nocturne sun
oh wait nvm I read the thing wrong entirely
#help-10
finite vector
ok so u have a specific q as context?
finite vector
esentially I want to find a point on a line
or if u could find one, would be better
alpine flare
hmm
I'm lost ,but basted on that stack post it could be this > ```
auto solveStraight = [](const QVector2D &start, const QVector2D &end, const float t) -> QVector2D {
std::sqrt(std::pow(end.x() - start.x(), 2) + std::pow(end.y() - start.y(), 2));
const float xt = (1 - t) * (float) start.x() + (t * end.x());
const float yt = (1 - t) * (float) start.y() + (t * end.y());
return {xt, yt};
};
whoops i only know a bit of python
alpine flare
I think the relevant parts are >
const float xt = (1 - t) * (float) start.x() + (t * end.x());
const float yt = (1 - t) * (float) start.y() + (t * end.y());
t being distance % (I think) from 0-1
so t=0.5, should return point in the middle between the 2 points
alpine flare
see what
alpine flare
you mean this ?
finite vector
t*d
alpine flare
ok
finite vector
ok wait
alpine flare
ant (1-t) is it pow(1-t,a) ?
finite vector
do u know the section formula?
alpine flare
no...
finite vector
ok then..
alpine flare
sorry
finite vector
hold on
alpine flare
33yold dumbo here
finite vector
hey dw about it
alpine flare
I'm learning how to "handle" curves in 3d, so straight, cubic, quadratic,
finite vector
ok wait could u look it up? i'll try searching for a vid which explains well
finite vector
alpine flare
OH NEAT that is it! Thanks 😄
finite vector
then u could get back to me (or anyone)
for the 't' part
tho i believe u would realize what's happening once u get a gist of it
alpine flare
yep, I think I'm good with this equation
will watch vid to be more "sure" but so far its "working"
finite vector
oh lol ok
alpine flare
I'm moving now slowly to figuring out a quadratic curve xy point now
finite vector
ohk
rough bough
obtuse pebbleBOT
rough bough
obtuse pebbleBOT
@rough bough Has your question been resolved?
alpine flare
what was the question
rough bough
Integrate
alpine flare
Feels like hes a doctor
rough bough
Not really, more like a cross examiner
royal basin
write better 4s, your's looks like a u
their handwriting has been shit for a while
weak quiver
Hi, I have a theory question, i don't understand what the difference is between derivative and differential, but i know they are quit related with each other ? can someone explain me ? thankyou!
obtuse pebbleBOT
Hi, I have a theory question, i don't understand what the difference is between ...
@weak quiver Has your question been resolved?
drowsy iris
obtuse pebbleBOT
novel knoll
What have you tried?
eternal bloom
For the first inequality, consider the area under 1/x^p from k to k+1. For the bounds, consider approximating this area with rectangles.
Then sum it from k=1 to n to obtain the second
obtuse pebbleBOT
@drowsy iris Has your question been resolved?
drowsy iris
lemme try
drowsy iris
What have you tried?
integrate and differentiate
drowsy iris
For the first inequality, consider the area under 1/x^p from k to k+1. For the b...
should i convert the integral into riemann sum
eternal bloom
should i convert the integral into riemann sum
No.
For the first inequality, consider the area under 1/x^p from k to k+1. For the bounds, consider approximating this area with rectangles.
Then sum it from k=1 to n to obtain the second
drowsy iris
how should i present it tho
novel knoll
integrate and differentiate
Look at the inequality you need to prove
See any relation between the inequality and the integral?
drowsy iris
the bounds
novel knoll
Is the function you are integrating monotone?
drowsy iris
yes?
novel knoll
So it attains minimum at k+1
So compare it to having area under 1/(k+1)^p
And width (k+1)-k=1
Which is exactly the inequality you then get
drowsy iris
i understand the general picture here
but like how should i write the steps to my teacher
novel knoll
If f has a lower bound, m, in the area you are integrating you have
int from a to b of f(x) dx>=m(b-a)
drowsy iris
ok
novel knoll
Yes
drowsy iris
but how do i express the integral part
drowsy iris
??
drowsy iris
Try to figure out what to do if n>1
split the integral
novel knoll
Si
drowsy iris
?
novel knoll
?
novel knoll
Means yes in a few languages
obtuse pebbleBOT
@drowsy iris Has your question been resolved?
ashen dove
Why is there a negative sign over there?? Ik it's a physics question..
obtuse pebbleBOT
Why is there a negative sign over there?? Ik it's a physics question..
@ashen dove Has your question been resolved?
weak wind
hello! if V is a vector space, a is an element of V, and U is a subspace of V
how is a set a + U := { a+u | u from U } called? i'm trying to find the term in english
it might have something to do with topology
obtuse pebbleBOT
hello! if **V** is a vector space, **a** is an element of V, and **U** is a subs...
brave bramble
It's called a coset
Or maybe an affine space?
cosets are more common in group theory, so if you google them, you'd need to go "coset vector space"
obtuse pebbleBOT
@weak wind Has your question been resolved?
obtuse pebbleBOT
obtuse pebbleBOT
Available help channel!
main cargo
obtuse pebbleBOT
spark hinge
3x^2 + 3[y^2 + 4y] = 0 **
main cargo
3x^2 + 3[y^2 + 4y] = 0 yes
3x^2 + 3((y+2)^2 -4) = 0
3x^2 + 3(y+2)^2 = 12
where have i went wrong
radius should be 2
not root(12)
high lily
your error is assuming that equation you have at the end implies the radius is sqrt(12)
spark hinge
Well There's a common factor across the whole equation
high lily
algebraically what you did so far was fine
obtuse pebbleBOT
obtuse pebbleBOT
Available help channel!
fringe orbit
Hello I have geometry sequences question
An
I have A1> 0 and 0<q<1
Bn
B1= a6
And Cn = (an+ 5)/ bn
I have to prove that all of those sequences r positive
obtuse pebbleBOT
Hello I have geometry sequences question
An
I have A1> 0 and 0<q<1
Bn
B1= a6
An...
fringe orbit
An is positive cuz a1>0 and the q>0
Cn will be positive cuz + / + will be positive
But how I prove that Bn is positive ?
Is my question clear xd ?
tardy epoch
do you even have an equation for Bn ?
fringe orbit
Only thing I have is b1 = a6 xd
About b
And the q thingy for it is r
Like b2 = b1 * r xd
fringe orbit
do you even have an equation for Bn ?
I don’t have 1 I guess
tardy epoch
show the original question
tardy epoch
,rotate
warm shaleBOT
tardy epoch
translate if you want help.
fringe orbit
I don’t understand this language ether 😂 but An is infinite sequences and the thingy is q given that 0<q<1 and 0<a1
And the thingy for Bn is r and b1= a6
And Cn = (an+ 5)/bn
Prove that all those sequences r positive
tardy epoch
...
fringe orbit
😅
fringe orbit
That’s the country language not mine /:
My country occupied
And I’m forced to learn with this language
ashen dove
Huh?? Occupied
fierce lagoon
Oh
fringe orbit
Math already hard we have to learn it with another language ;-;
That’s the problem xd
Is there any idea how to solve this question xd
lean wigeon
does it say what to do with $a_n$?
warm shaleBOT
opfromthestart
timid silo
Still kinda stuck on this question i asked a while ago
obtuse pebbleBOT
Still kinda stuck on this question i asked a while ago
timid silo
I understand that rank(T)=rank(L_A) means the dimension of the image of T is the same as the dimension of the image of L_A
But i'm not sure how to go about proving it
strong vale
who can help me with geometry
slate cosmos
who can help me with geometry
This isn't your channel please go to available channels to ask a question :)
obtuse pebbleBOT
Available help channel!
bronze talon
determine the equation for a rational function that has asymptotes at x=5/2 and y= -2 hole x=1, no x-intercepts
obtuse pebbleBOT
determine the equation for a rational function that has asymptotes at x=5/2 and...
bronze talon
how do i do this question
tardy epoch
Finding Horizontal Asymptote of a Rational Function
https://www.cuemath.com/calculus/horizontal-asymptote/
and
Vertical Asymptotes of Rational Function
https://www.cuemath.com/calculus/vertical-asymptote/
you'll likely have to trial and error
obtuse pebbleBOT
@bronze talon Has your question been resolved?
bronze talon
i still don't understand
tardy epoch
i still don't understand
did you read the definition of horizontal and vertical asymptotes of rational functions?
bronze talon
yes but how would i know the aslymtotes when i have 5/2
of the horizontal and vertical
tardy epoch
you'll likely have to trial and error
.
obtuse pebbleBOT
@bronze talon Has your question been resolved?
timid silo
obtuse pebbleBOT
hybrid gyro
damn, feels like a bit of try-and-error to me as there's more than 2 prime numbers between 4 and 18
timid silo
damn, feels like a bit of try-and-error to me as there's more than 2 prime numbe...
yes its 5,7,11,13,17
do you know how I should start cause I was just guessing before and it got me nowhere
mystic wedge
what topic does this fall under?
timid silo
numebr theory
hybrid gyro
do you know how I should start cause I was just guessing before and it got me no...
I'm probably dumb but I don't see a way without more information
there might be properties about product and sum of primes that I just never thought of though
mystic wedge
you probably have to introduce some variables
gloomy basin
How do u make -x into x
solar gorge
How do u make -x into x
by transferring it to the other side (LHS) in this case
It will be ** x = -8y - 3 **
gloomy basin
by transferring it to the other side (LHS) in this case
It will be ** `x = -8y ...
ok
gloomy basin
by transferring it to the other side (LHS) in this case
It will be ** `x = -8y ...
i didnt times 3 by 2
x= -8y -6
.close
loud forum
obtuse pebbleBOT
loud forum
whats a and ?
obtuse pebbleBOT
@loud forum Has your question been resolved?
hazy rose
Why can the projection of a on b = the projection of a on the projection of b on c
obtuse pebbleBOT
Why can the projection of a on b = the projection of a on the projection of b ...
hazy rose
This being
a ↓b = a ↓(b ↓c)
I waited an hour on the previous channel, which I closed and opened this one so I’m going to flag <@&286206848099549185>
nocturne minnow
I waited an hour on the previous channel, which I closed and opened this one so ...
I mean technically you opened a new channel so you still have to wait at 15 minutes before pinging helpers
hazy rose
Ok, my bad, just a bit frustrated, idk what opening a new channel would have done anyways but thx for the heads up ig
hazy rose
Oh my bad
It’s late, the question says
a ↓c = a ↓(b ↓c)
These all being vectors, which I suppose is similar to what you were saying
But my question is why is this true
hazy rose
Ah, but how do we know that the angle between a and b is smaller than that of c and b
Although this clarifies a lot
hazy rose
Well I think that explains it, thanks for the help
opal trench
ye olde switcharoo
hazy rose
Many thanks
Haha I was writing out a whole essay and then reread my question and it said to draw a diagram
.close
obtuse pebbleBOT
opal trench
lmaooooo
obtuse pebbleBOT
Available help channel!
obsidian isle
what is div F
obtuse pebbleBOT
what is div F
obsidian isle
$\text{div} \vec{F}$
warm shaleBOT
EndTimes
gilded needle
divergence of a vector field?
gilded needle
obsidian isle
ok so let me get this straight
$\text{div} \vec{F} = \nabla \vec{F} \cdot \vec{F}$
warm shaleBOT
EndTimes
vapid hamlet
curl div F does not make any sense because div F is a scalar field and curl only acts on vector fields
And btw div curl F = 0
obsidian isle
always???
vapid hamlet
Yes
warm shaleBOT
Zanarcane
obsidian isle
wait ok i'm confused
do you take the gradient vector of F or of curlF there
what is nabla
vapid hamlet
nabla is a differential operator
And depending on whether you use \cdot or \cross or nothing at all, it's the divergence, curl, and gradient, respectively
vapid hamlet
And btw div curl F = 0
As for this relation, you can try to compute it by hand, and you will get 0
obsidian isle
$ \text{div curl} \vec{F} = \nabla(\vec{F}) \cdot (\nabla(\vec{F}) \times \vec{F}) $
why isn't it working
$ \text{div curl} \vec{F} = \nabla(\vec{F}) \cdot (\nabla(\vec{F}) \times \vec{F}) $
vapid hamlet
No idea
obsidian isle
my confusion is that how do you know what vector function to take the gradient of? like for the div curl operation do you take the nabla of curlF or just the nabla of F
vapid hamlet
I take the nabla dot (divergence) of the nabla cross (curl) of F
Never used the gradient
obsidian isle
yeah but which partial derivative vector do you use
the one of F itself or the one of curlF
obsidian isle
ok yeah that makes sense
so it's the partial derivative vector of G dot G
what do you call the partial derivative vector if not the gradient though
does gradient only apply to scalar functions
vapid hamlet
The "partial derivative vector" does not depend on the function you apply it to
vapid hamlet
does gradient only apply to scalar functions
Yes, but you can also define the gradient of a vector-valued function by the gradient of each of its components
And write it as a matrix, which corresponds to the transpose of the Jacobian
vapid hamlet
nabla is just a notation, which in physics can be thought of as the "partial derivative vector"
It's more important to know what happens if you apply the gradient (just nabla), the curl (nabla cross), and the divergence (nabla dot) to some function
warm shaleBOT
EndTimes
obsidian isle
?
vapid hamlet
No
vapid hamlet
I'm on phone, so it'll take a while until I tex the following
vapid hamlet
Yes
warm shaleBOT
EndTimes
obsidian isle
does that work?
obsidian isle
but the gradient vector is <dF/dx, dF/dy, dF/dz>
vapid hamlet
Divergence is the trace of the Jacobian matrix
obsidian isle
oh i see it's different because there's actually three different functions
vapid hamlet
Yeah
obsidian isle
right
that makes sense
ok
it's still kind of weird seeing d/dx by itself
so it's not really a vector
obsidian isle
but an operation vector
vapid hamlet
Yeah
obsidian isle
so derivatives are fractions then
vapid hamlet
Uhh what?
obsidian isle
yeah.
<d/dx, d/dy, d/dz> . <Fx, Fy, Fz> = dFx/dx + dFy/dy + dFz/dz
vapid hamlet
It's not a fraction
obsidian isle
then how does multiplication make any sense
vapid hamlet
If you continue your studies, maybe you will come across the fact that d/dx can be seen as a derivation / partial differential operator
vapid hamlet
then how does multiplication make any sense
Again, this is only a notation
The differential operator <d/dx, d/dy, d/dz> • maps a vector field F to the scalar field dFx/dx + dFy/dy + dFz/dz
obsidian isle
Right ok
So it's not a fraction but merely something we treat as a fraction in every conceivable context
Checks out
vapid hamlet
It does not behave like a fraction
But it's linear
$\frac{\dd}{\dd x}\colon C^1(\mathbb{R})\to C(\mathbb{R}),, f\mapsto f'$
warm shaleBOT
Zanarcane
obsidian isle
wdym by C
vapid hamlet
Maps a continuously differentiable function f to a continuous function f' (its derivative)
So, you can see d/dx as a mapping between function spaces
Better than seeing it as fractions
C1(R) is the space of continuously differentiable functions on R
C(R) is the space of continuous functions on R
obsidian isle
right yeah that checks out
so if there is such an issue with treating it as a fraction though then how would you 'formally' write the divergence operation
vapid hamlet
**Zanarcane**
obsidian isle
ok so yeah i guess that makes sense. the dot product is a useful mnemonic i guess
vapid hamlet
Yeah
vapid hamlet
Good
obsidian isle
oh shit green's theorem
vapid hamlet
What about it?
obsidian isle
it's fine lol i'll figure it out
vapid hamlet
Alr
obtuse pebbleBOT
@obsidian isle Has your question been resolved?
timid silo
Hello. I'm working through some factoring examples in my college algebra book and I'm confused about an example.
obtuse pebbleBOT
Hello. I'm working through some factoring examples in my college algebra book an...
timid silo
,tex
\begin{math}
x^2 + 6x = 0\
x(x+6) = 0\
x = 0 \text{ or } x + 6 = 0\
x = 0 \text{ or } x = -6
\end{math}
warm shaleBOT
Noneuclidian
timid silo
How were they able to remove the x outside the parens? I should mention that at this part in the book they haven't introduced us yet to radicals.
vapid hamlet
It's called factoring out
Follows from the distributive law
And you don't need radicals for that
timid silo
I understand factoring x^2 + 6x into x(x+6), but I don't understand how they can just drop that x and be left with x+6. Don't you have to divide both sides by(x+6) to get that x on its own? (or is taht what's going on?)
timid silo
What about getting x+6 on its own without the other x in front of it? How does that work?
short spruce
are you talking about line 2 -> line 3
timid silo
The last two lines basically
short spruce
as i mentioned, that's the zero product property
if you're multiplying multiple terms together and that product is equal to zero, that can only happen if at least one of the terms is equal to zero
for example, if you had $x\cdot y\cdot z = 0$, this can only be true if $x, y,$ or $z$ is equal to zero.
warm shaleBOT
a disappointing son
short spruce
since that would make the entire product zero
obsidian isle
Right
timid silo
I understand anything multiplied by anything 0 is 0
obsidian isle
No
timid silo
?
timid silo
But in the example I posted, there are two solutions, x = -6 and x = 0 . In the branch that determines x=-6, it looks like they just drop that first x? Or is "or" part of the statement itself?
If you say x=0, then you shouldn't be able to do further evaluation of the x+6 inside the x(x+6)?
obsidian isle
f(x) = x(x+6)
f(0) = 0(6) = 0
f(-6) = -6(0) = 0
Only one of the factors is 0 at a time. When we use the zero product property we are implicitly making the assumption that one of the factors is not zero and then solving for x
Because obviously we can't divide by 0
timid silo
Hmm I'm not sure if I'm fully satisfied with these answers. Thanks though. I'll work more of these examples
obsidian isle
Ok look at it this way.
We have that:
x(x+6) = 0
Now assume that (x+6) is not 0
So divide both sides by (x+6)
Then we get that x=0
timid silo
Okay, that I understand
obsidian isle
And same thing for the other solution
x(x+6) = 0
Assume that x is not 0
So divide both sides by x
x+6 = 0
x = -6
timid silo
Okay, that understand
timid silo
x(x+6) = 0
Assume that x is not 0
So divide both sides by x
x+6 = 0
x = -6
Thanks!
high lily
0 * stuff = 0
0 * x = 0
0 * (x+6) = 0
obsidian isle
From a logical standpoint it does rely on casework and making these assumptions
timid silo
When I gave this problem to symolab it just whipped out the quadratic formula
obsidian isle
Try WolframAlpha instead, it's a bit smarter
timid silo
What other webapps do you guys recommend besides Symbolab and WolframAlpha for checking answers?
timid silo
Thanks
obtuse pebbleBOT
Available help channel!
mental mortar
mental mortar
can someone tell me the first step in solving this
crimson berry
plucky ether
Help
plucky ether
How do I do this 😭
weary spruce
How do I do this 😭
you should know formulas
try converting cos into sin or sin into cos
obtuse pebbleBOT
@plucky ether Has your question been resolved?
plucky ether
try converting cos into sin or sin into cos
But they're cubed and not squared, I could've converted them if they were squared. How do I do it if they're cubed?
weary spruce
But they're cubed and not squared, I could've converted them if they were square...
There are some formulas for cube too, but the formulas ive done arent enough to solve this question.
So cant say
So
Just leave that question i guess
It could be wrong
weary spruce
There are only 3 that ive done
💀💀
Its
Sin^3A = 3sinA - sin3A/idk sumn i dont remember
💀💀💀💀💀💀
But thats bot enough obviously
Not*
sage iron
Help
wait I think this is a troll
LHS is 16.11 and RHS 4.26 according to my calculator
either that or I accidentally punched in the wrong keys
plucky ether
Hol up
sage iron
actually the equality holds but the 4 should be inside the parentheses
so it should be 4 * sin^3(25) + 4*cos^3(5)
sage iron
Help
yeah I ended up solving it
it's fun but complicated
but yeah there's a typo in the question
plucky ether
yeah I ended up solving it
Yo, how did you do it?
sage iron
well you have to get rid of the powers right
so I simply used the power reducing formulas
plucky ether
Wait idk that formula
plucky ether
Okay
sage iron
it's a complicated question btw so don't worry if how to solve it isn't immediately apparent
plucky ether
https://en.wikipedia.org/wiki/List_of_trigonometric_identities
Ah neat
a
Stuck again
plucky ether
it's a complicated question btw so don't worry if how to solve it isn't immediat...
What do I do next?
compact jewel
Btw, cos can turn to sin easily
Add 90 to the angle and it becomes sin
-90 you go from sin to cos
+90 cos to sin
plucky ether
Oh ya forgot about that
sage iron
yeah those 2 basically cancel out
last step requires a bit of thinking
(at least for me it did)
plucky ether
Alright converted them to sin but what now
sage iron
what do you have rn
plucky ether
3sin25 + 3sin95 + sin105 - sin75
Wait could I have cancelled out cos 15 by turning -sin75 into -cos15?
sage iron
you know that sin(75) = cos(15) right
sage iron
imagine a right triangle with angles 75, 15, and 90
sage iron
nice
plucky ether
Hmmmmm
sage iron
you're basically there
have you heard of sum to product formulas
maybe a product would be more useful
because the LHS is a product
plucky ether
The RHS, yes...
sage iron
yeah I meant RHS, sorry
sage iron
there's a sin + sin formula
sage iron
just convert your cosine to sin (90-x)
sage iron
great
sand pilot
How can you prove that a shape is a parabola
obtuse pebbleBOT
How can you prove that a shape is a parabola
royal basin
by proving that it satisfies the definition of a parabola
i.e. that your shape consists of precisely those points which are the same distance away from a focus and a directrix
sand pilot
So the method of proving it should be the same as the method of deriving the parabola focus and directrix formula
Is that all??
royal basin
i don't know, cause you didn't say what kind of 'shape' you're looking at or how exactly the shape is given to you
sand pilot
Let's say it's on the graph
Or we can't prove it unless it's in a shape like a cone
???
royal basin
on the graph? so all you're given is a drawing?
and asked to prove that it's a drawing of a parabola?
obtuse pebbleBOT
@sand pilot Has your question been resolved?
sand pilot
and asked to _prove_ that it's a drawing of a parabola?
Yes
royal basin
then it's impossible
fervent comet
obtuse pebbleBOT
fervent comet
I need help with the first question
I don’t understand how or what I need to do to prove it
fervent comet
Yeah thought so. What is it
royal basin
the triangles are not ABX and ABY
royal basin
why is segment XY not shown?
fervent comet
Do I have to draw that too?
fervent comet
also ABX hardly looks equilateral
Don’t matter right now
I just want a rough idea on how to do proofs
fervent comet
AXZ and BYZ
Ahh
Smart
Not sure how to prove AZ = 2ZB tho
@royal basin
Yo?
You there?
fervent comet
okay so you have proved the triangles are similar, yes?
Uh no
Can’t we just prove the triangles are similar by using SAS
?
royal basin
what sides and angles are you planning to use for SAS?
it might be that your idea will work, but there's no telling until you actually articulate it
fervent comet
it might be that your idea will work, but there's no telling until you actually ...
We had same idea or different?
royal basin
i had a different idea than yours, but i will not let you interpret that as saying your idea is wrong and should be rejected out of hand.
fervent comet
ZAX is similar to ZBY, ZB is similar to AZ, ZX is similar to ZY
royal basin
ZAX is similar to ZBY? how do you know?
fervent comet
Not entirely sure if it’s right cause I haven’t done this stuff in like a year or two
royal basin
isn't that our goal?
fervent comet
ZAX is similar to ZBY? how do you know?
Guess
royal basin
fervent comet
royal basin
are you serious?
royal basin
do you realize that guessing is as far as possible from mathematical proof
fervent comet
I forgot a lot about it and the textbook doesn’t help
fervent comet
do you realize that *guessing* is as far as possible from mathematical proof
Yes
Give me a source to learn how to proof pls
I gtg soon, pls
Can you go through what you would’ve done to prove it
royal basin
i would observe that angles YBZ and ZAX are equal, both being 60°,
and angles YZB and AZX are also equal, being a pair of vertical angles
two pairs of corresponding equal angles suffice to prove similarity for two triangles
i think your textbook (or any geometry textbook really) should teach you how to prove things in geometry
fervent comet
i think your textbook (or any geometry textbook really) should teach you how to ...
Teachers supposed to do that
I’m currently not taking the course for the stuff I’m doing in the textbook. I just do it for fun. I don’t have a person to teach me proofs so yea. I’m wanting sources where it explains how to do math proofs and such. Can you pls give me a link to a source if you know.
My textbook has questions but barely any explanation. You sorta got to figure it out by yourself if no one is teaching you
royal basin
strange
i dont know of any resource for elementary proofs like this
like maybe How To Prove It by velleman but i don't think it covers geometry specifically
errant sinew
obtuse pebbleBOT
errant sinew
how is this formula derived?
also, is the formula for the magnitude of an n-dimensional vector:
sqrt(a^2_1 + a^2_2 + a^2_3 + ... + a^2_n)
royal basin
how is this formula derived?
pythagorean theorem
and yes what you wrote is the euclidean norm in n dimensions
errant sinew
pythagorean theorem
its hard for me to visualize it
royal basin
imagine a cuboid with known length, width and height in which you want to find the length of the space diagonal
errant sinew
and yes what you wrote is the euclidean norm in n dimensions
what is a norm?
royal basin
fancy name for "function that assigns a length to vectors"
hybrid wing
how to draw figure for this?
obtuse pebbleBOT
how to draw figure for this?
hybrid wing
link to question https://www.teachoo.com/3334/730/Ex-8.1--10---Find-area-bounded-by-x2--4y-and-line-x--4y---2/category/Ex-8.1/
i actually understand that x square =4y is parabola but how to know where the lin intersects parabola
gloomy valve
how to draw figure for this?
if you solve both equations for y, you should have standard equations for a parabola and a line, which you should know how to sketch
to find the intersection points you can then set them equal
hybrid wing
cn u show me?
gloomy valve
which part?
hybrid wing
the getting intesection points part
i know how to solve the sum after getting the grah diagram but the hard part is to know where you ust draw the intersecting line on the parabola
in the link i sent they somehow got the graph from the equations in the beginning itself
gloomy valve
$$x^2 = 4y \implies y = \frac{1}{4} x^2$$
$$x = 4y - 2 \implies y = \frac{x+2}{4} = \frac{1}{4} x + \frac{1}{2}$$
warm shaleBOT
nvx
gloomy valve
Now you simply set them equal and solve for x to find the intersection points
daring kite
could someone explain what is going here? I am a bit confused with hypothesis testing? Are we testing that our sample mean is equal to a prespecified mean? Also doesn't T give an estimate of a parameter such as the mean? So why do you choose the test statistic to be of that form Zn = ..(X_bar = sampling dbn mean i believe)?
obtuse pebbleBOT
could someone explain what is going here? I am a bit confused with hypothesis te...
daring kite
i am not sure about this second slide either
novel knoll
could someone explain what is going here? I am a bit confused with hypothesis te...
We are testing for the true mean
daring kite
is true mean mu_0?
daring kite
so we want to test whether mu = 5 or mu != 5? mu_0 is the value that we think mean can be?
novel knoll
In hypothesis testing you have a null hypothesis say mu=5
Then you have an alternative hypothesis which is either of the 3 written there
novel knoll
Its written right there?
daring kite
i am not understanding why thats the case. Like what does the test statistic give? Because in the significance level point it talks about obtaining a value of test statistic at least as exterme under H_0
and i am not getting these parts
novel knoll
You have it has distribution N(0,1)?
You should have worked with normal dist before and figure out how to get prob given an observation
daring kite
standard normal mean is 0, why are values of Zn is tails (which i assume it means values far away from mean) unlikely under null hypothesis?
novel knoll
Just looking at formula we do sample mean -mu_0 (and some other stuff)
But if they are close we get a lower value than if they are far away
obtuse pebbleBOT
@daring kite Has your question been resolved?
main cargo
obtuse pebbleBOT
main cargo
i aim in painn
what i tried was subbing y = vx in straight
then i know that dy/dx = v
hybrid gyro
it seems that the content you do are very similar to mine, are you doing a-level further maths?
main cargo
so you get v = (1/x) ((4+v)(1+v)/(x))
main cargo
it seems that the content you do are very similar to mine, are you doing a-leve...
yep, year 2 exams in a month doing exam board OCR B (MEI)
hybrid gyro
same, but mine is pearson edexcel
hybrid gyro
then i know that dy/dx = v
is dy/dx = v though?
v is more like a variable than a constant
main cargo
igggg so alright time to do sum product rule then
hybrid gyro
yup
main cargo
alright that literally solves everything lol thanks
hybrid gyro
main cargo
.close
timid silo
obtuse pebbleBOT
timid silo
I started by differentiating the equation with respect to x. Now I can't eliminate the constant 'a' between the initial equation and the equation we got after differentiating
obtuse pebbleBOT
@timid silo Has your question been resolved?
obtuse pebbleBOT
@timid silo Has your question been resolved?
slim leaf
@timid silo No need to eliminate "a".
Now just take the negative reciprocal of dy/dx
then integrate the new dy/dx
obtuse pebbleBOT
@timid silo Has your question been resolved?
timid silo
The answer should be x² + y² -2a² logx= c
gentle badger
try eliminating the parameter lambda and then replacing dy/dx with the negative reciprocal
obtuse pebbleBOT
@timid silo Has your question been resolved?
obtuse pebbleBOT
@timid silo Has your question been resolved?
timid silo
I still am not able to get the answer
gentle badger
@timid silo you've missed a 'y' in the step two after "putting this in eq (i)"
brazen warren
The answer should be x² + y² -2a² logx= c
can you replace dy/dx by -dx/dy?
Am I missing something or that's totally wrong
gentle badger
you gotta eliminate the parameters first
@timid silo which book are you following for DEs?
limber quartz
don't change dy/dx, just reciprocate and negate the other side
(same thing, actually)
you have to solve for your parameter (lambda), and then plug in the expression for it after you find dy/dx.
I expected a lot of things to cancel but you can see it was a sloppy experience for me.
where it says "orthogonal family", I flipped and negated the RHS
then you just get your y-business on the dy-side and your dx on the x-side
Understand what is happening with the constants of integration because I didn't write everything
when you integrate both sides, both sides have a unique constant, since this is indefinite integration
so like C_1 on the left, C_2 on the right
but we can move either of those to the other side and combine them so that C_2 - C_1 = C
...because, "why not"? It's just a constant.
and if I multiply the equation through by 2, then 2-times-C is still just an arbitrary constant. So I rewrote it as "C" again in the last line.
@timid silo
summer stone
obtuse pebbleBOT
summer stone
how do i do this one?
rugged kite
If a polynomial P is such that P(2)=0, how can you factor P?
royal basin
If a polynomial P is such that P(2)=0, how can you factor P?
you don't even need that for most of the exercise
@summer stone do you still need help with this?
royal basin
have you worked with the concept of 'subspace' before?
royal basin
what is this
summer stone
have you worked with the concept of 'subspace' before?
yes
summer stone
like the axioms?
royal basin
the definition of a subspace.
summer stone
oh
royal basin
let V be a vector space. a subspace of V is a ____
summer stone
it is a vector space inside of another vector space
summer stone
let V be a vector space. a subspace of V is a \_\_\_\_
vector space?
royal basin
no
well, ok, sure, subspaces are vector spaces in their own right
but first and foremost a subspace of V is a subset of V which obeys some properties that make it a vector space in its own right.
what are those properties?
summer stone
um
royal basin
brb
summer stone
c(vu) = (cv)u
c(v+u) = cv + cu
V(0) = 0
royal basin
no, that is not correct
these look like axioms of a vector space in general, 2 of which you have poorly stated
do the words "closure under addition" and "closure under scaling" ring any bells?
summer stone
do the words "closure under addition" and "closure under scaling" ring any bells...
yes
royal basin
ok well that's what i expected to hear from you
summer stone
what are those properties?
umm
i'm not sure what u mean
it should be in the span of the vector space? like the subspace in our question should be in P^2 too?
royal basin
i have a feeling you're throwing words around with little idea of what they should mean
your question says $H := { p \in \mathbb{P}_2 \mid p(2) = 0 }$. it says this in words rather than symbols, but that is what it says.
warm shaleBOT
Ann
summer stone
yeah i understand that
royal basin
$H$ is defined as a subset of $\mathbb{P}_2$. there is no need to check that it is a subset of $\mathbb{P}_2$ --- it is already known to be one.
warm shaleBOT
Ann
royal basin
what we are interested in is showing that $H$ is a sub\textbf{space} of $\mathbb{P}_2$ --- namely:
\begin{itemize}
\item $H$ is nonempty,
\item for every $p \in H$ and $q \in H$ we have $p + q \in H$,
\item for every $p \in H$ and $c \in \bR$ we have $cp \in H$.
\end{itemize}
this is the \textbf{definition} of what the word ``subspace'' \textbf{means}, and so this is something you should know off the top of your head if you are doing linear algebra (which you no doubt are, since this is a linear algebra problem).
warm shaleBOT
Ann
royal basin
do you understand what to do? Y/N
summer stone
the question says we may assume that H is a subspace of P^2, so we don't need to prove that
royal basin
ok, so then we want a basis for H, yes
first off
let's think about how many elements we should be going for in our basis
summer stone
3
royal basin
can you tell me what the dimension of P_2 is?
summer stone
3?
royal basin
is that "i think it's 3 but i'm uncertain" or "i like to put question marks after my answer for no reason"
summer stone
i think its 3, but that the fact that you've asked this question the 10th time, i'm unsure whether that's true
royal basin
i only asked you about the dimension of P_2 now. hardly the tenth time.
but yes, the dimension of P_2 is 3, that is correct. you could have corroborated your answer by presenting a basis for P_2 consisting of three elements, such as {1, x, x^2}
summer stone
okay well its 3
royal basin
now think: can dim(H) be 3? clearly H isn't the whole of P_2
summer stone
now think: can dim(H) be 3? clearly H _isn't_ the whole of P_2
OH
okay yeah yeah yeah
wait wait just a second more
no it could be three
royal basin
how can dim(H) be 3 without H being the whole space?
summer stone
it could be a part of the space but with dim 3
royal basin
??
summer stone
like
royal basin
okay so let me get this straight: you think that there exists a space V with a subspace W ⊆ V such that dim(V) = dim(W) yet V ≠ W. do i understand you correctly?
yes or no
summer stone
yes
royal basin
can you give me an example of such a subspace for R^3?
royal basin
||because, to put it simply, you are wrong about this. and coming up with such an example is bound to fail.
a proper subspace necessarily has strictly lower dimension than its parent space. and pertinent to your problem, this alone means dim(H) ≤ 2.||
then maybe do not make claims you're not 100% certain of lmao
summer stone
||because, to put it simply, you are wrong about this. and coming up with such a...
oh okay okay
makes sense
so H would have dim 2 or lower
royal basin
yes exactly
so now that we know dim(H) ≤ 2
can you give me some elements of H
preferably ones that don't form linearly dependent sets
royal basin
these are not elements of H
royal basin
when i ask for elements of H i ask for elements of H
there is also no need to wrap these in braces
now ask yourself: is 1 ∈ H?
royal basin
is x ∈ H?
summer stone
yes
royal basin
is x an element of H?
royal basin
i don't care what letter you use for your variable
i mean ok fine i was using x while the problem was using t
doesnt matter
what remains is that i'm trying to get you to name things that are in H, yet the things you are naming are not in H and you for some reason cannot see that
whether you call it x or t, that polynomial is still not an element of H
summer stone
that's why i said it
royal basin
the value of t at t=2 is 2, not 0. H consists of those polynomials whose value at t=2 is 0.
summer stone
because it doesn't matter
summer stone
the value of t at t=2 is 2, not 0. H consists of those polynomials whose value a...
yes that's why i added -2
summer stone
yes
royal basin
as if those were two different things which are individually elements of H
which they are not
summer stone
i meant p(t) = t - 2
royal basin
then say p(t) = t-2!
royal basin
yet that pesky comma kept getting in the way
but ok, great, t-2 is an element of H.
can you name another element of H that is not a scalar multiple of t-2?
royal basin
that is not a scalar multiple of t-2
summer stone
oh my b
royal basin
the zero polynomial is in H, but it can't go in a basis.
summer stone
can you name another element of H that is not a scalar multiple of t-2?
something like p(t) = t^2 - 4?
no thats the same thing
uh
rugged kite
No that works
royal basin
no, t^2 - 4 is not the same thing as t - 2
summer stone
oh okay yeah
royal basin
t^2 - 4 is in H, and is not a scalar multiple of t-2.
it is in fact exactly what i asked for.
summer stone
oh great
royal basin
you could have given other answers
but ok
{t-2, t^2-4}
this set is linearly independent, as should be clear.
can you now explain why every polynomial in H can be expressed as a linear combination of these two?
summer stone
can you now explain why every polynomial in H can be expressed as a linear combi...
since they are linearly independent, so everything else would be a linear combination of the two, and hence in the span of the two
royal basin
no
'linear independence' and 'spanning' are different notions that should not be conflated.
summer stone
oh why?
royal basin
wym why
just because a set is linearly independent does not automatically mean it spans the space it's in
summer stone
okay but when it is linearly independent and has the same no. of vectors as the dimension of the space
then it spans it, right?
royal basin
{x} is a linearly independent subset of P_5 yet you would not claim every polynomial in P_5 is a linear combination of {x} would you?
yes
but it appears you're expected not to use that
and instead show directly that every polynomial in H is a linear combination of your basis (in our case {t-2, t^2-4})
royal basin
warm shaleBOT
ibzi
royal basin
and now bad notation + apparent eagerness to plunge headfirst into equation-bashing
i mean ok like. sure i guess
you could show that if a + 2b + 4c = 0 then there exist d and k such that what you wrote is true
if you want to do it like that, who am i to stop you
royal basin
i would say that any polynomial in H can be written as (at+b)(t-2)
and that in turn can be written as a(t+2) + (b-2a) = a(t+2)(t-2) + (b-2a)(t-2)
which is a linear combination of your basis
summer stone
oh okay okay
summer stone
~~and now bad notation + apparent eagerness to plunge headfirst into equation-ba...
and also how is it bad notation
royal basin
i personally don't like it, that's how 
in this particular case i was not being very objective about it
royal basin
the d and k do not strike me as good name choices for two constants that should appear related