewBOT
thorny flameBOT
lapis oak
please?
topaz sinewBOT
@lapis oak Has your question been resolved?
sweet nexus
What class is this for?
Is it trig or pre trig?
Or are you familiar with how to break it up into components? (Like turning a vector + an angle into horizontal and vertical parts, etc etc)
lapis oak
Is it trig or pre trig?
vectors
lapis oak
Or are you familiar with how to break it up into components? (Like turning a vec...
i tried graphing it so it's easier but i am very confuesed
sweet nexus
Well you can break them up into components then, by making both those vectors two right triangles
And add up the horizontal and verticals components to get there
For example, the horizontal component of your first vector would be 30km * sin(35°) and that’s moving east.
thorny flameBOT
lapis oak
And add up the horizontal and verticals components to get there
like this?
sweet nexus
No, they’d be tip to tip, One second and I’ll show
lapis oak
No, they’d be tip to tip, One second and I’ll show
oh okay that would be soooo helpful
sweet nexus
So you can break them down into components, getting the different sides of the triangles will tell you the amount moved south and the amount moved east/west (you can choose which one is negative), and that will give you the final position
Then taking the magnitude of the new vector will give you distance from the origin
I tried to show it together and then just broke them separately to try and demonstrate the idea of component form, which you can find with some not too bad trig :)
sweet nexus
Sure :)
lapis oak
sweet nexus
No, you don't know any of the exact angles inside of the triangle. You could find the angle between your two vectors if you really wanted to, but you can use simple angle identities to get to your answer by breaking your vectors up into components.
So for example, to find how far east or west you've gone you can take the sine of these triangles, so 30sin(35)-25sin(62)
(The second one is subtracted because it's going in the opposite direction, so we're saying east is like positive x and west is like negative x)
So we know that they've gone 4.86639... km west of the origin
You can use a similar method to find the distance they've gone south, by finding that side of each triangle and adding them
Then you can find the total distance traveled by finding the magnitude of both of these vector components (which is just the Pythagorean theorem, because you'd have the leg that goes west and a leg that goes south)
lapis oak
So we know that they've gone 4.86639... km west of the origin
then i can add this to the 35 and i have that angle?
sweet nexus
No no, that's the horizontal distance traveled. Have you learned about vector components? Like <x,y>?
sweet nexus
Okay, so this would be your x value
lapis oak
and sin to find y
sweet nexus
Well it depends on where your angle is from your original triangle, we just used sin to find x, but we found one of the two components, so this is the x component, so you find y
And then once you have the y component you have the distance traveled west and the distance traveled south, and you can use those components to get the magnitude of the new vector
lapis oak
Well it depends on where your angle is from your original triangle, we just used...
how do find y?
sweet nexus
It would be the cos for these triangles
sweet nexus
If you look at this, you can see that the side of the triangle going in the south direction is adjacent to our angle
lapis oak
so 30cos(35)+25cos(62)?
sweet nexus
We subtracted before because it was going in the opposite direction (East then west), but for this one they are going in the same direction
lapis oak
We subtracted before because it was going in the opposite direction (East then w...
so +?
sweet nexus
Yep
Or they could both be negative and you subtract them, as long as they are going in the same direction, it's your choice for what direction you would want to be positive or negative :)
lapis oak
So for example, to find how far east or west you've gone you can take the sine o...
this is the east and west component which is x?
sweet nexus
Yep
So in that we chose east to be postive and west to be negative, but when we got an answer that was negative east, we said it was positive west
Because if you went backwards in the east direction it would be going forwards in the west
lapis oak
Because if you went backwards in the east direction it would be going forwards i...
so the y is 39.3km
that is the distance from the origin?
sweet nexus
I got 36.3km
And that would be your x component and your y component
For your new vector, and the total distance from the origin would be your magnitude
So you'd have a side of 36.3km and a side of 4.86km, and you can use the Pythagorean theorem to find the hypotenuse giving you your distance.
lapis oak
So you'd have a side of 36.3km and a side of 4.86km, and you can use the Pythago...
im confuesed about that part
where is this traingle we are finding the hypotenuse
sweet nexus
So we've discovered that you've moved 36.3km south, and 4.86km west, right?
sweet nexus
Do you see how this gives us a triangle where the hypotenuse would be the distance from the origin?
(Does the diagram help)
sweet nexus
So combine everything we did and then find the hypotenuse for the final triangle and you'll get the distance from the origin :)
lapis oak
thank you sooooo mucchchc
sweet nexus
I hope I made it clear what you're doing and why along the way. I try not to just give answers :P
lapis oak
I hope I made it clear what you're doing and why along the way. I try not to jus...
yes
it helped a lot
i understood the steps
sweet nexus
Good deal!
sweet nexus
Sure
thorny flameBOT
lapis oak
,rotate
sweet nexus
I can read it on my phone, it's fine. So are you confused on how to start this one?
topaz sinewBOT
@lapis oak Has your question been resolved?
sweet nexus
You'd use the same exact methods of taking it as components, and then the final velocity would be the magnitude
So one direction would be where the boat is moving and the other is where it's carried by the river. B and C you should be able to get if you just think about it :)
lapis oak
So one direction would be where the boat is moving and the other is where it's c...
part a is just making a right angle traingle right and finding the hypotenuse ?
sweet nexus
Yep
That'd get you the speed (the hypotenuse), you can use the inverse sine or cosine after finding the hypotenuse to get the angle, which will give you the direction
lapis oak
part b is adding the speed of the river and the boat and the direction is east?
and part c the river speed - boat directed is west? cause it would be slow going against the water flow?
sweet nexus
Precisely :)
lapis oak
Precisely :)
okok,
lapis oak
That'd get you the speed (the hypotenuse), you can use the inverse sine or cosin...
which angle can i use?
for the direction
the angle adjecent to the river?
sweet nexus
You'll find it from using either the inverse cosine or sine
Oh, that's up to you
As long as you specify
Since the boat is traveling south and it's getting moved from the west, I'd use the angle from south towards the direction the boat is getting pushed in
Which would give it a similar form to the question before
thorny flameBOT
lapis oak
so something like this?
sweet nexus
Yep
sweet nexus
The earlier questions gave compass
So this is how I would personally draw it, but if you look, your triangle and chosen theta is actually equivalent to this one
This one just makes it a little more clear that it's going S(theta)E
lapis oak
This one just makes it a little more clear that it's going S(theta)E
okok
make sense
easier
sweet nexus
But if you wanted to you could make it go east and then south and then your theta on the inside would be E(theta)S instead, and still correct.
S just went with S(theta)E both because that's what you drew and that's what the earlier question provided.
lapis oak
S just went with S(theta)E both because that's what you drew and that's what the...
thank you soooo much
if u wanted just explain this question too?
last one
i would really much appreciate it
sweet nexus
That one is worded tricky but basically the same thing we've been doing
So you're given F2 and you know that F1 + F2 = (we'll call it F3)
We can use the components again and with some very simple algebra you can see that F1 = F3 - F2
So you can convert F2 and that third vector into components, subtract the components in a way that follows the easy algebra, and then turn your new vector back into the form they want it in
lapis oak
We can use the components again and with some very simple algebra you can see th...
so F1 is 20N on a bearing of 036
sweet nexus
I don’t think it works like that, I’d break it down into the x and y components and then subtract those, but I’m not 100% sure that’s wrong.
lapis oak
I don’t *think* it works like that, I’d break it down into the x and y component...
how do you break it down into the x and y component?
sweet nexus
The same way as you did two problems ago with south and east/west. Using the sine and cosine with the angles, multiplied by the magnitude
lapis oak
so 40sin(89)-20sin(53) and then 40cos(89)-20cos(53) ?
sweet nexus
Almost, I think it’d be minus for both of those, but I didn’t draw a diagram to be sure.
lapis oak
in the question it says include a diagram?
sweet nexus
It’s always a good idea to draw a diagram. That way you can know if you’re adding or subtracting your vectors
sweet nexus
Yes, the angle starts from your x-axis and moves counter-clockwise
So the 53° would be lower than 89°( and 90° is the direction of the positive y axis), but this still helps us see that they’re moving in the same direction
So we know that we just subtract normally
sweet nexus
Sin = opposite/hypotenuse
Rearrange to get
Hypotenuse * sin = opposite
With the angle where it is, we can see that opposite will be up and down, along the y axis, does that make sense?
lapis oak
With the angle where it is, we can see that opposite will be up and down, along ...
kind of
sweet nexus
Just make them into triangles and use SOH CAH TOA if you’re familiar with that and you should get through it
Once you have the vectors into their components just subtract and make a new vector, but I am falling asleep so someone else will have to help if you have more questions, I hope I’ve been helpful :)
lapis oak
Once you have the vectors into their components just subtract and make a new vec...
i just have a test tomorrow these are the practice question and that is the last one left
i wanted to get it done and go to sleep
lapis oak
Just make them into triangles and use SOH CAH TOA if you’re familiar with that a...
im familiar in SOH CAH TOA
lapis oak
Once you have the vectors into their components just subtract and make a new vec...
you been really helpfull thank you sooo much
@next tree
next tree
,rotate
lapis oak
hi
thorny flameBOT
lapis oak
next tree
Do you know what is F1 in your image?
lapis oak
,rotate
thorny flameBOT
lapis oak
Do you know what is F1 in your image?
no
next tree
Okay, recall
lapis oak
Okay, recall
to what?
next tree
lapis oak
so we get a right angle traingle?
lapis oak
Adding vectors together using tip to tail method
im familiar with that
lapis oak
No sure, you have to calculate
we just use pythagorean theorem to find F1?
next tree
Then can use find the angle between F2 and F2+F1?
next tree
we just use pythagorean theorem to find F1?
And I don't think it's a right angle, but I'm not sure.
lapis oak
And I don't think it's a right angle, but I'm not sure.
oh ok
next tree
Have you heard of cosine formula?
lapis oak
yes
next tree
Magnitude of F1 can use cosine formula
And for the true bearing
You may have to split those F2 and F2+F1 into vertical and horizontal parts
lapis oak
Magnitude of F1 can use cosine formula
but we need an angle for that?
next tree
The angle is just 89°-53°
next tree
You may have to split those F2 and F2+F1 into vertical and horizontal parts
And btw, if you use this method, you can find the horizontal and vertical components of F1, then you can find the bearing and magnitude
lapis oak
Magnitude of F1 can use cosine formula
i got 18.14N
lapis oak
And btw, if you use this method, you can find the horizontal and vertical compon...
im not sure how to do that
next tree
Yea, but I gtg soon, so I don't have enough time to explain
topaz sinewBOT
@lapis oak Has your question been resolved?
wanton bolt
@drifting swift I'm back at the problem,#help-2 message
wanton bolt
<@!183668144404037632> I'm back at the problem,https://discord.com/channels/268...
I just found 2 things:
shadow kayak
why?
sour bloom
this channel is for bandbee
wanton bolt
Thanks Element for the screenshot
keen venture
@shadow kayak
Please read #❓how-to-get-help
drifting swift
@neon iron @shadow kayak please move from this channel to somewhere free.
@wanton bolt ok so what did you find?
wanton bolt
- f(x+1) >= f(x)
- f(x+1) - f(x) <= 1
wanton bolt
<@876550410767323156> ok so what did you find?
Only those 2 points
sour bloom
1) f(x+1) >= f(x)
2) f(x+1) - f(x) <= 1
try combining them:
f(x)<=f(x+1)<=f(x)+1
so does that mean if f satisfies this for all x, then it is in S?
wanton bolt
try combining them:
f(x)<=f(x+1)<=f(x)+1
Thanks
wanton bolt
so does that mean if f satisfies this for all x, then it is in S?
Yes
drifting swift
i said no such thing.
wanton bolt
I thought she meant that
wanton bolt
i said no such thing.
Whoops sorry if I misunderstood
drifting swift
at no point did i assert S is uncountable
just because i asked you to prove your claim, doesn't mean your claim is false
wanton bolt
Yeah I thought you meant that
sour bloom
okay let's stick with bandbee's guess it's uncountable, now the easiest way to start would probably be to assume it is countable, and arrive at a contradiction
so assume you can list down all the functions, then the next step would be to construct some function not on the list
wanton bolt
Assuming it is countable, it would exist an injection from it to N
sour bloom
yeah
wanton bolt
I don't intuitively see how to find a function not on the list
drifting swift
||there is a way to biject S to a set whose cardinality is easy to find||
wanton bolt
Into R or R^2 again?
sour bloom
can you sort of "count" the number of such functions?
drifting swift
if you can find a nice bijection between S and R or R^2 then power to you
wanton bolt
can you sort of "count" the number of such functions?
I don't think so
median temple
||A diagonal argument works neatly here I might add||
wanton bolt
||A diagonal argument works neatly here I might add||
Would you have a hint of how to apply it?
It's not like counting sequences of 0 and 1
drifting swift
you'd be surprised
median temple
nope, but it doesn't have to
well
the heart of the diagonal argument doesn't lie in 0 and 1
sour bloom
construct a function that disagrees with the first function on the list at some point, disagrees with the second function at some other point, disagrees with the third function at some third point... and so on
drifting swift
but you actually CAN biject S to the set of 0-1 sequences
median temple
oh yeah
median temple
but you actually CAN biject S to the set of 0-1 sequences
I see now
wanton bolt
construct a function that disagrees with the first function on the list at some ...
Oh well
median temple
eugh, i forgot y'all don't include 0 in the naturals
drifting swift
map f ∈ S to the sequence (f(0), f(1)-f(0), f(2)-f(1), f(3)-f(2), ...)
and for the inverse, map a 0-1 sequence (a_n) to the function f(n) = sum[k=0,n] a_k
median temple
(the codomain would be ℕx{0,1}^ℕ in this case)
wanton bolt
map f ∈ S to the sequence (f(0), f(1)-f(0), f(2)-f(1), f(3)-f(2), ...)
Yeah one could define like that
median temple
which is a funny set, but an uncountable one
sour bloom
try sketching out the argument
wanton bolt
I'll take a paper
wanton bolt
map f ∈ S to the sequence (f(0), f(1)-f(0), f(2)-f(1), f(3)-f(2), ...)
Every element in the sequence is less or equal to 1
median temple
My diagonal argument went like this :
||Suppose f_n was a sequence of functions that reached every element of S. Then you can construct the function g as follows.||
||g(1) = f_1(1)+1||
||g(n+1) = g(n) if f_{n+1}(n+1)=g(n)+1, g(n)+1 otherwise||
||g can increase by at most 1 at every step, so it's in S (the actual proof isn't hard).||
||And by construction g is different from every f_n in at least one point (namely at the n-th value), which would mean g isn't in S, a contradiction.||
wanton bolt
Can we decide to "round" it to 0 or 1?
drifting swift
what do you mean by "round"
every element in the sequence is 0 or 1, because f(x+1) - f(x) ≥ 0.
and S consists of natural-valued functions.
median temple
- ≤ 1
wanton bolt
Alright got it
wanton bolt
every element in the sequence _is 0 or 1_, because f(x+1) - f(x) ≥ 0.
Because the codomain is N
drifting swift
@median temple both are true lol
wanton bolt
My diagonal argument went like this :
||Suppose f_n was a sequence of functions...
I'm looking at it
Ann's technique works because then one can apply the classical diagonalisation
median temple
not even that, you just proved a bijection between a set and an uncountable one
no need to actually pull out a diagonalisation
wanton bolt
map f ∈ S to the sequence (f(0), f(1)-f(0), f(2)-f(1), f(3)-f(2), ...)
If one defines a function in this way, it is an infinite sequence of 0 or 1, isn't it?
And then:
median temple
save for the first term which can be any natural number
wanton bolt
One chooses a function such that f(0) disagrees with the 1st function
drifting swift
If one defines a function in this way, it is an infinite sequence of 0 or 1, isn...
yes that was exactly my point
wanton bolt
yes that was exactly my point
Thanks a lot!
drifting swift
oh but yes of course f(0) could be anything
median temple
A much trickier exercise I had in a sheet was somewhat similar
It was proving that the set of all bijections between from ℕ to ℕ was uncountable
It's hard
wanton bolt
So what should one define for f(0) of the function, just saying that it is not f(0) of the 1st one?
median temple
If you're intent on using a diagonal argument, then yeah, just use f(0)+1 or something
wanton bolt
Thanks Ann Syst3ms and Element118 for your help!
How do you see solutions so intuitively?
drifting swift
probably just years of experience
wanton bolt
Do you have books/exercise sets recommendations?
median temple
diagonal arguments work wonders as far as uncountability goes, especially with sequences involved
although i studied countable sets more recently in class
so yknow
wanton bolt
Yeah you're right, I thinked about diagonal argument
wanton bolt
map f ∈ S to the sequence (f(0), f(1)-f(0), f(2)-f(1), f(3)-f(2), ...)
I mean mostly how to see intuively that this is a good idea for instance
Is there any good discrete maths practice exercises set you would recommend?
median temple
A way you can conceptualize it here is that the definition makes it so on each step, the function can at most increase by 1
So you start with an initial value, and at each step you decide whether to stay put (0) or increment (1)
wanton bolt
A way you can conceptualize it here is that the definition makes it so on each s...
Which is to say either 0 or 1
median temple
yup
Then what comes with some habit is noticing that this entirely describes the function (+ the first term)
wanton bolt
So you start with an initial value, and at each step you decide whether to stay ...
Thanks! I'll remember this technique for next times
median temple
These exercises are often about finding a way to represent your set in a different, more advantageous way
median temple
It was proving that the set of all bijections between from ℕ to ℕ was uncountabl...
oh my god standard solutions for this are far easier than what i came up with
wanton bolt
Hahaha
wanton bolt
With the exercise about functions
How to prove that the thing we used was injective?
{0, 1} ^infinite -> (f(0), f(1)-f(0), f(2)-f(1),...)
topaz sinewBOT
@wanton bolt Has your question been resolved?
wanton bolt
Or at least write our method as an injection {0, 1} ^infinite -> S
median temple
if you have two equal outputs
and corresponding functions f and g
you have f(0)=g(0) since the first term is equal
then f(1)-f(0)=g(1)-g(0), but since f(0)=g(0) that just simplifies to f(1)=g(1)
and so on
hence f and g are equal
wanton bolt
and hence their representation as a sequence are equal?
median temple
we started with the assumption that their representation as a sequence was equal
then ended up proving that they were equal
that's injectivity
wanton bolt
Yeah but injectivity is that f(x1) = f(x2) implies x1 = x2
I think you proved the contrary
x1 = x2 implies f(x1) = f(x2)
topaz sinewBOT
@wanton bolt Has your question been resolved?
wanton bolt
the sequence representation is f(x1)
For an injection {0, 1} ^infinite -> S, {0, 1} ^infinite is the domain and hence x1, x2, ...
median temple
oh, you took it in the other direction
also it's not {0,1}^ℕ
the first term can be any natural number
so it's
ℕx{0,1}^ℕ
wanton bolt
oh, you took it in the other direction
Yeah
So you mean the proof is like what you said before but in the other direction?
median temple
i mean, it shouldn't be too hard to prove that if two members of S are equal then their representation is the same
in fact that's easier
drifting swift
you can do something silly
to make the map actually go to {0,1}^N
such as starting with a run of floor(log_2(f(0))) zeros, then the binary representation of f(0), and then the values of f(x+1)-f(x)
topaz sinewBOT
@wanton bolt Has your question been resolved?
wanton bolt
such as starting with a run of floor(log_2(f(0))) zeros, then the binary represe...
Thanks!
I think the more simple version would fit better though
Because this one might be too complicated for an exercise of this kind
Oh and btw a quick question
How to formally prove span {f1, f2} = span {f3, f4}?
wooden matrix
Write the span out explicitly then apply log laws
drifting swift
$f_1 = f_3 + f_4$ and $f_2 = 2f_4$
thorny flameBOT
Ann
wanton bolt
Thanks!
wanton bolt
**Ann**
I mean formally what tells that span {f1, f2} = span {f3, f4} from this? Is there a theorem that derives this direct consequence?
drifting swift
you can use the formulas i just wrote to express any linear combination of f3 and f4 as a linear combination of f1 and f2
you can also express f3 and f4 in terms of f1 and f2 to go the other way around
wanton bolt
you can also express f3 and f4 in terms of f1 and f2 to go the other way around
You're right, I had a reasoning bug on this part
Thx a lot
.close
topaz sinewBOT
topaz sinewBOT
Available help channel!
topaz sinewBOT
Available help channel!
frank crag
Hello, can someone help me with this exercise (see image) about the measure theory . I don't know how to prove the statement above 
topaz sinewBOT
Hello, can someone help me with this exercise (see image) about the measure theo...
@frank crag Has your question been resolved?
topaz sinewBOT
@frank crag Has your question been resolved?
opaque agate
can anyone help me out with this exercise? we need to demonstrate this afirmation:
topaz sinewBOT
can anyone help me out with this exercise? we need to demonstrate this afirmatio...
opaque agate
grim jacinth
You can probably google a proof, but I've seen the following:
If you add two copies of the LHS together, you can combine the terms into groups of (n+1). You will get n copies of n + 1, or n(n+1). This is twice the sum, so if you divide both sides by 2, you get your result
topaz sinewBOT
@opaque agate Has your question been resolved?
grim jacinth
flat owl
can someone help?
topaz sinewBOT
can someone help?
gloomy orchid
I need help with number 9. It's asking to find dy/dx by implicit differentiation.
topaz sinewBOT
I need help with number 9. It's asking to find dy/dx by implicit differentiation...
gloomy orchid
Here's my work. I don't where I am stuck at exactly. All I know is that my answer is wrong and I made a mistake or mistakes somewhere.
topaz sinewBOT
@gloomy orchid Has your question been resolved?
topaz sinewBOT
gloomy orchid
.reopen
topaz sinewBOT
✅
gloomy orchid
<@&286206848099549185>
gloomy orchid
I don't understand
tacit skiff
,w implicit derivative x^4(x+y)=y^2(3x-y)
thorny flameBOT
tacit skiff
wolfram just multiplied it by (-1)/(-1) for some reason
but you can see you really didn't it wrong
gloomy orchid
I'm still confused. Is my solution right or wrong?
empty sail
It's not fully simplified
gloomy orchid
So my solution is correct but I need to simplify it?
tacit skiff
So my solution is correct but I need to simplify it?
yeah
untold cypress
need help finding the area of the figure
topaz sinewBOT
need help finding the area of the figure
untold cypress
this will be the translation in english of the problem:
The area of the figure enclosed by y=…., y=…. and y=….. is equal to
echo siren
are you studying calculus? you know about integration?
untold cypress
little bit
so do i make the bounds x/2 at top and -x/2 at bottom?
for numerator and denominator?
echo siren
not really, i was thinking this: Notice that the area enclosed is symmetric respect the y axis
so, let focus in the first half of the figure. Then if you can find the intersection of $y=\frac{1}{1+x^{2}}$ and $y=\frac{x}{2}$
thorny flameBOT
BillyElKid
untold cypress
intersects at (1, 0.5)
echo siren
nice, so what can you say about the next integral ? what area is enclosing ?
echo siren
$\int_{0}^{1} \frac{1}{1+x^{2}}-\frac{x}{2}$
thorny flameBOT
BillyElKid
echo siren
this time for y=-x/2?
yes but can you tell me what area is representing that integral?
untold cypress
well, the limit is from 0 to 1
echo siren
those are the limits of integration, i'm refering to the area, remember that the integral "sum" the area under the curve
so for example if i'm integrating 0 to 1 of the function $f(x)=x$ i.e $\int_{0}^{1} x $
thorny flameBOT
BillyElKid
untold cypress
$\int _0^1\frac{1}{1+x^2}dx-\int _0^1\frac{x}{2}dx
so basically this
just make one as u-substitution
echo siren
the area i'm calculating is this one
yes, the value of the integral is right, that is not what i'm concerned about, i'm asking you if you can tell me what represents that value, what area
"Remember that the integral represents the area under the curve"
So this integral $\int_{0}^{1} \frac{1}{1+x^{2}}$
thorny flameBOT
BillyElKid
echo siren
